some matlab Task with biomedical science
Honest139
BIOMEDICAL OPTICS
B I C E N T C N N I A I
|| 1 8 O 7 r|®WILEY ÜJ2 0 0 7
B I C E N T E N N I A L
T H E WILEY BICENTENNIAL-KNOWLEDGE FOR GENERATIONS
//~)ach generation has its unique needs and aspirations. When Charles Wiley first opened his small printing shop in lower Manhattan in 1807, it was a generation of boundless potential searching for an identity. And we were there, helping to define a new American literary tradition. Over half a century later, in the midst of the Second Industrial Revolution, it was a generation focused on building the future. Once again, we were there, supplying the critical scientific, technical, and engineering knowledge that helped frame the world. Throughout the 20th Century, and into the new millennium, nations began to reach out beyond their own borders and a new international community was born. Wiley was there, expanding its operations around the world to enable a global exchange of ideas, opinions, and know-how.
For 200 years, Wiley has been an integral part of each generation's journey, enabling the flow of information and understanding necessary to meet their needs and fulfill their aspirations. Today, bold new technologies are changing the way we live and learn. Wiley will be there, providing you the must-have knowledge you need to imagine new worlds, new possibilities, and new opportunities.
Generations come and go, but you can always count on Wiley to provide you the knowledge you need, when and where you need it!
iLLu^M. (2, WILLIAM J . P E S C E
PRESIDENT A N D CHIEF EXECUTIVE OFFICER PETER B O O T H WILEY
CHAIRMAN DF T H E BOARD
BIOMEDICAL OPTICS
PRINCIPLES AND IMAGING
Lihong V. Wang
Hsin-i Wu
B I C E N T E N N I A L ·
; | 1 8 0 7
©WILEY 2 OO 7
B I C E N T E N N I A L
WILEY-INTERSCIENCE A John Wiley & Sons, Inc., Publication
Copyright © 2007 by John Wiley & Sons, Inc. All rights reserved.
Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada.
No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permission.
Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.
For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002.
Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com.
Wiley Bicentennial Logo: Richard J. Pacifico
Library of Congress Cataloging-in-Publication Data:
Wang, Lihong V. Biomedical optics : principles and imaging / Lihong V. Wang, Hsin-i
Wu. p. ; cm.
Includes bibliographical references and index. ISBN: 978-0-471-74304-0 (cloth) 1. Imaging systems in medicine. 2. Lasers in medicine. 3. Optical
detectors. I. Wu, Hsin-i. II. Title. [DNLM: 1. Optics. 2. Diagnostic Imaging—methods. 3. Light.
4. Models, Theoretical. 5. Tomography, Optical—methods. WB 117 W246b 2007] R857.06W36. 2007 616.0754-dc22
2006030754
Printed in the United States of America.
109 8 7
To our families, mentors, students, and friends
CONTENTS
Preface xiii
1. Introduction 1
1.1. Motivation for Optical Imaging 1
1.2. General Behavior of Light in Biological Tissue 2
1.3. Basic Physics of Light-Matter Interaction 3
1.4. Absorption and its Biological Origins 5
1.5. Scattering and its Biological Origins 7
1.6. Polarization and its Biological Origins 9
1.7. Fluorescence and its Biological Origins 9
1.8. Image Characterization 10
Problems 14
Reading 15
Further Reading 15
2. Rayleigh Theory and Mie Theory for a Single Scatterer 17
2.1. Introduction 17
2.2. Summary of Rayleigh Theory 17
2.3. Numerical Example of Rayleigh Theory 19
2.4. Summary of Mie Theory 20
2.5. Numerical Example of Mie Theory 21
Appendix 2A. Derivation of Rayleigh Theory 23
Appendix 2B. Derivation of Mie Theory 26
Problems 34
Reading 35
Further Reading 35
vii
VIII CONTENTS
Monte Carlo Modeling of Photon Transport in Biological Tissue
3.1. Introduction
3.2. Monte Carlo Method
3.3. Definition of Problem
3.4. Propagation of Photons
3.5. Physical Quantities
3.6. Computational Examples
Appendix 3A. Summary of MCML
Appendix 3B. Probability Density Function
Problems
Reading
Further Reading
Convolution for Broadbeam Responses
4.1. Introduction
4.2. General Formulation of Convolution
4.3. Convolution over a Gaussian Beam
4.4. Convolution over a Top-Hat Beam
4.5. Numerical Solution to Convolution
4.6, Computational Examples
Appendix 4A. Summary of CONV
Problems
Reading
Further Reading
Radiative Transfer Equation and Diffusion Theory
5.1. Introduction
5.2. Definitions of Physical Quantities
5.3. Derivation of Radiative Transport Equation
5.4. Diffusion Theory
5.5. Boundary Conditions
5.6. Diffuse Reflectance
37
37
37
38
39
50
55
58
60
60
62
62
67
67
67
69
71
72
77
77
80
81
81
83
83
83
85
88
101
106
CONTENTS IX
5.7. Photon Propagation Regimes 114
Problems 116
Reading 117
Further Reading 118
Hybrid Model of Monte Carlo Method and Diffusion Theory 119
6.1. Introduction 119
6.2. Definition of Problem 119
6.3. Diffusion Theory 119
6.4. Hybrid Model 122
6.5. Numerical Computation 124
6.6. Computational Examples 125
Problems 132
Reading 133
Further Reading 133
Sensing of Optical Properties and Spectroscopy 135
7.1. Introduction 135
7.2. Collimated Transmission Method 135
7.3. Spectrophotometry 139
7.4. Oblique-Incidence Reflectometry 140
7.5. White-Light Spectroscopy 144
7.6. Time-Resolved Measurement 145
7.7. Fluorescence Spectroscopy 146
7.8. Fluorescence Modeling 147
Problems 148
Reading 149
Further Reading 149
Ballistic Imaging and Microscopy 153
8.1. Introduction 153
8.2. Characteristics of Ballistic Light 153
X CONTENTS
8.3. Time-Gated Imaging 154
8.4. Spatiofrequency-Filtered Imaging 156
8.5. Polarization-Difference Imaging 157
8.6. Coherence-Gated Holographic Imaging 158
8.7. Optical Heterodyne Imaging 160
8.8. Radon Transformation and Computed Tomography 163
8.9. Confocal Microscopy 164
8.10. Two-Photon Microscopy 169
Appendix 8A. Holography 171
Problems 175
Reading 177
Further Reading 177
9. Optical Coherence Tomography 181
9.1. Introduction 181
9.2. Michelson Interferometry 181
9.3. Coherence Length and Coherence Time 184
9.4. Time-Domain OCT 185
9.5. Fourier-Domain Rapid-Scanning Optical Delay Line 195
9.6. Fourier-Domain OCT 198
9.7. Doppler OCT 206
9.8. Group Velocity Dispersion 207
9.9. Monte Carlo Modeling of OCT 210
Problems 213
Reading 215
Further Reading 215
10. Mueller Optical Coherence Tomography 219
10.1. Introduction 219
10.2. Mueller Calculus versus Jones Calculus 219
10.3. Polarization State 219
10.4. Stokes Vector 222
CONTENTS XI
10.5. Mueller Matrix 224
10.6. Mueller Matrices for a Rotator, a Polarizer, and a Retarder 225
10.7. Measurement of Mueller Matrix 227
10.8. Jones Vector 229
10.9. Jones Matrix 230
10.10. Jones Matrices for a Rotator, a Polarizer, and a Retarder 230
10.11. Eigenvectors and Eigenvalues of Jones Matrix 231
10.12. Conversion from Jones Calculus to Mueller Calculus 235
10.13. Degree of Polarization in OCT 236
10.14. Serial Mueller OCT 237
10.15. Parallel Mueller OCT 237
Problems 243
Reading 244
Further Reading 245
11. Diffuse Optical Tomography 249
11.1. Introduction 249
11.2. Modes of Diffuse Optical Tomography 249
11.3. Time-Domain System 251
11.4. Direct-Current System 252
11.5. Frequency-Domain System 253
11.6. Frequency-Domain Theory: Basics 256
11.7. Frequency-Domain Theory: Linear Image Reconstruction 261
11.8. Frequency-Domain Theory: General Image Reconstruction 267
Appendix 11 A. ART and SIRT 275
Problems 276
Reading 279
Further Reading 279
12. Photoacoustic Tomography 283
12.1. Introduction 283
12.2. Motivation for Photoacoustic Tomography 283
XII CONTENTS
12.3. Initial Photoacoustic Pressure 284
12.4. General Photoacoustic Equation 287
12.5. General Forward Solution 288
12.6. Delta-Pulse Excitation of a Slab 293
12.7. Delta-Pulse Excitation of a Sphere 297
12.8. Finite-Duration Pulse Excitation of a Thin Slab 302
12.9. Finite-Duration Pulse Excitation of a Small Sphere 303
12.10. Dark-Field Confocal Photoacoustic Microscopy 303
12.11. Synthetic Aperture Image Reconstruction 307
12.12. General Image Reconstruction 309
Appendix 12A. Derivation of Acoustic Wave Equation 313
Appendix 12B. Green Function Approach 316
Problems 317
Reading 319
Further Reading 319
13. Ultrasound-Modulated Optical Tomography 323
13.1. Introduction 323
13.2. Mechanisms of Ultrasonic Modulation of Coherent Light 323
13.3. Time-Resolved Frequency-Swept UOT 326
13.4. Frequency-Swept UOT with Parallel-Speckle Detection 329
13.5. Ultrasonically Modulated Virtual Optical Source 331
13.6. Reconstruction-Based UOT 332
13.7. UOT with Fabry-Perot Interferometry 335
Problems 338
Reading 339
Further Reading 339
Appendix A. Definitions of Optical Properties 343
Appendix B. List of Acronyms 345
Index 347
PREFACE
Biomedical optics is a rapidly growing area of research. Although many universi- ties have begun to offer courses on the topic, a textbook containing examples and homework problems has not been available. The need to fill this void prompted us to write this book.
This book is based on our lecture notes for a one-semester (45 lecture hours) entry-level course, which we have taught since 1998. The contents are divided into two major parts: (1) fundamentals of photon transport in biological tissue and (2) optical imaging. In the first part (Chapters 1 -7) , we start with a brief introduc- tion to biomedical optics and then cover single-scatterer theories, Monte Carlo modeling of photon transport, convolution for broadbeam responses, radiative transfer equation and diffusion theory, hybrid Monte Carlo method and diffusion theory, and sensing of optical properties and spectroscopy. In the second part (Chapters 8-13), we cover ballistic imaging, optical coherence tomography, dif- fuse optical tomography, photoacoustic tomography, and ultrasound-modulated optical tomography.
When the book is used as the textbook in a course, the instructor may request a solution manual containing homework solutions from the publisher. To ben- efit from this text, students are expected to have a background in calculus
® and differential equations. Experience in MATLAB or C/C++ is also helpful. Source codes and other information can be found at ftp://ftp.wiley.com/public/ scLtech_med/biomedical_optics.
Although our multilayered Monte Carlo model is in the public domain, we have found that students are able to better grasp the concept of photon transport in biological tissue when they implement simple semiinfinite versions of the model. For this reason, we encourage the use of simulations whenever appropriate.
Because a great deal of material beyond our original lecture notes has been added, two semesters are recommended to cover the complete textbook. Alterna- tively, selected chapters can be covered in a one-semester course. In addition to serving as a textbook, this book can also be used as a reference for professionals and a supplement for trainees engaged in short courses in the field of biomedical optics.
We are grateful to Mary Ann Dickson for editing the text and to Elizabeth Smith for redrawing the figures. We appreciate Sancy Wu's close reading of
xiii
XIV PREFACE
the manuscript. We are also thankful to the many students who contributed to the homework solutions. Finally, we wish to thank our students Li Li, Manojit Pramanik, and Sava Sakadzic for proofreading the book.
LIHONG V. WANG, PH.D. HSIN-I Wu, PH.D.
CHAPTER 1
Introduction
1.1. MOTIVATION FOR OPTICAL IMAGING
The most common medical imaging modalities include X-ray radiography, ultra- sound imaging (ultrasonography), X-ray computed tomography (CT), and mag- netic resonance imaging (MRI). The discovery of X rays in 1895, for which Roentgen received the first Nobel Prize in Physics in 1901, marked the advent of medical imaging. Ultrasonography, which is based on sonar, was introduced into medicine in the 1940s after World War II. The invention of CT in the 1970s, for which Cormack and Hounsfield received the Nobel Prize in Medicine in 1979, initiated digital cross-sectional imaging (tomography). The invention of MRI, also in the 1970s, for which Lauterbur and Mansfield received the Nobel Prize in Medicine in 2003, enabled functional imaging with high spatial resolution. Optical imaging, which is compared with the other modalities in Table 1.1, is currently emerging as a promising new addition to medical imaging.
Reasons for optical imaging of biological tissue include
1. Optical photons provide nonionizing and safe radiation for medical appli- cations.
2. Optical spectra—based on absorption, fluorescence, or Raman scatter- ing—provide biochemical information because they are related to molec- ular conformation.
3. Optical absorption, in particular, reveals angiogenesis and hyperme- tabolism, both of which are hallmarks of cancer; the former is related to the concentration of hemoglobin and the latter, to the oxygen satura- tion of hemoglobin. Therefore, optical absorption provides contrast for functional imaging.
4. Optical scattering spectra provide information about the size distribution of optical scatterers, such as cell nuclei.
5. Optical polarization provides information about structurally anisotropic tissue components, such as collagen and muscle fiber.
Biomedical Optics: Principles and Imaging, by Lihong V. Wang and Hsin-i Wu Copyright © 2007 John Wiley & Sons, Inc.
1
INTRODUCTION
TABLE 1.1. Comparison of Various Medical Imaging Modalities
Characteristics
Soft-tissue contrast Spatial resolution Maximum imaging depth Function Nonionizing radiation Data acquisition Cost
X-ray Imaging
Poor Excellent Excellent None No Fast Low
Ultrasonography
Good Good Good Good Yes Fast Low
MRI
Excellent Good Excellent Excellent Yes Slow High
Optical Imaging
Excellent Mixed" Good Excellent Yes Fast Low
"High in ballistic imaging (see Chapters 8-10) and photoacoustic tomography (see Chapter 12); low in diffuse optical tomography (see Chapter 11).
6. Optical frequency shifts due to the optical Doppler effect provide infor- mation about blood flow.
7. Optical properties of targeted contrast agents provide contrast for the molecular imaging of biomarkers.
8. Optical properties or bioluminescence of products from gene expression provide contrast for the molecular imaging of gene activities.
9. Optical spectroscopy permits simultaneous detection of multiple contrast agents.
10. Optical transparency in the eye provides a unique opportunity for high- resolution imaging of the retina.
1.2. GENERAL BEHAVIOR OF LIGHT IN BIOLOGICAL TISSUE
Most biological tissues are characterized by strong optical scattering and hence are referred to as either scattering media or turbid media. By contrast, optical absorption is weak in the 400-1350-nm spectral region. The mean free path between photon scattering events is on the order of 0.1 mm, whereas the mean absorption length (mean path length before photon absorption) can extend to 10-100 mm.
Photon propagation in biological tissue is illustrated in Figure 1.1. The light source is spatially a pencil beam (an infinitely narrow collimated beam) and temporally a Dirac delta pulse. The optical properties (see Appendix A) of the tissue include the following: refractive index n — 1.37, absorption coefficient [ia =z 1.4 cm- 1 , scattering coefficient [is = 350 cm"1, and scattering anisotropy g = 0.8. The mean free path equals 28 μιτι, corresponding to a propagation time of 0.13 ps. The transport mean free path equals 140 μηι, corresponding to a propagation time of 0.64 ps. Note how widely the photons spread versus time in relation to the two time constants mentioned above. This diffusion-like behavior of light in biological tissue presents a key challenge for optical imaging. Various techniques have been designed to meet this challenge.
BASIC PHYSICS OF LIGHT-MATTER INTERACTION 3
Air I Laser beam
Tissue 1 x
ΙΟΟμπι/div ~ 1 ί ί ί ί 1 1 1 1 i
Geometry
0.15 ps
0.55 ps
1.55 ps
0.05 ps
0.35 ps
1.05 ps '.,, _ JULU ,
\ /
2.05 ps
Figure 1.1. Snapshots of the simulated photon density distribution in a piece of biological tissue projected along the y axis, which points out of the paper.
1.3. BASIC PHYSICS OF LIGHT-MATTER INTERACTION
Absorption of a photon can elevate an electron of a molecule from the ground state to an excited state, which is termed excitation. Excitation can also be caused by other mechanisms, which are either mechanical (frictional) or chemical in nature. When an electron is raised to an excited state, there are several possi- ble outcomes. The excited electron may relax to the ground state and give off luminescence (another photon) or heat. If another photon is produced, the emis- sion process is referred to as fluorescence or phosphorescence, depending on the lifetime of the excited electron; otherwise, it is referred to as nonradiative relax- ation. Lifetime is defined as the average time that an excited molecule spends in the excited state before returning to the ground state. The ratio of the number of photons emitted to the number of photons absorbed is referred to as the quantum yield of fluorescence. If the excited molecule is near another molecule with a sim- ilar electronic configuration, the energy may be transferred by excitation energy transfer—the excited electron in one molecule drops to the ground state while the energy is transferred to the neighboring molecule, raising an electron in that molecule to an excited state with a longer lifetime. Another possible outcome is photochemistry, in which an excited electron is actually transferred to another
4 INTRODUCTION
Excited state
hvA w \ +
Absorption
Internal conversion
Nonradiative relaxation
I
Vibrational energy levels
\ \ Intersystem crossing \ \ Excited \ triplet state
Fluorescence
Virtual state N
Phosphorescence
hvA hvR
Raman
Ground state
Figure 1.2. Jablonski energy diagram showing excitation and various possible relaxation mechanisms. Each hv denotes the photon energy, where subscripts A, F, P, and R denote absorption, fluorescence, phosphorescence, and Raman scattering, respectively.
molecule. This type of electron transfer alters the chemical properties of both the electron donor and the electron acceptor, as in photosynthesis.
A Jablonski energy diagram describing electronic transitions between ground states and excited states is shown in Figure 1.2. Molecules can absorb photons that match the energy difference between two of their discrete energy levels, provided the transitions are allowed. These energy levels define the absorption and the emission bands.
Fluorescence involves three events with vastly different timescales. Excita- tion by a photon takes place in femtoseconds (1 fs = 10-15 s, about one optical period). Vibrational relaxation (also referred to as internal conversion) of an excited-state electron to the lowest vibrational energy level in the excited state lasts for picoseconds (1 ps = 10~12 s) and does not result in emission of a photon (nonradiative). Fluorescence emission lingers over nanoseconds (1 ns = 10~9 s). Consequently, fluorescence lifetime is on the order of a nanosecond.
Phosphorescence is similar to fluorescence, but the excited molecule further transitions to a metastable state by intersystem crossing, which alters the electron spin. Because relaxation from the metastable state to the ground state is spin- forbidden, emission occurs only when thermal energy raises the electron to a state where relaxation is allowed. Consequently, phosphorescence depends on temperature and has a long lifetime (milliseconds or longer).
Two types of photon scattering by a molecule exist: elastic and inelastic (or Raman) scattering. The former involves no energy exchange between the photon and the molecule, whereas the latter does. Although both Raman scattering and fluorescence alter the optical wavelength, they have different mechanisms. In
ABSORPTION AND ITS BIOLOGICAL ORIGINS 5
Raman scattering, the molecule is excited to a virtual state; in fluorescence, the molecule is excited to a real stationary state. In both cases, the excited molecule relaxes to an energy level of the ground state and emits a photon. The molecule may either gain energy from, or lose energy to, the photon. If the molecule gains energy, the transition is known as a Stokes transition. Otherwise, the transition is known as an anti-Stokes transition. The scattered photon shifts its frequency accordingly since the total energy is conserved. Raman scattering can reveal the specific chemical composition and molecular structure of biological tissue, whereas elastic scattering can reveal the size distribution of the scatterers.
1.4. ABSORPTION AND ITS BIOLOGICAL ORIGINS
The absorption coefficient μα is defined as the probability of photon absorp- tion in a medium per unit path length (strictly speaking, per unit infinitesimal path length). It has a representative value of 0.1 cm - 1 in biological tissue. The reciprocal of \ia is referred to as the mean absorption length.
For a single absorber, the absorption cross section oa, which indicates the absorbing capability, is related to its geometric cross-sectional area og through the absorption efficiency Qa : σα = Qaog. In a medium containing many absorbers with number density Na, the absorption coefficient can be considered as the total cross-sectional area for absorption per unit volume:
ν>α = Νασα. (1.1)
Here, absorption by different absorbers is considered to be independent. According to the definition of the absorption coefficient, light attenuates as it
propagates in an absorbing-only medium according to the following equation:
— = - μ β Λ χ , (1.2)
where / denotes the light intensity and x denotes the distance along the light propagation direction. Equation (1.2) means that the percentage of light being absorbed in interval (JC, X -f- dx) is proportional to the product of \ia and dx; the negative sign is due to the decrease of / as x increases. Integrating Eq. (1.2) leads to the well-known Beer law
I(x) = / 0 βχρ( -μ β χ ) , (1.3)
where IQ is the light intensity at x — 0. Beer's law actually holds even for a tortuous path. The transmittance is defined by
rw = T ,̂ (1.4)
which represents the probability of survival after propagation over x.
6 INTRODUCTION
Wavelength (nm)
Figure 1.3. Molar extinction coefficients of oxy- and deoxyhemoglobin versus wave- length.
Optical absorption in biological tissue originates primarily from hemoglobin, melanin, and water. Hemoglobin has two forms: oxygenated and deoxygenated. Figure 1.3 shows the molar extinction coefficients—the extinction coefficient divided by ln(10) (see Section 7.3) per unit molar concentration—of oxy- and deoxyhemoglobin as a function of wavelength, where the extinction coefficient is defined as the probability of photon interaction with a medium per unit path length. Although extinction includes both absorption and scattering, absorption dominates scattering in hemoglobin. The molar extinction spectra of oxy- and deoxyhemoglobin are distinct but share a few intersections, termed isosbestic points. At these points, the absorption coefficient of an oxy- and deoxyhe- moglobin mixture depends only on the total concentration, regardless of the oxygen saturation.
The absorption coefficients of some primary absorbing biological tissue com- ponents are plotted as a function of wavelength in Figure 1.4. Melanin absorbs ultraviolet (UV) light strongly but longer-wavelength light less strongly. Even water can be highly absorbing in some spectral regions. At the 2.95-μπι water absorption peak, the penetration depth is less than 1 μτη since \xa — 12,694 cm- 1 .
The absorption coefficients of biological tissue at two wavelengths can be used to estimate the concentrations of the two forms of hemoglobin based on the following equations:
μα(λ,) = 1η(10)εοχ(λ1)Γοχ + 1η(10)εα6(λ1)Γα6, (1.5)
μα(λ2) = ln(10)8ox(X2)Cox + ln(10)ede(X2)Cde. (1.6)
Here, λι and λ2 are the two wavelengths; εοχ and ede are the known molar extinction coefficients of oxy- and deoxyhemoglobin, respectively; Cox and Cde
SCATTERING AND ITS BIOLOGICAL ORIGINS 7
W 10 3 104
Wavelength (nm)
105
Figure 1.4. Absorption coefficients of primary biological absorbers.
are the molar concentrations of oxy- and deoxyhemoglobin, respectively, in the tissue. Once Cox and Cde are obtained, the oxygen saturation (SO2) and the total concentration (CHÖ) of hemoglobin can be computed as follows:
C0 so2 =
CfJb = Cox + Cde-
(1.7)
(1.8)
This principle provides the basis for pulse oximetry and functional imaging. Angio- genesis can increase Cnb, whereas tumor hypermetabolism can decrease SO2.
1.5. SCATTERING AND ITS BIOLOGICAL ORIGINS
Scattering of light by a spherical particle of any size can be modeled exactly by the Mie theory, which reduces to the simpler Rayleigh theory if the spherical particle is much smaller than the wavelength. In a scattering medium containing many scatterers that are distributed randomly in space, photons usually encounter multiple scattering events. If scatterers are sparsely distributed (where the mean distance between particles is much greater than both the scatterer size and the wavelength), the medium is considered to be loosely packed. In this case, scat- tering events are considered to be independent; hence, single-scattering theory applies to each scattering event. Otherwise, the medium is considered to be densely packed. In this case, scattering events are coupled; thus, single-scattering theory does not apply. In this book, we consider only loosely packed scattering media. Keep in mind that one must differentiate a single coupled-scattering event (which involves multiple particles) from successive independent-scattering events (each of which involves a single particle).
8 INTRODUCTION
The scattering coefficient [is is defined as the probability of photon scattering in a medium per unit path length. It has a representative value of 100 cm- 1 in bio- logical tissue. The reciprocal of μν is referred to as the scattering mean free path.
For a single scatterer, the scattering cross section σν, which indicates the scattering capability, is related to its geometric cross-sectional area og through the scattering efficiency Qs : as = Qsog. For a medium containing many scatterers with number density Ns, the scattering coefficient can be considered as the total cross-sectional area for scattering per unit volume:
\is = Nsas. (1.9)
The probability of no scattering (or ballistic transmittance T) after a photon propagates over path length x can be computed by Beer's law:
T(x) = ε χ ρ ( - μ , χ ) . (1.10)
Optical scattering originates from light interaction with biological structures, which range from cell membranes to whole cells (Figure 1.5). Photons are scat- tered most strongly by a structure whose size matches the optical wavelength and whose refractive index mismatches that of the surrounding medium. The indices of refraction of common tissue components are 1.35-1.36 for extracellular fluid, 1.36-1.375 for cytoplasm, 1.38-1.41 for nuclei, 1.38-1.41 for mitochondria and organelles, and 1.6-1.7 for melanin. Cell nuclei and mitochondria are primary scatterers. The volume-averaged refractive index of most biological tissue falls within 1.34-1.62, which is greater than the refractive index of water (1.33).
The extinction coefficient μ,, also referred to as the total interaction coefficient, is given by
μ, = μ« + μ*. ( 1 . 1 1 )
The reciprocal of μ, is the mean free path between interaction events.
Cells
Nuclei
Mitochondria
Lysosomes, vesicles
Striations in collagen fibrils Macromolecular aggregates
Membranes
ΙΟμπι
1 μηι
0.1 μπ\
0.01 μπι
Figure 1.5. Biological structures of various sizes for photon scattering.
FLUORESCENCE AND ITS BIOLOGICAL ORIGINS 9
1.6. POLARIZATION AND ITS BIOLOGICAL ORIGINS
Linear birefringence (or simply birefringence), which is also known as double refraction, is the most important polarization property. A linearly birefringent material has dual principal indices of refraction associated with two linear polar- ization states of light (orientations of the electric field). The index of refraction for light polarization that is parallel with the optical axis of the material (e.g., the orientation of collagen fibers) is commonly denoted by ne, while the light is referred to as the extraordinary ray. By contrast, the index of refraction for light polarization that is perpendicular to the optical axis is commonly denoted by n0, while the light is referred to as the ordinary ray. If ne > n0, the birefringence is said to be positive. Conversely, if ne < n0, the birefringence is said to be negative.
Similarly, a circularly birefringent material has dual principal indices of refrac- tion associated with the left and the right circular polarization states of light; as a result, it can rotate a linear polarization. The amount of rotation depends on the properties and the concentration of the active material, the optical wavelength, and the path length. If the other parameters are known, the amount of rotation can reveal the concentration.
Collagen, muscle fibers, myelin, retina, and keratin have linear birefringence. Collagen I is intensely positively birefringent, whereas collagen III is weakly negatively birefringent. Amino acids and glucose have circular birefringence; amino acids are levorotatory (exhibit left rotation) to linearly polarized light, whereas glucose is dextrorotatory (exhibits right rotation).
1.7. FLUORESCENCE AND ITS BIOLOGICAL ORIGINS
Fluorescence has the following characteristics:
1. Fluorescence light is red-shifted (wavelength is increased or frequency is reduced) relative to the excitation light; this phenomenon is known as the Stokes shift. The primary origins include the initial vibrational relaxations and the subsequent inclined fluorescence transitions to higher vibrational energy levels of the ground state. Other origins include excited-state reac- tions, complex formations, and resonance energy transfers.
2. Emission wavelengths are not only longer than but also independent of the excitation wavelength. Although the initial excited state is related to the excitation wavelength, a vibrational relaxation to the same intermediate state terminates the memory of such a relationship.
3. Fluorescence light is incoherent even if the excitation light is coherent because the uncertain delays in the vibrational relaxations spread over more than one light period.
4. Fluorescence spectrum, when plotted against the frequency, is generally a mirror image of the absorption spectrum for the following reasons: (a) before excitation, almost all the molecules are at the lowest vibra- tional energy level of the ground state; (b) before emission, almost all the
10 INTRODUCTION
molecules are at the lowest vibrational energy level of the first excited state; (c) the least photon energy for excitation equals the greatest emis- sion photon energy; (d) the vibrational energy levels in the ground and first excited states have similar spacing structures; and (e) the probability of a ground-state electron excited to a particular vibrational energy level in the first excited state is similar to that of an excited electron returning to a corresponding vibrational energy level in the ground state.
The properties of some endogenous fluorophores are listed in Table 1.2 (where \a denotes maximum absorption wavelength; ε denotes molar extinction coefficient; \ x denotes maximum excitation wavelength; \m denotes maximum emission wavelength; Y denotes quantum yield of fluorescence). Fluorescence can provide information about the structure, dynamics, and interaction of a bioassembly. For example, mitochondrial fluorophore NADH (nicotinamide adenine dinucleotide, reduced form) is a key discriminator in cancer detection; it tends to be more abundant in cancer cells owing to their higher metabolic rate. NAD(P)H (nicoti- namide adenine dinucleotide phosphate, reduced form) has a lifetime of 0.4 ns when free but a longer lifetime of 1-3 ns when bound.
1.8- IMAGE CHARACTERIZATION
Several parameters are important in the characterization of medical images. In this section, the discussion is limited primarily to two-dimensional (2D) images, but the principles involved can be extended to one-dimensional (ID) or three- dimensional (3D) images.
When a high-contrast point target is imaged, the point appears as a blurred blob in the image because any practical imaging system is imperfect. The spatial distri- bution of this blob in the image is referred to as the point spread function (PSF). The PSF is sometimes called the impulse response (or Green's function) because
TABLE 1.2. Properties of Endogenous Fluorophores at Physiologic pH
Fluorophore
Ceroid
Collagen, elastin FAD Lipofuscin
NAD+
NADH
Phenylalanine Tryptophan Tyrosine
Xö(nm)
—
— — —
260 260 340 260 280 275
e(cm_1M_1)
—
— — —
18 x 103
14.4 x 103
6.2 x 103
0.2 x 103
5.6 x 103
1.4 x 103
λ*(ηηα)
340-395
325 450
340-395
— 290 340 — 280 —
Xw(nm)
430-460 540-640
400 515
430-460 540-540
— 440 450 280 350 300
Y
—
— — —
— — —
0.04 0.2 0.1
IMAGE CHARACTERIZATION 11
a geometric point can be represented by a spatial Dirac delta function (an impulse function). When two point targets are too close to each other, the combined blob in the image can no longer be clearly resolved into two entities. The full width at half maximum (FWHM) of the PSF is often defined as the spatial resolution. Even though an ideal geometric point target cannot be constructed or detected in reality, a point target needs only to be much smaller than the spatial resolution.
Sometimes, a line spread function (LSF), which is the system response to a high-contrast geometric line, is measured instead of a PSF. For a linear system, an LSF can be related to a PSF on the (JC, y) plane by
LSF( - /
PSF(;t, y)dy. (1.12)
Likewise, an edge spread function (ESF), which is the system response to a high- contrast semiinfinite straight edge, can be measured as well. For a linear system, an ESF can be related to an LSF as follows (Figure 1.6):
ESF(JC) -Γ J-c
LSF(x)dx\
LSF(x) = —ESF(x). dx
(1.13)
(1.14)
In a linear, stationary, and spatially translation-invariant system, image function i(r) equals the convolution of object function o(r) with point spread function PSF(r):
/ ( r ) = o ( r ) * * P S F ( r ) , (1.15)
1
0.8
0.6
0.4
0.2
n
-
—r
J - ** S
__ —
/ /
/ /
/ /
1
™ "" " Lol' ESF 1
H
, x» -"""
\
1 ^ " * * *
-0.5 0.5
Figure 1.6. Illustration of an LSF and an ESF.
12 INTRODUCTION
where r — (x, y) and ** represents 2D spatial convolution. Equation (1.15) can be expressed in several forms:
i(r)= if o(r')PSF(? -r')dr'
o(x, / ) P S F ( J C -x',y- y')dx'dy' (1.16)
o(r-r")PSF(r")dr".
Taking the 2D Fourier transformation of Eq. (1.15) yields
/(ρ,ξ) = 0(ρ,ξ)//(ρ,ξ). (1.17)
Here, p and ξ represent the spatial frequencies; / represents the image spectrum; O represents the object spectrum; and H represents the PSF spectrum, which is the system transfer function (STF). The amplitude of the STF is referred to as the modulation transfer function (MTF):
ΜΤΡ(ρ,ξ) = |//(ρ,ξ)|. (1.18)
Similarly, for an LSF, the MTF is based on the ID Fourier transformation:
(1.19) /
+oo exp(-j2npx)[LSF(x)]dx
-oo
Most imaging systems act as lowpass filters, resulting in blurring of the fine structures.
The visibility of a structure in an image depends on, among other factors, the contrast C:
Δ / C = — . (1.20)
(/)
While (/) is the average background image intensity, Δ / is the intensity variation in the region of interest (Figure 1.7).
Contrast does not represent a fundamental limitation on visualization since it can be artificially enhanced by, for example, subtracting part of the background (thresholding) or raising the intensity to some power. Statistical noise does, however, represent a fundamental limitation. The signal-to-noise ratio (SNR) is defined as
SNR= — , (1.21)
IMAGE CHARACTERIZATION 13
/ ▲
L-JT" τ
Figure 1.7. Illustration of image contrast.
where σ/ denotes the standard deviation of the background intensity, that is, the noise representing the root-mean-squared (rms) value of the intensity fluctuations.
Ultimately, the ability to visualize a structure depends on the contrast-to-noise ratio (CNR), which is defined as
Δ / C N R = — , (1.22)
07
which can be rewritten as
CNR = C SNR. (1.23)
The field of view (FOV) in an image refers to the extent of the image field that can be seen all at once. A tradeoff often exists between FOV and spatial resolution. For example, "zooming in" with a camera compromises the FOV for resolution.
The maximum imaging depth in tomography is the depth limit at which the SNR or the CNR is acceptable. A tradeoff often exists between maximum imaging depth and depth resolution. The ratio of maximum imaging depth to depth reso- lution, referred to as the depth-to-re solution ratio (DRR), represents the number of effective pixels in the depth dimension. A DRR of 100 or greater is considered to indicate high resolution in terms of pixel count.
The frame rate is defined as the number of frames of an animation that are displayed per second, measured in frames per second (fps); it measures how rapidly an imaging system produces consecutive 2D images. At or above the video rate (30 fps), the human eye cannot resolve the transition of images; hence, the animation appears smooth.
In this book, the object to be imaged is typically a scattering medium, which can be a biological tissue phantom, a sample (specimen) of biological tissue, or an insitu or in vivo biological entity. Sometimes, "sample" refers broadly to the object to be imaged.
14 INTRODUCTION
Example 1.1. Derive Eq. (1.13).
On the basis of ID convolution followed by a change of variable, we derive
E S F U ) = / LSFU - x') dx = / LSF(x")d(-x")
= j LSF(x')dx'. (1.24) J — oo
PROBLEMS
1.1 Derive the following relationship between electromagnetic wavelength λ in the unit of μπ\ and photon energy hv in electron volts (eV): \hv — 1.24, where h denotes the Planck constant and v denotes the electromagnetic frequency.
1.2 In a purely absorbing (nonscattering) medium with absorption coefficient μα, what percentage of light is left after a lightbeam propagates a length of L? Plot this percentage as a function of L in MATLAB.
1.3 In a purely absorbing (nonscattering) medium with absorption coefficient μα, derive the average length of survival of a photon.
1.4 In a purely scattering (nonabsorbing) medium with scattering coefficient μs, what percentage of light has not been scattered after the original light- beam propagates a length of L?
1.5 In a purely scattering (nonabsorbing) medium with scattering coefficient μν, derive the average length of survival of a photon.
1.6 In a scattering medium with absorption coefficient μα and scattering coef- ficient μ5, what percentage of light has survived scattering and absorption after the original lightbeam propagates a length of L? Of the percentage that has been absorbed and scattered, what is the percentage that has been absorbed?
1.7 In MATLAB, draw a 2D diagram to simulate a random walk by follow- ing the subsequent steps: (1) start the point at (0,0); (2) sample a random number x\ that is evenly distributed in interval (0,1]; (3) determine a step size by s = lOOlnQcj); (2) sample a random number X2 that is evenly dis- tributed in interval (0,1]; (4) determine an angle by α = 2π*2; (5) move the point by step size s along angle a; (6) repeat steps 2-5 20 times to obtain a trajectory; (7) repeat steps 1-6 3 times to trace multiple trajec- tories.
1.8 Derive the oxygen saturation SO2 and the total concentration of hemo- globin CHÖ based on Eqs. (1.5) and (1.6).
FURTHER READING 15
1.9 Download the data for the molar extinction coefficients of oxy- and deoxy- hemoglobin as a function of wavelength from the Web (URL: http://omlc. ogi.edu/spectra/) and plot the two curves in MATLAB.
1.10 Download the data for the molar extinction coefficients of oxy- and deoxy- hemoglobin as a function of wavelength from the Web (URL: http://omlc. ogi.edu/spectra/). Download the data for the absorption coefficient of pure water as a function of wavelength as well. Using physiologically repre- sentative values for both oxygen saturation SO2 and total concentration of hemoglobin CHÖ* compute the corresponding absorption coefficients. Plot the three absorption spectra on the same plot in MATLAB. Identify the low-absorption near-IR window that provides deep penetration.
READING
Drezek R, Dunn A, and Richards-Kortum R (1999): Light scattering from cells: Finite- difference time-domain simulations and goniometric measurements, Appl. Opt. 38: 3651-3661. [See Section 1.5, above (in this book).]
Jacques SL (2005): From http://omlc.ogi.edu/spectra/ and http://omlc.ogi.edu/classroom/. (See Sections 1.5 and 1.6, above.)
Richards-Kortum R and Sevick-Muraca E (1996): Quantitative optical spectroscopy for tissue diagnosis, Ann. Rev. Phys. Chem. 47: 555-606. (See Section 1.7, above.)
Wang LHV and Jacques SL (1994): Animated simulation of light transport in tissues. Laser-tissue interaction V, SPIE 2134: 247-254. (See Section 1.2, above.)
Wang LHV, Jacques SL, and Zheng LQ (1995): MCML—Monte Carlo modeling of pho- ton transport in multi-layered tissues, Comput. Meth. Prog. Biomed. 47(2): 131-146. (See Section 1.2, above.)
Wang LHV (2003): Ultrasound-mediated biophotonic imaging: A review of acousto- optical tomography and photo-acoustic tomography, Disease Markers 19(2-3): 123-138. (See Section 1.1, above.)
FURTHER READING
Hecht E (2002): Optics, Addison-Wesley, Reading, MA. Lakowicz JR (1999): Principles of Fluorescence Spectroscopy, Kluwer Academic/Plenum,
New York. Macovski A (1983): Medical Imaging Systems, Prentice-Hall, Englewood Cliffs, NJ. Mourant JR, Freyer JP, Hielscher AH, Eick AA, Shen D, and Johnson TM (1998): Mech-
anisms of light scattering from biological cells relevant to noninvasive optical-tissue diagnostics, Appl Opt. 37(16): 3586-3593.
Shung KK, Smith MB, and Tsui BMW (1992): Principles of Medical Imaging, Academic Press, San Diego.
Tuchin VV (2000): Tissue Optics: Light Scattering Methods and Instruments for Medical Diagnosis, SPIE Optical Engineering Press, Bellingham, WA.
Welch AJ and van Gemert MJC (1995): Optical-Thermal Response of Laser-Irradiated Tissue, Plenum Press, New York.
CHAPTER 2
Rayleigh Theory and Mie Theory for a Single Scatterer
2.1. INTRODUCTION
Both the Rayleigh and the Mie theories, which are based on the Maxwell equations, model the scattering of a plane monochromatic optical wave by a single particle. Even if the particle size is much greater than the optical wave- length, the wave is diffracted by the particle with an effective cross section that is usually different from the geometric cross section. The Rayleigh theory is applicable only to particles that are much smaller than the optical wavelength, whereas the Mie theory is valid for homogeneous isotropic spheres of any size. The Mie theory reduces to the Rayleigh theory when the particle is much smaller than the wavelength.
2.2. SUMMARY OF RAYLEIGH THEORY
The Rayleigh theory (Appendix 2A) models the scattering of light by particles that are much smaller than the optical wavelength. Figure 2.1 shows the spherical polar coordinates used for light scattering. The incident light propagates along the z axis; the scatterer is located at the origin; field point P is located at (r, θ, φ). The distribution of the scattered light intensity for unpolarized incident light is given by
( l+cos29)fc4 |q | 2 , /<>, Θ) = —2 /0. (2.1)
Here, a denotes the polarizability of the particle; /o denotes the incident light intensity; k denotes the propagation constant (also referred to as the magnitude of the wavevector or the angular wavenumber) in the background medium. We have
k=2-^, (2.2)
Biomedical Optics: Principles and Imaging, by Lihong V. Wang and Hsin-i Wu Copyright © 2007 John Wiley & Sons, Inc.
17
18 RAYLEIGH THEORY AND MIE THEORY FOR A SINGLE SCATTERER
* y
Figure 2.1. Spherical polar coordinates used for light scattering, where Θ denotes the polar angle and φ denotes the azimuthal angle.
where nt, denotes the refractive index of the background medium and λ denotes the wavelength in vacuum. Substituting Eq. (2.2) into Eq. (2.1), we obtain /(r, Θ) a l/λ4. This strong wavelength dependence explains the blue sky in broad day- light because blue light is scattered much more strongly than red light.
The scattering cross section is given by
Snk4\a\2 (2.3)
The polarizability of a sphere with radius a is given by
4i + 2 (2.4)
where nre\ is the relative refractive index of the particle:
"rel = — · nb
(2.5)
Here, ns is the refractive index of the sphere and nt, is the refractive index of the background. Substituting Eq. (2.4) into Eq. (2.3), we obtain
8πα x 2 V 4 a. =
<ι - 1 Λ«ι + 2
(2.6)
where size parameter x is defined as
x = ka. (2.7)
NUMERICAL EXAMPLE OF RAYLEIGH THEORY 19
Substituting Eqs. (2.7) and (2.2) into Eq. (2.6), we obtain os oc a6/\4. From Eq. (2.6), the scattering efficiency is given by
0* = ΊΓ Sx4 ' * ' l2 »rel ~ 1
i L + 2 'rel (2.8)
The scattering efficiency depends only on x and ηκ\. If nre\ is close to unity, Eq. (2.8) reduces to
32JC4
Qs = ^ - K e i - l | 2 · (2.9)
Note that rcrei can be complex, in which case the imaginary part is responsible for absorption.
2.3. NUMERICAL EXAMPLE OF RAYLEIGH THEORY
The Rayleigh theory can compute scattering cross section os and scattering effi- ciency Qs. As an example, the following parameters are given:
1. Diameter of sphere: 2a = 20 nm 2. Wavelength in vacuum: λ = 400 nm 3. Refractive index of sphere: ns = 1.57 4. Refractive index of background: n^ — 1.33 5. Specific weight of sphere: ρ̂ = 1.05 g/cm3
6. Specific weight of background: p^ = 1.00 g/cm3
7. Concentration of spheres in background by weight: Cwt = 1 x 10~5
We compute in SI units as follows:
1. Propagation constant in background medium: k = Inn^/X = 2.09 x 107 m-1.
2. Relative refractive index of sphere: nre\ — ns/ni, — 1.18. 3. Size parameter: x = ka = 0.209. 4. From Eq. (2.6), σν = 2.15 x 10~20 m2. 5. From Eq. (2.8), Qs = 6.83 x 10~5. 6. Compute the number density of scatterers Ns. For a sphere, the mass density is
Ps = ms/Vs, where ms denotes the mass and Vs denotes the volume—Vs = (4/3)πα3. For the background, the mass density is p^ = my, j V&, where m^ is the total mass of the background and V& is the total volume of the background.
2 0 RAYLEIGH THEORY AND MIE THEORY FOR A SINGLE SCATTERER
The concentration by weight is Cwt — (NsVb)ms/mi,. Therefore, we have Ns = Cwt(9h/ps)/Vs = 2.27 x 1018 m~3.
7. From Section 1.5, μ5 = Nsos = 0.0488 m _ l .
We can implement the Rayleigh theory with the following MATLAB script:
% Rayleigh scattering % Use SI units
diameter=input('Diameter of sphere (e.g., 20 nm):')*1e-9; radius=diameter/2; lambda=input('Wavelength (e.g., 400 nm):')*1e-9; n_sphere=input('Refractive index of sphere (e.g., 1.57):'); n_background=input('Refractive index of background (e.g., 1.33):'); w_sphere=input('Specific weight of sphere(e.g., 1.05 g/cc):')*1e3; w_background=input("Specific weight of background(e.g., 1 g/cc):')*1e3; concentration=input('Concentration by weight (e.g., 1e-5):');
k=2*pi*n_background/lambda x=k*radius n_rel=n_sphere/n_background
Qs = 8*x A 4 /3*abs( (n_rer2 - 1 ) / (n_re l A 2 + 2) )Λ2 sigma_s=Qs*pi*radiusΛ2
vol_sphere = 4*p i /3* rad ius"3 N_s=concentration*w_background/(vol_sphere*w_sphere) mu_s=N_s*sigma_s
% Output resu l t s {'wavelength(nm) ' ,'Qs ( - ) V m u s ( /cm) ' ; lambda*1e9, Qs, mu_s/1e2}
2.4. SUMMARY OF MIE THEORY
The Mie theory (Appendix 2B) models the scattering of light by a spherical particle of any radius a. The sphere is made of homogeneous and isotropic material and is irradiated by a plane monochromatic wave. In practice, we can treat the incident wave as a plane wave if the wavefront is much wider than both the wavelength and the particle size.
Application of the Mie theory is straightforward. The scattering efficiency Qs and the scattering anisotropy g (defined by g = (cosG)) can be computed as follows:
2 °°
Qs = -2 Σ ( 2 / + 1}(ι^ι2 + ι^/i2)' (2·10)
^ ^ Ϊ Σ [ Τ Τ Γ R e ^ > + * ' * ' + > ) + mh^01^} (2·η)
NUMERICAL EXAMPLE OF MIE THEORY 21
Size parameter x = ka. Coefficients ai and b[ are given by
Ψ/ΟΟΨ/00 - Ππ;ΐΨ/Ο0Ψ/(*) 0/ =
bi =
ΨϋΟζ/(*)-Λι*ιΨ/ϋΟζ(*) (2.12)
πι*ιΨί(3θζ/(*)-Ψ/ϋΟζί(*) '
where superscript prime denotes first-order differentiation and size parameter y is defined by
2nnsa y = nrdx = . (2.13)
The Riccati-Bessel functions are defined by
' π ζ \ ΐ / 2
< π ζ \ ΐ / 2
Ψ/(ζ) - zji(z) = ( y ) ' 7/+i/2(z) - St(z), (2.14)
X/(z) = -zyiiz) = - ( y ) F/+i/2(z) = Q(z), (2.15)
ζ/ω = Ψ/ω + ιχ/ω = zApfe) = ( y ) , / 2 #/(+ί/2ω. (2·16> Here, / and I + \ are the orders; ji( ) and y/( ) denote the spherical Bessel functions of the first and second kind, respectively; 7/( ) and K/( ) denote the (2) Bessel functions of the first and second kind, respectively; h) {) denotes the spherical Hankel function of the second kind; //, }{ ) denotes the Hankel function of the second kind; and 5/() and C/() are alternative symbols that are commonly used. Note that
A/(2)() = 7/( ) - i > ( ) , (2.Π)
/ / / 2 ) ( ) = J / O - i T / O . (2.18)
If ttrei is complex, the extinction instead of scattering efficiency that also contains a component representing absorption can be computed.
2.5. NUMERICAL EXAMPLE OF MIE THEORY
For a spherical particle of any size, the Mie theory can compute the scattering efficiency Qs, the scattering anisotropy g, and the scattering cross section os. For a scattering medium, we can further compute the scattering coefficient μ5 and the reduced scattering coefficient μ^. When the Mie theory is implemented in MATLAB or another high-level computer language, the following derivative
2 2 RAYLE1GH THEORY AND MIE THEORY FOR A SINGLE SCATTERER
identities are useful:
J,'(z) = --Ji(z) + Ji-dz). (2 z
y;(z) = --Yi(z) + Yi-\(z)· (2
An example MATLAB script is shown below:
% Mie theory % Use SI units
diameter = input('Diameter of sphere (e.g., 579 nm):')*1e-9; radius = diameter/2; lambda = input('Wavelength (e.g., 400 nm):')*1e-9; n_s = input('Refractive index of sphere (e.g., 1.57):'); n_b = input('Refractive index of background (e.g., 1.33):'); w_s = input('Specific weight of sphere(e.g., 1.05 g/cc):')*1e3; w_b = input('Specific weight of background(e.g., 1.0 g/cc):')*1e3; concentration = input('Concentration by weight (e.g., 0.002):');
k = 2*pi*n_b/lambda x = k*radius n_rel = n_s/n_b y = n_rel*x
% Calculate the summations err = le-8; Qs = 0; gQs = 0; for n = 1:100000
Snx = sqrt(pi*x/2)*besselj(n+0.5,x); Sny = sqrt(pi*y/2)*besselj(n+0.5,y); Cnx = -sqrt(pi*x/2)*bessely(n+0.5,x); Zetax = Snx+i*Cnx;
% Calculate the first-order derivatives Snx_prime = - (n/x)*Snx+sqrt(pi*x/2)*besselj(n-0.5,x); Sny_prime = - (n/y)*Sny+sqrt(pi*y/2)*besselj(n-0.5,y); Cnx_prime = - (n/x)*Cnx-sqrt(pi*x/2)*bessely(n-0.5,x); Zetaxprime = Snx_prime + i*Cnx_prime;
an_num = Sny_prime*Snx-n_rel*Sny*Snx_prime; an_den = Sny_prime*Zetax-n_rel*Sny*Zetax_prime; an = an_num/an_den;
bn_num = n_rel*Sny_prime*Snx-Sny*Snx_prime; bn_den = n_rel*Sny_prime*Zetax-Sny*Zetax_prime; bn = bn_num/bn_den;
Qs1 = (2*n+1)*(abs(an)^2+abs(bn)A2); Qs = Qs+Qs1;
APPENDIX 2A. DERIVATION OF RAYLEIGH THEORY 2 3
if Π > 1 gQsl = (n-1)*(n+1)/n*real(an_1*conj(an)+bn_1*conj(bn))...
+(2*n-1)/((n-1)*n)*real(an_1*conj(bn_1)); gQs = gQs+gQs1;
end
an_1 = an; bn_1 = bn;
if abs(Qs1)<(err*Qs) & abs(gQs1)<(err*gQs) break;
end end
Qs = (2/xA2)*Qs; gQs = (4/x"2)*gQs; g = gQs/Qs;
vol_s = 4*pi/3*radiusA3 N_s = concentration*w_b/(vol_s*w_s) sigma_s = 05*ρί*Γ3αίυ8Λ2; mu_s = N_s*sigma_s
mu_s_prime = mu_s*(l-g);
% Output resul ts {'wavelength(nm)','Qs ( - ) V g ( - ) V m u s ( /cm)' , 'mus_prime(/cm)' ; . . .
lambda*1e9,Qs,g,mu_s*1e-2,mu_s_prime*1e-2}
Below, we present a numerical example with the following inputs:
1. Diameter of sphere: 2a = 579 nm 2. Wavelength: λ = 400 nm 3. Refractive index of sphere: ns == 1.57 4. Refractive index of background: η\, = 1.33 5. Specific weight of sphere: ps = 1.05 g/cm3
6. Specific weight of background: p^ = 1.0 g/cm3
7. Concentration of spheres in the solution by weight: Cwt = 0.002
The MATLAB script gives the following outputs: Qs = 2.03, g = 0.916, \is = 100 cm"1, and μ̂ = 8.40 cm"1.
In Figure 2.2, Qs and g, which are calculated using a modified version of the MATLAB program presented above, are plotted against ka, where ns = 1.40 and nh = 1.33. Note that g is less than unity even for large x values.
APPENDIX 2A. DERIVATION OF RAYLEIGH THEORY
The Rayleigh theory is derived here. The polarizability a is defined as the pro- portionality constant between the induced oscillating dipole moment ρβχρ(ιωί)
2 4 RAYLEIGH THEORY AND MIE THEORY FOR A SINGLE SCATTERER
(b) x - ka
Figure 2.2. (a) Qs versus ka, where the dashed line represents the asymptote from the Rayleigh theory; (b) anisotropy g versus ka.
and the electric field of the incident linearly polarized wave Εο^χρ(ίωί), where ω denotes angular frequency and t denotes time:
p = VLEQ. (2.21)
On the basis of the dipole radiation theory for a <^ λ, the electric field of the scattered wave in the far field (r > λ) is
k2puxvy ikr t, = e (2.22)
APPENDIX 2A. DERIVATION OF RAYLEIGH THEORY 2 5
where γ is the angle between the directions of the scattered light propagation and the dipole oscillation. From Eq. (2.21) and (2.22), the scattered light intensity is
9 £4|/?|2sin2 γ £4 |a |2sin2y 7 &4|al2sin2y / = \E\2 = — ^ - Y- = —L» L|£0|2 = _Ji L/0f ( 2 2 3 )
where IQ is the incident light intensity. The incident light is assumed to propagate in the positive z direction. Thus, its
electric field lies on the xy plane. We express both the unit vector of polarization p and the unit vector that points to the field point P from the scatterer at the origin r in terms of the unit vectors of the Cartesian coordinates, (ex, ey, ez):
p = ex cos ψ + ey sin ψ, (2.24)
r = ex sin Θ cos φ -f ey sin Θ sin φ -f ez cos Θ. (2.25)
Thus, we obtain
cosy = p · r = sinθcos(φ — ψ), (2.26)
which leads to
sin2 Y = 1 - cos2 γ = 1 - sin2 Θ c o s 2 ^ - ψ). (2.27)
If the incident light is unpolarized, sin2 y needs to be averaged over angle ψ:
(sin2 y) = 1 - - sin2 Θ = - (1 + cos2 Θ). (2.28)
Hence
£ 4 | a | 2 ( l+cos 2 0) /(r, Θ) - 2 r 2 -/o, (2.29)
which is Eq. (2.1). The total scattering cross section σ̂ is defined as
os = - I 7(Γ,θ)Γ2ί/Ω. (2.30)
Evaluating Eq. (2.30) with differential solid angle element dQ = sin θί/θ^φ, we obtain
a , f l + c o s 2 0 8π^4|α|2 σ, = 2πΓ |α | ζ / sinOJO = — , (2.31) t4|a|2 Γ Jo
which is Eq. (2.3).
2 6 RAYLEIGH THEORY AND MIE THEORY FOR A SINGLE SCATTERER
APPENDIX 2B. DERIVATION OF MIE THEORY
The Mie theory is an exact solution of the Maxwell equations for a plane monochromatic electromagnetic wave scattered by a homogeneous sphere of radius a with an isotropic relative index of refraction nrt\. An abbreviated deriva- tion of the Mie theory is presented here. The general idea is to (1) solve the Maxwell equations inside and outside the sphere with undetermined coefficients in the solution expansions and (2) determine these coefficients by applying bound- ary conditions on the spherical surface. The Mie theory for a cylindrical scatterer is beyond the scope of this book.
We assume that all conditions for the following Maxwell equations are met:
V · E = 0, (2.32)
V · B = 0, (2.33)
r) R
S/xE = , (2.34) dt
V x * = ( - ) - . (235)
Here, E and B are the electric and magnetic fields, respectively; n is the refractive index of the medium; c is the speed of light in vacuum.
2B.1. Vector and Scalar Wave Equations
The following vector wave equation for both E and B can be obtained from Eqs. (2.32)-(2.35):
/n\2 32A
= Ü w ,236) V2A where A represents either E or B. Each vector component satisfies the following scalar wave equation:
ν2ψ = (ö) w i237)
Example 2.1. Derive Eq. (2.36) for E.
Operating Vx from the left on both sides of Eq. (2.34), we obtain V x (V x E) = -3 (V x B)/dt. From V x (V x E) = V(V · E) - V2E = - V 2 £ , we get
9 - dV x B V2E = . (2.38) dt
Substituting Eq. (2.35) into Eq. (2.38), we obtain Eq. (2.36) for E.
APPENDIX 2B. DERIVATION OF MIE THEORY 2 7
2B.2. Solution of Scalar Wave Equation
The standard procedure for solving Eq. (2.37) is by separation of the variables. Let Ψ(χ, 0 = X(x)T(t), where x represents the spatial coordinates. Substituting this expression into Eq. (2.37) and dividing both sides by XT, we obtain
V2X /n\2 1 /d2T\
The left-hand side of this equation is a function of only x, whereas the right-hand side is a function of only t\ thus, the two sides must equal a constant, which is termed a separation variable constant', this constant is denoted by — β2. The time-dependent part can be expressed as
(d2T\ + ω2Τ = 0, (2.40)
where ω = ßc/n. Since ω is the angular frequency of the wave, we have β = k = nko, where k and ko are the propagation constants in the medium and in vacuum, respectively. The solution of Eq. (2.40) is
C O S ( 0 / ) . (2.41) sinooi J
The pair of braces represents a linear combination of the two functions inside. The spatially dependent part is a scalar Helmholtz equation
V2X + k2X = 0, (2.42)
which can be expressed in spherical coordinates as
- x — [r2— + - (sinO— I + = X + k2X = 0. r2dr\ drj r2 sind dd \ 3Θ/ r2 sin2 Θ d<\>2 J
(2.43)
Letting X = /?(λ-)Θ(θ)Φ(φ), substituting it into Eq. (2.43), and then dividing the equation by /?(Γ)Θ(Θ)Φ(Φ), we obtain
1 a / 2dR\ 1 3 / . dS\ 1 32Φ f 2 Λ -Ϊ \r2— ) + -T ( s inG— 1 + = T + * = 0. r2Rdr\ dr ) r 2 0 s in0aOV 3Θ / Γ2Φδίη2θ3φ2
(2.44)
Multiplying Eq. (2.44) by r2 sin2 Θ, we see that the third term on the left-hand side depends only on φ, whereas the other terms are independent of φ. Thus, we let
1 32Φ φ " ^
2 constant = — m . (2.45)
2 8 RAYLEIGH THEORY AND MIE THEORY FOR A SINGLE SCATTERER
Thus, we have
92Φ 3φ2
which has the following solutions:
+ m2cD = 0, (2.46)
Φ cos(m(|)) sin(m$) I * (2.47)
From condition Φ(0) = Φ(2π), we have m = 0, ± 1 , ±2, The r0-dependent part from Eq. (2.44) is
l a / 7dR\ i a / . a © \ —^ r2— I + -^ sinG— ) - - r2Rdr\ dr r 2 0 s i n 0 a 0 V 9Θ / r 2sin29
+ r = 0. (2.48)
Multiplying this equation by r2, we see that the first and fourth terms depend only on r, whereas the other terms are independent of r. Letting
i a 0sin0äÖ
we have
s i n0— -
r2 dr \ dr
—7Γ- = constant = —/(/ + 1), sin20
+ .2 K / + D
R = 0
(2.49)
(2.50)
and
d-z2)- d2@ Λ ^ Θ Γ / ( /+ ! ) - m dz2 dz \ \-z2
where z = cos θ. The solutions of Eq. (2.50) are
1θ = 0, (2.51)
R jl(kr) | _ | ji(nk0r) yiihr) I I yi(nkQr)
(2.52)
Here, / is an integer; ji(x) and yi(x) are the spherical Bessel functions of the first and second kinds, respectively, given by
JiM = J^JIM\/2)W,
yiM = J 2^Yi+(\/2)(x)-
(2.53)
(2.54)
APPENDIX 2B. DERIVATION OF MIE THEORY 2 9
The solutions of Eq. (2.51) are
Θ ~ ( ^ ( ; 0 8 θ > } , (2.55) l Ö/,m(cOS0) j
where P/m (cosΘ) and Q\m (cosΘ) are the associated Legendre polynomials of the first and second kinds, respectively.
Recalling that Ψ ( ί , 0 = X(x)T(t) and X = / ? (Γ)Θ(Θ)Φ(Φ) , we obtain the solution of the scalar wave equation [Eq. (2.37)] in spherical polar coordinates (see Figure 2.1):
ψ _ | cos(ü>f) ] ί cos(/w<|>) I | ji(nk0r) J J P/,m(cosO) | { sin(ü>r) J [ sin(/w<|>) J { yi(nk0r) J { ß/,m(cos9) J
We use expO'cof) from the linear combination of the time-dependent factors for the phasor expressions of the waves. Of course, we can also use exp(—ι'ωί) consistently. Function Qi m (cosO) is dropped off because it has singularities at Θ € {0, π}. For a wave inside the sphere, function ylinker) is dropped off because ylinker) -» —oo when r -> 0. For an outgoing spherical wave outside
(2) the sphere, the spherical Hankel function of the second kind h\ {nk^r) is chosen because of its asymptotic behavior:
(2)
h) (nkor) — exp(—ink$r). (2.57) nkor
Therefore, we use
*~«HÄ)H*?w>},,-<a,,')· ("8) If exp(—ΐωί) is used, the spherical Hankel function of the first kind h) \nkor) should be used instead for the outgoing spherical wave.
2B.3. Theorem Relating Solutions of Scalar and Vector Wave Equations
From Eq. (2.37), the time-independent scalar wave equation (the scalar Helmholtz equation) is given by
V2X + rc2^X = 0, (2.59)
which has the following solutions:
cos(m(|)) 1 [ ji(nkor) } ( 4 ^ ( 2 · 6 0 ) sin(ra<t>) I I hi(nk0r)
3 0 RAYLEIGH THEORY AND MIE THEORY FOR A SINGLE SCATTERER
Similarly, from Eq. (2.36), the time-independent vector equation (the vector Helmholtz equation) is given by
V2A+n2klA = 0. (2.61)
The solution to the vector Helmholtz equation can be found from the following theorem. If X satisfies the scalar wave equation [Eq. (2.59)], vectors Μχ and Νχ, defined by
Mx = V x (rX) and nk0Nx = V x Mx, (2.62)
must satisfy the vector Helmholtz equation [Eq. (2.61)], where Μχ and Νχ are related by
nk0Mx = V x Nx. (2.63)
The full components of Μχ and Νχ are
Μθ
Μφ
nkoNr
nk0Ne
nk0N^
Equations (2.67) can also be expressed as
1 d / dX\ 1 d2X nk0Nr = — - s i n 0 — - — - j - - ^ . (2.70)
If u and v are two solutions to the scalar wave equation, and Mu, Nu, Mv, and Nv are the derived vector fields, the spatial components of the solutions of the vector wave equations are
E = Mv + iNu and B = -(-Mu + iNv). (2.71) c
o, 1 d(rX)
rsinG 3φ 1 d(rX)
~~r 9Θ ' i a * 2HrX)~ r
1 32(rX)
~r 3r39 ' 1
r sin Θ d2(rX)
+ n2k^r X,
(2.64)
(2.65)
(2.66)
(2.67)
(2.68)
(2.69)
APPENDIX 2B. DERIVATION OF MIE THEORY 3 1
The components of E and B can thus be written in terms of the scalar solutions u and v and their first and second derivatives.
Example 2.2. Show that vector Μχ defined by Eq. (2.62) satisfies the vector wave equation given by Eq. (2.61).
Multiplying r and then operating Vx from the left through Eq. (2.59), we obtain
V x (?V2X) + n2klMx = 0. (2.72)
From the vector identity
V[V · (VX x r)] = r(V x VX) - VX(V x r) = 0, (2.73)
we derive
V x V x (VX x r) = V[V · (VX x r)] - V2(VX x r)
= -V 2 (VX x r) . (2.74)
Also
V x [V x (VX x r)] = V x [(r · V)VX - ?V2X - (VX · V)r + VX(V · r)]
= V x [(r · V)VX] - V x [rV2X] - V x [(VX · V)r]
+ V x [ V X ( V . r ) ] , (2.75)
The first, third, and fourth terms on the right-hand side can be evaluated as follows:
r · V = r— => V x [(r · V)VX] = V x ( r —VX ) = 0, (2.76) Or \ dr )
(VX .V)? = V X 4 V x [(VX · V)?] = V x VX = 0, (2.77)
V · r = 3 => V x [VX(V · r)] = 0. (2.78)
Therefore, Eq. (2.75) becomes
V x [V x (VX x r)] = - V x (rV2X). (2.79)
From Eqs. (2.74) and (2.79), we obtain
V2(VX x r) = V x (?V2X). (2.80)
We also obtain
V2MX = V2[V x (rX)] = V2[VX x r 4- X(V x r)] = V2(VX x r). (2.81)
3 2 RAYLEIGH THEORY AND MIE THEORY FOR A SINGLE SCATTERER
Merging Eqs. (2.72), (2.80), and (2.81) yields
V2Mx+n 2klMx = 0, (2.82)
which shows that Μχ is a valid solution of the vector wave equation.
Example 2.3. Verify the relationships in Eqs. (2.64)-(2.66).
From the vector operation of curl in spherical coordinates
V X V :
we obtain
f
r sin0 9 9Ve — (snGVd.) 9Θ φ 9φ
φ Γ 9 dVr~ + - - ( r V e ) - - ^ r \_dr 9Θ _
Θ + -
r
'
M x = V x (rX) ■■ Θ
sin 3X
Θ9φ
" 1 3Vr _sin9 9φ
.ax - Φ — ,
- έ(^φ)] (2.83)
(2.84)
which can be rewritten as Eqs. (2.64)-(2.66).
2B.4 Solution for Coefficients from Boundary Conditions
The origin of the coordinates is set at the center of the spherical scatterer. The positive z axis is set along the propagation direction of the incident wave. The x axis is set in the plane of electric vibration of the linearly polarized incident wave. Solutions u and v are chosen in the following forms:
1. For the incident plane wave outside the spherical particle, we have
°° 2/ + 1 u =η^6χρ( /ωΟοο8φ^(- / ) / —-—-P/ , i (cose)7 / ( / : r ) , (2.85)
/= i /(/ + 1)
2/ + 1 v = η^χρ^ωΟύη^Υ^^ί)1 -—— P/j(cosO)y/(*r), (2.86)
/=!
where k represents the propagation constant of the background. 2. For the scattered wave outside the spherical particle, we have
°° 2/ + 1 u = rib exp(io)t) cosφ 22, ~~ai^~^ fyiicosOJAj (fcr), (2.87)
/=! / ( /+!) 00 2/4-1
v = n^exp(i^0 s i n φ ^ ] - & / ( - / ) ' PlA[cosü)hf\kr), (2.88) / = i
APPENDIX 2B. DERIVATION OF MIE THEORY 3 3
where a\ and b\ are coefficients to be determined. 3. For the wave inside the spherical particle, we have
00 2/ + 1 u = η5εχρ(ίωί)οο$φΣαι(-ί)
1 ——--Pij (cos d)jt(nre\kr), (2.89)
°° 2/ + 1 υ = π,βχρίιωθβίηφ Υ ] φ(-ί)1 ——— P/,i(cos0).//(ftrei*r), (2.90)
where Q and d/ are also coefficients to be determined.
To determine these coefficients, we substitute solutions E and B from Eqs. (2.85) -(2.90) into the following boundary conditions:
än x (E0 - Ei) = unx (B0 - Bi) = 0. (2.91)
Here, un is the unit vector perpendicular to the boundary surface; subscripts o and i represent the outside and inside, respectively. The obtained coefficients a\ and b\ are shown in Eq. (2.12).
2B.5 Scattering Efficiency and Anisotropy
Substituting the asymptotic expression of the spherical Hankel function of the second kind into Eqs. (2.87) and (2.88) leads to
exp(—ikr -f ίωί) v-> 2/ -f 1 u = -i-^—, c o s φ Γ ο , — — Ρ,,i(cosΘ), (2.92)
kr ^ /(/ + 1)
exp(—ikr + ιωί) ^ 2/ -f- 1 υ = - ί F . - s i ^ V ^ / — — f t , , ( c o s e ) . (2.93)
fcr ^ /(/ + 1) The resulting field components are
exO(—ikr + ιωί) ΕΘ = Β^ = ~i
FV y - cos φ52(θ), (2.94)
exp(—/fcr -f /ωί) - £ φ = £θ = - t y βίηφΑΚΘ). (2.95)
kr The amplitude functions are given by
2/4-1 5ι(β) = ΣΖ 77ϊ^-ΤΤ[α'π'(°08θ> + *ix/(cose)], (2.96)
S2Q) = Z^JTTT—[&/^(cose)+a,T/(cose)], (2.97)
3 4 RAYLEIGH THEORY AND MIE THEORY FOR A SINGLE SCATTERER
where
π,(cosΘ) = J . Λ , (2.98) sinO
x/(cose) = ^ P / j ( c o s e ) . (2.99) αυ
The scattering efficiency Qs, defined as the ratio of the scattering cross-sectional area as to the physical cross-sectional area πα2, can be expressed in terms of the amplitude functions:
Qs = ^ = - ^ f ( |5,(θ) |^θ82φ + |52(θ)|28ίη2φ)^Ω. (2.100)
Following the integration over φ, Eq. (2.100) becomes
Qs = \ f (|5,(θ)|2 + |52(θ)|2)8ίηθ^θ. (2.101) x2 Jo
Likewise, the scattering anisotropy g — (cos Θ) can be evaluated by
gQs = \ f (|51(0)|2 + |52(0)|2)cosesineJ6. (2.102)
The integrations over Θ in Eqs. (2.101) and (2.102) can be completed using the orthogonality relations of π/ and x/, which yields Eqs. (2.10) and (2.11).
PROBLEMS
2.1 Show that
2.2 Show that
2.3 Show that
Sji(kr) k . 0 = -zr—rVji-dkr) - (I + l)ji+\(kr)]. or 21 + 1
= —ψ/(ζ) + ψ/-ι(ζ). dz z
— — = — ζ / ( ζ ) + ζ / - ι ( ζ ) . dz z
2.4 Derive the net radiation force exerted by light on a spherical particle. (Hint: The photon momentum equals the photon energy divided by the speed of light.)
2.5 Plot the angular distribution of the scattered light in the Rayleigh scattering regime and calculate the scattering anisotropy.
FURTHER READING 3 5
2.6 Implement the Mie theory using an alternative program to calculate the scattering efficiency, scattering anisotropy, scattering coefficient, and reduced scattering coefficient of spherical particles suspended in a back- ground medium. (a) Use the example in Section 2.5 to verify your program. (b) Duplicate Figure 2.2. (c) Summarize the asymptotic dependence of the scattering cross section
on the particle size as ka varies. 2.7 Derive coefficients a\ and b\ shown in Eqs. (2.12) by completing the
derivations in Appendix 2B. 2.8 (a) Extend the Mie theory to absorbing scatterers.
(b) Extend the Rayleigh theory to absorbing scatterers.
READING
Van de Hülst HC (1981): Light Scattering by Small Particles, Dover Publications, New York. (See Sections 2.2, 2.4, 2.5, and 2.5, above.)
FURTHER READING
Bohren CF and Huffman DR (1983): Absorption and Scattering of Light by Small Particles, Wiley, New York.
Ishimaru A (1997): Wave Propagation and Scattering in Random Media, IEEE Press, New York.
CHAPTER 3
Monte Carlo Modeling of Photon Transport in Biological Tissue
3.1. INTRODUCTION
Photon transport in biological tissue can be numerically simulated by the Monte Carlo method. The trajectory of a photon is modeled as a persistent random walk, with the direction of each step depending on that of the previous step. By contrast, the directions of all the steps in a simple random walk are independent. By tracking a sufficient number of photons, we can estimate physical quantities such as diffuse reflectance.
3.2. MONTE CARLO METHOD
Although widely used in various disciplines, the Monte Carlo method defies a succinct definition. Here, we adopt the description provided by Lux and Koblinger (1991):
In all applications of the Monte Carlo method, a stochastic model is constructed in which the expected value of a certain random variable (or of a combination of several variables) is equivalent to the value of a physical quantity to be determined. This expected value is then estimated by averaging multiple independent samples representing the random variable introduced above. For the construction of the series of independent samples, random numbers following the distribution of the variable to be estimated are used.
It is important to realize that the Monte Carlo method estimates ensemble- averaged quantities.
The Monte Carlo method offers a flexible yet rigorous approach for simulating photon transport in biological tissue. An ensemble of biological tissues is mod- eled for the averaged characteristics of photon transport; the ensemble consists of all instances of the tissues that are microscopically different but macroscop- ically identical. Rules are defined for photon propagation from the probability
Biomedical Optics: Principles and Imaging, by Lihong V. Wang and Hsin-i Wu Copyright © 2007 John Wiley & Sons, Inc.
37
3 8 MONTE CARLO MODELING OF PHOTON TRANSPORT IN BIOLOGICAL TISSUE
distributions of, for example, the angles of scattering and the step sizes. The statistical nature requires tracking a large number of photons, which is compu- tationally time-consuming. Multiple physical quantities can be simultaneously estimated, however.
In this chapter, photons are treated as waves at each scattering site but as clas- sical particles elsewhere. Coherence, polarization, and nonlinearity are neglected. Structural anisotropy—not to be confused with scattering angular anisotropy—in tissue components, such as muscle fibers or collagens, is neglected as well.
3.3. DEFINITION OF PROBLEM
The problem to be solved begins with an infinitely narrow photon beam, also referred to as a pencil beam, that is perpendicularly incident on a multilay- ered scattering medium (Figure 3.1); various physical quantities are computed as responses. The pencil beam can be represented by an impulse (Dirac delta) function of space, direction, and time; thus, the responses are termed impulse responses or Green's functions. The layers are infinitely wide and parallel to each other. Each layer is described by the following parameters: thickness d, refractive index n, absorption coefficient μβ, scattering coefficient μ.ν, and scat- tering anisotropy g. The top and the bottom ambient media are each described by a refractive index. Although never infinitely wide in reality, a layer can be so treated if it is much wider than the photon distribution.
Three coordinate systems are defined. A global Cartesian coordinate system (JC, y, z) is used for tracking photons (Figure 3.1); the xy plane coincides with the surface of the scattering medium; the z axis is along the pencil beam.
A global cylindrical coordinate system (r, φ', ζ), which shares the z axis with the Cartesian coordinate system, is used for recording photon absorption as a function of r and z. The photon absorption distribution has cylindrical symmetry
Photon beam
Layer 1
Layer 2
Layer Nj
Figure 3.1. Schematic of a scattering medium with Ni layers. The y axis points out of the paper.
PROPAGATION OF PHOTONS 3 9
because of the homogeneity of each layer and the axial alignment of the pencil beam. The r coordinate is also used for recording both the diffuse reflectance and the diffuse transmittance that are functions of r as well as a, where a is the polar angle of the propagation direction of a reemitted photon with respect to the normal vector of the exit surface of the scattering medium (—z axis for the top surface and +z axis for the bottom surface). One can further resolve reemitted photons with the azimuthal angle φ'.
A local moving spherical coordinate system whose z axis is dynamically aligned with the propagation direction of the photon is used for sampling the scattering angles. Once the polar angle Θ and the azimuthal angle φ are sampled, they are converted to direction cosines in the global Cartesian coordinate system.
The physical quantities to be computed include relative specific absorption, rel- ative fluence, relative diffuse reflectance, and relative diffuse transmittance, all of which are relative to the incident energy. The relative specific absorption A(r, z) represents the probability of photon absorption per unit volume by the medium. From A(r, z), the relative fluence F(r,z)—which is the probability of photon flow per unit area—can be computed. The unscattered absorption from the first photon interaction events, which always take place on the z axis, is recorded sepa- rately. The relative diffuse reflectance Rd(r, a) for the top surface and the relative diffuse transmittance 7^(r, a) for the bottom surface—collectively referred to as the relative diffuse reemittance—are defined as the probability of photon reemis- sion from the surfaces per unit area at r per unit solid angle around a, where a solid angle has the unit of steradians (sr). Like unscattered absorption, specularly reflected and unscattered transmitted photons are recorded separately. Physical quantities of lower dimensions can be computed from those of higher dimen- sions. For brevity, relative physical quantities are written as physical quantities in this chapter unless otherwise noted.
Simulated physical quantities are represented in grids on the coordinate sys- tems. For photon absorption, a 2D homogeneous grid system is set up in the r and z directions. The grid element sizes in the r and z directions are ΔΓ and Δζ, respectively, and the total numbers of grid elements are Nr and Nz, respec- tively. For reemitted photons, a ID grid system in the a direction is further set up with Na grid elements. Since a has a range of π/2, the grid element size is Δα = π/(2Να). For convenience, the grid elements that should appear after a physical quantity are sometimes represented in this chapter by coordinates.
For consistency, centimeters (cm) are used as the basic unit of length through- out the simulation. For example, the thickness of each layer and the grid element sizes in the r and z directions are measured in cm. The absorption and scattering coefficients are measured in reciprocal centimeters (cm-1).
3 A PROPAGATION OF PHOTONS
This section presents the rules that govern photon propagation. Figure 3.2 shows a basic flowchart for the photon tracking part of the Monte Carlo simulation of
4 0 MONTE CARLO MODELING OF PHOTON TRANSPORT IN BIOLOGICAL TISSUE
/ Launch photon Udimensionless s_ = 0y
Figure 3.2. Flowchart for tracking photons in a scattering medium by the Monte Carlo method, where s_ denotes the dimensionless step size (to be discussed) and Y and N represent yes and no, respectively.
light transport in multilayered scattering media. The Monte Carlo simulation was written in ANSI (American National Standards Institute) Standard C as a software package entitled MCML (Appendix 3A). This software can be executed on any computer platform that supports ANSI Standard C. The following subsections implement many of the boxes in the flowchart.
PROPAGATION OF PHOTONS 41
3.4.1. Sampling of a Random Variable
The Monte Carlo method relies on the sampling of random variables from their probability distributions. The probability density function (PDF) /?(χ) defines the distribution of χ over interval (a, b). The interval can also be closed or half- closed in some cases, which usually makes no practical difference. For readers unfamiliar with PDF, a brief review is given in Appendix 3B.
To sample χ, we choose a value repeatedly based on its PDF. First, a pseu- dorandom number ξ that is uniformly distributed between 0 and 1 is generated by computer. Then, χ is sampled by solving the following equation:
/ ' Ja
ρ ( χ ) ^ χ = ξ. (3.1)
Since the left-hand side represents the cumulative distribution function (CDF) ^ (x ) , we have
P(lO = *. (3.2)
This equation means that if Ρ(χ) is sampled uniformly by ξ between 0 and 1, the inverse transformation correctly samples χ as illustrated in Figure 3.3:
χ = />"ι(ξ). (3.3)
This sampling method, referred to as the inverse distribution method (IDM), is invoked repeatedly below.
Figure 3.3. Illustration of the inverse distribution method (IDM) for sampling a random variable χ based on a random number ξ uniformly distributed between 0 and 1.
4 2 MONTE CARLO MODELING OF PHOTON TRANSPORT IN BIOLOGICAL TISSUE
Example 3.1. Prove Eq. (3.1)
We show that when χ is sampled according to Eq. (3.1), χ follows its intended CDF Ρ(χ) .
If χ is sampled according to Eq. (3.1), the probability that a selected χ is less than xi can be expressed by
/>ιθΜ{χ<Χι} = />{ξ<ξι} · (3.4)
Here, PIDM{ } denotes the probability in the IDM of the event in the braces; P{} denotes the true probability of the event in the braces based on the CDF of the random variable; χι is related to ξι through Eq. (3.1):
*;■= Γ%(χ)<ίχ. (3-5) Ja
Because ξ is equidistributed between 0 and I, we have Ρ{ξ < ξι} = ξ\. Thus, we obtain
^ I D M { X < X I } = [ ' PdOdm. (3.6) Ja
Since the right-hand side is the CDF Ρ(χ) , we have
A D M ( X < X I } - / > ( X I ) , (3.7)
which means that the sampled χ indeed follows its intended CDF.
3.4.2. Representation of a Photon Packet
A simple variance reduction technique, referred to as implicit photon capture, is used to improve the efficiency of the Monte Carlo simulation. This technique enables many photons to propagate as a packet of equivalent weight W along a particular trajectory.
Related parameters are structured in C to make the program easier to write, read, and modify. Thus, parameters for a photon packet are grouped into a single structure defined by
typedef struct { double x, y ,z ; /* Cartesian coordinates of photon packet. */ double ux, uy, uz ; / * d i rect ion cosines of photon propagation. */ double w; /* weight of photon packet. */ char dead; /* 0 i f photon is propagating, 1 i f terminated. */ double s_; /* dimensionless step size. */ long scatters; /* number of scatter ing events experienced. */ short layer; /* index of layer where photon packet resides. */
} PhotonStruct;
PROPAGATION OF PHOTONS 4 3
Structure members x, y, and z represent the coordinates of a photon packet, (JC, y, z), respectively. Structure members ux, uy, and uz represent the direction cosines of the propagation direction of the photon packet, {μχ, μ>;, μ^}, respectively. Structure member w represents the weight of the photon packet, W.
Structure member dead represents the status of the photon packet and is initialized to zero when the photon packet is launched. If the photon packet has reemitted from the scattering medium or has not survived a Russian roulette (to be discussed), this structure member is set to unity, which signals the program to stop tracking the current photon packet.
Structure member s_ represents the dimensionless step size, which is defined as the integration of extinction coefficient μ, over the trajectory of the photon packet. In a homogeneous medium, the dimensionless step size is simply the physical path length multiplied by the extinction coefficient.
Structure member s c a t t e r s stores the number of scattering events expe- rienced by the photon packet. If this structure member is zero when the weight of a photon packet is recorded, the weight contributes to the unscat- tered physical quantities.
Structure member l aye r is the index of the layer in which the photon packet resides; it is updated only when the photon packet leaves the current layer.
3.4.3. Launching of a Photon Packet
One photon packet is launched at a time orthogonally onto the scattering medium. For each new photon packet, the coordinates (x, y, z) are initialized to (0,0,0), the direction cosines {μ*, μγ,μζ] are initialized to (0,0,1), and the weight W is ini- tialized to unity. Several other structure members—including dead, s c a t t e r s , and layer—are also initialized.
If the upper ambient medium and the first layer have mismatched indices of refraction (no and n\, respectively), specular (Fresnel) reflection occurs. Specular reflectance^ in normal incidence is given by
K S P = ( ^ V . 0.8)
If the first layer of refractive index n\ is a clear medium, it causes multiple spec- ular reflections and transmissions. If interference effect is neglected, an effective specular reflectance can be computed by
Asp = flspl + - j · (3.9)
Here, Rsp\ and Rsp2 are the specular reflectances on the two boundaries of the first layer:
*spi = (**^λ) , (3.10)
4 4 MONTE CARLO MODELING OF PHOTON TRANSPORT IN BIOLOGICAL TISSUE
( Π\ — ηι\ *sp2= \ - ~ - - , (3.11)
where n2 denotes the refractive index of the second layer. When the photon packet enters the scattering medium, the weight is decreased
by #sP:
W=l- Rsp. (3.12)
If Eq. (3.8) is applicable, structure member l ayer is set to the first layer. If Eq. (3.9) is applicable, it is set to the second layer, and member z is set accord- ingly.
Example 3.2. Compute the specular reflectance in normal incidence between air and water, glass, and soft tissue, respectively.
For water, n\ = 1.33. For an air-water interface, Rsp = 2.0%, where HQ = 1 for air.
For glass, n\ — 1.5. For an air-glass interface, Rsp — 4.0%. For soft tissue, n\ — M.37. For an interface between air and soft tissue,
Rsp = -2 .4%.
Example 3.3. Compute the effective specular reflectance in normal incidence for a glass slide that is sandwiched between air and water.
For air, no = 1. For glass, n\ — 1.5. For water, n2 = 1.33. Thus, R$p\ = 0.04, and /?sp2 = 0.0035. The effective specular reflectance Rsp = 0.0432. In this case, Rsp is quite close to the direct sum of Rsp\ and /?sp2<
3.4.4. Stop Size of a Photon Packet
The step size of a photon packet is sampled by the IDM [Eq. (3.3)], based on the PDF of the dimensional free-path length s (0 < s < oo)) of a photon. We first consider a homogeneous medium. According to the definition of the extinction coefficient μ,, we have
~dP{s > s') 1 . . . . . μ — (3.13) ^ P{s>s'} dsf
where P{] denotes the probability of the event in the braces. The first frac- tion on the right-hand side represents the probability of interaction in interval (s\ s' + ds'), and the second fraction represents normalization by the path length. Rearranging Eq. (3.13) yields
d[\n(P{s > s'})] = - μ , ds'. (3.14)
PROPAGATION OF PHOTONS 4 5
Integrating this equation over s' in interval (0, s\), we obtain
P{s > 5ι} = βχρ(-μ , ί ι ) . (3.15)
This is a form of the well-known Beer law. From Eq. (3.15), the probability that an interaction occurs within s\ is given by
P{s < s\] — 1 - exp(—\its\). (3.16)
The corresponding PDF is given by dP{s <s\]
P(s\) = = μ, cxp(-\itsi). (3.17) ds\
According to the IDM, the CDF of s in Eq. (3.16) can be equated to ξ to yield the sampling equation for the step size
ln(l ~ %) n 15n s\ = , (3.18)
where 1 — ξ can be replaced by ξ for simplicity because ξ is uniformly distributed between 0 and 1:
„ = - ί ^ . (3.19) μ*
We then consider a multilayered medium where the photon packet may experience a free path over multiple layers before an interaction occurs. In this case, the counterpart of Eq. (3.15) becomes
-Y^VtiSi \ . P{s > st) = exp I - 2 ^ VtiSi 1 . (3.20)
Here, the summation is over all the segments that the photon packet has traversed before an interaction occurs; μπ is the extinction coefficient for the ith segment, Si is the length of the ith segment, and st is the total step size:
st Σ*· <3·21)
Equating Eq. (3.20) to ξ, we obtain the sampling equation
£ > „ * , = - 1 η ( ξ ) , (3.22)
which is a generalized form of Eq. (3.19). The left-hand side of Eq. (3.22) is the total dimensionless step size. Note that photon paths in a clear medium do not change the dimensionless step size because the extinction coefficient is zero.
4 6 MONTE CARLO MODELING OF PHOTON TRANSPORT IN BIOLOGICAL TISSUE
Equation (3.22) is used to sample the step size in MCML, where the dimen- sionless step size s_ is initialized to — 1η(ξ). A photon packet may travel multiple substeps of size s, in a multilayered scattering medium before reaching an interac- tion site. Only when the photon packet has completed — 1η(ξ) in dimensionless step size does an interaction occur. In an interaction event, the entire photon packet must experience both absorption and scattering. Since the step size is modeled, the simulation is intrinsically time-resolved.
3.4.5. Movement of a Photon Packet
Once the dimensional substep size s; is determined, the photon packet is moved. The coordinates of the photon packet are updated by
x^-x + [ixSj, y<^y + VLySi, ζ « - ζ + μζ5/ , (3.23)
where the arrows indicate quantity substitutions. The variables on the left-hand side have the new values, and the variables on the right-hand side have the old values. In C/C 4- +, an equal sign (=) is used for this purpose.
3.4.6. Absorption of a Photon Packet
Once the photon packet reaches an interaction site, a fraction of the weight (AW) is absorbed:
AW = — W. (3.24)
If the photon packet has not been scattered, AW is recorded into an array for unscattered absorption. Otherwise, it is recorded into A(r, z) at the local grid element:
A(r,z) <~ A(r,z) + AW. (3.25)
The weight must be updated by
W+-W-AW. (3.26)
The photon packet with the new weight then undergoes scattering at the interac- tion site.
3.4.7. Scattering of a Photon Packet
For scattering, the polar angle Θ (0 < θ < π) and the azimuthal angle φ (0 < φ < 2π) are sampled by the IDM. The probability distribution of cos Θ is commonly given by the Henyey-Greenstein phase function that was originally proposed for galactic scattering:
P(cos Θ) - * - ^ . (3.27) 2(1 + gl — 2 gcosO)-5/2
PROPAGATION OF PHOTONS 4 7
The anisotropy g, defined as (cos0), has a value between —1 and 1. A value of zero indicates isotropic scattering, and a value close to unity indicates dom- inantly forward scattering. For most biological tissues, g is M).9. In addition to the Henyey-Greenstein phase function, the Mie theory can also provide a phase function. Note that the phase function is irrelevant to the phase of an electromagnetic wave.
Applying the IDM [Eq. (3.1)] to Eq. (3.27), we sample cosG as follows:
COS0 = (ΐΐ4ξ)} lf * * ° . (3.28) 2% - 1 if g = 0
The azimuthal angle φ, which is assumed to be uniformly distributed over interval [0, 2π), is sampled with another independent pseudorandom number ξ:
φ = 2πξ. (3.29)
Once the local polar and azimuthal angles are sampled, the new propagation direction of the photon packet can be represented in the global coordinate sys- tem as
/ s in0^j^zcos( | ) — μν sinc|)) μ* = , l· μ* cosO,
iz cos φ 4-μν = p = h μν cos Θ, (3.30) ι2
μζ = -J\ — μ2 8 ΐηθ^8φ + μ^θ8θ .
If the photon direction is sufficiently close to the z axis (e.g., |μζ | > 0.99999), the following formulas are used instead so that division by a small number is avoided:
μχ = sin Θ cos φ,
μ^ = sin Θ sin φ, (3.31)
μ̂ = s g n ^ ) c o s 0 ,
where sign function sgn^ z ) returns 1 when μ̂ is positive and —1 when μ̂ is negative.
Note that the direction cosines are in the global Cartesian coordinate system, whereas the polar and azimuthal angles are in the local spherical coordinate system. Since trigonometric operations are computationally intensive, alternative algebraic operations are encouraged in the program when possible.
4 8 MONTE CARLO MODELING OF PHOTON TRANSPORT IN BIOLOGICAL TISSUE
Example 3.4. Derive Eq. (3.28) for g φ 0 using the IDM [Eq. (3.1)].
Here, χ = cos6 and χ e [—1, 1]. Therefore, we have
/ x /»cose
ρ(χ) d\ = / p(COS0') dcOSd'
= izi! / » L·^ 2g V A / I + ^ - ^ ^ C O S G 1 + g /
which leads to Eq. (3.28) for g φ 0.
3.4.8. Boundary Crossing of a Photon Packet
During a step of dimensionless size s_, the photon packet may hit a boundary of the current layer. Several steps are involved in boundary crossing.
Step 1. The distance df, between the current location (x, y, z) of the photon packet and the boundary of the current layer in the photon propagation direction is computed by
db =
zo-z if μζ < 0
oo if μ, = 0 , (3.32) Ζ\ — Ζ
μ< if μζ > 0
where zo and z \ are the z coordinates of the upper and lower boundaries of the current layer. If μζ approaches zero, the distance approaches infinity, which is represented by constant DBL-MAX in C. Step 2. The dimensionless step size s_ and the distance db are compared as follows:
db[Lt < * - , (3-33)
where μ, is the extinction coefficient of the current layer. If Eq. (3.33) holds, the photon packet is moved to the boundary; s_ is updated by s_ <— s_ — ^ μ , ; the simulation proceeds to step 3. Otherwise, the photon packet is moved by s-/\it to reach the interaction site, s- is set to zero to signal the generation of the next dimensionless step size, and the photon packet experiences absorption and scattering. Since division by μ, is avoided, Eq. (3.33) is applicable to clear layers (μ, = 0) as well. Step 3. If the photon packet hits a boundary, the specular reflectance is computed. The angle of incidence a, of the photon packet is first calculated by
a , = c o s - V z l ) . (3.34)
PROPAGATION OF PHOTONS 4 9
The angle of transmission at is then computed by Snell's law
rii sin a,· = nt sin a,, (3.35)
where Π[ and nt represent the refractive indices of the incident and transmit- ted media, respectively. If n, > nt and a, is greater than the critical angle sin_1(nr/n/), the local reflectance /?,((*,) equals unity. Otherwise, /?,·(α,·) is cal- culated by Fresnel's formula:
*/(««)= 2 sin2 (a,· — a,) tan2 (a,; — at) sin2 (a, + ar) tan2 (a,· + a , )
(3.36)
which is an average of the reflectances for two orthogonal linear polarization states because light is assumed to be randomly polarized. Step 4. To determine whether the photon packet is reflected or transmitted, a pseudorandom number ξ is generated. If ξ < /?/(α,·), the photon packet is reflected; otherwise, it is transmitted. If reflected, the photon packet stays on the boundary and its direction cosines are updated by reversing the z component:
{μχ, μ^, μζ} <r- [[ix, μν, - μ , } . (3.37)
If transmitted into a neighboring layer, the photon packet continues to propagate with an updated direction and step size. The new direction cosines are
s i n o t ' n M v [i =μχ- , (3.38)
sin a,· S i n 0 t ' / - J i m
μ Y = Vy- , (3.39) y sin a,
μζ = s g n ^ ) cos a,. (3.40)
On the basis of Snell's law, Eqs. (3.38) and (3.39) can be computed more effi- ciently by
μχ=μχ —, (3.41) nt
f Hi
μ ^ μ ^ —. (3.42)
If transmitted into an ambient medium, the photon packet contributes to reemit- tance. If the photon packet has not been scattered, its weight is recorded into the unscattered reemittance; otherwise, its weight is recorded into either the diffuse reflectance Rd(r, at) or the diffuse transmittance 7^(r, at):
Rd(r,at) *-Rd(r,at) + W if z = 0; (3.43)
7^(r, at) <- Td(r, at) -f W if z is at the bottom of the medium.
5 0 MONTE CARLO MODELING OF PHOTON TRANSPORT IN BIOLOGICAL TISSUE
When reemitted from the scattering medium, the photon packet completes its history (or Markov chain).
Reemission at an interface can be modeled alternatively. Rather than making the reflection of the photon packet an all-or-none event, the photon packet can be partially reflected and partially transmitted. A fraction 1 - /?/((*,) of the current weight of the photon packet is reemitted from the scattering medium and recorded to the local reemittance. Then, the weight is updated by W <— W/?/(a;). The photon packet with the new weight is reflected and further propagated.
3.4.9. Termination of a Photon Packet
A photon packet can be terminated from the scattering medium automatically by reemission as discussed above. If the weight of a photon packet has been sufficiently decreased by many interaction events, further propagation of the photon packet yields little useful information unless interest is focused on a late stage of photon propagation. However, photon packets must be properly terminated so that energy is conserved.
A technique called Russian roulette is used in MCML to terminate a photon packet when the weight falls below a threshold W^ (e.g., W^ — 0.0001). This technique gives the photon packet one chance in m (e.g., m — 10) of surviving with a weight of m W. In other words, if the photon packet does not survive the Russian roulette, it is terminated with the weight set to zero; otherwise, the photon packet increases in weight from W to mW. This technique is mathematically summarized as
ί mW if ξ < 1 , W <- \ m (3.44)
0 if %> -,
where ξ is a uniformly distributed pseudorandom number (0 < ξ < 1). This method terminates photons in an unbiased manner while conserving the total energy.
3.5. PHYSICAL QUANTITIES
In this section, the process of obtaining physical quantities is discussed in detail. The units for some of the physical quantities are given at the end of their respec- tive expressions.
Physical quantities are stored in arrays. Although photon packets propagate in infinite continuous space (limited only by computational precision), weights are recorded in finite discrete space (limited by grid element sizes). When a photon packet is recorded, its physical location may not fit into the grid system. In this case, the last grid element in the direction of the overflow collects the weight. Therefore, the last grid elements in the r and z directions do not reflect the actual values at the corresponding locations. Angle a, however, does not overflow.
PHYSICAL QUANTITIES 51
As a rule of thumb for most problems, each spatial grid element should mea- sure about 10% of the penetration depth or the transport mean free path. If the grid elements are too small, the relative errors—which are determined by the number of events occurring in each grid element—will be too large. If the grid elements are too large, the dependence of the physical quantities will not be represented with good resolution.
3.5.1. Reflectance and Transmittance
The diffuse reflectance and the diffuse transmittance are represented in MCML by two arrays, Rd-ralhJa] and Td_r0L[ir, ia], respectively, where ir and ia are the indices of r and a (0 < ir < Nr — 1 and 0 < ia < Na — 1), respectively. The unscattered reflectance and the unscattered transmittance are stored in Rd_r[— 1] and Td_r[— 1], respectively.
It can be shown that the optimal coordinates of the simulated physical quanti- ties for the grid elements in the radial and angular directions are as follows (see Problem 3.1):
r(ir) = H) 1 + Ar (cm), (3.45) 12(/r + i ) αθ'α) = ( ι'α + - j Δα + 1 - -Aaco t l - A a j cot L· + - j Aa (rad).
(3.46)
The deviation of the optimized point from the center is 25% for the first radial grid element but decreases as the index of the grid element increases. Because the optimized coordinates are computed only after all photon packets are simulated, this optimization does not increase simulation time but improves accuracy.
After multiple photon packets (N) have been tracked, raw data Rd^rAh, ia] and Td_ra[ir, ia] represent the total accumulated weights in each grid element. To compute the integrated weights in each direction of the 2D grid system, we sum the 2D arrays in the other dimension:
Na-\
Rd-rUA = Σ Rd-rAir, l a] , (3.47) ία=0
Nr-\
Rd^aUa] = Σ Rd-raUr, I 0 ] , (3 .48) i r=0
Na-\
Td_rVr]= £ Td_ra[ir,ial (3.49) ί'α=0
Nr-\
Td-oilia] = Σ Td-raUrJal (3-50) ,7=0
5 2 MONTE CARLO MODELING OF PHOTON TRANSPORT IN BIOLOGICAL TISSUE
To compute the total diffuse reflectance and the total diffuse transmittance, we sum the ID arrays:
Nr-\
Rd = Σ Rd-r[ir]> ( 3 5 1 )
/r=0
N,--\
Τα=Σ Td-rVr]. (3.52) /r=0
These arrays describe the total weights in each grid element on the basis of N initial photon packets of unit weight. Raw data /?</_ra[/r, *<*] and 7j_m[/ r , ia] are converted into probabilities of reemission per unit cross-sectional area per unit solid angle as follows:
Rd-ra[h,ia] <- - -prr: (cm sr ), (3.53)
Td_r*[ir,i*\+- J d~rAlr\l^AT ( cm-V" 1 ) . (3.54)
AacosaAilN The area Aa and the solid angle ΔΩ are given by
= 2π(,ν + 1)
ΔΩ = 4π sin I ia + ~ I Δα sin I - Δ α 1 (sr). (3.56)
Raw Rd-rUr] and Td_r[iA are converted to probabilities of reemission per unit area as follows:
* r f - r [ i V ] ^ % ^ (cm"2), (3.57) N Aa
7 i _ r [ i V ] ^ - % x ^ (cm-2). (3.58) N Aa
Raw Rd_r[— 1] and 7^_Γ[—1] are converted to total unscattered reflectance and total unscattered transmittance, respectively, by dividing them by N. Then, Rd-r[—l] is augmented by the specular reflectance or the effective specular reflectance. Raw Rd^aUa] and 7^_α[/α] are converted to the probabilities of reemission per unit solid angle as follows:
Ärf-J/α] ^ - £ χ 7 ~ (sr"'), (3.59) ΝΑΩ
Td^Va)^7^^ ( s r 1 ) . (3.60) ΝΑΩ
Aa = 2π ( ir + - ) (Ar)2 (cm2), (3.55)
PHYSICAL QUANTITIES 5 3
Raw Rj and 7^ are converted to probabilities as follows:
Rd Rd < (dimensionless), (3.61)
N T
Td < (dimensionless). (3.62) N
3.5.2. Absorption and Fluence
At each interaction, the absorbed weight is stored in the specific absorption array Arz[ir, iz], where ir and iz are the indices of grid elements in the r and z directions, respectively (0 < ir < Nr — 1 and 0 < iz < Nz — 1). The unscattered absorption is stored in Arz[—l,iz]. Whereas the optimal coordinate for ir is shown in Eq. (3.45), the coordinate for iz is simply
A) = (/z + \) zdz) = \iz + 2 ) Az' ( 3 ' 6 3 )
Raw Arz[ir, iz] represents the total accumulated weight in each grid element. The total weight in each grid element in the z direction is computed by summing the 2D array in the r direction:
Nr-\
Az[iz]= J^Arz[ir,izl (3.64) i r = 0
Next, the total weight absorbed in each layer, A/[//], can be computed by
A/[//] = ^ A J i J , (3.65)
where // is the index of a layer, and the summation includes any iz that leads to a z coordinate in layer //. Further, the total weight absorbed in the scattering medium A can be computed by
Nz-\ Α = Σ A*vJ' ( 3 · 6 6 )
i z=0
Then, these raw quantities are converted into the final physical quantities as follows:
Arz[irJz]+- A'z[h'*z] (cm"3), (3.67)
NAaAz
Ard-hit]*- An[J*'il] (cm-3), (3.68)
NAz
AzVz]^^r ( c m _ l ) ' ( 3 6 9 > NAz
5 4 MONTE CARLO MODELING OF PHOTON TRANSPORT IN BIOLOGICAL TISSUE
Mh] AiUi] < (dimensionless), (3.70)
N A
A < (dimensionless). (3.71)
The ID array Λ/[//] represents the probability of photon absorption in each layer. The quantity A represents the probability of photon absorption by the entire scattering medium. The 2D array Arz[ir, iz] represents the probability of photon absorption per unit volume, which can be converted into fluence Frz[ir,iz] as follows:
FrzUrJz)^ a U (cm"2), (3.72)
where \ia denotes the local absorption coefficient. This equation breaks down in a non-absorbing medium.
The ID array Az[iz] represents the probability of photon absorption per unit length in the z direction, which can be converted to a dimensionless quantity Fz[iz] as follows:
M'z] Fz[iz] = (dimensionless). (3.73) μ«
Fz[iz] represents the internal fluence as a function of z apart from a constant factor. This equation also breaks down in a non-absorbing medium.
Example 3.5. Show the equivalence of Eq. (3.73) to the convolution over an infinitely wide uniform beam (see Chapter 4).
According to Eqs. (3.64), (3.67) and (3.69), the final converted Az[iz] and Arz [ir, iz] have the following relationship:
Nr-\
Az[iz]= Σ Α Γ Ζ [ Ι Γ , / 2 ] Δ Ο ( Ι Γ ) , (3.74) ir=0
where Aa(ir) is computed by Eq. (3.55). Employing Eqs. (3.72) and (3.73), we convert Eq. (3.74) to
Nr-\ F^z]= £ F r z [ / r , / J A a ( / r ) . (3.75)
/r=o
This equation is a discrete representation of the following integral: poo
Fz(z)= / Frz(r,z)2nrdr. (3.76) Jo
This integration is essentially the convolution over an infinitely wide uniform beam of unit fluence. Therefore, Fz[iz] represents the fluence distribution along the z axis in response to such a beam.
COMPUTATIONAL EXAMPLES 5 5
3.6. COMPUTATIONAL EXAMPLES
As computational examples, MCML simulation results are compared with results from other theories and other investigators' Monte Carlo simulations. These com- parisons partially validate the MCML.
3.6.1. Total Diffuse Reflectance and Total Transmittance
The total diffuse reflectance Rd and the total transmittance Tt (sum of both the unscattered and the scattered transmittances) are computed for a slab of scatter- ing medium with the following optical properties: relative refractive index nre\ = 1, absorption coefficient μα = 10 cm- 1 , scattering coefficient μ5 = 90 cm- 1 , anisotropy g = 0.75, and thickness d = 0.02 cm. The relative refractive index is defined as the ratio of the refractive index of the scattering medium to that of the ambient medium. If nTe\ = 1, the boundaries are termed refractive-index- matched. After 10 Monte Carlo simulations of 50,000 photon packets each are completed, the averages and the standard errors (standard deviations of the averages) of the total diffuse reflectance and total transmittance are computed and compared with the data from van de Hulst's (1980) table and Prahl et al.'s (1989) Monte Carlo simulations (Table 3.1). Because the unscattered trans- mittance is exp[—(μα + [Ls)d] = e~2 = 0.13534, the total diffuse transmittance equals 0.66096-0.13534 = 0.52562. All results are in good agreement. It is worth noting that standard errors are expected to decrease proportionally with the square root of the number of photon packets tracked owing to the central limit theorem.
For a semiinfinite scattering medium that has a refractive-index-mismatched boundary (nre\ φ 1), the total reflectance is computed similarly and compared in Table 3.2 with the data from Giovanelli (1955) and Prahl et al.'s (1989) Monte Carlo simulations. The scattering medium has the following optical properties: nTe\ = 1.5, \ia = 10 cm"1, [is = 90 cm- 1 , g = 0 (isotropic scattering). Then, 10 Monte Carlo simulations of 5000 photon packets each are completed to compute the average and the standard error of the total diffuse reflectance.
TABLE 3.1. Comparison of Results from MCML with van de Hulst's Table and the Monte Carlo Data of Prahl et al. (1989)α
Source
MCML van de Hust (1980) Prahl et al. (1989)
Rd Average
0.09734 0.09739 0.09711
Rd Error
0.00035
0.00033
Tt Average
0.66096 0.66096 0.66159
Tt Error
0.00020
0.00049 a"Rd average" and "/?</ error" list the averages and the standard errors of the total diffuse reflectance, respectively. Columns 'Ύ, average" and 'T, error" list the averages and the standard errors of the total transmittance, respectively.
5 6 MONTE CARLO MODELING OF PHOTON TRANSPORT IN BIOLOGICAL TISSUE
TABLE 3.2. Comparison of Total Reflectance0 from MCML with Data from Giovanelli (1955) and Prahl et al. (1989)
Source Rj Average Rj Error
MCML 0.25907 0.00170 Giovanelli 1995 0.2600 — Prahl et al. (1989) 0.26079 0.00079
"The total reflectance includes both the specular reflectance (0.04) and the diffuse reflectance
3.6.2. Angularly Resolved Diffuse Reflectance and Transmittance
The angularly resolved diffuse reflectance and transmittance are also computed for a slab of scattering medium with the following optical properties: nre\ — 1, \xa = 10 cm- 1 , [is = 90 cm- 1, g = 0.75, and d = 0.02 cm. In this simulation, 500,000 photon packets are tracked, and the number of angular grid elements is 30. The results from MCML are compared in Figure 3.4 with the data from van de Hülst's (1980) table. Because van de Hülst used a different definition of reflectance and transmittance and also used an incident flux of π, his data are multiplied by cos a and then divided by π before the comparison.
3.6.3. Depth-Resolved Fluence
As an example, the depth-resolved internal fluence Fz[iz] is simulated by MCML for two semiinfinite scattering media with refractive-index-matched and refractive-index-mismatched boundaries, respectively (Figure 3.5). The optical parameters are nrf>\ = 1.0 or 1.37, μα = 0.1 cm- 1 , μ5 = 100 cm- 1, and g = 0.9. One million photon packets are tracked in each simulation. The grid element size and the number of grid elements in the z direction are 0.005 cm and 200, respectively.
As can be seen, the fluence near the surface is greater than unity because scattered light augments the fluence. Furthermore, the internal fluence in the scattering medium with a refractive-index-mismatched boundary is greater than that in the medium with a refractive-index-matched boundary, because photons can be bounced back into the scattering medium by the mismatched boundary.
When z is greater than the transport mean free path lr diffusion theory predicts that the internal fluence distribution is
F(z) = KF0exp(~y (3.77)
Here, K is a scaling factor that depends on the relative index of refraction, F() is the incident irradiance (unity in MCML), and δ is the penetration depth. Consequently, the two curves in the diffusive regime should be separated by a constant factor only, which means that the tails of the two curves should be
COMPUTATIONAL EXAMPLES 57
0.025
0.02
a: 0.015
C 0.01
Q 0.005
- D D D O
^ n n ° · α ~ D ° u □ a D D Rfc
D MCML • van de Hülst
D
0π Ο.ΐπ (a)
0.2π 0.3π Exit angle a (rad)
0.4π 0.5π
0.8
0.7
'fe 0.6
£ 0 . 5
0.4
1 0.3
0.2
0.1
m Γ D
r D
i ■ ■ ■■ T ■ l
ü MCML 1 1 • van de Hülst H
J I D J
I D J D
Γ D Ί D
l · -J Γ D | D D D D D * ^ D
Οπ Ο.ΐπ (b)
0.2π 0.3π Exit angle a (rad)
0.4π 0.5π
Figure 3.4. Angularly resolved (a) diffuse reflectance Rd(a) and (b) diffuse transmittance Td(OL).
parallel in a log-linear plot (Figure 3.5). The two curves shown here are parallel when z > l't = [μα + μ*(1 - g)]~l ^ 0.1 cm.
We fit the parallel portions of the two curves with exponential functions. The decay constants for the curves are approximately 0.578 cm for the refractive- index-matched boundary and 0.575 cm for the refractive-index-mismatched boundary. Both values are close to the one predicted by the diffusion theory
δ = 1
ν 3 μ α [ μ 0 + μ 5 ( 1 - g)] = 0.574 cm, (3.78)
which is independent of the relative index of refraction.
5 8 MONTE CARLO MODELING OF PHOTON TRANSPORT IN BIOLOGICAL TISSUE
10
c 3
1
0 0.2 0.4 0.6 0.8 1
z (cm)
Figure 3.5. Comparison of the internal fluence as a function of depth z for two semiin- finite scattering media with a refractive-index-matched boundary and a refractive-index- mismatched boundary, respectively. Minus (—) represents a dimensionless unit.
APPENDIX 3A. SUMMARY OF MCML
The entire source code for MCML can be found on the Web at ftp://ftp.wiley.com/ public/sci_tech_med/biomedical_optics. The whole program for MCML is divided into several files. Header file mcml. h defines the data structures and some con- stants. File mcmlmain.c contains the main function and the status-reporting function. File mcmlio.c deals with reading and writing data. File mcmlgo.c contains most of the photon-tracking code. File mcmlnr. c contains several func- tions for dynamical data allocations and error reports. Readers should read the main function first.
In MCML, ID and 2D physical quantities are stored in ID or 2D arrays, respectively. These arrays are dynamically allocated so that users are allowed to specify the array sizes at runtime without wasting computer memory, an advan- tage that static arrays do not provide.
The following list is generated by command cf low -d3 -n --omit-arguments --omit-symbol-names mcml*.c, which shows the structure of the program (MCML 1.2.2) with the nesting depth limited to 3:
1 main() < i n t () at MCMLMAIN.C:186>: 2 ShowVersion() <void () a t MCMLI0.C:71>: 3 C t rPu ts ( ) <void () at MCMLI0.C:48>: 4 p r i n t f ( ) 5 pu ts ( ) 6 GetFnameFromArgvO <void () at MCMLMAIN.C:130>:
APPENDIX 3A. SUMMARY OF MCML 5 9
7 strcpy() 8 GetFileO <FILE * () at MCMLIO.C:111>: 9 printf() 10 scant() 11 strlen() 12 exit() 13 fopen() 14 CheckParm() <void () at MCMLIO.C:531>: 15 ReadNumRuns() <short () at MCMLIO.C:222>: 16 printf() 17 ReadParm() <void () at MCMLIO.C:442>: 18 FnameTaken() <Boolean () at MCMLIO.C:504>: 19 sprintf() 20 free() 21 nrerrorO <void () at MCMLNR.C:17>: 22 FreeFnameListO <void () at MCMLI0.C:517>: 23 rewind() 24 ReadNumRunsO <short () at MCMLIO.C:222>: 25 FindDataLine() <char * () at MCMLIO.C:201>: 26 strcpy() 27 nrerror() <void () at MCMLNR.C:17>: 28 sscanf() 29 ReadParm() <void () at MCMLIO.C:442>: 30 ReadFnameFormatO <void () at MCMLIO.C:242>: 31 ReadNumPhotons() <void () at MCMLI0.C:260>: 32 ReadDzDr() <void () at MCMLIO.C:277>: 33 ReadNzNrNa() <void () at MCMLIO.C:293>: 34 ReadNumLayers() <void () at MCMLI0.C:316>: 35 ReadLayerSpecs() <void () at MCMLIO.C:390>: 36 CriticalAngleO <void () at MCMLIO.C:421>: 37 DoOneRun() <void () at MCMLMAIN.C:145>: 38 InitProfileO 39 cecho2file() 40 InitOutputDataO <void () at MCMLIO.C:562>: 41 Rspecular() <double () at MCMLG0.C:116>: 42 PunchTimeO <time_t () at MCMLMAIN.C:59>: 43 printf() 44 PredictDoneTimeO <void () at MCMLMAIN.C:94>: 45 LaunchPhoton() <void () at MCMLGO.C:140>: 46 HopDropSpin() <void () at MCMLGO.C:734>: 47 exit() 48 ReportResultO <void () at MCMLMAIN.C:115>: 49 FreeData() <void () at MCMLIO.C:598>: 50 fcloseO
6 0 MONTE CARLO MODELING OF PHOTON TRANSPORT IN BIOLOGICAL TISSUE
APPENDIX 3B. PROBABILITY DENSITY FUNCTION
The probability density function (PDF), expressed as p(x), is a function that gives the probability that random variable x assumes a value between jti and X2 as follows:
P{x] < x < xi) = / p(x) dx. (3.79)
The PDF has the following properties:
"+00
/ " p(x) dx = 1 (3.80)
and p(x) > 0 for x e (-oo, +oo). (3.81)
The relationship between the PDF and the cumulative distribution function (CDF) P(x) is
P(x) = I p{x') dx J—oo
(3.82)
or
PROBLEMS
p(x) = —P(x). (3.83) ax
3.1 Derive Eqs. (3.45) and (3.46).
3.2 Derive Eq. (3.30). Note that the formula depends on the choice of the local moving coordinate system. The Monte Carlo simulation, however, leads to the same result.
3.3 Find the effective specular reflectance in normal incidence from a water layer (n — 1.33) placed between air and biological tissue (n = 1.37).
3.4 Show that the number of scattering events occurring in path length s follows the Poisson distribution. Neglect absorption.
3.5 Show that the mean and standard deviation of the free path between scat- tering events are both equal to 1/μΛ. Neglect absorption.
3.6 An alternative to Eq. (3.28) when g φ 0 is
cos Θ = — 2g
14V- \\+g-2g%)
Why?
PROBLEMS 61
An absorbing slab of thickness d has absorption coefficient \ia with neg- ligible scattering. A collimated laser beam of intensity Iin is normally incident on the slab. (a) Assume that the mismatch between the refractive indices of the ambi-
ent media and the absorbing medium is negligible. Calculate the trans- mitted light intensity /out.
(b) Assume that the mismatch between the refractive indices of the ambi- ent media and the absorbing medium causes a specular reflection R on each surface of the slab. Recalculate the transmitted light intensity /out·
A pencil beam is incident on a semiinfinite reference scattering medium. The spatially and temporally resolved diffuse reflectance Ro(r, t) is known for the following optical parameters: absorption coefficient μαο, scattering coefficient μ5ο, and scattering anisotropy go, where r denotes the radial distance between the observation point and the point of incidence, and t denotes time. The speed of light in the medium is c. (a) Write the expression for the new diffuse reflectance /?i(r, /) assuming
that the absorption coefficient is changed from μαο to μα\ but the other optical parameters are unchanged.
(b) Write the expression for the new diffuse reflectance Ri(r, t) where both the absorption coefficient and the scattering coefficient are scaled by the same factor C and the other optical parameter is unchanged: μ«2 = C\iao and \xs2 = Cvs0-
Consider a Gaussian beam with a radius of R, a total energy of 1 mJ, and a radial energy density distribution of S(r) = (2/nR2) exp(—2r2/R2). Derive the sampling expression for the radius r based on the random number ξ that is uniformly distributed between 0 and 1.
Implement a Monte Carlo simulation of photon transport in a semiinfi- nite scattering medium in C/C++ or another programming language. The Henyey-Greenstein function is assumed to be the phase function. Inputs include nrei, μα, μ*, and g. Outputs from the program include the total reflectance R and the depth-resolved fluence. (a) Use the results in Table 3.2 to verify your program. (b) Calculate for nK\ — 1.37, μα = 0 . 1 cm- 1 , μ5 = 100 cm- 1 , and g =
0.9. (c) Reproduce Figure 3.5.
Use the original Monte Carlo code written for Problem 3.10 and a mod- ified version to verify that the following two algorithms are statistically identical. Compare the total diffuse reflectances and the total absorptions using a statistical test. Use nre\ — 1, μα = 0.1 cm- 1 , μ5 = 100 cm- 1 , and 8 - 0.9. (a) Sample the step size with s — — 1η(ξ)/μ, and calculate the weight loss
at each step by AW = ^-W.
6 2 MONTE CARLO MODELING OF PHOTON TRANSPORT IN BIOLOGICAL TISSUE
(b) Sample the step size with s = —1η(ξ)/μν and calculate the weight loss at each step by AW = W[\ - exp(—μα5)].
READING
Giovanelli RG (1955): Reflection by semi-infinite diffusers, OpticaActa 2: 153-162. (See Section 3.6, above.)
Lux I and Koblinger L (1991): Monte Carlo Particle Transport Methods: Neutron and Photon Calculations, CRC Press, Boca Raton. (See Section 3.2, above.)
Marquez G, Wang LHV, Lin SP, Schwartz JA, and Thomsen SL (1998): Anisotropy in the absorption and scattering spectra of chicken breast tissue, Appl. Opt. 37(4): 798-804. (See Section 3.2, above.)
Prahl SA, Keijzer M, Jacques SL, and Welch AJ (1989): A Monte Carlo model of light propagation in tissue, in Dosimetry of Laser Radiation in Medicine and Biology, Müller GJ and Sliney DH, eds., SPIE Press, IS5: 102-111. (See Section 3.6, above.)
van de Hülst HC (1980): Multiple Light Scattering: Tables, Formulas, and Applications, Academic Press, New York. (See Section 3.6, above.)
Wang LHV, Jacques SL, and Zheng LQ (1995): MCML—Monte Carlo modeling of light transport in multi-layered tissues, Comput. Meth. Prog. Biomed. 47: 131-146. (All sections in this chapter.)
FURTHER READING
Ahrens JH and Dieter U (1972): Computer methods for sampling for the exponential and normal distributions, Commun. ACM 15: 873-882.
Baranoski GVG, Krishnaswamy A, and Kimmel B (2004): An investigation on the use of data-driven scattering profiles in Monte Carlo simulations of ultraviolet light prop- agation in skin tissues, Phys. Med. Biol. 49(20): 4799-4809.
Bartel S and Hielscher AH (2000): Monte Carlo simulations of the diffuse backscattering Mueller matrix for highly scattering media, Appl. Opt. 39(10): 1580-1588.
Boas DA, Culver JP, Stott JJ, and Dunn AK (2002): Three dimensional Monte Carlo code for photon migration through complex heterogeneous media including the adult human head, Opt. Express 10(3): 159-170.
Carter LL and Cashwell ED (1975): Particle-Transport Simulation with the Monte Carlo Method USERDA Technical Information Center, Oak Ridge, TN.
Cashwell ED and Everett CJ (1959): A Practical Manual on the Monte Carlo Method for Random Walk Problems, Pergamon Press, New York.
Churmakov DY, Meglinski IV, Piletsky SA, and Greenhalgh DA (2003): Analysis of skin tissues spatial fluorescence distribution by the Monte Carlo simulation, J. Phys. D—Appl. Phys. 36(14): 1722-1728.
Cote D and Vitkin IA (2005): Robust concentration determination of optically active molecules in turbid media with validated three-dimensional polarization sensitive Monte Carlo calculations, Opt. Express 13(1): 148-163.
Ding L, Splinter R, and Knisley SB (2001): Quantifying spatial localization of optical map- ping using Monte Carlo simulations, IEEE Trans. Biomed. Eng. 48(10): 1098-1107.
FURTHER READING 6 3
Flock ST, Patterson MS, Wilson BC, and Wyman DR (1989): Monte-Carlo modeling of light-propagation in highly scattering tissues. 1. Model predictions and comparison with diffusion-theory, IEEE Trans. Biomed. Eng. 36(12): 1162-1168.
Flock ST, Wilson BC, and Patterson MS (1989): Monte-Carlo modeling of light- propagation in highly scattering tissues. 2, Comparison with measurements in phan- toms, IEEE Trans. Biomed. Eng. 36(12): 1169-1173.
Gardner CM and Welch AJ (1994): Monte-Carlo simulation of light transport in tis- sue—unscattered absorption events, Appl. Opt. 33(13): 2743-2745.
Hendricks JS and Booth TE (1985): MCNP variance reduction overview, Lecture Notes Phys. (AXV64) 240: 83-92.
Henniger J, Minet O, Dang HT, and Beuthan J (2003): Monte Carlo simulations in com- plex geometries: Modeling laser light transport in real anatomy of rheumatoid arthritis, Laser Phys. 13(5): 796-803.
Igarashi M, Gono K, Obi T, Yamaguchi M, and Ohyama N (2004): Monte Carlo simula- tions of reflected spectra derived from tissue phantom with double-peak particle size distribution, Opt. Rev. 11(2): 61-67.
Jacques SL, Later CA, and Prahl SA (1987): Angular dependence of HeNe laser light scattering by human dermis, Lasers Life Sei. 1: 309-333.
Jacques SL (1989): Time resolved propagation of ultrashort laser-pulses within turbid tissues, Appl. Opt. 28(12): 2223-2229.
Jacques SL (1989): Time-resolved reflectance spectroscopy in turbid tissues, IEEE Trans. Biomed. Eng. 36(12): 1155-1161.
Jaillon F and Saint-Jalmes H (2003): Description and time reduction of a Monte Carlo code to simulate propagation of polarized light through scattering media, Appl. Opt. 42(16): 3290-3296.
Kahn H (1950): Random sampling Monte Carlo techniques in neutron attenuation prob- lems I, Nucleonics 6: 27-37.
Kalos MH and Whitlock PA (1986): Monte Carlo Methods, Wiley, New York. Keijzer M, Jacques SL, Prahl SA, and Welch AJ (1989): Light distributions in artery
tissue—monte-carlo simulations for finite-diameter laser-beams, Lasers Surg. Med. 9(2): 148-154.
Keijzer M, Pickering JW, and Gemert MJCv (1991): Laser beam diameter for port wine stain treatment, Lasers Surg. Med. 11: 601-605.
Kienle A and Patterson MS (1996): Determination of the optical properties of turbid media from a single Monte Carlo simulation, Phys. Med. Biol. 41(10): 2221-2227.
Li H, Tian J, Zhu FP, Cong WX, Wang LHV, Hoffman EA, and Wang G (2004): A mouse optical simulation environment (MOSE) to investigate bioluminescent phenomena in the living mouse with the Monte Carlo method, Acad. Radiol. 11(9): 1029-1038.
Lu Q, Gan XS, Gu M, and Luo QM (2004): Monte Carlo modeling of optical coherence tomography imaging through turbid media, Appl. Opt. 43(8): 1628-1637.
MacLaren MD, Marsaglia G, and Bray AT (1964): A fast procedure for generating expo- nential random variables, Commun. ACM 7: 298-300.
Marsaglia G (1961): Generating exponential random variables, Ann. Math. Stat. 32: 899-900.
McShane MJ, Rastegar S, Pishko M, and Cote GL (2000): Monte carlo modeling for implantable fluorescent analyte sensors, IEEE Trans. Biomed. Eng. 47(5): 624-632.
Meglinski IV (2001): Monte Carlo simulation of reflection spectra of random multilayer media strongly scattering and absorbing light, Quantum Electron. 31(12): 1101 -1107.
6 4 MONTE CARLO MODELING OF PHOTON TRANSPORT IN BIOLOGICAL TISSUE
Meglinski IV and Matcher SD (2001): Analysis of the spatial distribution of detector sen- sitivity in a multilayer randomly inhomogeneous medium with strong light scattering and absorption by the Monte Carlo method, Opt. Spectrosc. 91(4): 654-659.
Mroczka J and Szczepanowski R (2005): Modeling of light transmittance measurement in a finite layer of whole blood—a collimated transmittance problem in Monte Carlo simulation and diffusion model, Optica Applicata 35(2): 311-331.
Nishidate I, Aizu Y, and Mishina H (2004): Estimation of melanin and hemoglobin in skin tissue using multiple regression analysis aided by Monte Carlo simulation, J. Biomed. Opt. 9(4): 700-710.
Patwardhan SV, Dhawan AP, and Relue PA (2005): Monte Carlo simulation of light-tissue interaction: Three-dimensional simulation for trans-illumination-based imaging of skin lesions, IEEE Trans. Biomed. Eng. 52(7): 1227-1236.
Plauger PJ and Brodie J (1989): Standard C, Microsoft Press, Redmond, WA. Plauger PJ, Brodie J, and Plauger PJ (1992): ANSI and ISO Standard C : Programmer's
Reference, Microsoft Press, Redmond, WA. Qian ZY, Victor SS, Gu YQ, Giller CA, and Liu HL (2003): "Look-ahead distance" of a
fiber probe used to assist neurosurgery: Phantom and Monte Carlo study, Opt. Express 11(16): 1844-1855.
Ramella-Roman JC, Prahl SA, and Jacques SL (2005): Three Monte Carlo programs of polarized light transport into scattering media: Part I, Opt. Express 13(12): 4420-4438.
Sharma SK and Banerjee S (2003): Role of approximate phase functions in Monte Carlo simulation of light propagation in tissues, J. Opt. A 5(3): 294-302.
Swartling J, Pifferi A, Enejder AMK, and Andersson-Engels S (2003): Accelerated Monte Carlo models to simulate fluorescence spectra from layered tissues, J. Opt. Soc. Am. A 20(4): 714-727.
Tycho A, Jorgensen TM, Yura HT, and Andersen PE (2002): Derivation of a Monte Carlo method for modeling heterodyne detection in optical coherence tomography systems, Appl Opt. 41(31): 6676-6691.
van de Hülst HC (1980): Multiple Light Scattering: Tables, Formulas, and Applications, Academic Press, New York.
Wang LHV and Jacques SL (1993): Hybrid model of Monte Carlo simulation diffusion theory for light reflectance by turbid media, J. Opt. Soc. Am. A 10: 1746-1752.
Wang LHV and Jacques SL (1994): Optimized radial and angular positions in Monte Carlo modeling, Med. Phys. 21: 1081-1083.
Wang LHV, Nordquist RE, and Chen WR (1997): Optimal beam size for light delivery to absorption-enhanced tumors buried in biological tissues and effect of multiple-beam delivery: A Monte Carlo study, Appl. Opt. 36(31): 8286-8291.
Wang LHV (2001): Mechanisms of ultrasonic modulation of multiply scattered coherent light: A Monte Carlo model, Opt. Lett. 26(15): 1191-1193.
Wang RKK (2002): Signal degradation by coherence tomography multiple scattering in optical of dense tissue: A Monte Carlo study towards optical clearing of biotissues, Phys. Med. Biol. 47(13): 2281-2299.
Wang XD, Yao G, and Wang LHV (2002): Monte Carlo model and single-scattering approximation of the propagation of polarized light in turbid media containing glucose, Appl. Opt. 41(4): 792-801/
Wang XD, Wang LHV, Sun CW, and Yang CC (2003): Polarized light propagation through scattering media: Time-resolved Monte Carlo simulations and experiments, J. Biomed. Opt. 8(4): 608-617.
FURTHER READING 6 5
Wang XY, Zhang CP, Zhang LS, Qi SW, Xu T, and Tian JG (2003): Reconstruction of optical coherence tomography image based on Monte Carlo method, J. Infrared Millim. Waves 22(1): 68-70.
Wilson BC and Adam GA (1983): Monte Carlo model for the absorption and flux distri- butions of light in tissue, Med. Phys. 10: 824-830.
Wilson BC and Jacques SL (1990): Optical reflectance and transmittance of tissues: Prin- ciples and applications, IEEE J. Quantum Electron. 26: 2186-2199.
Wong BT and Menguc MP (2002): Comparison of Monte Carlo techniques to predict the propagation of a collimated beam in participating media, Num. Heat Transfer B—Fundamentals 42(2): 119-140.
Xiong GL, Xue P, Wu JG, Miao Q, Wang R, and Ji L (2005): Particle-fixed Monte Carlo model for optical coherence tomography, Opt. Express 13(6): 2182-2195.
Yadavalli VK, Russell RJ, Pishko MV, McShane MJ, and Cote GL (2005): A Monte Carlo simulation of photon propagation in fluorescent poly(ethylene glycol) hydrogel microsensors, Sensors Actuators B—Chemical 105(2): 365-377.
Yang Y, Soyemi OO, Landry MR, and Soller BR (2005): Influence of a fat layer on the near infrared spectra of human muscle: Quantitative analysis based on two-layered Monte Carlo simulations and phantom experiments, Opt. Express 13(5): 1570-1579.
Yao G and Wang LHV (1999): Monte Carlo simulation of an optical coherence tomogra- phy signal in homogeneous turbid media, Phys. Med. Biol. 44(9): 2307-2320.
Yao G and Haidekker MA (2005): Transillumination optical tomography of tissue- engineered blood vessels: A Monte Carlo simulation, Appl. Opt. 44(20): 4265-4271.
CHAPTER 4
Convolution for Broadbeam Responses
4.1. INTRODUCTION
The Monte Carlo program MCML, introduced in Chapter 3, computes responses to a pencil beam normally incident on a multilayered scattering medium. These responses are referred to as Green 's functions or impulse responses. When a collimated photon beam is of finite width, the Monte Carlo method is still able to compute the responses by distributing the incident positions over the cross section of the beam. Each broad beam, however, requires a new time-consuming Monte Carlo simulation, even if the other parameters are unchanged. Convolu- tion of Green's functions for the same multilayered scattering medium, however, can efficiently compute the responses to a broad beam. Such convolution was implemented in a program named CONV (Appendix 4A). Like MCML, CONV is written in ANSI Standard C and hence can be executed on various computer platforms. Although convolution is applicable to collimated beams of any inten- sity distribution, only Gaussian and top-hat (flat-top) beams are considered in CONV version 1.
4.2. GENERAL FORMULATION OF CONVOLUTION
Convolution is applicable to a system that is stationary (time-invariant), lin- ear, and translation-invariant. The system here consists of horizontal layers of homogeneous scattering media that have stationary properties (see Figure 3.1 in Chapter 3). The input to the system is a collimated photon beam perpendicu- larly incident on the surface of the scattering medium. The responses can be any observable physical quantities, such as specific absorption, fluence, reflectance, or transmittance. The linearity implies that (1) the responses increase by the same factor if the input intensity increases by a constant factor and (2) any response to two photon beams together is the sum of the two responses to each photon beam alone. The translation invariance in space here means that if the photon beam is
Biomedical Optics: Principles and Imaging, by Lihong V. Wang and Hsin-i Wu Copyright © 2007 John Wiley & Sons, Inc.
67
68 CONVOLUTION FOR BROADBEAM RESPONSES
shifted in any horizontal direction by any distance, the responses are also shifted in the same direction by the same distance. The translation invariance in time indicates that if the photon beam is delayed by a given time, the responses are also shifted by the same delay. Therefore, responses to spatially and temporally broad beams can be computed using the convolution of the impulse responses; only spatial convolution is described in this chapter, however.
Impulse responses to a normally incident pencil beam are first computed using MCML, where a Cartesian coordinate system is set up as described in Chapter 3. The origin of the coordinate system is the incident point of the pencil beam on the surface of the scattering medium, and the z axis is along the pencil beam; hence, the xy plane is on the surface of the scattering medium.
We denote a particular response to a collimated broad photon beam as C(x, y, z) and denote the corresponding impulse response as G(x, y, z). If the broad collimated light source has intensity profile S(x, y), the response to this broad beam can be obtained through the following convolution
/
OO pOO
I G(x - xf, y - yf, z)S(x', y) dx dy, -co J—oo
(4.1)
which can be reformulated with a change of variables x" = x — x' and y" = y-y':
/
CO /»OO
/ G(x'\y",z)S(x~x",y-y")dx"dy". (4.2) -CO «/— OO
Because the multilayered structure has planar symmetry and the photon beam is perpendicularly incident on the surface of the scattering medium, G(x, y,z) possesses cylindrical symmetry. Consequently, the Green function in Eq. (4.1) depends only on the distance ros between the source point (xf, yf) and the obser- vation point (JC, y), rather than on their absolute locations:
ros = > / ( * - * ' ) 2 + ( y - / ) 2 . (4.3)
If S(x', y') also has cylindrical symmetry about the origin, it becomes a function of only the radius r'\
r' = vV2 + y'1. (4.4)
On the basis of these symmetries, we reformulate Eqs. (4.1) and (4.2) to
C(JC, y,z)= ί ί G (y(x-x')2 + (y-y')2, z) S (V*'2 + / 2 ) dx'dy\
( 4
/
OO pOO / \ / \
/ G Ux"2 + y"2, z) S U(x - x")2 + (y- y")2) dx"dy".
(4.5)
(4.6)
CONVOLUTION OVER A GAUSSIAN BEAM 6 9
Because C(x, y, z) has the same cylindrical symmetry, Eqs. (4.5) and (4.6) can be rewritten in cylindrical coordinates (r, φ):
C(r, z) = I S(r')r' ί G (y/r2 + r'2 - Irr'costy, z) dtf\ dr', (4.7)
C(r, z)= I G{r", z)r" j S i^Jr2 + r"2 - 2rr" cos φ") </φ" dr".
(4.8) In Eq. (4.8), the integration over φ" is independent of z and hence needs to be computed only once for all z values. In some cases, the integration over φ" can be solved analytically; thus, the 2D integral in Eq. (4.8) is reduced to a compu- tationally more efficient ID integral. Therefore, Eq. (4.8) is more advantageous computationally than Eq. (4.7).
Example 4.1. Derive Eq. (4.7) from Eq. (4.5).
The differential area element can be changed from dxdy to rdrd§, and the corresponding limits of the integrations are from 0 to -f oo for r and from 0 to 2π for φ. In the polar coordinates aligned with (JC, y), (JC, y) is represented by (r, 0) and (x\ yf) by (rf, φ'). We convert Eq. (4.3) into ros = yjr2 -f r'2 — 2rrf cos φχ. Thus, Eq. (4.7) can be obtained from Eq. (4.5).
4.3. CONVOLUTION OVER A GAUSSIAN BEAM
For a Gaussian beam, the convolution can be further simplified. The intensity profile of the beam is given by
S(r') = S0 exp <ff (4-9) Here, R denotes the l/e2 radius of the beam; So denotes the intensity at the center of the beam, and So is related to the total power PQ by
So = 2P0 π/?2 '
(4-10)
Substituting Eq. (4.9) into Eq. (4.8), we obtain
C(r,z) = S(r)jC°G(r",z)exp\-2('^\
D f-2* /4rr"cos( | )" \ „1 „ ,, (4.11)
7 0 CONVOLUTION FOR BROADBEAM RESPONSES
The inner integral in the square brackets resembles the integral representation of the zeroth-order modified Bessel function, which is defined by
/oW 1 ί2π
2π Jo exp(;c sin φ) d<\>
or
1 [2π
2π Jo εχρ0^θ8φ)<Ζφ.
(4.12)
(4.13)
By using Eq. (4.13), we can rewrite Eq. (4.11) as
ί°° {r"Y\ / 4 r r " \ C(r,z) = 2nS(r)j G(r",z)exp - 2 i - J 70 ί - ^ - J r"dr". (4.14)
Example 4.2. Derive Eq. (4.10).
The total power
P0 = ί S(r')2n/dr' = 2nS0 f exp - 2 ( - )
- 5Ό exp (4.15)
which leads to Eq. (4.10).
Example 4.3. Show that Eqs. (4.12) and (4.13) are equivalent.
Letting φ = φ' + (π/2) and splitting the integral in Eq. (4.12) into two parts, we obtain
/»2π /·3π/2 I exp(xsin<\))d$ — / exp(Jccosφ/)ί/φ/
JO J-n/ ir/2
/
0 /*3π/2
exp(x cos φ') d φ/ + I exp(;c cos φ') d(\>' -π/2 JO
1/2
Letting φ' = φ" + 2π in the first integral on the right-hand side gives
Γ·2π
/
0 /»Z71
exp(jt ^8φ')οίφ' = I exp(jc cos φ^) άφ". -n/2 hn/2
(4.16)
(4.17)
CONVOLUTION OVER A TOP-HAT BEAM 71
Substituting Eq. (4.17) into Eq. (4.16) and merging the two integrals on the right- hand side of Eq. (4.16), we obtain
/»2π /·2π /»3π/2
/ expU sin φ) dφ = / exp(x cos φ") d<\>" + / exp(jt cos φ') d§' JO J3n/2 JO
ρ2π
=■ I βχρί^οοβφ) */φ. (4.18) Jo
Since both φ' and φ" are dummy variables, they are both replaced with φ above.
4 A CONVOLUTION OVER A TOP-HAT BEAM
For a top-hat beam of radius R, the source function becomes
s<H° ϊ '»£· <4i9) where So denotes the intensity inside the beam. We have
Po So = ~ , (4.20)
nRz
where Po denotes the total power of the beam. Substituting Eq. (4.19) into Eq. (4.8), we obtain
C{r, z) = 2nSQ / G(r", Ζ)/Φ(Γ, r")r"' dr\ (4.21) Jo
where
1 if R > r + r"
^ c o s - ^ ^ + ^ V ^ 2 ) if V-r"\ < R<r + r". (4.22) /φ(Γ,Γ") =
0 if R < \r - r"\
From Eq. (4.22), the limits of integration in Eq. (4.21) can be changed to a finite range
rr+R it \ j / J i \ > ' j ' f C(r, z) = 2nS0 [ G{r"\ Ζ)/Φ(Γ, r")r" dr", (4.23)
Ja
where
a = max(0, r - R). (4.24)
Function max( ) takes on the greater of the two arguments.
7 2 CONVOLUTION FOR BROADBEAM RESPONSES
If R tends to infinity, Eq. (4.23) becomes /»OO
C(r, z) = 2nSo / G{r\ z)r" dr". (4.25) Jo
This equation implies that in MCML, if the absorbed weights in all r grid ele- ments are summed and then divided by the total number of tracked photon packets, the result represents the specific absorption as a function of z for an infinitely wide beam of unit intensity.
4.5. NUMERICAL SOLUTION TO CONVOLUTION
As described in Chapter 3, a grid system is used in the Monte Carlo model. A 2D homogeneous grid system is set up in the r and z directions. The grid element sizes are Ar and Az in the r and z directions, respectively; the total numbers of grid elements in the r and z directions are Nr and Nz, respectively.
When the photon beam is Gaussian or top-hat, the 2D convolution becomes ID. Because the Monte Carlo simulation assigns physical quantities to discrete grid elements, an appropriate integration algorithm is based on the extended trapezoidal rule. This algorithm is ideal for a nonsmooth integrand that is linearly interpolated between available data points; it is implemented in C as a function named t r a p z d ( ) , which is called by another function named q t r a p ( ) .
Another method of integration is to evaluate the integrand at the original grid points. This approach, however, does not offer any control over the integration accuracy. For a top-hat beam, for example, Nr is 50, and R is about 5Ar. If C(0, z) is computed from Eq. (4.23), the integration interval [0, R] covers only 5Ar. Thus, only five function evaluations contribute to the integration and may yield unacceptable accuracy. By contrast, the extended trapezoidal rule continues to perform function evaluations until a user-specified accuracy is reached.
The sequence of integrand evaluations in the extended trapezoidal rule is shown in Figure 4.1a. Subsequent calls to t r apzd( ) incorporate the previous evaluations and evaluate the integrand only at the new points. To integrate f(x) over interval [a,b], we evaluate f(a) and f(b) in the first step as noted by 1 and 2 in Figure 4.1a. To refine the grid, we evaluate / ( | (a + b)) in the second step as noted by 3. This process is repeated until the integral evaluation reaches a specified accuracy. The bottom line shows all function evaluations after four calls.
4.5.1. Interpolation and Extrapolation of Physical Quantities
The physical quantities under discussion have been computed using MCML over a grid system. As discussed in Chapter 3, the optimal r coordinate is
r(ir) = 2 / l2(/r + ±)
Ar, (4.26)
NUMERICAL SOLUTION TO CONVOLUTION
(a)
O
Original data Interpolation Extrapolation
' I ' I ' I ' I ' I " I ' I 1 2 3 4 5 6 7
(b) r/Ar N -0.5
Figure 4.1. (a) Sequence of integrand evaluations in the extended trapezoidal rule of integration; (b) interpolation and extrapolation of the physical quantities. In this example, Nr = 8. Symbols a and b denote the integration limits, and i denotes the iteration index. Arrows point to where the integrand is evaluated. Solid circles represent the original data points. Dashed and solid lines represent linear interpolation and extrapolation, respectively.
where ir is the index of the grid element (0 < ir < Nr — 1). For ir = 0, r(0) is | Ar instead of ^Ar. The offset between the optimized and the centered coordi- nates in each grid element decreases as ir increases.
In qt rap (), the integrand—of which G is only a part—is evaluated at points that may not fall on the original grid as illustrated in Figure 4.1. Linear interpo- lations are used for those points that fall between two original grid points, and linear extrapolations are used for those points that are located beyond the origi- nal grid system (Figure 4.1b). Extrapolation is extended only up to (Nr - 0.5) Ar because further extrapolation is unreliable. In MCML, the last cells in the r direc- tion are used to collect contributions from photon packets that do not fit into the
74 CONVOLUTION FOR BROADBEAM RESPONSES
grid system and thus do not represent the true local physical quantities. There- fore, the upper limit for extrapolation is (Nr — 0.5)Ar instead of (Nr + 0.5)Ar, and the physical quantity beyond (Nr — 0.5) Ar is set to zero. We denote
rmax = (Nr-0.5)Ar. (4.27)
4.5.2. Integrand Evaluation for a Gaussian Beam
Although the integration in Eq. (4.14) must converge for physical reasons, it may cause overflow in a computer because the modified Bessel function increases rapidly as the argument increases. Therefore, a proper reformulation of the inte- grand is necessary.
We note that the modified Bessel function has the following asymptotic approx- imation for large x values:
expOc) / o W « - ^ . (4.28)
y/2nx
We define the following new function on the basis of IQ
Ioe(x) = Io(x)exp(-x) (4.29)
or
/o(x) = /o*U) expU), (4.30)
where IQ€ is always well bounded. Substituting Eqs. (4.9) and (4.30) into Eq. (4.14), we obtain
poo
C(r, z) = 2π5ο / G(r", z) exp Jo
Because both the exponential and the I$e terms are well bounded, the integrand can be computed without overflow. Since Eq. (4.28) is not actually used in the computation, Eq. (4.31) does not carry any asymptotic approximation.
Computation speed is another issue. The evaluation of exp( )Ioe() in Eq. (4.31) is a major part of the computation, which can take up to 90% of the total time. For multi-dimensional physical quantities (e.g., the fluence as a function of r and z), the convolution may repeatedly evaluate exp( )Ioe( ) at the same r coordi- nate as the integration is computed for different z coordinates. Therefore, if the values of exp( )/o*( ) are stored during the convolution for one z coordinate, computation time can be reduced. Because the number of function evaluations is unknown in advance, the function evaluations should be saved with dynamic data allocation. Since the evaluation sequence in q t r a p O resembles a binary tree as shown in Figure 4.1a, a binary tree can be used to store the function evaluations. Although the first two nodes are out of balance, the subtree below node 3 is perfectly balanced.
^ Ι Χ Ϊ Κ «->
NUMERICAL SOLUTION TO CONVOLUTION 7 5
4.5.3. Integration Limits for a Gaussian Beam
Although the upper integration limit for a Gaussian beam in Eq. (4.31) is infinity, it can be converted to a finite value by a change of variables; then, the integration can be computed by a function called midexp(). This approach, however, is not computationally efficient. Therefore, q t r a p ( ) is preferred; the upper integration limit, however, must be reduced to a finite value. To this end, the integrand is nonzero if
| r " - r | <KR (4.32)
or
r-KR<r" <r + KR, (4.33)
where A' is a constant that can be set in CONV. For example, if K is 4, the exponential term in Eq. (4.31) is exp(—32) ~ 10~~14.
The computation of G covers only interval [0, rm a x] , where rmax is as given by Eq. (4.27). Combining this limit and Eq. (4.33), we rewrite Eq. (4.31) as
fb Γ / r " - r \ 2 1 /4rr"\ C(r,z) = 2nS0j G(r",z)exp I —2 f—^—1 \l0ei ~^-\ r"dr"% (4.34)
where
a = max(0, r - KR), (4.35)
fc = min(rmax,r + KR). (4.36)
Functions max( ) and min( ) take on the greater and the lesser of the two arguments, respectively.
4.5.4. Integration for a Top-Hat Beam
The integrand for a top-hat beam can be evaluated more easily than that for a Gaussian beam, because the integration limits are finite and the integrand causes no overflow. However, evaluation of /ψ in Eq. (4.23) is also time-consuming. As in the integrand evaluation for Gaussian beams, the evaluated /φ values for q t r a p ( ) are stored in a binary tree for computational efficiency.
Since the physical quantities are computed only in interval [0, rm a x] , Eq. (4.23) can be expressed as
C(r, z) = 2JZSO f G{r\ Ζ)/Φ(Γ, r")r" dr", (4.37) Ja
76 CONVOLUTION FOR BROADBEAM RESPONSES
where
a = max(0, r - /?), (4.38)
fc = min(rmax,r + /?). (4.39)
4.5.5. First Interactions
In MCML, absorption from the first photon-tissue interactions is recorded sep- arately. The first interactions always occur on the z axis and hence contribute to the specific absorption or related physical quantities as a delta function. The total impulse response can be expressed in two parts
G(r, z) = G,(0, z)^- + G2(r, z), (4.40)
where the first term results from the first interactions and the second, from sub- sequent interactions.
For a Gaussian beam, substituting Eq. (4.40) into Eq. (4.34) yields
C(r,z) = G{(0,z)S(r) + 2nS0 f G2{r",z) Ja
- 2 (̂ Χ )̂'*-· x exp For a top-hat beam, substituting Eq. (4.40) into Eq. (4.37) yields
C(r, z) = Gi(0, z)S(r) + 2nS0 f G2(r\ Ζ)/Φ(Γ, r")r" dr". (4.42) Ja
The numerical results obtained with and without separately recording the first interactions are compared in the next section.
4.5.6. Truncation Error in Convolution
As shown in Eqs. (4.36) and (4.39), the upper integration limits may be bounded by rmax. For a top-hat beam, if
r < W - R, (4.43)
the limited grid coverage in the r direction does not affect the convolution; otherwise, it truncates the convolution and leads to error in the convolution for r > ''max — R- Thus, to convolve reliably for physical quantities at r in response to a top-hat beam, we must ensure that rmax in the Monte Carlo simulation is large enough that Eq. (4.43) holds.
APPENDIX 4A. SUMMARY OF CONV 7 7
For a Gaussian beam, no simple formula similar to Eq. (4.43) exists because a Gaussian beam theoretically extends to infinity. At r ^> R, a Gaussian beam and a top-hat beam of the same R and So have comparable convolution results. Therefore, Eq. (4.43) can be used approximately for Gaussian beams as well.
4.6. COMPUTATIONAL EXAMPLES
In this section, the error that is caused by not recording the first photon-matter interactions separately is illustrated, and a numerical example of convolution is presented. The impulse responses are computed by MCML for a scattering medium described in Table 4.1. The grid element sizes in the r and z directions are both 0.01 cm. The numbers of grid elements in the r and z directions are 50 and 40, respectively. One million photon packets are tracked.
The impulse fluence near the surface of the scattering medium (z = 0.005 cm) is shown in Figure 4.2a, where the first interactions are recorded separately. If they were recorded into the first r grid element instead, it would augment the fluence in the first grid element by 1.95 x 103 cm- 2 , which is significantly greater than the current value of 1.34 x 103 cm- 2 . For comparison, the impulse response is convolved over a top-hat beam of 1-nJ energy and 0.01-cm radius both with, and without, recording the first interactions separately (Figure 4.2b). The convolved results differ at r = 0.015 cm by as much as 120%.
The impulse response is also convolved over a Gaussian beam (1-nJ total energy, 0.1-cm radius), where the convolution error is set to 0.01. The contour lines of the fluence distribution before and after the convolution are shown in Figure 4.3.
APPENDIX 4A. SUMMARY OF CONV
The entire source code of CONV can be found on the Web at ftp://ftp.wuey. com/public/sci_tech_med/biomedical_optics. The program is divided into several files. Header file conv.h defines the data structures and some constants. File convmain.c contains primarily the main function. File convi .c handles data reading. File convo.c handles data writing. File convconv.c implements the
TABLE 4.1. Optical Properties and Structure of a Three-Layered Scattering Medium0
Layer
1 2 3
n
1.37 1.37 1.37
μ« (cm"1)
1.0 1.0 2.0
μ* f (cm" 1)
100.0 10.0 10.0
8
0.9 0 0.7
Thickness (cm)
0.1 0.1 0.2
aRefractive indices for top and bottom ambient media are both 1.0.
78 CONVOLUTION FOR BROADBEAM RESPONSES
0.2 0.3 Radius r (cm)
Figure 4.2. (a) Relative fluence at z = 0.005 cm in response to a pencil beam computed by MCML; (b) fluence at z = 0.005 cm in response to a top-hat beam computed by CONV. Circles and crosses represent data with and without, respectively, the first inter- actions scored separately.
actual convolution. File conviso .c handles calculation of contours. File con- vnr . c contains several functions for dynamical data allocations and error reports. Readers should read the main function first.
The following list is generated by command cf low -d4 -n - -omit-arguments - -omit - symbol- names conv* . c, which shows the structure of the program with
APPENDIX 4A. SUMMARY OF CONV 7 9
(a) 0.2 0.3 Radius r (cm)
0.2 0.3 Radius r (cm)
Figure 4.3. (a) Relative fluence distribution in response to a pencil beam computed by MCML; (b) fluence distribution in response to a Gaussian beam computed by CONV.
the nesting depth limited to 4:
1 main() <int () at C0NVMAIN.C:143>: 2 ShowVersionO <void () at C0NV0.C:37>: 3 pu t s ( ) 4 Cen te rS t r ( ) <char * () a t C0NV0.C:11>: 5 s t r len ( ) 6 strcpy()
8 0 CONVOLUTION FOR BROADBEAM RESPONSES
7 s t r c a t ( ) 8 p r i n t f ( ) 9 ge ts ( )
10 s t r l e n ( ) 11 BranchMainCmd() <void () at CONVMAIN.C:122>: 12 strlen() 13 BranchMaihCmdl() <void () at C0NVMAIN.C:63>: 14 toupper() 15 ReadMcoFileO <void () at CONVI.C:568>: 16 LaserBeam() <void () at C0NVC0NV.C:92>: 17 ConvResolution() <void () at C0NVC0NV.C:127>: 18 ConvError() <void () at CONVCONV.C:156>: 19 ShowMainMenu() <void () at CONVMAIN.C:25>: 20 QuitProgram() <void () at C0NVMAIN.C:44>: 21 puts() 22 BranchMainCmd2() <void () at C0NVMAIN.C:92>: 23 toupper() 24 0u tpu t0 r i gDa ta ( ) <void () at C0NV0.C:784>: 25 OutputConvData() <void () at CONVCONV.C:1017>: 26 ContourOrigData() <void () at C0NV0.C:893>: 27 ContourConvData() <void () at CONVCONV.C:1137>: 28 ScanOrigData() <void () a t CONVO.C:1211>: 29 ScanConvData() <void () at CONVCONV.C:1506>: 30 pu ts ( ) 31 pu ts ( )
PROBLEMS
4.1 Derive Eq. (4.11).
4.2 Derive Eq. (4.21).
4.3 Derive Eq. (4.23).
4.4 Derive Eq. (4.25).
4.5 Derive Eq. (4.31).
4.6 Derive Eqs. (4.41) and (4.42).
4.7 Write a computer program that can convolve the impulse responses over a flat beam.
4.8 Take the Fourier transformation of Eqs. (4.1) and (4.2) with respect to x and y.
4.9 Write the time-domain counterparts of Eqs. (4.1) and (4.2) for an arbi- trary pulse profile S(t) of the incident pencil beam. In this case, an
FURTHER READING 81
impulse response G(x, y, z, /) is defined as a time-resolved response to a temporally infinitely short δ(ί) photon beam. Assuming the response to S(t) to be experimentally measured, explain how to recover the impulse response G(JC, y, z, t) through deconvolution.
4.10 Take the Fourier transformation of the time-domain counterparts of Eqs. (4.1) and (4.2) with respect to t.
4.11 Assume that the incident photon beam is finite in both time and space and can be represented by S(x, y,z,t). Write the convolution over this beam.
4.12 Although Eqs. (4.9) and (4.19) are related to the total power of the light- beam, explain why the total energy can be used when, for example, the lightbeam is infinitely short-pulsed.
READING
Wang LHV, Jacques SL, and Zheng LQ (1997): CONV—convolution for responses to a finite diameter photon beam incident on multi-layered tissues, Comput. Meth. Prog. Biomed. 54(3): 141-150. (All sections in this chapter.)
FURTHER READING
Amelink A and Sterenborg H (2004): Measurement of the local optical properties of turbid media by differential path-length spectroscopy, Appl. Opt. 43(15): 3048-3054.
Carp SA, Prahl SA, and Venugopalan V (2004): Radiative transport in the delta-P-1 Approximation: Accuracy of fluence rate and optical penetration depth predictions in turbid semi-infinite media, J. Biomed. Opt. 9(3): 632-647.
Choi B, Majaron B, and Nelson JS (2004): Computational model to evaluate port wine stain depth profiling using pulsed photothermal radiometry, J. Biomed. Opt. 9(2): 299-307.
Diaz SH, Aguilar G, Lavernia EJ, and Wong BJF (2001): Modeling the thermal response of porcine cartilage to laser irradiation, IEEE J. Select. Topics Quantum Electron. 7(6): 944-951.
Ding L, Splinter R, and Knisley SB (2001): Quantifying spatial localization of optical map- ping using Monte Carlo simulations, IEEE Trans. Biomed. Eng. 48(10): 1098-1107.
Fried NM, Hung VC, and Walsh JT (1999): Laser tissue welding: Laser spot size and beam profile studies, IEEE J. Select. Topics Quantum Electron. 5(4): 1004-1012.
Gardner CM and Welch AJ (1994): Monte-Carlo simulation of light transport in tis- sue—unscattered absorption events, Appl. Opt. 33(13): 2743-2745.
Garofalakis A, Zacharakis G, Filippidis G, Sanidas E, Tsiftsis D, Ntziachristos V, Papa- zoglou TG, and Ripoll J (2004): Characterization of the reduced scattering coefficient for optically thin samples: theory and experiments, /. Opt. A 6(7): 725-735.
Giller CA, Liu RL, Gurnani P, Victor S, Yazdani U, and German DC (2003): Validation of a near-infrared probe for detection of thin intracranial white matter structures, J. Neurosurg. 98(6): 1299-1306.
8 2 CONVOLUTION FOR BROADBEAM RESPONSES
Hidovic-Rowe D and Claridge E (2005): Modelling and validation of spectral reflectance for the colon, Phys. Med. Biol. 50(6): 1071-1093.
Johns M, Giller CA, and Liu HL (2001): Determination of hemoglobin oxygen satu- ration from turbid media using reflectance spectroscopy with small source-detector separations, Appl. Spectrosc. 55(12): 1686-1694.
Johns M, Giller CA, German DC, and Liu HL (2005): Determination of reduced scat- tering coefficient of biological tissue from a needle-like probe, Opt. Express 13(13): 4828-4842.
Klavuhn KG and Green D (2002): Importance of cutaneous cooling during photothermal epilation: Theoretical and practical considerations, Lasers Surg. Med. 31(2): 97-105.
Laufer JG, Beard PC, Walker SP, and Mills TN (2001): Photothermal determination of optical coefficients of tissue phantoms using an optical fibre probe, Phys. Med. Biol. 46(10): 2515-2530.
Lee JH, Kim S, and Kim YT (2004): Diffuse-diffuse photon coupling via nonscattering void in the presence of refractive index mismatch on the void boundary, Med. Phys. 31(8): 2237-2248.
Marquez G, Wang LHV, Lin SP, Schwartz JA, and Thomsen SL (1998): Anisotropy in the absorption and scattering spectra of chicken breast tissue, Appl. Opt. 37(4): 798-804.
McShane MJ, Rastegar S, Pishko M, and Cote GL (2000): Monte carlo modeling for implantable fluorescent analyte sensors, IEEE Trans. Biomed. Eng. 47(5): 624-632.
Prahl SA, Keijzer M, Jacques SL, and Welch AJ (1989): A Monte Carlo model of light propagation in tissue, in Dosimetry of Laser Radiation in Medicine and Biology, Müller GJ and Sliney DH, eds., SPIE Press, IS5: 102-111.
Press WH, Flannery BP, Teukolsky SA, and Vetterling WT (1992): Numerical Recipes in C, Cambridge Univ. Press.
Reuss JL (2005): Multilayer modeling of reflectance pulse oximetry, IEEE Trans. Biomed. Eng. 52(2): 153-159.
Shah RK, Nemati B, Wang LHV, and Shapshay SM (2001): Optical-thermal simulation of tonsillar tissue irradiation, Lasers Surg. Med. 28(4): 313-319.
Wang LHV and Jacques SL (1994): Optimized radial and angular positions in Monte Carlo modeling, Med. Phys. 21: 1081-1083.
Wang LHV, Jacques SL, and Zheng LQ (1995): MCML—Monte Carlo modeling of photon transport in multi-layered tissues, Comput. Meth. Prog. Biomed. 47: 131-146.
CHAPTER 5
Radiative Transfer Equation and Diffusion Theory
5.1. INTRODUCTION
Photon transport in biological tissue can be modeled analytically by the radiative transfer equation (RTE), which is considered equivalent to the numerical Monte Carlo method covered in Chapter 3. Because the RTE is difficult to solve, it is often approximated to a diffusion equation, which provides solutions that are more computationally efficient but less accurate than those provided by the Monte Carlo method.
5.2. DEFINITIONS OF PHYSICAL QUANTITIES
Spectral radiance Lv, the most general physical quantity discussed in this chapter, is defined as the energy flow per unit normal area per unit solid angle per unit time per unit temporal frequency (temporofrequency) bandwidth, where the normal area is perpendicular to the flow direction. Radiance L is defined as the spectral radiance integrated over a narrow frequency range [v, v + Δν]:
L(r, 5, t) = Lv(r, s, t)Av (W nrV - 1 ) . (5.1)
Here, r denotes position, s denotes unit direction vector, t denotes time, and the parentheses enclose the unit of the physical quantity on the left-hand side of the equation. The amount of radiant energy dE that is transported across differential area element dA within differential solid angle element dQ during differential time element dt (Figure 5.1) is given by
dE = L(rJ,t)(S'fi)dAdQdt (J). (5.2)
Here, h denotes the unit outward normal vector of dA; s · h denotes the dot product of the two unit vectors, which equals the cosine of the angle between them. The radiance is the dependent variable in the RTE (to be derived). Several additional physical quantities can be derived from the radiance.
Biomedical Optics: Principles and Imaging, by Lihong V. Wang and Hsin-i Wu Copyright © 2007 John Wiley & Sons, Inc.
83
8 4 RADIATIVE TRANSFER EQUATION AND DIFFUSION THEORY
Figure 5.1. Schematic of energy flow through a differential area element dA within a differential solid angle element dQ.
Fluence rate (or intensity) Φ is defined as the energy flow per unit area per unit time regardless of the flow direction; it is expressed as the radiance integrated over the entire 4π solid angle:
φ(?, t) = I L(r,s, J4n
t)dQ (W/m2). (5.3)
An infinitesimal sphere of surface area dS receives power in the amount of Φ ( Γ , t)dS. In spherical coordinates, we have
/»π /»2π Φ ( Γ , 0 = / / L(r,S,r) sin θ</φέ/θ (5.4)
and
s — (sin Θ cos φ, sin Θ sin φ, cosO), (5.5)
where Θ and φ denote the polar and azimuthal angles, respectively. Fluence F is defined as the time-integrated fluence rate:
F(r) /
+oo Φ(Γ ,
-oo
t)dt (J/m2). (5.6)
Current density J is defined as the net energy flow per unit area per unit time; it can be expressed as
7 ( r , i ) = / sL(?J,t)dQ (W/m2), J4n
(5.7)
DERIVATION OF RADIATIVE TRANSPORT EQUATION 8 5
which is the vector counterpart of Eq. (5.3). Current density points to the direction of the prevalent flow since flows in opposite directions partially offset each other. Current density is also referred to as energy flux; the term flux, however, can also refer to a vector quantity integrated over a given area.
Energy density ue is defined as the energy of the propagating electromagnetic wave per unit volume; it can be obtained by
ue = - (J/m3), (5.8) c
where c is the speed of light in the medium. Photon density U is defined as the number of propagating photons per unit
volume; for monochromatic light, it can be expressed as
ue Φ i U = IT = ~T <m~ >' <5'9)
nv cnv where h is the Planck constant and hv is the energy of a single photon.
Specific power deposition (or specific absorption rate) Ap is defined as the optical energy absorbed by the medium per unit volume per unit time; it can be expressed as
Αρ = μαΦ (W/m3), (5.10)
where μα is the absorption coefficient of the medium. Specific energy deposition (or specific absorption) Ae is defined as the time-
integrated specific power deposition:
/
+oo A„(r,t)dt (J/m3). (5.11)
-00
5.3. DERIVATION OF RADIATIVE TRANSPORT EQUATION
With the quantities defined above, we now heuristically derive the RTE from the principle of conservation of energy, where coherence, polarization, and non- linearity are neglected. The optical properties—including refractive index n, absorption coefficient μα, scattering coefficient μ5, and scattering anisotropy g—are assumed to be time-invariant but space-variant. Only elastic scattering is considered in this chapter.
Consider a stationary differential cylindrical volume element as shown in Figure 5.2. Here, ds is the differential length element of the cylinder along pho- ton propagation direction s ; dA is the differential area element perpendicular to direction s. Below, we consider all possible contributions to the energy change in this volume element within differential solid angle element dQ around direction s. In addition, dQ' is a differential solid angle element around direction s'.
8 6 RADIATIVE TRANSFER EQUATION AND DIFFUSION THEORY
Figure 5.2. Schematic of a stationary differential cylindrical volume element.
5.3.1. Contribution 1: Divergence
If the local photon beam is not collimated, divergence is nonzero. Energy diverg- ing out of the volume element or the solid angle element per unit time is given by
dL(r,s,t) dL(r,s,t) dPdiy = — v } dsdAdQ = —V } dQdV, (5.12)
ds ds
where dV = dAds. This contribution is positive for actual divergence and neg- ative for actual convergence.
In divergence form, Eq. (5.12) becomes
dPdiw = § · VL(?, s, t)dQdV = V · [L<7, s, t)s]dQdV. (5.13)
This contribution is due to local "noninteractive" beam propagation; thus, it can exist even in a nonscattering medium. Scattering elsewhere, however, can affect the local divergence. It can be seen later that this contribution still exists when the absorption and scattering coefficients are set to zero in the radiative transfer equation.
5.3.2. Contribution 2: Extinction
Energy loss per unit time in the volume element within the solid angle element due to absorption and scattering is given by
rfPext = (lLtds)[L(r,s,t)dAdto], (5.14)
where \^tds represents the probability of extinction—by either absorption or scattering—in ds. Light scattered from all directions into solid angle element dQ is considered in the next subsection.
DERIVATION OF RADIATIVE TRANSPORT EQUATION 8 7
5.3.3. Contribution 3: Scattering
Energy incident on the volume element from any direction s' and scattered into d£2 around direction s per unit time is given by
dP,ca = (NsdV) I L(?, ί', t)P(s', s)os άΐϊ dQ. (5.15)
Here, Ns denotes the number density of the scatterers and os denotes the scattering cross section of a scatterer. Thus, NsdV denotes the number of scatterers in the volume element; L(r,s' ,t)osd£l' denotes the energy intercepted by a single scatterer within solid angle dQ' per unit time. The phase function P(s\ s) is a PDF:
L P(sJ)dQ= 1. (5.16) 4π The product P(s\s)dQ represents the probability of light with propagation direction s' being scattered into dQ around direction s. Often, the phase func- tion depends only on the angle between the scattered and incident directions, that is
P(s'J) = P(s' s), (5.17)
where s' · s equals the cosine of the angle between the two unit vectors. We limit our consideration to this case. The scattering anisotropy can be expressed as
g= [ (S''§)P(S'-§)da. (5.18)
From μ5 = Nsos and Eq. (5.17), Eq. (5.15) can be rewritten as
dP^ = (\isdV)\ I L(rJ',t)P(s' 'S)dQ'\ da. (5.19)
5.3.4. Contribution 4: Source
Energy produced by a source in the volume element within the solid angle element per unit time is given by
dPSTC = S ( r , i , 0 dV dQ, (5.20)
where 5 carries the unit of W/(m3 sr).
8 8 RADIATIVE TRANSFER EQUATION AND DIFFUSION THEORY
5.3,5. Conservation of Energy
The change in energy in the volume element within the solid angle element per unit time is given by
dP = — - 1 - dVdtt, (5.21) dt
where L/c represents the propagating energy per unit volume per unit solid angle. This rate of change is a result of the balance among the two negative and two positive contributions described above. The principle of conservation of energy requires
dP = -dPdiv - dPexl + dP,ca + dP$rc. (5.22)
Substituting Eqs. (5.13), (5.14), and (5.19)-(5.21) into Eq. (5.22), we obtain
dL(r,s,t)/c — J/ = -s · VL(r, 5, t) - [itL(r, S, /)
*' (5.23) + μ, / L(rJ',t)P(s' - s)d& + S(r, s, t),
J4n
which is the well-known RTE (or the Boltzmann equation). For time-independent responses, the left-hand side of Eq. (5.23) is zero:
—V = 0. (5.24) dt
To reach a time-independent state requires the use of a time-invariant light source, that is, a constant-power continuous-wave lightbeam. For a pulsed light source, time-independent responses are still applicable to time-integrated physical quan- tities such as specific energy deposition.
5.4. DIFFUSION THEORY
The RTE is difficult to solve since it has six independent variables (JC, y, z, θ, φ, t). Usually, the RTE is simplified in the diffusion approximation. The diffusion approximation assumes that the radiance in a high-albedo (μα <£ μ5) scattering medium is nearly isotropic after sufficient scattering.
5.4.1. Diffusion Expansion of Radiance
Spherical harmonics Y,hm form a basis set, on which the radiance can be expanded. In the diffusion approximation, the radiance is expanded to the first order
DIFFUSION THEORY 8 9
1 n
L(r, 3,ί)*ΣΣ L " ^ ' OYn.m(S), (5.25) n=0 m=—n
where Ln,m are the expansion coefficients. The term for n = 0 and m — 0 on the right-hand side represents the isotropic component, whereas the terms for n = 1 and m = 0, ± 1 represent the anisotropic component.
In terms of the associated Legendre polynomials Pnm and a periodic function of φ, we have
Yn,m(s)'= ΚΠιΐη(θ,φ) = ( - 1 Γ / ( 2 w + 1 ) ( n ^ ) ! F n , m ( c o s e y ^ , (5.26)
where
(Ί _ x2\m/2 jm+n Pnm(x) = !t ί _ ( J C 2 - 1)". (5.27)
' 2nrc! Jjcm+n
When m = 0, P n m reduces to the (unassociated) Legendre polynomials Pn. If the expansion of L in Eq. (5.25) continues to n = /V, the approximation is known as the PH approximation. Hence, the diffusion approximation is also known as the Pi approximation.
The spherical harmonics for n — 1 are
ΙΌ,οίθ, Φ) = 1
(5.28) Κι,_ι(θ,φ) = λ /^-5ίηθ^- 'Φ, V 8π
1Ί.ο(θ,φ) = ^ ο ο 8 θ ,
ίΊ.ι(θ,φ) = - ^ 8 ί η θ β ί φ .
The following symmetry and orthogonality (or orthonormality) exist:
r„,-m(e, Φ) = ( - i r r ; , m ( G , φ), (5.29)
Yn,m(S)Y:,m,(s)dn = S n n W . (5.30) /
Here, * denotes complex conjugation; hnn',mm' denotes the Krönecker delta func- tion, which equals 1 if both n — n' and m —m! hold and 0 otherwise. The integral in Eq. (5.30) is referred to as the inner product, analogous to the dot product of two vectors.
9 0 RADIATIVE TRANSFER EQUATION AND DIFFUSION THEORY
Substituting Eq. (5.25) into Eq. (5.3), we obtain
Φ(Γ,0 = 4πΔ0,ο(Γ,0^ο(δ) (5.31)
or
4π
which means that the isotropic term in Eq. (5.25) is equal to the fluence rate divided by the entire 4π solid angle.
Unit vector s can also be expressed in terms of the spherical harmonics:
s = (sin Θ cos φ, sin Θ sin φ, cos Θ)
= V Y (^ ι . - ι^) - Yu(S), *m, - i ( i ) + Yu(s)], V2F,,0(i)) · (5.33)
Multiplying Eq. (5.25) by s and substituting the result into Eq. (5.7), we obtain
4. [
7(r, t)-S = ^- Σ Li.™<?' OYumiS) (5.34) 3
m = - i
or 1
4π Σ L ^ ( ? ' 0Yum(S) = ^ - J ( r , 0 · 5. (5.35)
m = - l
Note that J(r,t) · s = \J(r, t)\ cos a, where a denotes the angle between J(r, t) and s. Therefore, the anisotropic term in Eq. (5.25) is proportional to the projec- tion of 7(r, t) onto s.
Substituting both Eqs. (5.32) and (5.35) into Eq. (5.25), we obtain
1 3 - _ L(r, S, t) = — Φ ( Γ , 0 + — 7 ( r , i) · s, (5.36)
4π 4π
which is illustrated in Figure 5.3.
Example 5.1. Derive Eq. (5.31).
Substituting Eq. (5.25) into Eq. (5.3), we obtain
Φ ( Γ , 0 = [ Lo,o(r,i)r0.o(S)dn+ f L,,_,(r, O l ^ K ^ r f n (5.37)
+ f L,,o(r,t)Yi.0mdii+ [ Lu(r,t)Yu(S)da. J An J4n
DIFFUSION THEORY 9 1
Figure 5.3. Illustration of the effect of current density J on the radiance. Solid line represents the isotropic term; dashed line represents the total radiance.
Noting that L„ m is independent of s and using Eq. (5.28), we have
/ L0,o(r, 0*b,o(i) dQ = L0lo(r, 0*b,o(S) / dQ = 4nL0,o(r, t)Y0t0(s),
(5.38)
/ Li,-i(r,t)Yl,-l(S)dn = J— L,
4π V 8π
/»2π /»π
, - Ι ( Γ , 0 / β" , - φέ/φ/ 8ίη2θέ/θ = 0, Jo Jo
X Li 0(r, ΟΚιο(ί)^Ω = 2πΛ/ — L , 0(r, 0 / cosΘsinθί/θ = 0, 4π ' V 4π ' Jo (5.39)
(5.40)
f Lix(r,t)Yii(s)da = -J—Lii(r,t)[ e^</φ f sm2Qdd = 0. J4n ' ' V 8π ' Jo Jo
Finally, we obtain
Φ ( Γ , 0 = 4π^,ο(Γ,ΟΪ"ο,ο(5).
Example 5.2. Derive Eq. (5.34).
(5.41)
(5.42)
Substituting Eqs. (5.33) and (5.25) into Eq. (5.7) and using Eqs. (5.29) and (5.30), we obtain
J(7, 0 = ^ y ( - L u + Ll t_i , - / ( L l f l 4- L,f_i), V2LK 0) . (5.43)
9 2 RADIATIVE TRANSFER EQUATION AND DIFFUSION THEORY
From Eqs. (5.33) and (5.43), we obtain
4 i
s · J(r, 0 = y £ L,,m(r, 0*Vm($). (5-44) m = -\
5.4.2. Source
The source 5 is assumed to be isotropic; that is, S(r, s, t) is independent of s:
4π
A collimated source can be approximately converted into an isotropic source (to be discussed).
5.4.3. Scalar Differential Equation
Substituting the diffusion expansion of L(r,s,t) [Eq. (5.36)] into the RTE [Eq. (5.23)] and integrating over the full 4π solid angle, we obtain the following scalar differential equation:
V + M.fl*(r, t) + V · J(r , 0 - S(r, t). (5.46) cat
Example 5.3. Derive Eq. (5.46).
We substitute Eq. (5.36) into Eq. (5.23), integrate over the full 4π solid angle (scalar sum), and then evaluate each term as follows:
1. For the left-hand side, on the basis of Eq. (5.3), we obtain
f 8LCr,U)/cdas=B*£j)^ ( 5 4 ? ) Λπ d' Cdt
2. For the first term on the right-hand side, on the basis of the vector identities s · VL = V · (sL) - LV · § and V · s = 0 and then Eq. (5.7), we obtain
- / S-VL(rJj)dn = -' I V .[sL(r,s,t)]dQ
= - V · / SL(r, i , 0 dQ = -V · J(r, 0-
(5.48)
3. For the second term, from Eq. (5.3), we obtain
J4i\ -μ, / Δ(Γ,5 ,0^Ω = ~μΓΦ(Γ,0. (5-49)
/4π
DIFFUSION THEORY 9 3
4. For the third term, we have
</Ω J4n U4
L{?,s,t)P(s s)dti 4π
= ψ- ί I l4>(r,t) + 3J(r,t)-s']P(S''S)dada. (5.50) 4π J4n J4?l
We first evaluate the following two integrals:
f \f ΦίΓ ,Ο^ί ί ' s)dQ'\ £/Ω = Φ ( Γ , 0 f | f Ρ ( 5 ' . ί ) ί / Ω Ί Λπ 1_./4π J «Μπ ίΛπ J
= Φ(Γ,0 ί J4
ΛΩ
</Ω = 4 π Φ ( Γ , ί ) 4π
(5.51) and
ί ί [J(r,t)sf]P(s' s)dQ'dQ */4π J4n
= \J(r,t)\[ \f P(S,'3)dQ\cos&da' J4n U4K J (5.52)
= |7 ( r ,0 l [ costfdQ' J4n
= 0.
Here, / is aligned with the τ! axis, and dQ! — sinQ'dd'dfy'. Therefore
[is f \ I L(?,s',t)P(sf s)dQf ί/Ω = μ ,Φ(? ,0 · (5.53)
5. For the last term, using Eq. (5.45), we obtain
/ S(rJ,t)dQ = — I S(r,t)da = S(r,t). (5.54) J4n 4 π J4n
Combining these five parts completes the proof, where μ, — μ5 = μα is used.
5.4.4. Vector Differential Equation
Substituting the diffusion expansion of L(r, 5 ,0 [Eq. (5.36)] into the RTE [Eq. (5.23)], multiplying both sides by s, and integrating over the full 4π solid angle, we obtain the following vector differential equation:
\ + (Ha + W,)J(r, t) + - νΦ(Γ, 0 - 0, (5.55) cdt 3
9 4 RADIATIVE TRANSFER EQUATION AND DIFFUSION THEORY
where
μ ; = μ , ( 1 - £ ) (5.56)
is referred to as the transport (or reduced) scattering coefficient. The sum μα -f μ̂ . is referred to as the transport (or reduced) interaction coefficient μ[:
μ; = μ0 + μ,· (5.57)
The reciprocal of μ, is referred to as the transport mean free path lt:
/,' = Λ · (5.58)
Example 5.4. Derive Eq. (5.55).
We substitute Eq. (5.36) into Eq. (5.23), multiply both sides by s, integrate over the full 4π solid angle (vector sum), and then evaluate each term as follows.
1. On the left-hand side, on the basis of Eq. (5.7), we obtain
f JL(rJ, ̂ , Ω = ^ Μ . (5.59) cdt
2. For the first-term on the right-hand side, we have
s(s VL)dQ
= — f s(s · ν Φ ) Λ Ω + — / s[s V(J s)]dQ. (5.60)
It can be shown that the two integrals on the right-hand side here have the following results (see Problems 5.1 and 5.2):
/ J4n
4π ί ( ί · ν Φ ) έ / Ω = — ν Φ , (5.61)
/ s[sV(Js)]dQ = 0. (5.62)
3. For the second-term on the right-hand side, from Eq. (5.7), we obtain
SL(r,s,t)dQ = VLtJ(r,s,t). (5.63) μ, / s J4n
/4π
s[<b(r,t)P(S''§)da']dQ 4π
+ j - l s\l U(r,t) -s']P(s' s)dQ'\ dQ. (5.64)
(5.65)
DIFFUSION THEORY 9 5
4. For the third-term on the right-hand side, we have
I I s[L(r,s',t)P(s' s)dQ']dQ «/4π J4n
--[ I
ΙΊΙ J4n {J4
For the first integral, we have
/ / s[<b(r,t)P(s' s)d JAn J4n
= Φ(?,0 I s\ I P(s' s)dQ'\ ί/Ω = Φ(Γ,0 / sdQ = 0 J4n U4n J J4n
(
For the second integral, we have
f Si [ [J(r,t)-5']P(§'-s)da'\da J4n [J4n J
= [ \f SP(s'-s)da\[J(r,t) s]dQ!. (5.66)
On the basis of the identity
s = s{s · s) + s x (s x s), (5.67)
the inner integral in Eq. (5.66) is split into two integrals. The first one is
[ §'(§' s)P(s' s)dQ = s'g. (5.68) J4n
The second one is
f s' x 0? x §')P(§' s)dQ = sf x\([ sP(s' · s)da\ x 5Ί . (5.69)
Since P(sf · S) is azimuthally symmetric about s, f4n sP(s' · £)*/Ω is par- allel with s'; hence, its cross-product with s' is zero. Therefore, Eq. (5.66)
9 6 RADIATIVE TRANSFER EQUATION AND DIFFUSION THEORY
becomes
H/ J4n
[ i ( r , 0 s']P(sf s)drt\ dQ
[J(r,t)'S']dQf
4π - _ = —gJ(rj),
(5.70)
where the last step is based on Problem 5.1. 5. For the last term on the right-hand side, from Eq. (5.45), we obtain
L S(r,t) f sS(r,s,t)dtt = —— I sdtt = 0. 4π 4π / J4n (5.71) Combining these five parts completes the proof.
5.4.5. Diffusion Equation
We notice that Eqs. (5.46) and (5.55) do not contain s as does Eq. (5.23) but contain two physical quantities J(r,t) and Φ(Γ , t). We now aim to obtain a single differential equation containing Φ(Γ , 0 only.
We further assume that the fractional change in J{r,t) within lt is small, specifically
!L 1
\J(rj)\
dJ(r,t)
dt « 1, (5.72)
where the first pair of parentheses contains the time duration for photons to traverse lt (which may be referred to as the transport mean free time) and the second pair of parentheses contains the fractional change in the current density per unit time. Equation (5.72) can be rewritten as
dJ(r,t)
cdt <<(μ* + [is)\J(?j)\. (5.73)
Under this condition, the time-dependent term in Eq. (5.55) is negligible, lead- ing to
7( r , i ) = -DV4>(; , / ) , (5.74)
DIFFUSION THEORY 9 7
which is referred to as Fick's law. A negative sign appears above because dif- fusion current density is always along the negative gradient. The constant D is referred to as the diffusion coefficient:
D=~( * , , · (5.75) 3(μα 4- μ5)
Fick's law describes the, diffusion of photons in a scattering medium. In fact, Fick's law can describe diffusion in many other forms such as pollutant diffusion in air, ink diffusion in water, and heat diffusion in metal. It is not, however, applicable to propagations driven by external forces, such as electron drift in an external electrical field and particle drift under external pressure.
Substituting Eq. (5.74) into Eq. (5.36), we obtain
1 - 3 L(r, 11) = —Φ(? , t) DVO(r, t) · s, (5.76)
4π 4π which expresses the radiance in terms of the fluence rate alone.
Substituting Eq. (5.74) into Eq. (5.46), we obtain
^ + μαΦ(?, 0 - V · [DVO(?, t)] = S(?, 0 , (5.77) cat
which is referred to as the diffusion equation. If the absorption coefficient is zero, this diffusion equation reduces to the heat diffusion equation. If the diffusion coefficient is space-invariant, we have a simpler version:
cdt + μαΦ(Γ, 0 - Ο Υ Ζ Φ ( Γ , 0 = S(r, t). (5.78)
The diffusion equation does not depend on vector s and hence has 4 instead of 6 degrees of freedom; it can be used to solve for the fluence rate instead of the radiance. Note that the diffusion equation does not depend on \is and g independently but on their combination μ^. This degeneracy is referred to as the similarity relation, which is valid in the context of the diffusion approximation.
Two approximations are made in the derivation of the diffusion equation from the RTE: (1) the expansion of the radiance is limited to the first-order spherical harmonics and (2) the fractional change in the current density in one transport mean free path is much less than unity. The interpretation of the first approxima- tion is that the radiance is nearly isotropic (omnidirectional) owing to directional broadening. The interpretation of the second approximation is that the photon current is temporally broadened relative to the transport mean free time. Both broadenings are caused by multiple scattering events. Consequently, these two approximations can be translated into a single condition μ̂ > μα, because all of the diffuse photons must have sustained a sufficient number of scattering events before being absorbed. In addition, we also require that the observation point be sufficiently far from sources and boundaries. However, boundary conditions can be applied to improve accuracy.
9 8 RADIATIVE TRANSFER EQUATION AND DIFFUSION THEORY
5.4.6. Impulse Responses in an Infinite Scattering Medium
For an infinitely short-pulsed point source, S(r, t) — h(r, t), the solution to the diffusion equation [Eq. (5.78)] for t > 0 is
c ( r2 \ Φ(Γ, t) = — exp uact , (5.79)
which is an impulse response, also referred to as a Green function, in an infinite homogeneous scattering medium. The exponential decay exp(—\iact) actually represents a form of Beer's law with respect to time due to absorption, whereas the other terms represent broadening due to scattering. Note that Eq. (5.79) incor- rectly predicts a nonzero fluence rate anywhere in space at time 0+ , which violates causality.
For an arbitrary infinitely short-pulsed point source located at r' and peaked at t\ Eq. (5.78) becomes
V ' ' ' ; + μΛΦ(Γ, t\ r\ t') - Ζ)ν2Φ(?, t\ r', t') = h(r - ?f)h(t - tf), cdt (5.80)
which yields a new form of Green's function for t > t'\
* ( r ' f ; r ' O = [ 4 T C D c ( r - r 0 p / 2 e x p
I r - r ' l 2
4Dc(t - /') \iac(t - t
f) (5.81)
We should note that this solution depends on the distance between source point r! and observation point r but that it is independent of the roles of the source and the detector, which indicates reciprocity. In other words, if the source and the observation points are exchanged, the solution remains the same. This is the well- known principle of reciprocity, which is applicable to many wave phenomena.
From the Green function, Green's theorem provides a solution for any arbitrary source in space and time, S(r'', tf):
Φ ( Γ , ί ) = / / Φ(ΐ,ί;Γ',ΐ)8(?',ΐ)άΓ'άΐ'. (5.82) JO JO
The integral represents a superposition of impulse responses weighed by the source distribution; it is actually a convolution here because the Green function is translation-invariant.
In a time-independent state, Eq. (5.78) becomes
1 7 - S(r) Φ(Γ) - - ^ ν 2 Φ ( ? ) = ——. (5.83)
Here, μ6ίί denotes the effective attenuation coefficient:
DIFFUSION THEORY 9 9
For a time-independent point source, S(r) = δ(7); the solution to Eq. (5.83) is
Φ(?) = T - j r - e x p i ^ e f f r ) , (5.85)
which is a time-independent 3D impulse response or a Green function in an infinite homogeneous scattering medium.
For an infinitely broad isotropic source in a ID case, S(z) = δ(ζ); Eq. (5.83) reduces to
1 α2ΦΙΌ(ζ) Hz) Φ\Ό(Ζ) 5 7 1 = ' <5·86)
Miff dl ^a
which yields the following solution in an infinite homogeneous scattering medium:
Φ ι ο ω = ^ χ ρ ( - μ 6 * | ζ | ) . (5.87) 2[La
The l/e decay constant in this equation is the penetration depth δ:
& = lMeff. (5.88)
Comparing Eq. (5.87) with the following Beer's law for a nonscattering medium
Φ ι ο ω = Φ(0)βχρ(-μΛ|ζ |) , (5.89)
we observe that the ratio μeff/μö = ·γ/3(μ0 4- μ^)/μα can be inteφreted as the ratio of the "mean" photon path length in the scattering medium to the depth. However, if μ5 = 0, we have μείί = \ /3μα, which implies erroneously that the "mean" photon path is greater than the depth even in a nonscattering medium. This breakdown of the diffusion theory occurs because condition μ̂ ^> μα is not satisfied.
Example 5.5. Derive Eq. (5.85).
Solution 1. The following Fourier transformation pair is used:
Φ(ί) = ί φ(?) exp(-i"jfc · r) dr, (5.90)
(2π)3 J
Since 5(r) = 5(r), Eq. (5.83) becomes
Φ ( Γ ) - - τ - ν 2 Φ ( ? ) = — . (5.92) M-eff Μ Ά
1 0 0 RADIATIVE TRANSFER EQUATION AND DIFFUSION THEORY
Taking the Fourier transformation of this equation, we obtain
ψ(£) = — — (5.93) \La(l+k2/\L2en)
Taking the inverse Fourier transformation of this equation yields Eq. (5.85):
Α/-Λ ] f exp(ifc-r) - <t>(r) — 5— / :—T-dk
(2π)3μ«7 l+*2Mrff 1 f°° k2dk fn
= — / —— / exp(ikr cos0)sin9J9 (2π)2μ« io 1 + k2/\L2ef{ io
- 1 f°° kdk = 7ΓΊΤ- / . , ,■>, 2 exp(f*rcos9)|0
2 Γ (2π)2μαΓ io
kün(kr)dk (5-94)
eff
μ ^ χ ρ ( - μ 6 ί ί τ ) 4πμαΓ
1 — - — β χ ρ ( - μ β ί τ Γ ) . 4nDr
Solution 2. The general solution to Eq. (5.92) is from the following homo- geneous equation:
Φ(Γ) r V 2 0 ( r ) = 0. (5.95)
Since 4>(r) is independent of Θ and φ, Eq. (5.95) can be rewritten as
r 2 ^ ) + 2 r ^ o _ r) = o (5%) drL dr
This is a transformed Bessel equation that has the following solution after the imaginary part of the solution is discarded
Φ(Γ) = exp(-^effr), (5.97)
where C is a constant. Substituting Eq. (5.97) into Eq. (5.92) yields C = M<eff/(^D), and thus Eq. (5.85) is derived.
BOUNDARY CONDITIONS 101
Example 5.6. Derive Eq. (5.87).
Solution 1. The general solution of Eq. (5.86) is Φιο(ζ) = Cexp(—|xeffl^D- Integrating Eq. (5.86) with respect to z from —oo to oo gives / ^ o Φ\Ό(ζ)αζ = 1/μ« since
/
00 ά2ΦΧΌ(ζ) . . dz = 0 dz2
and f^°ooh(z)dz = 1. Integrating the general solution gives
/
oo
* I D ( -oo
roo 2C >(z)dz = — ·
-oo l̂ eff
Therefore, C = (μβίτ/2μα) a n d Eq. (5.87) is derived. Solution 2. Integrating the 3D solution given by Eq. (5.85) over the source
plane yields
/•OO /»OO
Φ ι ο ( ζ ) = Ι Φ(Γ)2πρ^ρ= / <D(r)2:rcr</r, (5.98)
where z2 + p2 = r2. Completing the integration yields Eq. (5.87).
5.5. BOUNDARY CONDITIONS
5.5.1. Refractive-Index-Matched Boundary
If a nonscattering ambient medium and a scattering medium have the same index of refraction, the interface between them is referred to as a refractive-index- matched boundary. For example, an interface between water and soft tissue is approximately refractive-index-matched. At this kind of boundary, no light prop- agates into the scattering medium from the ambient medium (Figure 5.4). This boundary condition is mathematically expressed as
L(r, S , f ) = 0 for s · n > 0, (5.99)
where r denotes a point on the boundary and h denotes the unit normal vector of the interface pointing into the scattering medium. If the z axis is defined along /z, we have s · n = cosO, where Θ is the polar angle of s. Because the radiance is nonnegative, an equivalent boundary condition can be expressed as
/ L( r ,5 ,0S-Ärfn = 0, (5.100) Jsn>0
102 RADIATIVE TRANSFER EQUATION AND DIFFUSION THEORY
I
Extrapolated boundary Scattering medium
Real boundary
Figure 5.4. Schematic representation of the boundary condition.
which means that the direction-integrated radiance toward the scattering medium is zero.
In the diffusion approximation, the boundary condition becomes
Φ(?, t) + 2J(r, t) · h = 0 . (5.101)
Substituting Fick's law [Eq. (5.74)] into Eq. (5.101), we obtain
Φ(?, t) - 2DVO(?, 0 . h = 0 (5.102)
or
Φ(Γ , 0 - 2D ^ — = 0. (5.103) dz
This boundary condition mathematically falls into the category of homogeneous Cauchy boundary conditions because a linear combination of the fiuence rate and its normal derivative on the boundary is zero.
Using the Taylor expansion to the first order, we obtain
Φ(ζ = - 2 D , t) = Φ(ζ = 0, t) - ID K ; dz
= 0, (5.104) z=0
which means that the fiuence rate at z = —2D is approximately zero. On an extrapolated boundary at
zb = - 2 D , (5.105)
BOUNDARY CONDITIONS 103
the fluence rate is approximately zero. This boundary condition mathematically falls into the category of homogeneous Dirichlet boundary conditions because the fluence rate on the boundary is zero.
Example 5.7. Derive Eq. (5.101).
Substituting the diffusion expansion of the radiance [Eq. (5.36)] into Eq. (5.100), we obtain
/ L(r,s,t)s ηάΩ= / — Φ ( Γ , t) + —J(r , t) s\s -ηάΩ Λη>0 Jsn>0l^ 4π J
1 - f 3 = — Φ ( Γ , ί ) / s-ndQ +— (5.106)
4π JS.A>0 4π
x / [J(r,t)-s]s-ndn = 0. Js-n>0
For a smooth boundary, the first integral on the right-hand side equals π. The second integral can be evaluated as follows:
I [J(r,t) s]s ndQ= I I [Jx(r, i)sinOcos<|> + Jy(r, f)sin0sin<|> Jsn>0 JO JO
-f Jz (r, t) cos Θ] sin Θ cos Θ d<\> JO
•π/2 (5.107)
= 2π / Jo
Jz{r,t) cosz Θ sin0 dd
Therefore
/ Jsn>0
2π _ 2π - ̂ = j i , ( r , / ) = j 7 ( r , i ) · « .
L(r, S, t)S -ndQ = -<D(r, /) + - 7 ( r , t) - n = 0, (5.108)
which leads to Eq. (5.101).
5.5.2. Refractive-Index-Mismatched Boundary
When the ambient and scattering media have different indices of refraction, the interface between them is referred to as a refractive-index-mismatched bound- ary. For example, an interface between air and soft tissue is refractive-index- mismatched. In this case, the boundary condition is modified as follows owing to the Fresnel reflections
Φ(Γ , 0 - 2CRDV<&(7, t) · h = 0 (5.109)
104 RADIATIVE TRANSFER EQUATION AND DIFFUSION THEORY
or
Φ(Γ, t) - 2CRD I = 0, (5.110)
where
1 + # eff C* = - - ^ . (5.111)
The effective reflection coefficient /?eff represents the percentage of the outgoing radiance integrated over all directions pointing toward the ambient medium that is converted to incoming radiance integrated over all directions pointing toward the scattering medium. /?eff can be calculated as follows:
Reff= R* + Rj , (5.112)
Ι-Μφ + Rj
where
»π/2
/?Φ = / 2sin0cose/?/r(cose)d0, (5.113) - / '
Jo
Jo
3 sin 6(cos Θ)2 RF(cos Θ) dQ, (5.114)
1 /nrei cos θ' — cos θ \ 2 1 //trei cos Θ — cos θ'\ flF(cos0) = - — I + - I — for 0 < θ < ΘΓ, 2 V^rel COS θ ' - f COS Θ / 2 y*rel COS Θ + COS Θ ' /
(5.115)
/?/r(cos6) = 1 for dc < Θ < - . (5.116)
The angle of incidence is determined by
Q = cos~](s -n). (5.117)
The angle of refraction is determined by Snell's law as
e ' ^ s in^Ke j s inO) , (5.118)
where the relative refractive index nrt\ is the ratio of the refractive index of the scattering medium to that of the ambient medium. The critical angle is given by
ΘΓ = siiTI — . (5.119) Wrel
BOUNDARY CONDITIONS 1 0 5
Likewise, the distance between the extrapolated boundary and the actual boundary is modified to
Zb = ~2CRD. (5.120)
Example 5.8. Derive Eq. (5.109).
For a refractive-index-mismatched boundary, we have
f L(r,s,t)s-ndQ= I RF(s · n)L(r,s,t)s · ndQ, (5.121)
where s · n = cos Θ and the Fresnel reflection RF of the light—presumed to be unpolarized—at the boundary is given by Eqs. (5.115) and (5.116).
We define an effective reflection coefficient as
A Ä̂ n RF($ · n)L(r, s, t)s · h dQ Reff = r / . (5.122)
JSii<0L(rJ,t)s -ndQ
As in Example 5.7, /?eff is evaluated in the diffusive regime using the diffusion expansion of radiance [Eq. (5.36)]
Reff = 4 i \ , Λ (5.123)
where
R<P= f Jo
L
π/2 2sin0cos0/?/r(cos0)i/0, (5.124)
-π/2
Rj = / 3sine(cose)2/?/r(cos0)i/G. (5.125)
Similarly, boundary condition Eq. (5.121) leads to
1 - 1 - _ 1 - 1 - _ - Φ ( Γ , 0 + - / ( r , t) · n = - / ? Φ Φ ( Γ , 0 - Rj-J(7, t) · A (5.126)
in the diffusive regime. Merging Eq. (5.126) and Eq. (5.123) yields
R<t> -\- Rj Reff = 7Γ-^ —, (5.127)
2-R^ + Rj
which can be solved numerically. Fitting this equation can provide an empirical formula for /?eff.
106 RADIATIVE TRANSFER EQUATION AND DIFFUSION THEORY
Substituting Eq. (5.122) into Eq. (5.121) yields
/ L(rJ,t)s ndQ = Refi L(r, s,t)s · hdQ. (5.128) Jsn>0 Jsn<0
In the diffusive regime, this equation can be rewritten as
-Φ(? , t) + - / ( ? , t)-n = -fieff<Mr, t) - Reff-J(r, t) · h. (5.129)
Substituting Fick's law [Eq. (5.74)] into this equation yields boundary condition Eq. (5.109).
5.6. DIFFUSE REFLECTANCE
Measured diffuse reflectance can be used, for example, to determine the optical properties of biological tissue noninvasively. The relative diffuse reflectance (or simply diffuse reflectance) is defined here as the probability of photon reemis- sion per unit surface area from a scattering medium. Although the Monte Carlo method can predict diffuse reflectance accurately, it is computationally intensive. In particular, when the absorption coefficient is much less than the scattering coefficient, photons may propagate over long distances before being absorbed. Fortunately, the diffusion theory offers an alternative rapid approach although it is inaccurate near the light source.
The task here is to compute the diffuse reflectance in response to an infinitely narrow photon beam (a pencil beam) normally incident on a semiinfinite homogeneous scattering medium that has a refractive-index-matched boundary (Figure 5.5a). The problem is solved below by the diffusion theory along with the boundary condition. Key factors that affect the accuracy of the diffusion theory are also discussed.
5.6.1. Steps of Approximation
Three steps of approximation are involved in the solution (Figure 5.5): (1) the anisotropically scattering medium (Figure 5.5a) is converted into an isotropically scattering medium (Figure 5.5b), based on the similarity relation; (2) the unit- power pencil beam is converted into an equivalent isotropic point source at z = l't with a power equal to transport albedo a' (Figure 5.5c); see also Problem 5.3); and (3) the surface of the scattering medium is removed after an image source is added above the surface at z = —{l't + 2zb) to satisfy the boundary condition.
An image point source is mirror-symmetric with the original point source about the extrapolated boundary at z = — Zb [Eq. (5.105)]; it is added to satisfy the boundary condition so that the original single source in a semiinfinite medium can be converted into double sources in an infinite medium. The response to a single source in a semiinfinite medium (Figure 5.5c) can be approximated by a
DIFFUSE REFLECTANCE 107
I *
z μ,ι
μ«2=μαΐ μ,2=μ,ι( ΐ -^ι) #2 = 0
(c) Θ
(d)
■ Θ
λ Θ
μ«2 μ,2 82
μ«2 μ,2 82
Figure 5.5. Illustrations of the steps of approximation (the boxes represent the scatter- ing media): (a) a pencil beam incident on the original scattering medium with absorption coefficient μαι, scattering coefficient [is\, and nonzero anisotropy g\; (b) a pencil beam incident on an isotropically scattering medium with μα2 = μαι, μ*2 = P^iO — gi)> and g2 — 0; (c) an isotropic point source under the surface of the isotropically scattering medium; (d) an image point source added to approximately satisfy the boundary condi- tion—with this addition, the physical boundary (dashed line) is removed (circled signs indicate the polarities of the sources).
superposition of the two responses to each of the double sources in an infinite medium (Figure 5.5d). The latter problem can be solved easily because it is free of boundaries. This approach is akin to the common practice of solving electrostatic problems with a zero-potential conducting boundary. Therefore, instead of dealing with a pencil beam incident on a semiinfinite anisotropically scattering medium (Figure 5.5a), we deal with two isotropic point sources in an infinite scattering medium (Figure 5.5d).
5.6.2. Formulation
A cylindrical coordinate system (r, θ, z) is set up. The origin of the coordinate system is the point of light incidence on the surface of the scattering medium, and the z axis is along the pencil beam.
The fluence rate that is generated by a unit-power point source in an infinite scattering medium is described by Eq. (5.85) and is rewritten as follows in the
108 RADIATIVE TRANSFER EQUATION AND DIFFUSION THEORY
cylindrical coordinates
ΦΟΟ(Γ, θ, ζ\ r\ θ\ ζ') = — l — β χρ ί -μ^ρ ) , (5.130) 4nDp
where p is the distance between observation point (r, Θ, z) and source point (Λθ' ,ζ ' ) :
9 = Jr 2 + r'2 - 2rr' cos(0 - θ') + (z - z')2. (5.131)
A linear combination of the solutions for each of the two isotropic sources in Figure 5.5d, according to Eq. (5.130), yields approximately the fluence rate in response to the original isotropic point source in the original semiinfinite scat- tering medium:
Φ(Γ, Θ, Z\ r\ θ', zf) = fl'Oooir, Θ, z\ r \ θ', ζ') - Α'ΦΟΟ^, Θ, Z; r\ θ', -z' - 2zb), (5.132)
where z! = l't and a' denotes the transport albedo. According to Fick's law, the diffuse reflectance from the semiinfinite scattering
medium is approximately the current density projected to the surface normal:
3Φ Rd(r) = D —
dz (5.133)
2=0
Substituting Eq. (5.132) into Eq. (5.133), we obtain
, ζ ' (1+μβ ί τΡ ι )βχρ(-μβ ί ϊΡ ι ) , , (ζ' + 4D)(1 + \LeffQ2) exp(^e f fP2) Rd(r) = a —z + a — r .
4πρ| 4πρ2 (5.134)
Here, pi is the distance between observation point (r, 0, 0) and original source point (0, 0, z') and p2 is the distance between observation point (r, 0, 0) and image source point (0, 0, — z! — 2zb)·
Example 5.9. Derive Eq. (5.134).
From Eqs. (5.130) and (5.131), we derive
^Φηο 1 1 4- UeffP ^ μ - " / - -x (5.135)
(5.136)
exp(^effPi).
(5.137)
δρ 4nD
3p z-z! dz p
tierefore
ΘΦΟΟΟ',Θ,Γ,Γ' ,Θ' ,Ζ ') dz z=0
2 ^ Λ ^ P2
ν μ'βημ;»
ΘΦοο ΘΡ dp dz
ζ' 1 + M-effPi ζ=0 ~ Απϋ 9]
DIFFUSE REFLECTANCE 1 0 9
Likewise, we have
d<S>00(r,$,z\r',V,-z f -2zb)
dz z=0
z! + AD l + μ6ίΓΡ2 , , 3 6Χρ(-μ6ίίΡ2)· 4πΖ) P2
Combining Eqs. (5.137) and (5.138) leads to Eq. (5.134). (5.138)
(a)
Si - 0 . 4
(b)
— — - A: Monte Carlo D: Diffusion theory
0.2 0.3 Radius r(cm)
0.2 0.3 Radius r (cm)
Figure 5.6. (a) Diffuse reflectance in response to a pencil beam incident on a semi- infinite scattering medium. Curve A is from the Monte Carlo simulation for the case in Figure 5.5a. Curve D is from the diffusion theory for the case in Figure 5.5d. (b) Relative error between the two curves in part (a), which is the difference between curves D and Λ divided by curve A point-by-point.
1 1 0 RADIATIVE TRANSFER EQUATION AND DIFFUSION THEORY
5.6.3. Validation of Diffusion Theory
In this section, we evaluate each step of the approximation described above using the accurate Monte Carlo method. The following optical properties are used: nrei = 1, \ia\ =0 .1 cm- 1 , \is\ = 100 cm- 1 , and g\ = 0.9. As shown in Figure 5.6, the diffuse reflectance Rd(r) from the diffusion theory is accurate only when r is greater than — l\ (l't = 0 . 1 cm here).
103
102
101
Ö 10 r
1(T
10~
: I
LA
L ^ ^ r ^ ^ F ^ ^
1 1
1 1 1 d
— — — A: Pencil beam, g = 0.9 3
B'. Pencil beam, g — 0 M
1 1 1 ^ ^ " *
0.2
(a) 0.4 ■ 0.6 Radius r (cm)
0.8
(b) 0.4 0.6 Radius r (cm)
Figure 5.7. (a) Comparison between the diffuse reflectance distributions from the anisotropically (see Figure 5.5a) and isotropically (see Figure 5.5b) scattering media cal- culated using the Monte Carlo method; (b) relative error versus r.
DIFFUSE REFLECTANCE 111
£
(a)
(b)
-0.5
0.4 0.6
Radius r (cm)
0.4 0.6
Radius r (cm)
Figure 5.8. (a) Comparison between the diffuse reflectance distributions from an isotrop- ically scattering medium in response to a pencil beam (see Figure 5.5b) and an isotropic point source (see Figure 5.5c) calculated using the Monte Carlo method; (b) relative error versus r.
Deviations caused by each step of the approximation are illustrated in Figures 5.7-5.9. Curves A, B, and C are from the Monte Carlo method, whereas curve D is from the diffusion theory; curves A-D are associated with parts (a)-(d) in Figure 5.5.
The error due to the approximation of Figure 5.5a with 5.5b is shown in Figure 5.7. The scattering anisotropy is converted from g = 0.9 to g = 0 while
112 RADIATIVE TRANSFER EQUATION AND DIFFUSION THEORY
102
(a)
£ -0.02
(b)
C: Monte Carlo D: Diffusion theory
0.4 0.6 Radius r (cm)
0.4 0.6 Radius r (cm)
Figure 5.9. (a) Comparison between the diffuse reflectance distributions in response to an isotropic point source in a semiinfinite scattering medium (see Figure 5.5c) and a pair of isotropic point sources in an infinite scattering medium (see Figure 5.5d). Curves C and D are from the Monte Carlo method and the diffusion theory, respectively, (b) Relative error versus r.
μ̂ . is held constant. The relative error decreases with increasing r; it is >100% near r — 0 and -^20% near r = 2l't =0 .2 cm.
The error due to the approximation of Figure 5.5b with 5.5c is shown in Figure 5.8. This pencil beam is converted to a single isotropic point source at z = l't = 0.1 cm. Such a conversion causes a severe underestimation of Rdir) near r = 0.
DIFFUSE REFLECTANCE 1 1 3
The error due to the approximation of Figure 5.5c to 5.5d is shown in Figure 5.9. Curves C and D are calculated by the Monte Carlo method and the diffusion theory, respectively; they show relatively small systematic differences.
Although the diffusion theory is acceptable when the isotropic point source is far from the surface of the scattering medium as demonstrated in Figure 5.9, it becomes less accurate as the source approaches the surface (Figure 5.10). To demonstrate this point, we compare the results from the Monte Carlo method and
Monte Carlo (0.1//)
Diffusion theory (0.1//)
0.2 0.3 Radius r (cm)
104
103
102
B 101
<4-i
s
(b)
10°
10"
10"
i ι ι : i
Monte Carlo (0.01//) ]
Diffusion theory (0.01//) L
1 1 1
0.1 0.2 0.3
Radius r (cm)
0.4 0.5
Figure 5.10. Comparison between the diffuse reflectance distributions from the Monte Carlo method and the diffusion theory. An isotropic light source is placed at (a) z = 0.1 /,' and then (b) z = 0.01/^ in an isotropically scattering semiinfinite medium.
114 RADIATIVE TRANSFER EQUATION AND DIFFUSION THEORY
10" C: isotropic source, g = 0
- E: Isotropic source, g = 0.9
0.4 0.6 Radius r (cm)
Figure 5.11. Comparison between the diffuse reflectance distributions from the Monte Carlo method in response to an isotropic point source placed at z — l't in two scattering media whose optical properties are related by the similarity relation. For curve C, g = 0; for curve E, g = 0.9; for both, nre\ — 1, \ia = 0.1 cm-1, and μ̂ . = 10 cm"1.
the diffusion theory for the configurations in Figures 5.5c and 5.5d, respectively. The point source, however, is placed at z = 0.1/,' and then at z — 0.01/,' instead of at z = lfr As expected from the diffusion theory, data for z = 0.0\l't are less accurate than those for z = 0.1/,'.
Although the conversion from Figure 5.5a to 5.5b introduces considerable error in Rd(r) near the source as shown in Figure 5.7, it is acceptable if the pho- tons originate isotropically deep inside the scattering medium, as demonstrated below. In response to an isotropic point source at z = l't, the diffuse reflectance distributions from an isotropically scattering medium (as in Figure 5.5c) and an anisotropically scattering medium are computed by the Monte Carlo method; they are approximately equal to each other (Figure 5.11).
5.7. PHOTON PROPAGATION REGIMES
The cumulative effect of photon scattering by a medium can be loosely classified into four regimes. The term ballistic regime refers to photons that have undergone no scattering; quasiballistic regime refers to photons that have sustained a few scattering events but retain a strong memory of the original incidence direction. The term quasidiffusive regime refers to photons that have sustained many scat- tering events and retain only a weak memory of the original incidence direction; diffusive regime refers to photons that have suffered many scattering events that they have almost completely lost their memory of the original incidence direction.
PHOTON PROPAGATION REGIMES 1 1 5
In addition, nonballistic photons are those that have deviated from the ballistic path. The propagation regimes can be approximately related to the propagation time t through the mean free path lt and the transport mean free path l\.
On the basis of Beer's law, the probability of no scattering for a photon decays with time t is as follows:
P ( c O = e x p ( - ^ - ) . (5.139) = e x p ( - ^ ) .
Accordingly, we define the ballistic regime to cover ct <lt, within which the probability of no scattering is P(ct) > exp(—1) = 37%. We define the quasibal- listic regime to cover lt < ct < l't, within which the probability of no scattering falls between exp(—////,) and exp(—1) : exp(—1) > P(ct) > exp(—////,).
When a pencil beam is incident within an infinite scattering medium, the photons spread into a photon cloud. From cumulant expansion, it is found that the center of the photon cloud approaches l\ according to
/ ; _ Z c = / ; e x p ( - ^ V (5.140)
where zc is the distance between the weighted center of the photon cloud and the point of incidence. We define a new constant as
ε, = ^ . (5.141)
Thus, we have
er=exp(-yY (5.142)
Accordingly, we define the quasidiffusive regime to cover Vt < ct < 10/,', within which we have exp(—1) > εΓ > exp(—10) = 4.5 x 10~5. We define the diffusive regime to cover ct > 10//, within which we have εΓ < exp(—10). If lt — 0.1 mm and l't = 1 mm, the four scattering regimes are divided at path lengths of 0.1, 1, and 10 mm. The classification holds in scattering dominant media.
If μα « μ^, the mean number of scattering events that photons experience within the dividing path lengths (Ns) can be estimated. Within /, (the ballistic regime), Ns < 1 holds. Within l't (the quasiballistic regime), we have
/; i N5<±* - , (5.143)
U l -g
which equals 10, for example, if g = 0.9. Likewise, within 10// (the quasidiffu- sive regime), we have
io/; io Ns < —l- % , (5.144)
It l~g which equals 100, for example, if g — 0.9.
116 RADIATIVE TRANSFER EQUATION AND DIFFUSION THEORY
PROBLEMS
5.1 Show that f4 s(s · Α)άΩ — (4π/3)Α, where A is independent of s.
5.2 Show that f4n s[s · V(7 · s)]dQ = 0.
5.3 Show that a pencil beam normally incident on a semiinfinite medium can be approximated by an isotropic source placed one transport mean free path below the surface. Extend to a ID case, namely, an infinitely broad beam normally incident on a semiinfinite medium. Explain why this is an approximation.
5.4 Verify Eq. (5.85) using a Monte Carlo simulation for μα =0Λ cm- 1 , μ5 = 100 cm- 1, and g = 0.9. (Hint: Use spherical coordinates to record photon absorption.)
5.5 Duplicate Figures 5.6-5.11.
5.6 Derive the average number of scattering events in one transport mean free path given \ia = 0.
5.7 Derive Eq. (5.10).
5.8 Show that P(s' · s) — /?(COS0)/2TT, where p(cosO) is defined as in Chapter 3.
5.9 Plot Eq. (5.36) in polar coordinates as a function of a, where 7(r, t) ■ s = \J(r, t)\ cos a. Set <&(r,t) = 1 and plot for 3 | /(r , 01 = 3 , 1, 0.3, 0.1, and 0.03.
5.10 Plot the Henyey-Greenstein phase function in polar coordinates as a func- tion of Θ for g = 0, 0.1, 0.5, 0.9, and 0.99.
5.11 (a) Using the Monte Carlo method, compute L(r, s, t) integrated over time t as a function of the polar angle Θ at z = (0.1, 0.5, 1.0, 2.0)/rr below a pen- cil beam in an infinite medium, where μα — 0.1 cm- 1, μΞ = 100 cm- 1 , and g = 0.9. Plot the result in polar coordinates, (b) Use the least-squares fitting algorithm available in MATLAB to fit the derived distributions to a + b cos Θ. List b/a versus z in a table.
5.12 Modify the Monte Carlo code written for Chapter 3 to compute and plot the specific absorption distributions on the z axis in response to a pencil beam in two infinite scattering media of g = 0.9 and g = 0. Both media have \xa =0 .1 cm- 1 and μ̂ = 10 cm- 1 . The range of z should cover several transport mean free paths.
5.13 Integrate Eq. (5.79) over the entire space and explain the result. Then, set μα = 0 and explain the result.
5.14 Integrate Eq. (5.79) over time from 0 to +oo and explain the result.
READING 117
5.15 Using a Monte Carlo program, compute and plot the ID depth-resolved fluence rate as a function of z in response to a pencil beam incident normally on a semiinfinite scattering medium. Fit the curve for μ^ and compare with the value predicted by the diffusion theory. Compare the depth of the peak fluence rate with l'r
5.16 Derive Eq. (5.85) from Eq. (5.83) using an alternative method.
5.17 Derive Eq. (5.79) using the Fourier transformation.
5.18 Assuming that the absorption coefficient is zero, from Eq. (5.85), derive the current density and explain the conservation of energy.
5.19 (a) The phase function P(sf · s) is highly forward-directed in biological tissue. Explain why it is not expanded in spherical harmonics in the deriva- tion of the diffusion theory, (b) Explain that since P(s' · s) is azimuthally symmetric about s, f4jf sP(s' · s)dQ is parallel with s'.
5.20 One approximation in the diffusion theory is that the fractional change in the current density in one transport mean free path is much less than unity. Explain why this approximation can be translated to the statement that the reduced scattering coefficient must be much greater than the absorption coefficient.
5.21 The diffusion equation derived in this chapter does not conform to the postulate of causality. If a second-order temporal wave equation term is added, this problem can be corrected. The new equation is referred to as the telegraphy equation:
^ + μαΦ(Γ, t) - V · [DV<S>(r, t)] + 3D \ ' = S(r, t). cdt czdtz
Derive this equation.
5.22 Derive the RTE by considering a differential area that moves along photon propagation direction s. {Hint : (dL/ds) = {dL/ds) + [(dL/dt) (dt/ds)].)
READING
Boas DA (1996): Diffuse Photon Probes of Structural and Dynamical Properties of Scatter- ing Media, Ph.D. dissertation, Univ. Pennsylvania, Philadelphia. (See Sections 5.2-5.5, above.)
Cai W, Lax M, and Alfano RR (2000): Cumulant solution of the elastic Boltzmann transport equation in an infinite uniform medium, Phys. Rev. E 61(4): 3871-3876. (See Section 5.7, above.)
Haskell RC, Svaasand LO, Tsay TT, Feng TC, and Mcadams MS (1994): Boundary- conditions for the diffusion equation in radiative-transfer, J. Opt. Soc. Am. A 11(10): 2727-2741. (See Section 5.5, above.)
118 RADIATIVE TRANSFER EQUATION AND DIFFUSION THEORY
Ishimaru A (1978): Wave Propagation and Scattering in Random Media, Academic Press, New York. (See Sections 5.2-5.5, above.)
Wang LHV and Jacques SL (2000): Source of error in calculation of optical diffuse reflectance from turbid media using diffusion theory, Comput. Meth. Prog. Biomed. 61(3): 163-170. (See Section 5.6, above.)
FURTHER READING
Aronson R (1995): Boundary-conditions for diffusion of light, J. Opt. Soc. Am. A 12(11): 2532-2539.
Case KM and Zweifel PF (1967): Linear Transport Theory, Addison-Wesley, Reading, MA.
Chandrasekhar S (1960): Radiative Transfer, Dover Publications, New York. Cheong WF, Prahl SA, and Welch AJ (1990): A review of the optical-properties of
biological tissues, IEEE J. Quantum Electron. 26(12): 2166-2185. Faris GW (2002): Diffusion equation boundary conditions for the interface between turbid
media: A comment, J. Opt. Soc. Am. A 19(3): 519-520. Farrell TJ, Patterson MS, and Wilson B (1992): A diffusion-theory model of spatially
resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo, Med. Phys. 19(4): 879-888.
Flock ST, Wilson BC, and Patterson MS (1989): Monte-Carlo modeling of light- propagation in highly scattering tissues. 2. comparison with measurements in phantoms, IEEE Trans. Biomed. Eng. 36(12): 1169-1173.
Groenhuis RAJ, Ferwerda HA, and Tenbosch JJ (1983): Scattering and absorption of turbid materials determined from reflection measurements. 1. Theory, Appl. Opt. 22(16): 2456-2462.
Keijzer M, Star WM, and Storchi PRM (1988): Optical diffusion in layered media, Appl. Opt. 27(9): 1820-1824.
Kienle A, Patterson MS, Dognitz N, Bays R, Wagnieres G, and van den Bergh H (1998): Noninvasive determination of the optical properties of two-layered turbid media, Appl Opt. 37(4): 779-791.
Markel VA and Schotland JC (2002): Inverse problem in optical diffusion tomography. II. Role of boundary conditions, J. Opt. Soc. Am. A 19(3): 558-566.
Shafirstein G, Baumler W, Lapidoth M, Ferguson S, North PE, and Waner M (2004): A new mathematical approach to the diffusion approximation theory for selective photothermolysis modeling and its implication in laser treatment of port-wine stains, Lasers Surg. Med. 34(4): 335-347.
Wyman DR, Patterson MS, and Wilson BC (1989): Similarity relations for the interaction parameters in radiation transport, Appl. Opt. 28(24): 5243-5249.
You JS, Hayakawa CK, and Venugopalan V (2005): Frequency domain photon migration in the delta-P-1 approximation: Analysis of ballistic, transport, and diffuse regimes, Phys. Rev. E 72(2): 021903.
CHAPTER 6
Hybrid Model of Monte Carlo Method and Diffusion Theory
6.1. INTRODUCTION
The Monte Carlo method and the diffusion theory have complementary attributes for modeling photon transport in a scattering medium. The Monte Carlo method is accurate but computationally inefficient, whereas the diffusion theory is inac- curate but computationally efficient. A hybrid of the two approaches, however, is constructed to combine the advantages of both. The hybrid model computes as much as 100 times faster than the Monte Carlo method yet improves the accuracy of the diffusion theory.
6.2. DEFINITION OF PROBLEM
A pencil beam is normally incident on a slab of homogeneous scattering medium. The geometric and optical properties of the slab are described by thickness d, relative refractive index nrei, absorption coefficient μα, scattering coefficient μ5, and scattering anisotropy g, where nrt\ is the ratio of the refractive index of the scattering medium to that of the ambient medium. The Henyey-Greenstein phase function is assumed. Cylindrical coordinates (r, φ, z) are used; the origin is the point of incidence of the pencil beam on the top surface of the slab; the z axis points along the pencil beam. The diffuse reflectance and the diffuse transmittance versus r are computed.
6.3. DIFFUSION THEORY
The diffusion theory for a scattering slab with refractive-index-mismatched (nrel φ 1) boundaries is an extension of the theory for a semiinfinite scatter- ing medium with a refractive-index-matched (nrei = 1) boundary that was cov- ered in Chapter 5. The fluence rate Φ at observation point (r, φ, ζ) in response
Biomedical Optics: Principles and Imaging, by Lihong V. Wang and Hsin-i Wu Copyright © 2007 John Wiley & Sons, Inc.
119
120 HYBRID MODEL OF MONTE CARLO METHOD AND DIFFUSION THEORY
to an isotropic point source of unit power at (r \ φ', ζ') in an infinite scatter- ing medium is
^ < A > x* >\ λ exp(^effp) ΦοοΟ,φ,ζ; r , φ , ζ ) = — · (6.1)
4πυ ρ Here, p is the distance between the observation point and the source point, D is the diffusion coefficient, and μ^ is the effective attenuation coefficient:
p = yjr2 + r'2 - Irr' cos(<|> - φ') + (z - z')2, (6.2)
D = ! , (6.3) 3[μα + μ*0 - g ) ]
μείΤ = y/Va/D. (6.4)
To compute the fluence rate in response to an isotropic point source in a scattering slab on the basis of Eq. (6.1), we first theoretically convert the slab into an infinite medium by satisfying the boundary conditions with an array of image sources, akin to the infinite array of images seen by a person standing between two parallel mirrors. Two extrapolated boundaries, at which the fluence rate is approximately zero, are used; they are separated from the slab surfaces by a distance of Zb (Figure 6.1)
zb = 2CRD, (6.5)
where CR is related to the effective reflection coefficient /?eff- If nTe\ = 1, then CR = 1. Otherwise, CR is estimated by
CR = \±£L. (6.6) 1 - Aeff
Here, the following empirical formula is used (the exact formula can be found in Chapter 5):
Äeff = - 1 . 4 4 0 ^ + 0.710η",1 + 0.668 + 0.0636nreI. (6.7)
An original isotropic point source at (r', θ', ζ') and its images are shown in Figure 6.1. The images are caused by reflections from the two extrapolated boundaries, where each reflection alternates the polarity of the point source. The z coordinates of the ith source pair are given by
zi± = -zb + 2i(d + 2zh) ± (z + zt), (6.8)
where / = 0, ± 1 , ±2, — The source pair at zo± (the original and its image) straddles the top boundary of the slab. The source pair at z\± is the image of the
DIFFUSION THEORY 121
Θ
Θ ι = - 1
Θ <·=<>
Extrapolated boundary
Θ ι = 1
Θ Figure 6.1. Illustration of the original point source inside the scattering slab and the image point sources outside the slab. Circled signs indicate the polarities of the sources.
pair at zo± with respect to the bottom extrapolated boundary. The source pair at Z-\± is the image of the pair at z\± with respect to the top extrapolated boundary. Although infinite, the image series can be truncated after several source pairs.
With these image sources, the boundary conditions are satisfied; hence, the true boundaries can be removed. Consequently, the original point source in the scatter- ing slab is converted to an array of isotropic sources in an infinite homogeneous medium. The fluence rate from the original source in the slab is approximated by
Φ(Γ, φ, z; r\ φ', ζ') = £ [ΦοοίΓ, φ, ζ; Λ φ', ζ\+) - Φοο(ι\ Φ, ζ; Λ φ', *{_)], ι=ζ 'min
(6.9)
where / m j n and /max are the lower and upper indices, respectively, of the truncated source pair series. The diffuse reflectance and the diffuse transmittance from the
122 HYBRID MODEL OF MONTE CARLO METHOD AND DIFFUSION THEORY
slab are given by
'max
/?(/-, φ, 0; r\ φ', ζ') = Σ [ ^ ( r , Φ· ° : ^ Φ'- *ί+) ' — ' m i n
- Α Ο Ο ( Γ , Φ , 0 ; Γ ' , Φ ' , Ζ ; _ ) ] , (6.10)
'max
T(r, φ, ά\ r \ φ', ζ') = Σ [ΤΌοί̂ Φ, ^ ; Λ Φ', ζί+) ' = ' m i n
-Γοο(Γ,φ, < / ;Γ ' ,Φ ' , *;_)], (6.11)
where
ΑΟΟ(Γ,Φ,0 ;Γ ' ,Φ ' ,Ζ ' ) = D — ^ ζ = ο " 4 π Ρ 3
(6.12)
7oo(r, φ, d; Γ', φ', ζ) = -£>—■— 9ζ
6.4. HYBRID MODEL
_ (d - zf)(\ + μeffp)exp(-μeffp)
z=d ~ 4 ^ P 3
(6.13)
Accurate conversion of the incident pencil beam into an isotropically emitting light source deep in the scattering medium can improve the accuracy of the diffusion theory. Such a conversion can be provided by the Monte Carlo method. The combination of the Monte Carlo method and the diffusion theory is referred to as a hybrid model.
In the Monte Carlo step, the incident pencil beam is converted into a distributed isotropic source while reemitted photons are recorded. Since the diffusion theory is inaccurate when photons are within a critical depth zc from the two boundaries of the slab, photons are tracked by the Monte Carlo method until they reach the center zone defined by zc £ z < d — zc (Figure 6.2).
The Monte Carlo step is based on the conventional Monte Carlo method described in Chapter 3. A photon packet with an initial weight of unity is launched perpendicularly onto the surface along the z axis (Figure 6.2). If the boundary is refractive-index-matched (ηκ\ — 1), all photon weight enters the scattering medium. Otherwise, only a portion enters after the Fresnel reflection. Then, a step size s is chosen statistically by
, - ^ ψ - . (6.14) μβ + μ.9
where ξ is a pseudorandom number evenly distributed between 0 and 1 (0 < ξ < 1). The photon packet loses some weight owing to absorption at the end of
HYBRID MODEL 123
Incident beam
Isotropic source ·
zc y Scattering slab
Figure 6.2. Illustration of the conversion from an incident photon packet to an isotropic point source in the Monte Carlo step of the hybrid model. The last step of l't in length converts the photon packet into an isotropic point source.
each step; the loss is equal to the photon weight at the beginning of the step multiplied by 1 — a, where a denotes the albedo. The photon packet is then scattered in a new propagation direction that is statistically determined by the Henyey-Greenstein phase function with anisotropy g. When the photon weight is less than a threshold, the photon packet can be either terminated or continued as determined by Russian roulette. If reemitted into the ambient medium, the photon packet contributes to the diffuse reflectance /?MC(/) (where the subscript denotes Monte Carlo) or the diffuse transmittance. The process is then repeated with multiple (N) photon packets.
If scattered in the center zone, the photon packet is conditionally converted to an isotropic point source. If one transport mean free path l\ along the pho- ton propagation direction fits in the center zone, the conversion is implemented (Figure 6.2); otherwise, the Monte Carlo step continues.
The conversion is based on the similarity relation, which converts the scatter- ing medium from anisotropic to isotropic scattering while conserving the reduced scattering coefficient μ^. After taking the step of l\ in length, the photon packet interacts with the isotropic scattering medium according to the transport albedo a'. With the weight reduced by a factor of 1 — a' due to absorption, the photon packet experiences isotropic scattering. The scattered photon packet then becomes an isotropic point source; its weight is recorded into a source function S(r, z), which is guaranteed to be zero outside the center zone. Note that the step size for the conversion in a finite medium is slightly less than /,', but l\ is used for simplicity.
In the diffusion step, the additional contribution to the diffuse reflectance that is due to the converted source is calculated by the diffusion theory. After the Monte Carlo step tracks all photon packets, S gives the total accumulated weight distribution. Next, S is converted to a relative source density function Sj, which represents the source strength per unit volume. Then, Sd is used to compute the
124 HYBRID MODEL OF MONTE CARLO METHOD AND DIFFUSION THEORY
additional diffuse reflectance R^j based on the diffusion theory (DT)
poo poo p2n
Rm(r)= / / / Sd(r\z')R(r,0,0;r\tf,z')r'dtfdr'dz\ (6.15) Jo Jo Jo
where R is given by Eq. (6.10). Because of the cylindrical symmetry, /?DT is independent of the azimuthal angle φ. The final diffuse reflectance Rj is given by
Rd(r) = Ruc(r) + Rm(r). (6.16)
The diffuse transmittance can be similarly computed.
6.5. NUMERICAL COMPUTATION
A grid system is set up in the cylindrical coordinates (Figure 6.1). The grid element sizes in the r and z directions are Ar' and Δζ', respectively; the number of grid elements are Nr and Nz, respectively. The center coordinates of each grid element are given by
Γ'(ΙΓ) = ( « Γ + 0 . 5 ) Δ Γ ' , (6.17)
ζ'(ιζ) = ( ι ζ+0.5)Δζ ' , (6.18)
where ir = 0 ,1 , . . . Nr — 1 and iz = 0,1, . . . Nz — 1. We can also use the optimized version of Eq. (6.17) as shown in Chapter 3. For brevity, the array elements for the physical quantities are referenced by either the location of the grid element or the indices of the grid element.
At the end of the Monte Carlo step, raw 7?MC represents the total accumulated weight reflected into an annulus grid; it is converted to diffuse reflectance by
D r . , RMCUK]
NAa(ir)
where Δα denotes the area of the annulus:
Aa(ir) = 2nr\ir)Ar f.
Similarly, raw S is converted to Sj by
S[ir, iz] Sd\ir, iA — , c NAV(ir)
where AV denotes the grid volume:
AV(ir) = Aa(ir)Az.
(6.19)
(6.20)
(6.21)
(6.22)
COMPUTATIONAL EXAMPLES 1 2 5
The grid system for recording the source term S is also used to compute the integral over r' and z! in Eq. (6.15). From symmetry, the upper limit of the integral over φ' is lowered from 2π to π. Therefore, Eq. (6.15) is computed as follows:
Nr-2Nz-2
Rm(r) = 2 Y Y Sd[ir, ιζν'(ιΓ)Δτ'Δζ' / Ä(r, 0, 0; r'(ir), φ', z\iz)) d<\>\ ,V=o «z=o Jo
(6.23)
The last grid elements in each direction are not used in the summation because they record weight deposited outside the grid system in the Monte Carlo step. The integration over φ' in Eq. (6.23) is implemented with Gaussian quadratures.
6.6. COMPUTATIONAL EXAMPLES
In this section, we compare the hybrid model with both the pure diffusion theory and the pure Monte Carlo method. Unless otherwise specified, 100,000 photon packets are tracked in both the Monte Carlo and the hybrid simulations.
The diffuse reflectance and the diffuse transmittance in response to an isotropic point source at z! — l\ in a scattering slab computed by the pure Monte Carlo method and the pure diffusion theory are shown in Figure 6.3. One to three point source pairs are used in the diffusion theory to satisfy the boundary conditions. The single pair is at zo± 0 = 0); the double pairs are at zo± and z\±, the triple pairs are at z~\±, zo±, a nd z\±.
It is important to determine the number of point source pairs needed in the diffusion theory to accurately model diffuse reflectance and diffuse transmittance. As shown in Figure 6.3a, three pairs are required to achieve good accuracy in the diffuse reflectance. With fewer pairs, the accuracy is good until the radial distance is greater than approximately the slab thickness. As shown in Figure 6.3b, a single pair does not ensure accuracy to the diffuse transmittance because the boundary condition for the bottom surface is neglected altogether; two or three pairs, however, do provide accuracy. The number of pairs needed depends on the observation distance, the thickness of the slab, and the optical properties of the slab. In practice, more source pairs can be added until the new pair makes negligible contributions.
The diffuse reflectance and the diffuse transmittance in response to a pencil beam from both the pure Monte Carlo method and the pure diffusion theory are shown in Figure 6.4. The diffusion theory simulates an equivalent isotropic point source located at z! = l\ (see Chapter 5) using three source pairs (/ = —1,0, 1). The relative errors of the diffuse reflectance and the diffuse transmittance from the diffusion theory are shown in Figure 6.4c; they represent the differences between the results from the diffusion theory and the Monte Carlo method, divided point- by-point by the results from the Monte Carlo method. In this case, the diffuse reflectance from the diffusion theory is less than that from the accurate Monte
126 HYBRID MODEL OF MONTE CARLO METHOD AND DIFFUSION THEORY
I0l
rCT 10°
io-
sH
lo-
ur
10"
l·
° Monte Carlo Diffusion (1 pair)
— — Diffusion (2 pairs) 1
Diffusion (3 pairs)
(a) 0.5 1 1.5
Radius r (cm)
io-
6 10"
10-
iB 10-4
Q
(b)
0.5
: ' '
H ° Monte Carlo M Diffusion (1 pair) L . __ . _ Diffusjon (2 pairs)
| Diffusion (3 pairs) 1 1 H
i 4
1
1
Radius r (cm)
1.5
Figure 6.3. Comparison between the pure Monte Carlo method and the pure diffusion theory in terms of (a) the diffuse reflectance and (b) the diffuse transmittance in response to an isotropic source. The properties of the scattering slab are nre\ — 1, μα =0.1 cm"', μ, = 100 cm"1, g = 0.9, and d = 1 cm.
Carlo method by as much as 75% near the source, but it becomes more accurate far from the source (Figures 6.4a and 6.4c). The diffuse transmittance, however, is accurate at all distances from the source (Figures 6.4b and 6.4c).
The diffuse reflectance data in response to an isotropic point source at various depths (ζ' — 0.1//, 0.3/^, or 0.5/^) from both the pure Monte Carlo method and the pure diffusion theory are shown in Figure 6.5. Three source pairs (/ = —1,0, 1)
COMPUTATIONAL EXAMPLES 127
(a)
(b)
(c)
20
-20
-40
-60
-80
0.5 l Radius r (cm)
0.5 1 1.5 Radius r (cm)
\ I Rd \\ Td
0.5 1 Radius r (cm)
1.5
Figure 6.4. Comparison between the Monte Carlo method and the diffusion theory in terms of (a) the diffuse reflectance and (b) the diffuse transmittance in response to a pencil beam; (c) relative errors between the results. The properties of the scattering slab are described in Figure 6.3.
1 2 8 HYBRID MODEL OF MONTE CARLO METHOD AND DIFFUSION THEORY
lCT*
Monte Carlo Diffusion
0.2
(a) 0.4 0.6 Radius r (cm)
0.8
(b) 0.4 0.6 Radius r (cm)
Figure 6.5. (a) Comparisons between the Monte Carlo method and the diffusion theory in terms of the diffuse reflectance when an isotropic point source is placed at z! = 0.3/,'; (b) relative errors between the results when an isotropic point source is placed sequentially at z! — 0.1/,', 0.3/p 0.5/,'. The properties of the scattering slab are described in Figure 6.3.
are used in the diffusion theory. The relative errors between the diffusion theory and the Monte Carlo method in diffuse reflectance for these three source locations are shown in Figure 6.5b. Here, 0.5/,' and 0.3/,' give errors of 5% and 12%, respectively, whereas 0.1/,' gives an error of up to 25%. Clearly, critical depth represents a tradeoff between the computational accuracy and efficiency of the hybrid model. Increasing the critical depth improves the computational accuracy at the expense of computational efficiency.
COMPUTATIONAL EXAMPLES 1 2 9
Or
0.25 I
0.3 I 1 1 1 1 1 1 0 0.05 O.l 0.15 0.2 0.25 0.3
Radius r (cm)
Figure 6.6. Contours of Sd in response to a pencil beam from the initial Monte Carlo step of the hybrid model. The critical depth zc is set to 0.05 cm (~ 0.5/,'). The contour values are in the unit of cm-3. The properties of the scattering slab are described in Figure 6.3.
The contours of Sd from the Monte Carlo step of the hybrid model are shown in Figure 6.6. Since the critical depth is 0.05 cm (^0.5/,'), Sd is densely populated near z' — 1.5/,; it is also limited to within approximately 2Vt from the point of incidence in both the r and the z dimensions.
The diffuse reflectance data from both the pure Monte Carlo method and the hybrid model in response to a pencil beam, where Ar' = Αζ' = 0.01 cm and Nr = Nz = 30, are shown in Figure 6.7a. Figure 6.7b plots the relative error in the diffuse reflectance from the hybrid model, which is within ±6% including both statistical and systematic differences. The statistical error can be further reduced by using either more photon packets at the expense of computation time or larger grid elements at the expense of resolution, whereas the systematic error can be further reduced by using a larger critical depth at the expense of computation time. In this example, if one million photon packets are tracked in each model, the hybrid model is 23 times faster than the Monte Carlo method. In other words, the hybrid model is significantly faster than the Monte Carlo method and almost as accurate.
The diffuse reflectance data from both the pure Monte Carlo method and the hybrid model in response to a pencil beam at various \ia values, where Ar' = 0.005 cm, Δζ' = 0.003 cm, and Nr = Nz = 100, are shown in Figure 6.8. When μα becomes comparable with μ̂ (e.g., μα = 10 cm- 1 , and μ̂ = 10 cm- 1) , the accuracy of the hybrid model is poor because the diffusion theory is valid only when μα <£ μ^. Therefore, the hybrid model is not expected to be accurate when \ia <^ \i's is not satisfied. If the critical depth increases, the accuracy of the hybrid model improves—even in the case of strong absorption—at the expense
130 HYBRID MODEL OF MONTE CARLO METHOD AND DIFFUSION THEORY
102
£ 10°
s
i U 0 0.05 0.1 0.15 0.2 0.25 0.3 (a) Radius r (cm)
6
4
g 2
§
2 o
e* -2
-4
" 0 0.05 0.1 0.15 0.2 0.25 0.3 (b) Radius r (cm)
Figure 6.7. (a) Comparison between the pure Monte Carlo method and the hybrid model in terms of the diffuse reflectance in response to a pencil beam; (b) relative errors between the two approaches. The properties of the scattering slab are described in Figure 6.3.
of computation time, because the portion of the photon history tracked by the Monte Carlo step increases. The computer-dependent user times of the Monte Carlo method 7MC are 698, 583, and 136 s, respectively, for \ia = 0.1, 1, and 10 cm- 1 . With the critical depth set to 0.05 cm, the user times of the hybrid model TH are only 104, 99, and 76 s, respectively, for \ia = 0 . 1 , 1, and 10 cm- 1 . With the critical depth set to 0.1 cm, the user times of the hybrid model increase to 147, 142, and 114 s, respectively, for \xa — 0.1, 1, and 10 cm- 1 .
User times for both the Monte Carlo method and the hybrid model under var- ious conditions are listed in Table 6.1. While the other parameters are held con- stant, μα and d are varied. Here, zc = 0.05 cm ̂ 0.5/,;, Ar' = Az' = 0.01 cm,
"T i 1 i r
T I I I Γ
COMPUTATIONAL EXAMPLES 131
(a) 0.2 0.3 Radius r (cm)
0.2 0.3 Radius r (cm)
Figure 6.8. Comparisons between the Monte Carlo method and the hybrid model in terms of diffuse reflectance in response to a pencil beam when the critical depth is set to (a) 0.05 cm and (b) 0.1 cm. The absorption coefficient \ia varies among 0.1, 1, and 10 cm-1, while the other properties are held constant at nrt\ = 1.37, \is — 100 cm-1, g = 0.9, and d = 1 cm.
and Nr — Nz — 30. The computation time of the hybrid model is insensitive to the optical properties unless \ia becomes comparable with \i's as shown in the results associated with Figure 6.8. By contrast, the computation time of the Monte Carlo method is highly sensitive to the optical properties. A lower \ia lengthens photon tracking because the chance of photon absorption per scattering event is
132 HYBRID MODEL OF MONTE CARLO METHOD AND DIFFUSION THEORY
TABLE 6.1. User Times (Computer-Dependent) for Both Monte Carlo Method {TMc) and Hybrid Model (TH) and Their Ratio (TMC/TH) under Various Conditions.0
"rel
1.37 1.37 1.37 1.37 1.37 1.37 1.37 1.37 1.37 1 1 1 1 1 1 1 1 1
d (cm)
10 10 10 3 3 3 1 1 1 10 10 10 3 3 3 1 1 1
μα (cm ')
0.01 0.1 1 0.01 0.1 1 0.01 0.1 1 0.01 0.1 1 0.01 0.1 1 0.01 0.1 1
TMC (S)
6684 2589 679 2095 1961 679 696 698 583 3992 1611 468 1253 1201 468 415 416 382
TH (S)
23 23 23 23 23 23 23 23 23 19 19 19 19 19 19 19 19 19
TMC/TH
291 113 30 91 85 30 30 30 25 210 85 25 66 63 25 22 22 20
aThe fixed optical properties include \is — 100 cm ' and g = 0.9
reduced. Computation under the refractive-index-mismatched boundary condition takes longer because internal reflection at the boundary extends the lifetime of the photons in the scattering slab. In all cases, the hybrid model is faster than the Monte Carlo method by a factor of 20 to nearly 300, depending on the optical properties, the slab thickness, the number of photons tracked, the threshold for the Russian roulette, and the critical depth.
If the slab thickness reduces to several transport mean free paths, the diffusion theory becomes inaccurate. In this case, the pure Monte Carlo method should be used. Since the slab is relatively thin, 7MC is reasonably short. For example, T^c = 75 s when the slab parameters are nlt\ — 1, \ia — 0.1 cm- 1 , μ5 = 100 cm- 1, g = 0.9, and d = 0.2 cm; Tue increases to 136 s when ttrei = 1.37.
PROBLEMS
6.1 Derive Eqs. (6.12) and (6.13).
6.2 Find the z coordinates of the first three source pairs (i.e., / = 0, 1,-1) for nrei = 1.37, μα =0 .1 οιτ ι^ ,μ, = 100 cm~\ g = 0.9, z! = 0.1 cm, and d — 1 cm.
FURTHER READING 133
6.3 Calculate the weight of a photon packet that enters a scattering medium versus the angle of incidence if nre\ = 1.37.
6.4 Explain how the computation time of the hybrid model depends on the optical properties.
6.5 Explain how the computation time of the Monte Carlo model depends on the scattering anisotropy in a scattering slab.
6.6 Implement the hybrid model. Update Table 6.1 with new computation times.
6.7 Implement the hybrid model. Adjust the threshold weight for Russian roulette and compare the computation times and accuracies.
6.8 Implement the hybrid model. Vary the critical depth in the hybrid model and compare the computation times and accuracies.
6.9 Extend the diffusion theory to the case of an infinitely narrow photon beam obliquely incident on a semiinfinite scattering medium.
6.10 Implement a hybrid model for an obliquely incident pencil beam on a semiinfinite scattering medium and compare with the diffuse reflectance computed from the diffusion theory.
READING
Wang LHV and Jacques SL (1993): Hybrid model of Monte Carlo simulation and diffusion theory for light reflectance by turbid media, J. Opt. Soc. Am. A 10(8): 1746-1752. (All sections in this chapter.)
Wang LHV (1998): Rapid modeling of diffuse reflectance of light in turbid slabs, J. Opt. Soc. Am. A 15(4): 936-944. (All sections in this chapter.)
FURTHER READING
Alexandrakis G, Busch DR, Faris GW, and Patterson MS (2001): Determination of the optical properties of two-layer turbid media by use of a frequency-domain hybrid Monte Carlo diffusion model, Appl Opt. 40(22): 3810-3821.
Alexandrakis G, Farrell TJ, and Patterson MS (2000): Monte Carlo diffusion hybrid model for photon migration in a two-layer turbid medium in the frequency domain, App. Opt. 39(13): 2235-2244.
Carp SA, Prahl SA, and Venugopalan V (2004): Radiative transport in the delta-p-1 approximation: Accuracy of fluence rate and optical penetration depth predictions in turbid semi-infinite media, J. Biomed. Opt. 9(3): 632-647.
Farrell TJ, Patterson MS, and Wilson B (1992): A diffusion-theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical-properties invivo, Med. Phys. 19(4): 879-888.
Flock ST, Wilson BC, and Patterson MS (1989): Monte Carlo modeling of light- propagation in highly scattering tissues. 2. Comparison with measurements in phan- toms, IEEE Trans. Biomed. Eng. 36(12): 1169-1173.
Gardner CM and Welch AJ (1994): Monte Carlo simulation of light transport in tis- sue—unscattered absorption events, Appl. Opt. 33(13): 2743-2745.
1 3 4 HYBRID MODEL OF MONTE CARLO METHOD AND DIFFUSION THEORY
Gardner CM, Jacques SL, and Welch AJ (1996): Light transport in tissue: Accurate expres- sions for one-dimensional fluence rate and escape function based upon Monte Carlo simulation, Lasers Surg.Med. 18(2): 129-138.
Groenhuis RAJ, Ferwerda HA, and Tenbosch JJ (1983): Scattering and absorption of turbid materials determined from reflection measurements. 1. Theory, Appl. Opt. 22(16): 2456-2462.
Groenhuis RAJ, Tenbosch JJ, and Ferwerda HA (1983): Scattering and absorption of turbid materials determined from reflection measurements. 2. Measuring method and calibration, Appl. Opt. 22(16): 2463-2467.
Kim AD and Moscoso M (2005): Light transport in two-layer tissues, J. Biomed. Opt. 10(3): 031015.
Schweiger M, Arridge SR, Hiraoka M, and Delpy DT (1995): The finite-element method for the propagation of light in scattering media—boundary and source conditions, Med. Phys. 22(11): 1779-1792.
Spott T and Svaasand LO (2000): Collimated light sources in the diffusion approximation, Appl. Opt. 39(34): 6453-6465.
Tarvainen T, Vauhkonen M, Kolehmainen V, and Kaipio JP (2005): Hybrid radiative- transfer-diffusion model for optical tomography, Appl. Opt. 44(6): 876-886.
Tarvainen T, Vauhkonen M, Kolehmainen V, Arridge SR, and Kaipio JP (2005): Coupled radiative transfer equation and diffusion approximation model for photon migration in turbid medium with low-scattering and non-scattering regions, Phys. Med. Biol 50(20): 4913-4930.
Wang LHV and Jacques SL (1995): Use of a laser-beam with an oblique angle of incidence to measure the reduced scattering coefficient of a turbid medium, Appl. Opt. 34(13): 2362-2366.
Wang LHV, Jacques SL, and Zheng LQ (1995): MCML—Monte Carlo modeling of light transport in multilayered tissues, Comput. Meth. Prog. Biomed. 47(2): 131-146.
Wyman DR, Patterson MS, and Wilson BC (1989): Similarity relations for anisotropic scattering in Monte-Carlo simulations of deeply penetrating neutral particles, J. Com- put. Phys. 81(1): 137-150.
Wyman DR, Patterson MS, and Wilson BC (1989): Similarity relations for the interaction parameters in radiation transport, Appl. Opt. 28(24): 5243-5249.
Yoon G, Prahl SA, and Welch AJ (1989): Accuracies of the diffusion-approximation and its similarity relations for laser irradiated biological media, Appl. Opt. 28(12): 2250-2255.
CHAPTER 7
Sensing of Optical Properties and Spectroscopy
7.1. INTRODUCTION
Sensing the optical properties of biological tissue is important for diagnosis and therapy. After traversing biological tissue, reemitted light carries information about the optical properties of the scattering medium, which can be extracted using an inverse algorithm. Optical properties can be measured at multiple optical wavelengths for the production and investigation of spectra (spectroscopy).
7.2. COLLIMATED TRANSMISSION METHOD
The extinction coefficient μ,, which is defined as the sum of the absorption coefficient μα and the scattering coefficient μ5, can be measured by the colli- mated transmission method. In this method, a collimated lightbeam is incident perpendicularly on the surface of a sample. The sample can be a cuvette of liquid (e.g., an Intralipid® solution), a tissue-mimicking gel phantom (e.g., an agar gel containing polystyrene spheres), a solid phantom (e.g., a solidified resin containing T1O2 particles), or a piece of biological tissue. The collimated (or ballistic) portion of the transmitted light is selected by apertures and then mea- sured by a photodetector (Figure 7.1). First, a clear medium (e.g., water), the refractive index of which closely matches that of the sample to be tested, is measured to provide a reference ballistic-light signal /o. Then, the sample is measured, which yields a transmitted light signal Is. According to Beer's law, we have
/, = /οεχρί-μ,*/), (7.1)
where d denotes the sample thickness. Here, light absorption by the clear medium is neglected (see Problem 7.1). The ballistic transmittance T of the scattering
Biomedical Optics: Principles and imaging, by Lihong V. Wang and Hsin-i Wu Copyright © 2007 John Wiley & Sons, Inc.
135
136 SENSING OF OPTICAL PROPERTIES AND SPECTROSCOPY
Light source Sample
H M
d
ΘΛ
Detector
D Pinhole Pinhole
Figure 7.1. Schematic for the collimated transmission method.
medium is defined as
T = V (7.2)
Substituting Eq. (7.2) into Eq. (7.1), we obtain the extinction coefficient of the sample:
1 [it = - - l n 7 \
a (7.3)
A key assumption in Eq. (7.1) is that the detected scattered light is much weaker than the detected ballistic light. Three factors affect this assumption: (1) the scattering optical depth of the sample \isd, (2) the scattering phase func- tion /?, and (3) the acceptance angle (half-angle) of detection θ</. The first factor determines the ratio of the number of scattered photons to the number of unscat- tered transmitted photons. The next two factors determine the collection fraction of the scattered photons to be detected χ.
7.2.1. Distribution of Scattering Count
We first consider an ideal scattering slab with the following optical properties: relative refractive index nK\ — l and scattering anisotropy g = 1. In this slab, specular reflection does not exist, and scattering does not deflect the photon. If the number of incident photons is 7Vjn, the number of unscattered transmitted photons NQ can be computed from Beer's law:
N0 = Nmexp(-\itd). (7.4)
The number of transmitted photons that have experienced / scattering events N; is given by the following Poisson distribution (see Problem 7.2):
Ni = M ( μ ^ ν ' β χ ρ ί - Μ )
(7.5)
COLLIMATED TRANSMISSION METHOD 137
7.2.2. Collection Fraction
We next consider the collection fraction of singly scattered light in a real scat- tering medium. Distributed angularly, the singly scattered light is only partially received by the detector, which has a finite acceptance angle. The Henyey-Green- stein scattering phase function, which is the PDF of the cosine of the scattering polar angle Θ, is assumed here (see Chapter 3)
p(cos0) i - s 2
2 ( l + g 2 - 2 g c o s e ) 3 / 2 ' (7.6)
where Θ e [0, π]. Normalization requires that the integral of p(cosB) over cosO in the range of [—1, 1] be unity.
For a slab with nonunity nre\, θ^ in air can be converted into an acceptance angle in the sample #d by Snell's law:
sinG^ = ttrei sinO^, (7.7)
which can be simplified to θ^ = nre[dfd if θ^ < 1. Integrating the phase function given by Eq. (7.6) over cosO in interval [cos0^, 1] yields the collection fraction χ of the singly scattered light
1+8 2g
i-g J\+g2-2gcose'd
(7.8)
which can be simplified to
9J
2 ( 1 -g)2' (7.9)
if $d « 1 - g and 1 - g « 1 (i.e., g -> 1).
Example 7.1. Derive Eq. (7.9) from Eq. (7.8).
Because Q'd <?C 1 — g and 1 — g <5C 1, Eq. (7.8) can be approximated by repeti- tively using the Taylor expansion to the first order as follows:
1-g J(l-g)i + 2g(l-cosd'd)
1 g
i - s
J(i-g)2 + ge'-
138 SENSING OF OPTICAL PROPERTIES AND SPECTROSCOPY
1 -
yi+f(^2v(i-^)2]
1 l + ^[(^2)/( l-^)2]
i « # 2(1 - g ) 2
Θ '2
2(1 - g)2 (7.10)
7.2.3. Error Expression
While the signal is from the unscattered light, the unwanted bias is from the scattered light. The unscattered light, assumed to be collimated, is completely collected by the detector. The scattered light contains both single- and multiple- scattered light. When \isd < 1, multiple-scattered light is negligible; hence, only the single-scattered light is considered here. If g -> 1, the number of single- scattered photons can be estimated by N\ from Eq. (7.5). Thus, the relative error due to the detected bias is approximately
6r = No
(7.11)
where the numerator and denominator represent the numbers of received single- scattered and unscattered photons, respectively. Although specular reflections on both slab surfaces are neglected explicitly, the unscattered and the collected single-scattered photons experience similar specular reflections. Therefore, the contributions of specular reflections to the numerator and the denominator in Eq. (7.11) partially offset each other.
Substituting the expressions for No and N\ from Eq. (7.5) into Eq. (7.11), we obtain
Er = x[isd. (7.12)
Although derived using the preceding approximate analytical expressions, Eq. (7.12) can be validated by the accurate Monte Carlo method (see Prob- lem 7.3).
Substituting Eq. (7.9) into Eq. (7.12), we obtain
d == 2sr (l-g)2
μΛ2 (7.13)
SPECTROPHOTOMETRY 1 3 9
which is valid if both θ^ <^ \ ~ g and 1 — g <$C 1 hold. Therefore, the sample thickness must be limited on the basis of θ^ as well as the optical properties so that the relative error is controlled within a given level. Although unknown initially, μ5 can be first measured and then checked for error using Eq. (7.12). If the error is unacceptable, one can modify the sample thickness—or the concentration of scatterers if the sample is liquid—and repeat the measurement.
Example 7.2. Apply Eq. (7.13) to a realistic case. Known parameters include nre\ = 1.37, g = 0.99, and θ^ = 1 mrad (milliradian). Assume μ5 % 100 cm- 1 .
If er < 1% is desired, Eq. (7.13) leads to d < 0.037 cm.
7.3. SPECTROPHOTOMETRY
Spectrophotometry is based on the collimated transmission method. A spec- trophotometer measures μ, of a sample as a function of wavelength; if μ5 « μ α , then μ, ^ μα. The absorbance A, however, is typically reported; it is defined as
A = - l o g I 0 r . (7.14)
Substituting Eq. (7.3) into Eq. (7.14), we obtain
A = (log10 e)\it d = 0.4343μ, d (7.15)
or
μ, = (1η10)- = 2 . 3 0 3 - . (7.16) d d
The absorbance is also referred to as the optical density (OD), especially for a neutral-density filter, which has a nearly constant absorbance in a broad band. OD can be further related to dB (decibels) since the transmittance in dB is defined as — 101og107\ For example, an OD of 1 means a 10-dB or 10-times attenuation, and an OD of 2 means a 20-dB or 100-times attenuation. OD is sometimes defined as the absorbance per unit length, however.
The unit of dB/cm (decibels per centimeter) is also used for various coeffi- cients, such as the absorption and the extinction coefficients, although the unit of cm"1 (also expressed as nepers/cm in ultrasonics) is usually used in biomedical optics. The two units can be converted as follows:
1 cm"1 = (101og10*) dB/cm = 4.343 dB/cm, (7.17)
1 dB/cm = (0.1 In 10) cm"1 = 0.2303 cm - 1 . (7.18)
As a mnemonic aid for these two conversions, note that e23 % 10 and 10° 43 ^ e.
1 4 0 SENSING OF OPTICAL PROPERTIES AND SPECTROSCOPY
A typical spectrophotometer contains one or more light sources that provide a broad spectrum. For example, a tungsten lamp can provide visible and infrared light, and a deuterium lamp can provide ultraviolet light. A diffraction grating angularly disperses the light emanating from the lamp into a spectrum. A narrow portion of the dispersed spectrum passes through a slit opening. The wavelength of the selected light can be tuned by rotating the grating with a knob. The grating in combination with the slit is also referred to as a monochromator. The "monochromatic" light is incident on the sample, and the transmitted light is detected by an optical detector such as a photodiode, which converts the optical signal into an electrical signal.
7.4. OBLIQUE-INCIDENCE REFLECTOMETRY
An oblique-incidence reflectometer can rapidly measure both the absorption coef- ficient μα and the reduced scattering coefficient μ^, where \ια « μ ^ . As discussed in Chapter 5, a pencil beam normally incident on a semiinfinite scattering medium can be approximately represented by an isotropic point source (Figure 7.2a). The far diffuse reflectance—for which the observation points are beyond one trans- port mean free path l't from the point of incidence—in response to the pencil beam is well modeled using such an isotropic point source.
Similarly, an obliquely incident pencil beam can be approximated by an isotropic point source that is l[ away from the point of incidence along the unscat- tered transmission path, as illustrated in Figure 7.2b. As a result, the isotropic point source is horizontally shifted from the point of incidence. Here, a, and at are the angles of incidence and transmission, respectively; rcrei is the relative refractive index of the scattering medium. On the basis of Snell's law, we have
sin a/ = Jirei sin a,. (7.19)
From the geometry, we expect a horizontal shift of the far diffuse reflectance by
sin a, μ, + μβ'
(7.20)
A more accurate empirical expression is given below. The schematic of an experimental oblique-incidence reflectometer is shown in
Figure 7.3. A laser beam is incident on the object surface at an oblique angle. Diffusely reflected light is imaged by a CCD (charge-coupled device) camera. The CCD data are transferred to a computer and processed.
An experimentally measured diffuse reflectance distribution is shown in Figure 7.4. The midpoints of the left and right sides of curve M at all reflectance values are connected to form a centerline C. The shift xm of the vertical portion of curve C represents the horizontal shift of the far diffuse reflectance; it agrees well with the theoretically predicted xs from Eq. (7.20).
OBLIQUE-INCIDENCE REFLECTOMETRY 141
The diffuse reflectance versus x from the Monte Carlo method is shown in Figure 7.5. The shift of the vertical portion of curve C is xm = 0.174 ± 0.009 cm. From the optical properties used, Eq. (7.20) predicts xs =0.167 cm, which is in approximate agreement with the xm predicted by the Monte Carlo method. Although not accurate, Eq. (7.20) is validated both experimentally and numeri- cally.
Lightbeam
Air
(a)
Scattering medium
m Lightbeam
Air Scattering medium λί
xs =
U » a.· r 1
>o> 1 'V \ ^"Jr
If sin(a()
Mirror line
(b)
Figure 7.2. Lumped isotropic point sources for a pencil beam of (a) normal incidence (a, = 0) and (b) oblique incidence (α{· > 0).
CCD camera Computer
Mirror
Attenuator Scattering medium
Figure 7.3. Schematic of a CCD-based oblique-incidence reflectometer.
142 SENSING OF OPTICAL PROPERTIES AND SPECTROSCOPY
I
- l -0.5 0 0.5 1 Horizontal position JC (cm)
Figure 7.4. Experimentally measured diffuse reflectance as a function of x, where M represents the measured data and C represents the centerline.
6
-1.5 -0.5 0 0.5 Horizontal position JC (cm)
1.5
Figure 7.5. Curve M represents the Monte Carlo simulated diffuse reflectance of a 1-mW laser beam incident on a scattering medium with a, = 45°, and curve C represents the centerline of curve M. The optical properties of the scattering medium are Airei = 133, μ„ = 0.25 cm- 1 , μν = 20 cm- 1 , and g = 0.853.
OBLIQUE-INCIDENCE REFLECTOMETRY 143
When compared with the accurate Monte Carlo simulated shifts for various μα values, the shift from Eq. (7.20) is highly accurate when μα <ζ μ^, but it becomes progressively less accurate with increasing μα. The following empirical equation for the shift, however, significantly improves the accuracy:
sin at (7.21) s Ws + 0.35μα '
where 0.35μα is used instead of μα. Equation (7.21) predicts x's = 0.176 cm, which is in better agreement with the value predicted by the Monte Carlo method. For simplicity, we define the diffusion coefficient for this section as
D = . (7.22) 3(μ ;+0 .35μ α )
Merging Eqs. (7.21) and (7.22), we obtain
x's = 3D sin a,. (7.23)
To measure the optical properties of the scattering medium, we first estimate the center x's of the far diffuse reflectance from the experimental data. From Eq. (7.23), we have
D = s—. (7.24) 3 sin at
Since two independent optical properties are being measured, one more equation is needed.
The diffusion theory in Chapter 5 can be modified for the diffuse reflectance in response to an obliquely incident pencil beam:
(1 - Rsp)a' 4π
z's(\ H^effPi)exp(-^effPi) (z's + 2zb)(l + μ6ίίΡ2)6χρ(-μ6ΓίΡ2) 3 3
Pi P2 (7.25)
Here, Rsp denotes the specular reflectance; a' denotes the transport albedo; x denotes the distance between the observation point on the surface of the scatter- ing medium and the point of incidence; pi and P2 denote the distances from the two point sources (the original equivalent and the image sources) to the obser- vation point, respectively; Zb denotes the distance between the extrapolated and the actual boundaries; μ ^ denotes the effective attenuation coefficient; and zfs denotes the depth of the original point source:
z's=x'sco\.{*t). (7.26)
144 SENSING OF OPTICAL PROPERTIES AND SPECTROSCOPY
A nonlinear least-squares fit of the measured far diffuse reflectance to Eq. (7.25) yields μ^, which is defined as
μβίτ = yJ^a/D. {121)
We have now quantified both D and μείί from the relative profile of the far diffuse reflectance, which can be obtained more easily than its absolute counterpart. From Eqs. (7.24) and (7.27), we obtain
(7.28)
(7.29)
7.5. WHITE-LIGHT SPECTROSCOPY
A spectroscopic oblique-incidence reflectometer (Figure 7.6) can measure absorp- tion and reduced scattering spectra. White light from a lamp is coupled to a handheld probe made of 0.6-mm-diameter optical fibers. The source fiber is ori- ented at a 45° angle of incidence. Nine collection fibers, arranged in a linear array, collect the diffuse reflectance. Approximately 4.6 mW of white light is delivered
Probe Imaging spectrograph Computer
9 collection fibers
1 source fiber
Air
Scattering medium
n g \
Figure 7.6. Schematic of a spectroscopic oblique-incidence reflectometer.
TIME-RESOLVED MEASUREMENT 145
through the source fiber. The collection fibers are coupled to a connecting inter- face, the output of which is placed at the object plane of an imaging spectrograph. The imaging spectrograph spectrally disperses light from each detection fiber into a ID spectrum and subsequently projects the 2D spatiospectral distribution onto a CCD camera. The CCD camera, which is controlled by a personal computer, records the 2D spatiospectral distribution. The CCD matrix has 512 x 512-pixels and measures 9.7 x 9.7 mm2. With a 150 lines/mm grating, the CCD matrix is capable of accommodating a spectral range of 256 nm.
In the 2D spatiospectral distribution, the vertical and horizontal dimensions represent the spatial distribution of the diffuse reflectance at each wavelength and the spectral distribution of light from each collection fiber, respectively. The spatial distribution at each wavelength is used to fit for μα and μ^, according to the theory described in the preceding section. As discussed in Chapter 1, the absorption spectrum can be used to assess the concentrations of oxy- and deoxyhemoglobin; on the basis of the Mie theory, the reduced scattering spectrum can be used to estimate the size distribution of the scatterers.
7.6. TIME-RESOLVED MEASUREMENT
Time-resolved diffuse reflectance can be used to measure the optical properties of biological tissue as well. With a short-pulsed collimated narrow laser beam normally incident on a semiinfinite scattering medium, a fast time-resolved detec- tor—such as a streak camera or a single-photon counting system—measures the local diffuse reflectance Rd(r, t) or the total diffuse reflectance Rd(t). Here, r denotes the distance between the observation point and the point of incidence, and t denotes time. Under a simplified zero-boundary condition (the fluence rate on the real boundary is zero), the diffusion theory predicts
^ ( Γ ' ° = (4nDc)y2t5f2 e*P ( - ^ 4 ^ " ) e X P ( - ^ C i ) ' ( 7 3 0 )
Γ z' I z2 \ Rd(t) = / Rd(n t)2nrdr = exp - 7 7 — 1 exp(-jxacO, Jo VinDct3/2 \ 4Dct J
(7.31) where the source location z! — l\ and the diffusion coefficient D = l't/3. Whereas factors t5/2 and f3/2 dominate the early dynamics of the reflectance, exp(—\iact) dominates the later dynamics.
The absorption coefficient can be extracted by rewriting Eqs. (7.30) and (7.31) as follows:
d In Rd(r,t) 5 r 2 + z2
dinRdit) ^ 3 z'2
146 SENSING OF OPTICAL PROPERTIES AND SPECTROSCOPY
At large t values, each plot of d In R^ versus t approaches a straight line with a slope equal to — \iac. The second term in each of the equations above should be included for accuracy, while the third term can be neglected. For example, when μα =0 .1 cm- 1 and μ̂ = 10 cm"1, the second term is comparable with the first term for several nanoseconds, whereas the third term becomes negligible after only several hundred picoseconds. Therefore, the absorption coefficient can be estimated by either of the following expressions:
1 μ* ^ —
c
1
c
d In Rd(r, t) 5 , + - , (7.34) dt It
μ«
] ]
d\nRd(t) 3 , - ^ + ^ | . (7.35)
7.7. FLUORESCENCE SPECTROSCOPY
Fluorescence spectroscopy provides a means for measuring the concentrations, quantum yields, and lifetimes of fluorescent molecules. Concentrations can pro- vide morphologic information about biological tissue. Because quantum yields and lifetimes are related to the characteristics of biological molecules, they can provide biochemical information. These properties can reveal a variety of clinical problems such as epithelial neoplasia and atherosclerosis.
In a fluorescence spectroscopic system, light from a monochromatic excitation source is delivered through a flexible optical fiber bundle to the biological sample. The emitted fluorescent light from the sample is collected through another optical fiber bundle. The collected light is then separated into spectral components by a dispersing element. The dispersed fluorescence spectrum is finally detected by a detector array.
The system can be implemented with optical fibers or in free space. Whereas a handheld fiberoptic probe is typically used in contact with tissue, a free- space system is used in a noncontact fashion. A handheld probe may suffer from spectral dependence on the pressure applied by the probe on the in vivo tissue; a free-space system may suffer from intensity dependence on in vivo tissue motion. The contact approach is generally employed for small tissue areas, whereas the noncontact approach is more commonly used for relatively larger areas.
A fluorescence spectrum is related to both the excitation and the emission wavelengths. A fluorescence excitation spectrum can be produced by measuring the fluorescence intensity at a given emission wavelength for a range of excitation wavelengths. Conversely, a fluorescence emission spectrum can be produced by measuring the fluorescence intensity over a range of emission wavelengths at a given excitation wavelength. Ultimately, a fluorescence excitation-emission matrix (EEM) can be produced by measuring the fluorescence intensity over a range of emission wavelengths for a range of excitation wavelengths.
FLUORESCENCE MODELING 147
7.8. FLUORESCENCE MODELING
Although fluorescence is not discussed in Chapter 5, propagation of both excita- tion and fluorescent light in a scattering medium can be modeled by the diffusion theory. Assumed to be independent of fluorescence (actually, a Born approxima- tion), the diffusion equation for the excitation light is given by
~ < M r , t) + μβχ<ΜΓ, 0 - V · 0 , ν Φ , ( Γ , t) = Sx(r, 0 , (7.36) c at
where subscript x denotes the excitation wavelength. The other symbols are defined in Chapter 5.
The source term in Eq. (7.36) can be constructed from the first equivalent isotropic scattering events. From the similarity relation, an equivalent isotropic scattering medium is considered. Unscattered photons propagate along the bal- listic path and constitute the primary beam, which has the following fluence rate distribution according to Beer's law:
Φ Ρ * ( Γ , 0 = 0 - ^Ρ)Φο.(? ' , ί )βχρ(-μ; χ / ) . (7.37)
Here, the subscript p denotes the primary beam, Rsp denotes the specular reflect- ance from the surface of the scattering medium, Φο* denotes the incident fluence rate on the surface, and / denotes the ballistic path length into the scattering medium. The primary beam is converted into an isotropic source distribution, which serves as the source term for Eq. (7.36), by
5,(Γ,0 = μ;*Φ/«(Γ,0, (7.38)
where μ^ denotes the reduced scattering coefficient at the excitation wavelength. Once an excitation photon is absorbed by a fluorophore, the probability that a
fluorescence photon is emitted per unit time at time t (t > 0) can be modeled by
y(t) =-cxpi-^Y (7.39)
were Y denotes the quantum yield for fluorescence emission and τ denotes the fluorescence lifetime.
Once fluorescent light is generated, its propagation can be modeled using another diffusion equation
- —4>m(r, 0 + feOJr, 0 - V · Dm V4>m(r, t) = Sm(r, f). (7.40) c at
where subscript m denotes the fluorescence emission wavelength. The source term is derived from the excitation light distribution using the following convolution
Sm(r, t)= f y(t - ί ' ) μ β / χ [ 4 ν ( Γ , tf) + Φ Χ (Γ, t')]dt\ (7.41) Jo
148 SENSING OF OPTICAL PROPERTIES AND SPECTROSCOPY
where \iafx denotes the absorption coefficient of the fluorophores at the excitation wavelength.
PROBLEMS
7.1 (a) Prove Eq. (7.1). (b) Explain why it is important to measure a clear medium first, (c) Assuming that the absorption coefficient of the clear medium is significant, modify Eq. (7.1).
7.2 Prove Eq. (7.5).
7.3 Write a Monte Carlo program to validate Eq. (7.12).
7.4 If the transmittance is given by absorbance A, express the transmittance in dB and then calculate the absorption coefficient in terms of A if the thickness of the sample is known.
7.5 In the collimated transmission method, assuming that measurements /o, Is, and d have independent uncertainties that are quantified by standard devi- ations σο, σΛ, and σ^, respectively, derive the expected standard deviation in the predicted μ,.
7.6 In a collimated transmission measurement, if the sample is optically thin (d <£. 1/μ/), the number of particles along the path can fluctuate signifi- cantly as a result of, for example, Brownian motion. Estimate the standard deviation of the number of received photons due to this fluctuation.
7.7 Write a Monte Carlo program to simulate the oblique-incidence diffuse reflectance from a semiinfinite medium. Duplicate Figure 7.5.
7.8 Write a Monte Carlo program to simulate the total time-resolved dif- fuse reflectance Rd(t) from a semiinfinite scattering medium in response to a temporally ultrashort pencil beam. Compare it with the diffusion theory predicted values in response to an equivalent isotropic source located at (a) \/(μα + μ^), (b) 1/(0.35μα + μ^), (c) \/\jJs below the sur- face. Assume nre\ = 1.38, μα = 0.1 cm- 1 , μν — 100 cm- 1 , g — 0.9, and a,· = 45°.
7.9 Derive the time-resolved diffuse reflectance equations in Section 7.6 assuming a zero boundary condition. Then derive them again using the extrapolated virtual boundary described in Chapter 5.
7.10 A fluorescent point object is placed at (x',yf,zf) below the surface of a semiinfinite scattering sample, where the z axis starts at the sample surface and points into the sample. A normally incident continuous-wave pencil beam at (0, 0, 0) is used to excite the fluorophores. Use the diffusion theory to model the fluorescent reflectance measured on the sample surface at (jt,y,0). Assume that the optical properties of the medium and the quantum yield of the fluorophores are known.
7.11 Explain why dB is sometimes defined by 101ogI0 instead of 20 log,0.
FURTHER READING 149
READING
Farrell TJ and Patterson MS (2003): Diffusion modeling of fluorescence in tissue, in Handbook of Biomedical Fluorescence, Mycek MA and Pogue, BW, eds., Marcel Dekker, New York, pp. 29-60. (See Section 7.8, above.)
Jacques SL, Wang LHV, and Hielscher AH (1995): Time-resolved photon propagation in tissues, in Optical Thermal Response of Laser irradiated Tissue, Welch AJ and van Gemert MJC, eds., Plenum Press, New York, pp. 305-332. (See Section 7.6, above.)
Marquez G and Wang LHV (1997): White light oblique incidence reflectometer for measuring absorption and reduced scattering spectra of tissue-like turbid media, Opt. Express 1: 454-460. (See Sections 7.4 and 7.5, above.)
Ramanujam N (2000): Fluorescence spectroscopy of neoplastic and non-neoplastic tissues, Neoplasia 2(1-2): 89-117. (See Section 7.7, above.)
Richards-Kortum R and Sevick-Muraca E (1996): Quantitative optical spectroscopy for tissue diagnosis, Annu. Rev. Phys. Chem. 47; 555-606. (See Section 7.7, above.)
Wang LHV and Jacques SL (1994): Error estimation of measuring total interaction coef- ficients of turbid media using collimated light transmission, Phys. Med. Biol. 39: 2349-2354. (See Section 7.2, above.)
Wang LHV and Jacques SL (1995): Use of a laser beam with an oblique angle of inci- dence to measure the reduced scattering coefficient of a turbid medium, Appl. Opt. 34: 2362-2366. (See Section 7.4, above.)
FURTHER READING
Baker SF, Walker JG, and Hopcraft KI (2001): Optimal extraction of optical coefficients from scattering media, Opt. Commun. 187(1-3): 17-27.
Bevilacqua F and Depeursinge C (1999): Monte Carlo study of diffuse reflectance at source-detector separations close to one transport mean free path, J. Opt. Soc. Am. A 16(12): 2935-2945.
Chang SK, Mirabal YN, Atkinson EN, Cox D, Malpica A, Folien M, and Richards-Kortum R (2005): Combined reflectance and fluorescence spectroscopy for in vivodetection of cervical pre-cancer, J. Biomed. Opt. 10(2): 024031.
Collier T, Folien M, Malpica A, and Richards-Kortum R (2005): Sources of scattering in cervical tissue: Determination of the scattering coefficient by confocal microscopy, Appl. Opt. 44( 11): 2072-2081.
Dam JS, Pedersen CB, Dalgaard T, Fabricius PE, Aruna P, and Andersson-Engels S (2001): Fiber-optic probe for noninvasive real-time determination of tissue optical properties at multiple wavelengths, Appl. Opt. 40(7): 1155-1164.
Garcia-Uribe A, Kehtarnavaz N, Marquez G, Prieto V, Du vie M, and Wang LHV (2004): Skin cancer detection by spectroscopic oblique-incidence reflectometry: Classification and physiological origins, Appl. Opt. 43(13): 2643-2650.
Gobin L, Blanchot L, and Saint-Jalmes H (1999): Integrating the digitized backscattered image to measure absorption and reduced-scattering coefficients in vivo, Appl. Opt. 38(19): 4217-4227.
Hull EL and Foster TH (2001): Steady-state reflectance spectroscopy in the p-3 approxi- mation, J. Opt. Soc. Am. A 18(3): 584-599.
1 5 0 SENSING OF OPTICAL PROPERTIES AND SPECTROSCOPY
Intes X, Le Jeune B, Pellen F, Guern Y, Cariou J, and Lotrian J (1999): Localization of the virtual point source used in the diffusion approximation to model a collimated beam source, Waves Random Media 9(4): 489-499.
Jacques SL (1989): Time-resolved reflectance spectroscopy in turbid tissues, IEEE Trans. Biomed. Eng. 36: 1155-1161.
Johns M, Giller CA, German DC, and Liu HL (2005): Determination of reduced scat- tering coefficient of biological tissue from a needle-like probe, Opt. Express 13(13): 4828-4842.
Jones MR and Yamada Y (1998): Determination of the asymmetry parameter and scatter- ing coefficient of turbid media from spatially resolved reflectance measurements, Opt. Rev. 5(2): 72-76.
Kumar D and Singh M (2003): Characterization and imaging of compositional variation in tissues, IEEE Trans. Biomed. Eng. 50(8): 1012-1019.
Liebert A, Wabnitz H, Grosenick D, Moller M, Macdonald R, and Rinneberg H (2003): Evaluation of optical properties of highly scattering media by moments of distributions of times of flight of photons, Appl. Opt. 42(28): 5785-5792.
Lin SP, Wang LHV, Jacques SL, and Tittel FK (1997): Measurement of tissue optical properties by the use of oblique-incidence optical fiber reflectometry, Appl. Opt. 36(1): 136-143.
Lin WC, Motamedi M, and Welch AJ (1996): Dynamics of tissue optics during laser heating of turbid media, Appl. Opt. 35(19): 3413-3420.
Marquez G, Wang LHV, Lin SP, Schwartz JA, and Thomsen SL (1998): Anisotropy in the absorption and scattering spectra of chicken breast tissue, Appl Opt. 37(4): 798-804.
Mirabal YN, Chang SK, Atkinson EN, Malpica A, Folien M, and Richards-Kortum R (2002): Reflectance spectroscopy for in vivodetection of cervical precancer, J. Biomed. Opt. 7(4): 587-594.
Mourant JR, Bigio IJ, Jack DA, Johnson TM, and Miller HD (1997): Measuring absorption coefficients in small volumes of highly scattering media: Source-detector separations for which path lengths do not depend on scattering properties, Appl. Opt. 36(22): 5655-5661.
Mourant JR, Johnson TM, Los G, and Bigio LJ (1999): Non-invasive measurement of chemotherapy drug concentrations in tissue: Preliminary demonstrations of in vivomea- surements, Phys. Med. Biol. 44(5): 1397-1417.
Nichols MG, Hull EL, and Foster TH (1997): Design and testing of a white-light, steady- state diffuse reflectance spectrometer for determination of optical properties of highly scattering systems, Appl. Opt. 36(1): 93-104.
Nishidate I, Aizu Y, and Mishina H (2004): Estimation of melanin and hemoglobin in skin tissue using multiple regression analysis aided by Monte Carlo simulation, J. Biomed. Opt. 9(4): 700-710.
Papaioannou T, Preyer NW, Fang QY, Brightwell A, Carnohan M, Cottone G, Ross R, Jones LR, and Marcu L (2004): Effects of fiber-optic probe design and probe-to-target distance on diffuse reflectance measurements of turbid media: An experimental and computational study at 337 nm, Appl. Opt. 43(14): 2846-2860.
Patterson MS and Pogue BW (1994): Mathematical-model for time-resolved and frequency-domain fluorescence spectroscopy in biological tissue, Appl. Opt. 33(10): 1963-1974.
Patterson MS, Chance B, and Wilson BC (1989): Time resolved reflectance and trans- mittance for the noninvasive measurement of tissue optical properties, Appl Opt. 28: 2331-2336.
FURTHER READING 151
Pham TH, Bevilacqua F, Spott T, Dam JS, Tromberg BJ, and Andersson-Engels S (2000): Quantifying the absorption and reduced scattering coefficients of tissuelike turbid media over a broad spectral range with noncontact Fourier-transform hyperspectral imaging, Appl. Opt. 39(34): 6487-6497.
Rinzema K, Murrer LHP, and Star WM (1998): Direct experimental verification of light transport theory in an optical phantom, J. Opt. Soc. Am. A—Optics Image Sei. Vision 15(8): 2078-2088.
Sefkow A, Bree M, and Mycek MA (2001): Method for measuring cellular optical absorp- tion and scattering evaluated using dilute cell suspension phantoms, Appl. Spectrosc. 55(11): 1495-1501.
Seiden AC (2004): Photon transport parameters of diffusive media with highly anisotropic scattering, Phys. Med. Biol 49(13): 3017-3027.
Skala MC, Palmer GM, Zhu CF, Liu Q, Vrotsos KM, Marshek-Stone CL, Gendron- Fitzpatrick A, and Ramanujam N (2004): Investigation of fiber-optic probe designs for optical spectroscopic diagnosis of epithelial pre-cancers, Lasers Surg. Med. 34(1): 25-38.
Swartling J, Dam JS, and Andersson-Engels S (2003): Comparison of spatially and tempo- rally resolved diffuse-reflectance measurement systems for determination of biomedical optical properties, Appl. Opt. 42(22): 4612-4620.
Takagi K, Haneishi H, Tsumura N, and Miyake Y (2000): Alternative oblique-incidence reflectometry for measuring tissue optical properties, Opt. Rev. 7(2): 164-169.
Treweek SP and Barbenel JC (1996): Direct measurement of the optical properties of human breast skin, Med. Biol. Eng. Comput. 34(4): 285-289.
Utzinger U and Richards-Kortum RR (2003): Fiber optic probes for biomedical optical spectroscopy, J. Biomed. Opt. 8(1): 121-147.
Wan SK and Guo ZX (2006): Correlative studies in optical reflectance measurements of cerebral blood oxygenation, J. Quantit. Spectrosc. Radiative Transfer 98(2): 189-201.
Wu T, Qu JNY, Cheung TH, Lo KWK, and Yu MY (2003): Preliminary study of detecting neoplastic growths in vivowith real time calibrated autofluorescence imaging, Opt. Express 11(4): 291-298.
Yaroslavsky IV, Yaroslavsky AN, Goldbach T, and Schwarzmaier HJ (1996): Inverse hybrid technique for determining the optical properties of turbid media from integrating-sphere measurements, Appl. Opt. 35(34): 6797-6809.
Zijp JR and ten Bosch JJ (1998): Optical properties of bovine muscle tissue in vitro; a comparison of methods, Phys. Med. Biol. 43(10): 3065-3081.
Zonios G, Bykowski J, and KoUias N (2001): Skin melanin, hemoglobin, and light scattering properties can be quantitatively assessed in vivousing diffuse reflectance spectroscopy, J. Invest. Dermatol. 117(6): 1452-1457.
Zuluaga AF, Utzinger U, Durkin A, Fuchs H, Gillenwater A, Jacob R, Kemp B, Fan J, and Richards-Kortum R (1999): Fluorescence excitation emission matrices of human tissue: A system for in vivomeasurement and method of data analysis, Appl. Spectrosc. 53(3): 302-311.
CHAPTER 8
Ballistic Imaging and Microscopy
8.1. INTRODUCTION
Ideally, ballistic imaging is based on unscattered or singly backscattered ballistic photons. In reality, however, more-scattered quasiballistic photons are often mea- sured as well to increase the signal strength. For brevity, subsequent use of the term ballistic photons in this chapter also refers to quasiballistic photons unless otherwise noted. Ballistic imaging provides high spatial resolution but suffers from limited imaging depth.
8.2. CHARACTERISTICS OF BALLISTIC LIGHT
The intensity of unscattered light IT attenuates according to Beer's law:
/ Γ ω = /οβχρ(-μ,ζ) . (8.1)
Here, z denotes the ballistic path length in the scattering medium, μ, denotes the extinction coefficient, and /o denotes the fluence rate of the incident light if specular reflection is negligible. The intensity of singly backscattered light is given by
IR(z) = Ioexp(-2[L,z)Rb, (8.2)
where R^ denotes the percentage of the backscattered light to be received by the detector, and the factor of 2 in the exponent is due to round-trip propagation. In both cases, the strictly ballistic signals decay exponentially with the path length.
The objective of ballistic imaging is to reject nonballistic photons and to retain ballistic photons on the basis of the following characteristic differences between them:
1. Time of Flight. Transmitted ballistic photons take shorter paths and arrive at the detector earlier than do nonballistic photons. Time-gated imaging and coherence-gated holographic imaging are based on this difference.
Biomedical Optics: Principles and imaging, by Lihong V. Wang and Hsin-i Wu Copyright © 2007 John Wiley & Sons, Inc.
153
154 BALLISTIC IMAGING AND MICROSCOPY
Collimation. Transmitted ballistic light has better collimation (smaller divergence) than does nonballistic light. Spatial-frequency filtered imaging and optical heterodyne imaging are based on this difference. Polarization. Ballistic light retains the incident polarization in a nonbire- fringent scattering medium better than does nonballistic light. Polarization- difference imaging is based on this difference. Wavefront. Ballistic light possesses a better-defined wavefront than does nonballistic light and hence can be better focused. Confocal microscopy and two-photon microscopy are based on this difference. Note that wavefront and collimation are related.
8.3. TIME-GATED IMAGING
Time-gated imaging, also referred to as early-photon imaging, takes advantage of the difference in arrival time between ballistic and nonballistic photons to select the early arriving component from the transmitted light. Figure 8.1 shows a block diagram of an experimental configuration for ultrafast time-gated imaging. A collimated ultrafast laser beam irradiates the scattering medium. The time gate is turned on for a short time to allow only the early-arriving photons to pass to the detector. The early-arriving photons carry information about optical attenuation along the optical axis. If the imaging system is raster scanned transversely, a 2D projection image (also termed a shadowgram) of the medium can be acquired.
Both the gating delay (the time lapse from the laser pulse to the rising edge of the time gate) and the open duration of the gate affect the image quality; a tradeoff exists between the spatial resolution and the signal strength. If the scattering medium is optically thin (thinner than the mean free path), gating at the arrival of the ballistic photons yields the best spatial resolution with good signal strength. If the scattering medium is optically thick (thicker than the mean free path), nonballistic light becomes significant. As the gating delay or the open duration increases, more nonballistic light contributes to the signal, yet the image becomes more blurred.
A time gate can be constructed using the Kerr effect. In the Kerr effect, birefringence is induced by an electric field applied to an isotropic transparent substance. Inducing a half-wave retardation in such a substance with the electric field of an auxiliary lightbeam can provide a high-speed shutter (Figure 8.2). The auxiliary beam is obtained by splitting the incoming laser light so that the
Scattering medium
Ultrafast las(
w fc Ψ
Time gate D(
^ Ψ
Detector
D Figure 8.1. Experimental configuration for time-gated imaging.
TIME-GATED IMAGING 1 5 5
PH λ/2 at 45° Auxiliary ι ι beam \ . I I
k I \
\
pv
^ w
Figure 8.2. Schematic of a Kerr gate {PH represents a horizontal polarizer; Pyy a vertical polarizer; λ/2, an activatable half-wave plate).
half-wave retardation can be synchronized with the signal beam. The activatable retarder is sandwiched between two cross-polarized linear polarizers; its fast axis is oriented diagonally (±45°) between the orthogonal polarization axes of the two polarizers. When the retarder is not activated, no light can pass through the two polarizers; hence, the gate is closed. When the retarder is activated to provide half-wave retardation, the polarization orientation of the light ray is rotated π/2 and becomes aligned with the polarization axis of the second polarizer; hence, the gate is open. The shutter speed of such a gate can be as short as 100 fs.
A time gate can also be a single-photon counting system or a streak camera. The former is introduced in Chapter l l . A streak camera resolves the arrival time of ultrafast light by elegantly converting time into space (Figure 8.3). A photocathode plate converts incident photons into electrons by the photoelectric effect. The photoelectrons are accelerated toward a mesh and then deflected by a pair of fast sweep electrodes, the high voltage applied to which is synchronized to the incident light. Consequently, electrons arriving at different times bombard a microchannel plate (MCP) at different vertical locations. The MCP amplifies the current by generating secondary electrons. The amplified current then strikes the phosphor screen to produce photons. The vertical dimension of the phosphor screen thus provides the temporal resolution, which can reach about 200 fs. The horizontal dimension of the phosphor screen can provide either spatial or spectral resolution. For the former, a horizontal slit is added in front of the photocathode to form a narrow lightbeam. For the latter, a spectrometer is added in front of the photocathode to disperse the incident light horizontally into spectral components.
Photocathode Early
Phosphor screen
hv
Sweep electrodes
~T Mesh L a t e MCP
Figure 8.3. Basic components of a streak camera.
156 BALLISTIC IMAGING AND MICROSCOPY
Example 8.1. Estimate the spatial resolution using some practical open durations of the time gate.
If the open duration of the gate is 100 fs, the spatial resolution is on the order of 100 fs x 3 x 108 m/s = 30 μηη. If it is 5 ps, the resolution is on the order of 5 ps x 3 x 108 m/s = 1.5 mm.
8.4. SPATIOFREQUENCY-FILTERED IMAGING
Spatiofrequency-filtered imaging, also called Fourier space-gated imaging, takes advantage of the different spatiofrequency distributions between ballistic and nonballistic light to select the ballistic component from the transmitted light. Figure 8.4 is a schematic representation of two spatiofrequency-filtered imag- ing systems: (a) a narrowbeam scanning system and (b) a widebeam full- field system. A collimated laser beam—narrow in Figure 8.4a and broad in Figure 8.4b—irradiates the scattering medium. A lens focuses the ballistic com- ponent to a diffraction-limited point while dispersing the nonballistic component around the focus. A pinhole, placed at the focal plane of the lens, blocks most of the off-focus light and passes the ballistic component to the detector. In Figure 8.4a, optical attenuation through the medium is detected along one line at a time; transverse scanning yields a shadowgram. In Figure 8.4b, a 2D shad- owgram is formed through a second lens with a single exposure as in X-ray projection imaging. In either case, the image signal carries information about the integrated attenuation along the optical path.
From the perspective of Fourier optics, the 2D spatial Fourier transform of a lightbeam over a cross section provides the spatiofrequency spectrum in the same way that the 1D temporal Fourier transform of a lightbeam at an observa- tion point provides the temporal frequency spectrum. Different spatiofrequency
Scattering Pinhole medium Lens
(a)
Detector
(b)
Laser
Beam expand
t, w
Scattering ix medium
p, Ψ
Pinf
W M K
lole
>(Γ / \ /
Lens Len
r w
s
CCD
Figure 8.4. Schematic of spatiofrequency-filtered imaging; (a) a narrowbeam scanning system; (b) a widebeam full-field system.
POLARIZATION-DIFFERENCE IMAGING 157
components represent plane waves traveling in different directions, which can be focused to different points on the focal plane of a lens. Therefore, the lens functions as a spatial Fourier transformer, and the pinhole functions as a spatial filter. The major plane-wave component in the ballistic light propagates along the optical axis, whereas most plane-wave components in the nonballistic light propagate obliquely.
8.5. POLARIZATION-DIFFERENCE IMAGING
Polarization-difference imaging (PDI) takes advantage of the different polariza- tion states between ballistic and nonballistic components to select the ballistic component from the transmitted light. A nonbirefringent scattering medium does not alter the polarization state of the ballistic light, whereas the medium random- izes the polarization state of the nonballistic light. In the PDI system shown in Figure 8.5, a polarizer linearly polarizes the source beam. The transmitted light passes through a linear polarization analyzer that is sequentially aligned in two orthogonal directions. Then, the light is detected by a photodetector.
When the polarization axis of the analyzer is parallel to the incident polar- ization, the intensity measurement is denoted by /|J(JC, y), where (JC, y) represent the transverse Cartesian coordinates. Likewise, when the polarization axis of the analyzer is perpendicular to the incident polarization, the intensity measurement is denoted by I±(x, y). We can approximately express the two intensities as
1 /||(χ, y) = Ib(x, y) + -I„b(x, y),
1 i±(x,y) = ^inbix.y)-
(8.3)
(8.4)
Here, /&(*, y) and Inb(x, y) denote the ballistic and nonballistic intensities, respectively. The nonballistic light is assumed to be completely unpolarized and
Laser Scattering medium
4KHh Detector
(a)
Laser
Polarizer Analyzer
Scattering medium Detector
(b) Polarizer Analyzer
Figure 8.5. Schematic of a polarization-difference imaging system in two states. The polarization axis of the analyzer is (a) parallel and (b) perpendicular to the polarization axis of the incident polarizer.
1 5 8 BALLISTIC IMAGING AND MICROSCOPY
hence passes through the analyzer with a 50% transmittance, regardless of the polarization orientation of the analyzer.
A PDI system generates an image by
lm(x,y) = l\\(x,y)-l±(x,y)· (8.5)
Substituting Eqs. (8.3) and (8.4) into (8.5), we obtain
hD(x,y) = h(x,y). (8.6)
Therefore, the ballistic component is recovered. PDI is simple and fast; it can also be implemented in reflection mode. How-
ever, a small number of scattering events lead to only partial randomization of polarization, which affects the efficacy of PDI. Furthermore, if birefringence is present, more complex quantities such as the Stokes vector (see Chapter 10) need to be measured.
8.6. COHERENCE-GATED HOLOGRAPHIC IMAGING
Coherence-gated holographic imaging takes advantage of the difference in arrival times between ballistic and nonballistic photons to select the early-arriving com- ponent from the transmitted light. To fully appreciate this technique, readers should review the principle of conventional holography (see Appendix 8A). Here, coherence-gated holographic imaging is based on digital holography, in which both recording and reconstruction are accomplished digitally (Figure 8.6). In the object arm, the lightbeam is filtered by a pinhole (spatial filter) and then expanded and collimated by a lens before irradiating the scattering medium. The transmit- ted object beam is first spatiofrequency-filtered and then collimated before it is recombined with the reference beam, which is oblique with a small angle Θ (not shown in the figure). The path length of the reference beam is matched to that of the first-arriving ballistic light by adjusting a multimirror delay line. The inter- ference pattern is imaged onto a CCD camera to form an image-plane hologram. Multiple CCD images are averaged to reduce speckle noise because speckles in successive holograms are assumed to be uncorrelated.
For monochromatic light, the reference and the object fields can be expressed with phasor representations as follows:
£/?(ω, x) = Εο(ω) txp(ikxx — ίω(ί — tR)), (8.7)
Es(o),x, y) = Ε0(ω)[α\(χ, ^)βχρ(~ιω(ί - t\)) + a2(x, y)exp(-/a)(i - t2))). (8.8)
Here, subscripts R and S denote the reference and the object (sample) beams, respectively; ω denotes the angular frequency; (JC, y) denote the Cartesian coor- dinates on the detector surface; t denotes time; EQ denotes the electric field
COHERENCE-GATED HOLOGRAPHIC IMAGING 1 5 9
Light source
Beamsplitter
Lens <£> Delay
^\ Scattering medium
Mirror
Spatial filter
Spatial filter
n Bandwidth-limiting spatial filter
Beamcombiner
Relay x CCD
Figure 8.6. Schematic of a coherence-gated holographic imaging system. The reference beam is incident obliquely with a small angle Θ (not shown).
amplitude; kx denotes the JC component of the wavevector, which appears because the reference beam is assumed to be tilted with respect to the x axis; tR denotes the time delay in the reference beam; a\ and t\ denote the amplitude transmit- tance and the time delay of the ballistic light, respectively; and a2 and t2 denote the amplitude transmittance and the time delay of a representative group of non- ballistic light, respectively. In addition to the t2 component, more time-delay components can be added in a similar fashion. Since frequency tuning is needed, the frequency dependence of ER and Es is explicitly expressed here. The objec- tive of this imaging technique is to retain the a\ component and eliminate the a2 component.
Note that the phasor expression, also referred to as the complex expression, is a convenient mathematical convention for representing oscillations. The actual oscillations are the real part of the phasor expression. In linear mathematical operations, real-part operators Re{} on phasor expressions are implicit since real- part operators and linear operators are permutable. In nonlinear mathematical operations, however, one needs to exercise caution.
If tR = t\, the hologram can be expressed as
Ι(ω,χ,γ) = \Ε0(ω)\ 2{\ + \a{\
2 + \a2\ 2}
+ \Εο(ώ)\2{α\ exp(—ikxx) -f a* exp(ikxx)}
+ \E0((o)\ 2{a2exp(-ikxx + ιω(ί2 - *i)) (8.9)
-f a\ exp(ikxx — /ω(*2 — ^ι))}
+ \Εο(ω)\2{α*α2£χρ(ίω(ί2 - t\)) + axa\βχρ(-/ω(ί2 -* ι ) )} .
1 6 0 BALLISTIC IMAGING AND MICROSCOPY
The holographic image is reconstructed digitally as follows:
1. We take the spatial Fourier transformation of the hologram with respect to x:
/(ω, fc, v) - |Ε0(ω)|2{1 + kn I2 + \a2\2)M)
+ |Ε0(ω)|2{α,δ(/: - kx) + α*δ(* + kx)}
(8.10) + \Eo(o>)\2{a2Hk - kx)exp(ii»(t2 - tx))
+ a$h(k + kx) βχρ(-/ω(ί2 ~ *i))}
+ |Εο(ω)|2{α*α2εχρ(/ω(ί2 - fi))
-f αι«2 exp(—ί'ω(ί2 — ii))}8(fc).
Here, the terms containing h(k) represent the zero-frequency ("DC") com- ponent; the terms containing h(k — kx) represent the virtual image that has a spatial frequency of +kx, whereas the terms containing h(k -f kx) represent the real image that has a spatial frequency of —kx.
2. We filter the signal to retain the h(k — kx) terms (first-order diffraction terms) and to reject the other components.
3. We take the inverse Fourier transformation of the filtered signal and then drop exp(—ikxx), which results in
ΐ'(ω,χ,γ) = \Ε0(ω)\ 2{αι +a 2 exp(Mf 2 - *i))}. i8 ·1 1)
To separate the a\ and a2 terms in Eq. (8.11), we tune the laser frequency within a bandwidth Δω while acquiring a hologram at each frequency. If |£Ό(ω)|2 is slowly varying and Δω >> (f2 — i i ) _ 1 , integration of Eq. (8.11) over Δω approximately averages out the a2 term, which leads to
l"(x,y)= I ί(ω,χ, y)d(u ^ a\(x, y) I \Εο(ω)\ άω. (8.12) «/Δω J Αω
Thus, I" represents a ballistic image proportional to a\. As will be seen in Chapter 9, this frequency-swept gating is equivalent to coherence gating based on a wideband light source.
8.7. OPTICAL HETERODYNE IMAGING
Optical heterodyne imaging takes advantage of the different spatial frequencies between ballistic and nonballistic light to select the collimated component from the transmitted light. Optical heterodyne detection means the superposition of two coherent optical beams that have slightly different temporal frequencies and the subsequent detection of the beat-frequency component of the superposed beam.
OPTICAL HETERODYNE IMAGING 161
<\
>-: -D/2 "I \
KR T KS
T
ER
\kxD
D/2
z
Figure 8.7. Illustration of the relationship between the reference and the sample beams with respect to the detector.
The antenna property of optical heterodyne detection is the basis for the ballis- tic imaging here. Figure 8.7 shows a schematic relationship between the reference and the sample beams with respect to the detector. The antenna theorem can be stated as follows. In the ID case, |θ| ~ λ/Ζ), where Θ is the acceptance angle between the two beams, λ is the optical wavelength, and D is the width of the aperture (limited by the size of the optical beam or the detector, whichever is smaller). In the 2D case, Ω ~ λ2/Λ, where Ω is the acceptance solid angle and A is the area of the aperture. The limitation on the acceptance angle is simply due to the cancellation of interference fringes on the detector surface when the two waves propagate in different directions.
Ballistic light, which propagates along the optical axis, has a nearly zero spatiofrequency bandwidth, while nonballistic light has a broad spectrum of spa- tial frequencies. Therefore, low spatiofrequency ballistic light can be retained and high spatiofrequency nonballistic light rejected according to the antenna theorem.
An experimental setup for heterodyne imaging is shown in Figure 8.8. The laser beam from the source is divided into two different paths by a beamsplitter. Before illuminating the sample, the sample beam is passed through an acous- tooptic modulator to add a frequency shift of 80.0 MHz. The reference beam is passed through another acoustooptic modulator that introduces a slightly different frequency shift of 80.1 MHz. The two beams are then effectively superposed by a beam combiner before reaching a photomultiplier tube (PMT), which functions as a frequency mixer. The sum-frequency signal is filtered out automatically by the PMT, which is unable to respond to oscillations of optical frequencies. A beat-frequency signal of 0.1 MHz—the heterodyne frequency between the ref- erence and the sample beams—from the PMT is amplified, digitized, and finally transferred to a computer. This signal represents the ballistic photons and can be used to form an image by transversely scanning the system across the sample.
In general, heterodyne detection refers to the use of a local oscillator (refer- ence signal here) to mix a high-frequency signal (sample signal) with a more convenient intermediate-frequency signal (interference signal). The mixer (PMT)
1 6 2 BALLISTIC IMAGING AND MICROSCOPY
Mirror Acoustooptic modulator
x 40 lens
Ml· Mirror Aperture x 10 lens Scattering
medium
Beamcombiner τ Mirror /- ► ^L
Beamsplitter
Acoustooptic modulator
x 40 lens
Ml· Aperture x lOlens
Mirror PMT
Amplifier
He-Ne laser (single frequency) Computer Digitizer
Figure 8.8. Schematic of an optical heterodyne imaging system.
generates both upper and lower sidebands, either of which may be filtered out if desired. In this case, the upper sideband is filtered out because the photodetector cannot respond to signals of optical frequency, and the lower sideband provides the interference signal.
Example 8.2. Show that the acceptance angle in the antenna theorem for a ID case is approximately equal to λ/D.
In Figure 8.7, the electric fields of the sample and the reference waves can be expressed as
Es - Eso exp(ikxx - iMSt + ι'φίο).
ER = EROexp(-i(uRt + ίφκο),
(8.13)
(8.14)
respectively, where kx is the projection of the wavevector ks onto the x axis:
(8.15) L = — sin t λ
If we consider only the ID case, the light intensity on the detector surface (z = 0 plane) is
I(kx,x) - \ES + ER\2 = Ejo + E2m + 2ES0ER0cos(kxx - Δωί + Δφ0), (8.16)
RADON TRANSFORMATION AND COMPUTED TOMOGRAPHY 163
where Δω = ω$ — ω^ and Δφο = Φ50 — Φ/?ο· The first two terms represent the DC component while the last term represents the AC (interference) component. The AC photocurrent from the detector is proportional to the interference com- ponent integrated across the detector surface:
pD/2
IAC(kx) oc 2ESOERO / cos(kxx - Δωί + Δφ0)Λχ JrD/2
IESOERQ
kx
4ESQERQ
kx
sin I kx Δωί + Δφο I — sin f —kx Δωί + Δφο I
in('"f) cos i -Δωί + Δφ0)8ΐη \kx—) (8.17) = IDESOEROS'IUC I —— 1 cos(Aoo/ — Δφο),
V 2π /
where sinc(jc) = ύη(πχ)/(πχ), although sometimes the definition of sinc(jf) = sin(x)/x is used elsewhere. For effective detection, the argument of the sine function must be less than unity, specifically
\kxD\ < 2 π , (8.18)
which leads to
| s i n 0 | < ^ . (8.19)
If D » λ, then sin θ % Θ. Thus
|θ| < ~ . (8.20)
8.8. RADON TRANSFORMATION AND COMPUTED TOMOGRAPHY
In the aforementioned transmission-mode ballistic imaging, spatial resolution along the optical path can be achieved using the inverse Radon transformation, which is commonly used in X-ray CT for image reconstruction. Projections from multiple view angles are needed, however.
The Radon transform ρψ(χ') of a function /(JC, y) is defined as the integral of the function along a line that is parallel to the / axis at xr (Figure 8.9)
/
+00
/ ( χ ' α ^ φ — y βΐηφ, χ ^ ί η φ 4- y'costy)dy\ (8.21) -00
where φ denotes the view angle—the angle between the x and x' axes. The Radon transform, also referred to as the projection data or the sinogram, is the input for image reconstruction.
1 6 4 BALLISTIC IMAGING AND MICROSCOPY
Figure 8.9. Radon transformation.
In ballistic imaging, the Radon transformation is related to the generalized Beer law:
/
+00 \L,(z)dz), (8-22)
-00
where I(z) denotes the light intensity, and z denotes the optical (ballistic path) axis. The following reformulation of Eq. (8.22) shows that the absorbance along the optical axis equals the Radon transform:
Kz) Γ+σο -\n-^- = [it(z)dz. (8.23)
Various inverse algorithms can invert the Radon transformation for an image.
8.9. CONFOCAL MICROSCOPY
Confocal microscopy was invented in the 1950s but was not actively developed until the 1970s. A confocal microscope—in which both illumination and detec- tion are focused on the same point in the object—provides optical sectioning for high-resolution 3D imaging of scattering samples. By contrast, a conven- tional microscope (Figure 8.10a)—in which the illumination is broadened by a condenser lens and the illuminated area of the object is mapped onto the image plane by an objective lens—provides no optical sectioning of planar features; it does, however, form a full-field image at once.
CONFOCAL MICROSCOPY 165
Object
Source | Image
(a) Condenser Objective
(b) Objective
Figure 8.10. Schematic diagrams of (a) a conventional microscope and (b) a transmission confocal microscope.
In a transmission-mode confocal microscope (Figure 8.10b), a point source is imaged by a lens to a diffraction-limited spot to illuminate the 3D specimen. The illuminated spot is mapped by another lens to a pinhole to reject off-focus light. The filtered light is then detected by a photodetector. The detected signal is sensitive to the property of the sample at the illuminated point. Scanning point- by-point across the sample then forms an image. In other words, the focused light irradiates one tiny volume of the object at a time; the detector along with the pinhole collects light from the same region. The confocal illumination and detection effectively reject light from elsewhere. Consequently, both lenses play equally important roles in defining the spatial resolution.
A confocal microscope (Figure 8.11) can be implemented in reflection mode as well. As in the transmission mode, the illumination is focused by an objective lens to a spot. If elastically backscattered light is to be imaged, a beamsplitter is used to both partially reflect the source beam and partially transmit the reflected beam. If fluorescent light is to be imaged, a dichroic mirror can be used instead to both efficiently reflect the excitation light and efficiently transmit the fluorescent light. A reflection-mode confocal microscope can image hundreds of micrometers into scattering biological tissues.
The spatial resolution of a confocal microscope can be quantified by the PSF (the image of a point object). According to diffraction theory, the normalized
Pinhole
ollector
166 BALLISTIC IMAGING AND MICROSCOPY
Detector
Confocal aperture
Point source
Illuminating aperture
In-focus rays
Out-of-focus rays
Dichroic mirror
Objective lens
In-focus plane Sample
Figure 8.11. Schematic of a reflection-mode confocal microscope for fluorescence imaging.
optical coordinates u and v are defined by
8π8ΐη 2 (γ /2) u = z,
2Ksiny v = r.
λ
(8.24)
(8.25)
Here, λ denotes the optical wavelength in the object, z denotes the distance from the focal point along the optical axis (defocus distance), γ denotes the angle that defines the numerical aperture NA along with the refractive index n (NA = n sin γ), and r denotes the radial coordinate on the xy plane. If the system has circular symmetry, the field (or complex-amplitude) PSF of a lens in paraxial approximation is given by the following Hankel transform:
h(u, υ) = 2 /o ' e x p G M p 2 ) J0(pv)p dp. (8.26) Here, p denotes the radial coordinate at the pupil normalized by the pupil radius and Jo denotes the zeroth-order Bessel function of the first kind.
The PSF of a conventional microscope is
PSF(w,u) = |/i(w,u)|2, (8.27)
CONFOCAL MICROSCOPY 167
where the absolute squared operation is a consequence of conversion from com- plex amplitude to intensity. By contrast, the PSF of a confocal microscope is
PSF(M, v) = |A(w, v ) | \ (8.28)
where the absolute-to-the-fourth-power operation is a result of double conversions from complex amplitude to intensity for both illumination and detection. Here, we have implicitly invoked the principle of reciprocity, which means that when the source and the observation points are exchanged, the observed field remains the same.
The field PSF on the optical axis can be calculated as follows:
ft(ii,0) = 2 f expi^wp 2 ] p dp = exp( -u ] sine ^ V (8.29)
Therefore, the axial PSF of a conventional microscope is
PSFz(u) = |A(n, 0)|2 = sine2 (—) , (8.30) \ 4 π /
and the axial PSF of a confocal microscope is
PSFz(u) = |A(II, 0)|4 = sine4 (—) . (8.31) \ 4 π /
Likewise, the lateral field PSF on the in-focus plane can be calculated as follows:
,,„) = 2 / ' . - - 2 - 2 Jo
ft(0, v) = 2 / J0(pv)p dp = -[pJi(pv)]i = -Mv). (8.32) ' V V
Therefore, the lateral PSF of a conventional microscope is
PSFr(v) = |A(0, v)\2 = 2Ji(v)\2 (8.33)
and the lateral PSF of a confocal microscope is
PSFr(v) = |A(0, υ)|4 = 2 / ι (υ) | 4
(8.34)
If the sample has an arbitrary complex-amplitude reflectivity distribution o(u, v, Θ) (where Θ is the polar angle), the image intensity distribution from a reflection-mode conventional microscope with incoherent illumination can be modeled by inco- herent convolution |A|2 * |o|2. By contrast, the image intensity distribution from
168 BALLISTIC IMAGING AND MICROSCOPY
a reflection-mode confocal microscope can be modeled by coherent convolution \h2 * o\2. As expected, if the object reduces to a point, the convolution recovers the PSF.
If a fluorescent point object is imaged, the image intensity distribution from a conventional microscope remains \hm(um, vm)\2, where subscript m indicates that the optical coordinates are defined at the fluorescence emission wavelength. How- ever, the image intensity from a confocal microscope is modified to \hx(ux, vx) hm{um, vm)\
2, where subscript x indicates that the optical coordinates are defined at the fluorescence excitation wavelength.
If the sample contains an arbitrary fluorophore density distribution f(u, v, Θ), the image intensity distribution from a conventional microscope can still be modeled by incoherent convolution \hm\2 * / . The image intensity distribution
l
0.9
0.8
0.7
85 ° · 6 QU
3 0.4
0.3
0.2
O.l
0
— — - Conventional Confocal
10 (b)
Figure 8.12. (a) Axial and (b) lateral PSFs in confocal and conventional microscopes.
TWO-PHOTON MICROSCOPY 1 6 9
from a confocal microscope can be modeled by incoherent convolution \hxhm\2 * / instead.
Example 8.3. Plot the axial and lateral PSFs for both conventional and confocal microscopes.
The following MATLAB code produces Figure 8.12:
u = linspace(-1,1)*3*4*pi; subplot(2, 1, 1) plot(u, (sinc(u/4/pi)).Λ2, 'k--', u, (sinc(u/4/pi)).Λ4, 'k-') grid xlabel('u') ylabelCAxial PSF') legend('Conventional', 'Confocal')
v = linspace(0,10); subplot(2, 1, 2) plot(v, (2*besselj(1,v)./v).A2, 'k--', v, (2*besselj(1,v)./v).Λ4, 'k-') grid xlabel('v') ylabelC Lateral PSF') legend('Conventional', 'Confocal')
8.10. TWO-PHOTON MICROSCOPY
Two-photon microscopy was initially developed in the early 1990s. A two-photon microscope (TPM) (Figure 8.13) achieves sectioning by nonlinear optical exci- tation. Unlike a confocal microscope, a TPM does not use a pinhole, although a pinhole can enhance the spatial resolution at the expense of signal strength.
To understand why a pinhole is unnecessary in a TPM, we first examine the difference between one-photon and two-photon excitations of fluorescence
Dichroic mirror
Figure 8.13. Schematic of a reflection-mode two-photon microscope.
1 7 0 BALLISTIC IMAGING AND MICROSCOPY
Vibrational relaxation / A \'
T \
Excitation w ^
Fluorescence w \ *
Excitation V\A>
\
Fluorescence
One-photon excitation Two-photon excitation
Figure 8.14. Jablonski energy diagrams for a one-photon excitation and two-photon exci- tation of fluorescence.
(Figure 8.14). In one-photon excitation, an electron is boosted to an excited state by absorbing a single photon. After a brief vibrational relaxation, it returns to a ground state while emitting a fluorescence photon. The probability of one-photon absorption is proportional to the light intensity. In two-photon excitation, an elec- tron is pumped to an excited state by simultaneously absorbing two low-energy photons, followed by a similar process of fluorescence emission. Therefore, the probability of two-photon absorption is proportional to the square of the light intensity.
Compared with a confocal microscope, a TPM has the following characteristics in addition to the lack of a pinhole:
1. A more localized excitation volume defined by \hx(ux, vx)\4, rather than \hx(ux, vx)\
2 as in the case of a fluorescence confocal microscope, leads to reduced photo-bleaching.
2. A longer excitation wavelength leads to increased penetration because both the absorption and the reduced scattering coefficients are decreased in the typical spectral region.
3. An ultra-short-pulsed laser is used. 4. Scattering contrast is not directly measured.
The diffraction-limited PSF is \hx(ux, vx)hm(umj vm)\2 for a fluorescence con- focal microscope but becomes \hx{ux, vx)\4 for a TPM. In theory, when the two excitation wavelengths are the same and the single-photon emission wavelength is close to the single-photon excitation wavelength, the two PSFs are similar. However, when the two-photon excitation wavelength is twice as long as the single-photon excitation wavelength and the two-photon emission wavelength is the same as the single-photon emission wavelength, the PSF for a TPM is wider. If the sample contains an arbitrary fluorophore density distribution f(u, v, Θ), the image intensity distribution from a two-photon microscope can be modeled by incoherent convolution \hx\4 * / .
APPENDIX 8A. HOLOGRAPHY 171
Example 8.4. Estimate the number of excitation photons absorbed per fluo- rophore for a TPM.
Typically, the laser in a TPM has pulse duration τρ ~ 100 fs, pulse repetition rate fp ~ 80 MHz, and average excitation power Po ^ 50 mW. Thus, the pulse energy is
Po Ep = -j- * 0.6 nJ. (8.35) fp
The number of excitation photons absorbed per fluorophore Na can be estimated by
Na = j l · ^ . (8.36) p τη VP
Here, θ2Ρ denotes the two-photon absorption cross section (~10- 5 8 m4-s) and Jp denotes the photon flux per pulse (m~~2). We estimate Jp by
En JP = T - ^ - , (8.37) hvAf
where hv is the photon energy and Af is the area of the focused beam. We estimate Af by
2
^="(°"6^)=l44(^)· <OS) where λ is the excitation wavelength (~800 nm) and NA is the numerical aperture of the objective lens (M).9). According to these parameters, Na ^ 0.005 <$C 1. Only some of the absorbed photons are converted into fluorescence photons since the quantum yield ranges from —5% to ^90%. Nevertheless, fluorescence signals are detectable when the excitation volume contains enough fluorophores.
APPENDIX 8A. HOLOGRAPHY
The principle of conventional holography is described here. Holography records both the field amplitude EQ and the phase φ of a lightbeam, whereas conven- tional photography records only the intensity. A hologram presents the effect of stereovision. When a hologram is recorded (Figure 8.15), a source beam is split into two parts: one illuminates the object, and the other serves as a coherent ref- erence wave. When the object wave reaches the recording film, it interferes with the coherent reference wave. The intensity distribution of the combined beam forms an interferogram and is recorded on the film.
172 BALLISTIC IMAGING AND MICROSCOPY
Reference / wave χ y/
yo / \ x y
Object
Figure 8.15. Recording of a hologram. The spherical wave from a point on the (jto, yo) plane is illustrated as the object wave.
To simplify our discussion, we assume a monochromatic light source so that the phase difference between the object and the reference waves is time-invariant. The object and the reference waves are denoted in phasor expressions by Es(x, v) and ER(x,y), respectively, where (x, y) is the Cartesian coordinates on the recording plane. The recorded intensity I(x,y), referred to as a hologram, is then given by
/ (* , y) = \ES + ER\2 = (Es + ERKEs + ER)*
= \ES\ 2 + \ER\2 + ESER + E*SER. (8.39)
If ER is zero, a hologram reduces to a conventional photograph. Once the film is developed, the recorded hologram can be represented by the
complex-amplitude transmittance of the film as follows:
' / ( * , y) = tb + ß/(*, y) = tb + ß( |£5 | 2 + ESE*R + E*SER), (8.40)
where tb denotes the film-dependent baseline (background transmittance with zero exposure) and ß denotes the sensitivity that relates the transmittance to the recorded intensity. Since the reference beam contains no imaging information, the \ER\2 term is lumped with th\tb — tb-\- $\ER\2.
Reconstruction of a hologram recovers the object wave—in either the original or the conjugated form—by illuminating the hologram. If the reconstruction beam is the same as the reference beam, we multiply ER on both sides of Eq. (8.40) to obtain the field of the transmitted beam:
ERtf - ERtb + VER\ES\2 + tEs\ER\2 + ?>E*ERER. (8.41)
The last two terms on the right-hand side are important because they contain information about both the amplitude and phase of either the object wave Es or the complex-conjugated object wave E$.
Obiect wave
APPENDIX 8A. HOLOGRAPHY 173
Object Film
Source
(a)
Lens
Virtual image
Scattered Directly wave transmitted
wave
Hologram Real image
(b) Lens
Figure 8.16. (a) Recording and (b) reconstruction in Gabor holography.
To illustrate the recording and reconstruction, we first describe the original Gabor holography (Figure 8.16a). For recording, a plane wave—converted from a spherical wave by a lens—is normally incident on an object. The transmitted light consists of two emerging waves—one is a directly transmitted plane wave that serves as the reference, and the other is a scattered object wave that carries imaging information about the object. The interference between these two waves results in a hologram on the film.
For reconstruction, a plane wave is incident on the hologram (Figure 8.16b). The field of the transmitted beam is given by Eq. (8.41). The first term on the right-hand side represents a homogeneous background since ER for a plane wave is independent of x and y. The second term represents an intensity image of the object (a conventional photograph). The third term forms a virtual image because it replicates the original object wavefront and represents a divergent wave propagating from the hologram. Conversely, the fourth term forms a real image because it produces the complex conjugate of the original object wavefront and represents a converging wave propagating from the hologram. For a point object, the divergent and convergent waves for the virtual and real images are spherical as illustrated. The virtual image is so named because no photons actually reach the image location; by contrast, photons do reach the real image. Here, the real and virtual images appear together because their corresponding waves propagate in the same direction.
1 7 4 BALLISTIC IMAGING AND MICROSCOPY
Source
Film
Reference
Lens Figure 8.17. Schematic illustrating Leith-Upatnieks (offset-reference) holography.
To separate the real and the virtual images, Leith and Upatnieks used an obliquely incident reference wave for recording but a normally incident reference wave for reconstruction (Figure 8.17). For recording, a prism abutting the sample refracts the incident beam by an angle Θ. We choose the vertical direction as the y axis and express the reference and the object fields as
ER(y) = E0((u)cxp(ikyy - ίωί),
Es(x, y) = a(x, y)E0(u>) exp(-/a>0.
(8.42)
(8.43)
Here, EQ denotes the amplitude of the source beam, ω denotes the angular fre- quency, t denotes time, a(x, y) denotes the transverse distribution of the object wave, and ky denotes the y component of the wavevector
Ky 2π sin Θ
λ (8.44)
where λ is the optical wavelength. Because the reference wavefront is parallel to the x direction, ER is independent of x. Of course, Gabor holography is recovered if Θ = 0.
The recorded intensity distribution on the film is
I(x,y) = \Es + ER\ 2
(8.45) = £Q[1 + \a(x,y)\* + a(x,y)exp(-ikyy)+ a*(x,y)exp(ikyy)].
The reconstruction with a plane wave of amplitude E\ normally incident on the hologram is illustrated in Figure 8.18. The complex amplitude of the transmitted field is given by
tfEi = t'hEi + ßEg£/ki(jt, y)\2 + ^E^EMx, y)exp(-ikyy)
+ $Ε$Εια*(χ, y)txp(ikyy). (8.46)
As in Gabor holography, the last two terms on the right-hand side represent the virtual and the real images, respectively. The two complex-amplitude images,
PROBLEMS 175
Source
Figure 8.18. Reconstruction in Leith-Upatnieks holography.
however, have ^ikyy in the exponents, indicating different spatial frequencies. As a result, the virtual and the real images can be viewed in the =ρθ directions, respectively; thus, one image may be viewed at a time, which is essentially a filtering process. In digital holography, this filtering is implemented computa- tionally.
PROBLEMS
8.1 Use the Henyey-Greenstein phase function to compute the percentage of backscattered light R^ that can be received by the detector in a reflection- mode microscope. Assume that the diameter of the detector is 10 μπι and the distance between the scatterer and the detector is 2/,, where μ, = 100 cm- 1 . Set g to 0, 0.9, and 0.95 sequentially.
8.2 In transmission-mode ballistic imaging, assuming that the number of unscattered transmitted photons limits the maximum thickness of the bio- logical tissue that can be imaged, derive the increase in this maximum thickness when the power of the incident source beam is doubled. Given an original maximum thickness of 30//, compute the fractional improvement.
8.3 In time-gated transmission imaging, estimate the required temporal res- olution of the time gate if a resolution better than 0.3 mm through a 3-mm-thick tissue sample is desired.
8.4 In spatiofrequency-filtered imaging, the smaller the pinhole, the better the rejection of scattered light, but the worse the transmission of ballis- tic light owing to light diffraction. Derive the transmittance of ballistic light through a pinhole of radius rp. [Hints: (1) use [2J\(v)/v]2 for the
176 BALLISTIC IMAGING AND MICROSCOPY
diffracted intensity distribution and (2) use MATLAB to complete the Bessel function integration, e.g., syms v; i n t ( b e s s e l j ( 1 , ν) Λ 2/ν) .}
8.5 In polarization-difference imaging, assuming that the sample is equivalent to a half-wave retarder for the ballistic light due to tissue birefringence, modify the configuration for effective ballistic imaging.
8.6 If polarization-difference imaging is implemented in reflection mode, explain how to reject light scattered from deeper regions of the medium.
8.7 In Leith-Upatnieks holography, given a hologram recorded on a CCD camera with a pixel size of 5 μηι, compute the maximum offset angle below which the Nyquist criterion (>2 pixels per cycle) is satisfied. Assume an optical wavelength of 0.5 μπι.
8.8 In Leith-Upatnieks holography, assuming the reference beam to be tilted out of the xz plane, generalize the theory.
8.9 In coherence-gated holographic imaging, the a^ term contributes much less to the signal than does the a \ term when the tuned bandwidth of the laser is sufficiently wide. Explain how wide is considered sufficient.
8.10 Derive the ID antenna theorem in heterodyne detection using real values of the electric fields rather than phasor expressions. (Hint: In this case, time averaging over the response time of the detector is explicit.)
8.11 In heterodyne imaging, assuming that the source wavelength is tuned over a range, show that even late-arriving light normally incident on the detector can be rejected as in the coherence-gated holographic imaging.
8.12 For both confocal and conventional microscopes, find the radius r at which the lateral PSF is zero. The center area within this radius is referred to as the Airy disk. [Hint: The first zero of J\(v) is at ι; = 1.22π.]
8.13 Define spatial resolution as the FWHM of the PSF. Set refractive index n to 1.0 and 1.5 sequentially. For both confocal and conventional microscopes, plot the axial resolutions versus NA in the range of 0.40-0.99. Repeat for the lateral resolutions versus NA in a separate figure. Plot the ratio of the conventional microscopic resolution to the confocal microscopic resolution versus NA in a third figure. (Hint: In MATLAB, type help fzero.)
8.14 Use the Monte Carlo method to simulate time-resolved transmitted light through a scattering slab. An infinitely short-pulsed pencil beam is nor- mally incident on the slab from one side. Add a small pinhole on the other side in front of the detector. Set the thickness of the slab to 0.5, 1, 2, 4, . . . times the mean free path sequentially.
8.15 In confocal and conventional microscopes, if a planar target perpendicular to the optical axis is imaged, what are the axial PSFs?
FURTHER READING 177
READING
Cho ZH, Jones JP, and Singh M (1993): Foundations of Medical Imaging, Wiley, New York. (See Section 8.8, above.)
Denk W, Strickler JH, and Webb WW (1990): 2-photon laser scanning fluorescence microscopy, Science 248(4951): 73-76. (See Section 8.10, above.)
Dolne JJ, Yoo KM, Liu F, and Alfano RR (1994): IR Fourier space gate and absorption imaging through random media, Lasers Life Sei. 6: 131-141. (See Section 8.4, above.)
Goodman JW (2004): Introduction to Fourier Optics, Roberts & Co., publishers, Engle- wood, CO., (See Appendix 8A, above.)
Gu M (1996): Principles of Three Dimensional Imaging in Confocal Microscopes, World Scientific, Singapore/River Edge, NJ. (See Section 8.9, above.)
Leith E, Chen C, Chen H, Chen Y, Dilworth D, Lopez J, Rudd J, Sun PC, Valdmanis J, and Vossler G (1992): Imaging through scattering media with holography, J. Opt. Soc. Am. A 9(7): 1148-1153. (See Section 8.6, above.)
Rowe MP, Pugh EN, Tyo JS, and Engheta N (1995): Polarization-difference imaging—a biologically inspired technique for observation through scattering media, Opt. Lett. 20(6): 608-610. (See Section 8.5, above.)
Toida M, Kondo M, Ichimura T, and Inaba H (1991): 2-dimensional coherent detection imaging in multiple-scattering media based on the directional resolution capability of the optical heterodyne method, Appl. Phys. B—Photophys. Laser Chem. 52(6): 391-394. (See Section 8.7, above.)
Wang L, Ho PP, Liu C, Zhang G, and Alfano RR (1991): Ballistic 2-d imaging through scattering walls using an ultrafast optical kerr gate, Science 253(5021): 169-111. (See Section 8.3, above.)
Wilson T, ed. (1990): Confocal Microscopy, Academic Press, New York. (See Section 8.9, above.)
FURTHER READING
Alfano RR, Liang X, Wang L, and Ho PP (1994): Time-resolved imaging of translucent droplets in highly scattering turbid media, Science 264(5167): 1913-1915.
Bashkansky M and Reintjes J (1993): Imaging through a strong scattering medium with nonlinear-optical field cross-correlation techniques, Opt. Lett. 18(24): 2132-2134.
Bohnke M and Masters BR (1999): Confocal microscopy of the cornea, Progress Retinal Eye Res. 18(5): 553-628.
Cahalan MD, Parker I, Wei SH, and Miller MJ (2002): Two-photon tissue imaging: Seeing the immune system in a fresh light, Nature Rev. Immunol. 2(11): 872-880.
Chen H, Shih M, Arons E, Leith E, Lopez J, Dilworth D, and Sun PC (1994): Electronic holographic imaging through living human tissue, Appl. Opt. 33(17): 3630-3632.
Chen Y, Chen H, Dilworth D, Leith E, Lopez J, Shih M, Sun PC, and Vossler G (1993): Evaluation of holographic methods for imaging through biological tissue, Appl. Opt. 32(23): 4330-4336.
Das BB, Yoo KM, and Alfano RR (1993): Ultrafast time-gated imaging in thick tissues—a step toward optical mammography, Opt. Lett. 18(13): 1092-1094.
Demos SG and Alfano RR (1996): Temporal gating in highly scattering media by the degree of optical polarization, Opt. Lett. 21(2): 161-163.
178 BALLISTIC IMAGING AND MICROSCOPY
Demos SG and Alfano RR (1997): Optical polarization imaging, Appl. Opt. 36(1): 150-155.
Demos SG, Savage H, Heerdt AS, Schantz S, and Alfano RR (1996): Time resolved degree of polarization for human breast tissue, Opt. Commun. 124(5-6): 439-442.
Diaspro A (1999): Introduction to two-photon microscopy, Microsc. Res. Tech. 47(3): 163-164.
Dunn AK, Wallace VP, Coleno M, Berns MW, and Tromberg BJ (2000): Influence of optical properties on two-photon fluorescence imaging in turbid samples, Appl. Opt. 39(7): 1194-1201.
Emile O, Bretenaker F, and LeFloch A (1996): Rotating polarization imaging in turbid media, Opt Lett. 21(20): 1706-1708.
Gard DL (1999), Confocal microscopy and 3-D reconstruction of the cytoskeleton of Xenopus oocytes, Microsc. Res. Tech. 44(6): 388-414.
Gauderon R, Lukins PB, and Sheppard CJR (1999): Effect of a confocal pinhole in two-photon microscopy, Microsc. Res. Tech. 47(3): 210-214.
Guo YC, Ho PP, Savage H, Harris D, Sacks P, Schantz S, Liu F, Zhadin N, and Alfano RR (1997): Second-harmonic tomography of tissues, Opt. Lett. 22(17): 1323-1325.
Hebden JC and Delpy DT (1994): Enhanced time-resolved imaging with a diffusion-model of photon transport, Opt. Lett. 19(5): 311-313.
Hebden JC, Hall DJ, and Delpy DT (1995): The spatial-resolution performance of a time-resolved optical imaging-system using temporal extrapolation, Med. Phys. 22(2): 201-208.
Hee MR, Izatt JA, Jacobson JM, Fujimoto JG, and Swanson EA (1993): Femtosecond transillumination optical coherence tomography, Opt. Lett. 18(12): 950-952.
Hee MR, Izatt JA, Swanson EA, and Fujimoto JG (1993): Femtosecond transillumination tomography in thick tissues, Opt. Lett. 18(13): 1107-1109.
Horinaka H, Hashimoto K, Wada K, Cho Y, and Osawa M (1995): Extraction of quasi- straightforward-propagating photons from diffused light transmitting through a scat- tering medium by polarization modulation, Opt. Lett. 20(13): 1501-1503.
Kempe M, Genack AZ, Rudolph W, and Dorn P (1997): Ballistic and diffuse light detec- tion in confocal and heterodyne imaging systems, J. Opt. Soc. Am. A 14(1): 216-223.
Konig K (2000): Multiphoton microscopy in life sciences, 7. Microsc.—Oxford 200: 83-104.
Mahon R, Duncan MD, Tankersley LL, and Reintjes J (1993): Time-gated imaging through dense scatterers with a raman amplifier, Appl. Opt. 32(36): 7425-7433.
Minsky M (1988): Memoir on inventing the confocal scanning microscope, Scanning 10(4): 128-138.
Moon JA and Reintjes J (1994): Image-resolution by use of multiply scattered-light, Opt. Lett. 19(8): 521-523.
Moon JA, Battle PR, Bashkansky M, Mahon R, Duncan MD, and Reintjes J (1996): Achievable spatial resolution of time-resolved transillumination imaging systems which utilize multiply scattered light, Phys. Rev. E 53(1): 1142-1155.
Nakamura O (1999): Fundamental of two-photon microscopy, Microsc. Res. Tech. 47(3): 165-171.
Nie SM and Zare RN (1997): Optical detection of single molecules, Annu. Rev. Biophys. Biomolec. Struct. 26: 567-596.
Sappey AD (1994): Optical imaging through turbid media with a degenerate 4-wave- mixing correlation time gate, Appl Opt. 33(36): 8346-8354.
FURTHER READING 1 7 9
Schmidt A, Corey R, and Saulnier P (1995): Imaging through random-media by use of low-coherence optical heterodyning, Opt. Lett. 20(4): 404-406.
Sheppard CJR and Shotton DM (1997): Confocal Laser Scanning Microscopy, Springer- Verlag, New York.
Shuman H, Murray JM, and Dilullo C (1989): Confocal microscopy—an overview, Biotechniques 7(2): 154 FF.
So PTC, Dong CY, Masters BR, and Berland KM (2000): Two-photon excitation fluores- cence microscopy, Annw Rev. Biomed. Eng. 2: 399-429.
Tyo JS, Rowe MP, Pugh EN, and Engheta N (1996): Target detection in optically scattering media by polarization-difference imaging, Appl. Opt. 35(11): 1855-1870.
Wang L, Ho PP, and Alfano RR (1993): Time-resolved Fourier spectrum and imaging in highly scattering media, Appl. Opt. 32(26): 5043-5048.
Wang LM, Ho PP, and Alfano RR (1993): Double-stage picosecond Kerr gate for ballistic time-gated optical imaging in turbid media, Appl. Opt. 32(4): 535-540.
Wang QZ, Liang X, Wang L, Ho PP, and Alfano RR (1995): Fourier spatial filter acts as a temporal gate for light propagating through a turbid medium, Opt. Lett. 20(13): 1498-1500.
Watson J, Georges P, Lepine T, Alonzi B, and Brun A (1995): Imaging in diffuse media with ultrafast degenerate optical parametric amplification, Opt. Lett. 20(3): 231-233.
Wilson T and Sheppard C (1984): Theory and Practice of Scanning Optical Microscopy, Academic Press, London.
Yoo KM, Liu F, and Alfano RR (1991): Imaging through a scattering wall using absorp- tion, Opt. Lett. 16(14): 1068-1070.
Yoo KM, Xing QR, and Alfano RR (1991): Imaging objects hidden in highly scattering media using femtosecond 2nd-harmonic-generation cross-correlation time gating, Opt. Lett. 16(13): 1019-1021.
Zaccanti G and Donelli P (1994): Attenuation of energy in time-gated transillumination imaging—numerical results, Appl. Opt. 33(30): 7023-7030.
CHAPTER 9
Optical Coherence Tomography
9.1- INTRODUCTION
Optical coherence tomography (OCT), which was invented in the early 1990s, falls into the category of ballistic optical imaging. OCT is analogous to ultra- sonography. The transverse resolution results from the confocal mechanism, albeit with a small numerical aperture. The axial resolution results from the arrival times of echoes. The detection in OCT, however, is based on interferometry since light speed is five orders of magnitude greater than sound speed. The max- imum imaging depth in scattering biological tissue is 1-2 mm, and the spatial resolution ranges from 1 to 10 μπι. As a result, the depth-to-resolution ratio is greater than 100, which qualifies OCT as a high-resolution imaging modal- ity. The contrast originates primarily from backscattering (or backreflection) and polarization. Since the human eye provides an optically transparent window, noninvasive imaging of the retina is thus far the most competitive application of OCT.
9.2. MICHELSON INTERFEROMETRY
Michelson interferometry (Figure 9.1), the basis of OCT, is briefly introduced in this section. A monochromatic light source emits a source beam horizontally toward a beamsplitter that is inclined diagonally. The source beam is split into two halves. One half is reflected off the beamsplitter and then backreflected by a reference mirror. The other half is transmitted through the beamsplitter and then backreflected by an object surface (no internal backscattering exists here). These two backreflected beams are recombined by the beamsplitter and then received by a detector.
Biomedical Optics: Principles and Imaging, by Lihong V. Wang and Hsin-i Wu Copyright © 2007 John Wiley & Sons, Inc.
181
182 OPTICAL COHERENCE TOMOGRAPHY
Reference mirror
Light source
Beamsplitter
Sample
O Detector
Figure 9.1. Schematic of a Michelson interferometer.
If polarization is neglected, the two backreflected electric fields can be repre- sented by phasor expressions as follows:
ER = EROcxp(i(2kRlR - ωί)),
Es = Esoexp(i(2ksh - ωί)).
(9.1)
(9.2)
Here, subscripts R and S denote the reference and the sample arms, respectively; ERO and Eso denote the electric field amplitudes of the two beams; kR and ks denote the propagation constants in the two beams; IR and Is denote the two arm lengths measured from the splitting point at the beamsplitter to the backreflection surfaces; ω denotes the optical angular frequency; and t denotes time. The factor 2 in front of k arises from the round-trip light propagation in each arm. The electric field E of the recombined beam is a superposition of the two monochromatic electric fields:
E = ER + Es.
The photocurrent i(t) at the photodetector (a square-law detector) is given by
T I * ( | £ * + ESI2> i(t) = hv 2Z0
(9.3)
Here, η denotes the quantum efficiency of the detector (the ratio of the output number of electrons to the input number of photons), e denotes the electron
MICHELSONINTERFEROMETRY 183
charge, hv denotes the photon energy, ZQ denotes the intrinsic impedance of free space, and ( > denotes averaging over the response time of the detection system (e.g., 10"12 — 10~9 s, or ps to ns). The response-time averaging is equivalent to lowpass filtering; thus, the photocurrent can still be a function of time t. For brevity, we neglect the constant factors and simply write
I{t) = (\ER + Es\ 2). (9.4)
Here, /( /) denotes the short-time-averaged light intensity; it is used instead of i(t) from here on. For monochromatic light, we write
I(t) = \ER + Es\ 2, (9.5)
where another factor of | is neglected. Substituting Eqs. (9.1) and (9.2) into Eq. (9.5) yields
/(f) = E2R0 + E2S0 + 2ER0ES0 cos(2ksls ~ 2kRlR). (9.6)
The cosine term on the right-hand side results from the interference between the two lightbeams. We denote the phase difference between the two beams as Δφ:
A<\> = 2ksls-2kRlR. (9.7)
With a varying Δφ, this interference term becomes an alternating current (AC) that produces interference fringes; hence, the recorded / is also referred to as an interferogram.
If kR = k$ — k = 2πη/\ο, where n denotes the refractive index and λο denotes the optical wavelength in vacuum, we have
2nAl Δφ = 2k(ls - h) = 2π ——, (9.8) λο
where Al = ls-lR. (9.9)
From here on, Δ/ is termed the arm-length difference (or mismatch) between the sample and the reference arms; 2Δ/ is termed the (round-trip) path-length difference (or mismatch) between the two beams; 2nAl is termed the (round-trip) optical path-length difference (or mismatch) between the two beams. Therefore, the interference signal varies with Δ/ periodically. For monochromatic light, the fringes exhibit a sustained oscillation of constant amplitude.
184 OPTICAL COHERENCE TOMOGRAPHY
9.3. COHERENCE LENGTH AND COHERENCE TIME
The coherence length lc of light is defined as the spatial extent along the prop- agation direction over which the electric field is substantially correlated; it is related to the coherence time xc by lc — cxc, with c denoting the speed of light. In stationary states, where the statistical properties do not change with time, xc is defined as the FWHM of the autocorrelation function G\(x) of the electric field E(t):
/
+oo E(t)E(t + x)dt. (9.10)
-DO
Both the coherence length and the coherence time are inversely proportional to the frequency bandwidth for a given spectral shape according to the following Wiener-Khinchin theorem:
/ . G|(x)exp(/ü>x)</T = |£(ω)|2 , (9.11)
where Ε(ω) is the Fourier transform of E(t). Note that
|£(οο)|2 = £(ω), (9.12)
where Ξ(ω) is the power spectral density distribution of the light. The Wiener- Khinchin theorem, a special case of the cross-correlation theorem, states that the autocorrelation function of the electric field and the power spectrum are a Fourier transform pair.
If 5(ω) is Gaussian, we have
2 N
5 ( ω ) = ' e x p ( - < ! ^ Y (9.13) ν ^ σ ω V 2σ2 )
where ωο denotes the center angular frequency and σω denotes the standard deviation of ω. Since the profile is of key interest, 5(ω) is normalized to unit power:
/ S(<o)dü)=l. (9.14)
It can be shown (see Problem 9.1) that the coherence length is given by
41n2 λ2 lc = 7 7 , (9.15)
π Δλ
where λο denotes the center wavelength of the light source and Δλ denotes the FWHM bandwidth in wavelength. The broader the bandwidth, the shorter
TIME-DOMAIN OCT 185
the coherence length becomes. In an interferometer, the two beams are said to be coherent with each other when 2nAl < lc. Note that Eq. (9.15) is for a Gaussian Hneshape and the constant factor on the right-hand side varies with the spectral shape; XQ/AX, however, is sometimes used to estimate lc regardless of the spectral shape.
9.4. TIME-DOMAIN OCT
OCT is based on Michelson interferometry with a light source of short coherence length; it can be implemented either in free space or by using optical fibers (Figure 9.2). The optical fibers, however, must be single-mode because modal
Reference mirror
Low-coherence light source
r
t Δ / = / » - / Γ
/c
4—+-
Beamsplitter
ER + ES
Sample
(a) C?
Detector
Low-coherence light source Reference mirror
Axial
scanning
Transverse scanning
Computer Detector
(b) Sample
Figure 9.2. Schematic diagrams of (a) a free-space and (b) a fiberoptic OCT system.
186 OPTICAL COHERENCE TOMOGRAPHY
dispersions in multimode fibers broaden the axial resolution. A superluminescent diode (SLD) is commonly used as the light source because of its high radiance and relatively low cost. While the reference beam is reflected from a mirror, the sample beam is backscattered from various depths in the biological tissue sample.
In time-domain OCT (TD-OCT), the reference arm length oscillates period- ically, which causes a Doppler frequency shift in the reflected reference beam. The sample and the refererice beams are recombined and subsequently detected by the photodetector; the detection falls into the category of heterodyne detec- tion owing to the Doppler shift. The two beams coherently interfere only when their optical path-length difference is within the coherence length of the source, which is referred to as coherence gating, an effect that enables OCT to resolve the path-length distribution of the backscattered light. Therefore, the axial res- olution is determined by the coherence length of the source. In the ballistic or quasiballistic regime, the path length distribution can be directly converted into a physical depth distribution; in the quasidiffusive or diffusive regime, however, this conversion breaks down.
Recording the interference fringes or the envelope as a function of IR profiles the backscattering reflectance from the sample versus the depth. As a result, a ID image—referred to as an A-scan or A-line image—is produced. Multiple A-scan images acquired by transverse scanning form a 2D B-scan image or a 3D volumetric image.
Before a more rigorous theory is described, a simple approach is presented to illuminate the basic principle of OCT. We let Es = Ec + £/ , where Ec and Ei represent the components of the backscattered sample beam that are coherent and incoherent with the reference beam, respectively; thus, Eq. (9.4) becomes
/ ( 0 = <|£/? + £c + £/l2). (9.16)
Because E\ has a random phase difference relative to ER, it does not contribute to the AC signal. Therefore, we have
7(0 = E2R0 + E2a) + E% + 2EROEcocos ( ΐ π ^ ^ λ . (9.17)
Here, Eco and EJO denote the amplitudes of Ec and £/ , respectively; AICR denotes the arm-length difference between Ec and ER; λο denotes the center wavelength of the light source. In Eq. (9.17), the last term represents an AC signal /ACS whose amplitude is proportional to Eco- If the reference mirror is scanned axially, a depth-resolved distribution of Eco can be acquired; it provides an A-scan image with an axial resolution that is limited by the coherence length.
A more rigorous theory is presented below. Any low-coherence light field E(t) is a superposition of monochromatic waves of various frequencies by virtue of the inverse Fourier transformation:
1 r+ 0 0 E(t) = — / E(ü>)exp(-ia>i)d(D. (9.18)
2π J_00
TIME-DOMAIN OCT 187
Therefore, the electric fields of the reference and the sample beams can be expressed in the frequency domain as
Ε*(ω) = £Λ 0(ω)βχρ(ϊ(2Μω)/Α - ωί)), (9.19)
£5(ω) - £so(co) exp(i(2ks(a>)ls ~ ωί)). (9.20)
For brevity, only a single backscatterer along each A-line is considered for E$. The light intensity at angular frequency ω is
/(ω) = |Ε*(ω) + £*(ω)|2 = |£Α(ω) | 2 + |£5(ω) | 2 + 2Re {ΕΗ(ω)Ε*(ω)}, (9.21)
where the cross-term provides the interference signal at ω. Superposition of the interference signals at all angular frequencies yields the total interference signal /AC:
7AC = 2Re j ί Εκ(ω)Ε$(ω)αω\ . (9.22)
Substituting Eqs. (9.19) and (9.20) into Eq. (9.22), we obtain
7AC = 2Re j ί £/?0(ω)£*0(ω) exp(-iΔφ(ω)) </ω , (9.23)
where
Δφ(ω) = 2*5(ω)/5 ~ 2^(ω)Ικ. (9.24)
We have
5(ω) α ^ο(ω)£* 0 (ω) , (9.25)
where the proportionality constant is related to the amplitude reflectivities in the two arms. Thus, Eq. (9.23) can be rewritten as
1/: IAC oc Re { / S((U) exp(-i Αφ(ω)) αω (9.26) If (1) the spectrum of the source light is bandlimited around a center frequency ωο and (2) the sample and the reference arms consist of a uniform nondispersive material, we can approximately express the propagation constant as a first-order Taylor series around 000
Μ ω ) = Μ ω ) = *(ω) = *(ωο) + *'(ωο)(ω - ω0), (9.27)
188 OPTICAL COHERENCE TOMOGRAPHY
where k' denotes the derivative of k with respect to ω. Thus, Eq. (9.24) becomes
Δφ = *(ω0)(2Δ/) + *'(ω0)(ω - ω0)(2Δ/), (9.28)
which can be rewritten as
Δφ = ωοΑτρ + (ω - ωο)Δτ^. (9.29)
Here, Ατρ denotes the round-trip phase delay between the two arms
fc(ü)n) Ατρ = - ^ ( 2 Δ Ζ ) , (9.30)
ω0 and Δτ^ denotes the round-trip group delay between the two arms:
Δτ^ = *'(ω0)(2Δ/). (9.31)
From the definition of the phase velocity vp
ω0 P ΗωοΥ
Eq. (9.30) can be rewritten as
2Δ/ Vp
From the definition of the group velocity vg
1
(9.32)
Δτ ρ = ,. (9.33)
(9.34) * *'(ω<,)
Eq. (9.31) can be rewritten as
2Δ/ Δτ„ = . (9.35)
υ8
Substituting Eq. (9.29) into Eq. (9.26) yields
/AC <X Re |exp(—/ωοΔΧρ) / S((o)exp(—ι(ω - ωο)Δτ^)ί/ω J . (9.36)
If 5(ω) is symmetric about ωο, the integral in Eq. (9.36) is real; thus, Eq. (9.36) becomes
r»00
Γ /Ac OC cos^oAip) / S((o)exp(—/(ω - ωο)Δτ^)ί/ω. (9.37) «/ —oo
TIME-DOMAIN OCT 1 8 9
The cosine factor represents a carrier that oscillates with increasing Axp. The integral represents an envelope as a function of Δτ^; it determines the axial PSF of the interferometer. The envelope is equal to the inverse Fourier transform of 5(ω), which is an outcome of the Wiener-Khinchin theorem. Note that taking the envelope of /AC is a nonlinear operation.
If 5(ω) is Gaussian, substituting Eq. (9.13) into Eq. (9.37) yields
/ (Δτ„) 2 \ /AC OC exp I - \ - \ COS(Q>0AT:P). (9.38)
The temporal Gaussian envelope has a standard deviation στ:
στ = —. (9.39)
The source bandwidth is typically given by the FWHM in wavelength (Δλ). From ω = 2TTC/X, we have approximately
2TCC σω = - y a x , (9.40)
where σλ denotes the standard deviation of λ. For any Gaussian distribution of ξ with standard deviation σ^, its FWHM Δξ is given by
Δξ = ( 2 V 2 1 n 2 W (9.41)
Thus, we have
Δλ σλ = . (9.42)
2V21n2 In free space, we have k = ω/c and Δτ^ = Δτ^ = 2Al/c, where c denotes both the phase and the group velocities; thus, Eq. (9.38) becomes
(Δ/)2
2σ? /AC OC exp ( — — r cos(2fc0A/), (9.43)
Ji
where ko is the propagation constant at the center wavelength λο and the standard deviation σ/ is given by
oi = -^. (9.44)
The axial resolution of OCT in air AZR is commonly defined as the FWHM of the Gaussian envelope in Eq. (9.43). Using Eq. (9.41), we obtain
AZR = (2V21n2)o,. (9.45)
190 OPTICAL COHERENCE TOMOGRAPHY
Sequentially substituting Eqs. (9.44), (9.39), (9.40), and (9.42) into Eq. (9.45), we obtain
2ln2 λ^ Δζ* = ~f. (9.46)
π Δλ Comparing Eqs. (9.15) and (9.46), we find
Δζκ = \ . (9.47)
Therefore, the axial resolution in air equals half of the coherence length of the source owing to the round-trip propagation of the reference and the sample beams. If other factors are negligible, the axial resolution in biological tissue is the axial resolution in air divided by the index of refraction of the tissue.
The transverse resolution of OCT is commonly defined as the focal diameter of the incident sample beam, which is independent of the coherence length of the source. If the transverse distribution of the incident sample beam is Gaussian, the transverse resolution is given by
** = —£. (9.48) π D
Here, / denotes the focal length of the objective lens; D denotes the diameter of the beam on the lens or the diameter of the lens, whichever is smaller. From the following expression for the numerical aperture NA
N A ^ ^ , (9.49)
Eq. (9.48) can be rewritten as
π NA
The depth range within which the lateral resolution is approximately maintained is defined by the depth of focus Δζ/
πΔΓο Δ ζ , = — A (9.51)
Ζλο
which is twice the Rayleigh range of a Gaussian beam. This equation shows the tradeoff between the focal diameter and the focal zone of the sample beam—the smaller the focal diameter, the shorter the focal zone. Therefore, the use of a high-NA objective lens requires either transverse-priority scanning—as in en face (C-scan) imaging—or depth-priority scanning with dynamic focusing along the optical axis.
TIME-DOMAIN OCT 191
The preceding derivation for a single backscatterer can be extended to mul- tiple backscatterers that are distributed along the optical axis. The time window for the Fourier transformation, however, is truncated according to the axial resolution.
A block diagram of a complete OCT system, where the reference mirror is scanned axially, is shown in Figure 9.3. The detection of OCT signals by demodulation involves the following steps: highpass filtering to remove the DC background, rectification to reverse the signs of the negative AC signals, lowpass filtering to recover the envelope of the interference fringes, and analog-digital conversion to record the data.
A schematic of a representative experimental embodiment for the demodula- tion of the interference signals is shown in Figure 9.4. The first highpass filter removes the DC background and passes the AC interference signal. The active full-wave rectifier takes the absolute value of the AC signal. The lowpass filter then recovers the envelope of the rectified signal.
Low-coherence light source
t Ψ
Beamsplitte
I
. /
i
f
/
Sample
Ref mir
' 4 > 1
r
V t 1 r
t t
erence ror
IPZT 4 ^
"*—' ( )scillator
Detector
Highpass filter
Rectifier
Lowpass filter
A/D convertor
Computer
Figure 9.3. Block diagram of an OCT system, where PZT (lead zirconate titanate) represents a piezoelectric transducer that scans the reference mirror axially. A/D represents an analog-to-digital converter.
192 OPTICAL COHERENCE TOMOGRAPHY
\ / Highpass filter
..^mm^^. Lowpass filter \ /
Figure 9.4. Schematic of a representative experimental embodiment for demodulating the interference signals.
Example 9.1. Derive Eq. (9.38) from Eq. (9.36).
From the identity
I exp(—(ax + bx + c)) dx J-OQ
R /b2-4ac\
we derive
/
oo S(ü))exp(—/(ω — ωο)Δτ^)ί/ω
-oo
1 Γ™ ( ( ω - ω ο ) 2
= ε χ ρ ί - - ( σ ω Δ τ ^ ) 2 ) .
2ol — /(ω — ωη)Δτ« 1 αω (9.53)
TIME-DOMAIN OCT 1 9 3
From Eq. (9.36), we have
/Ac oc exp ί ω 8 j Re {exp(-i(u0ATp)}
( (σωΔχ^)2 \ =™p[——>
(9.54)
i COS(GL>OAX,,),
which can be reformulated to Eq. (9.38) by using Eq. (9.39).
Example 9.2. Assume that 5(ω) is Gaussian: (a) calculate AZR given λο = 830 nm and Δλ = 20 nm; (b) plot AZR as a function of Δλ at commonly used λο = 830 nm and λο = 1300 nm.
From Eq. (9.46), (a) AZR = 15.2 μπι, and (b) the axial resolution as a function of Δλ is plotted in Figure 9.5.
Example 9.3. Assume £(ω) to be Gaussian. Simulate the demodulation in TD- OCT in MATLAB. (a) Plot the interference signal versus A///c , where λο = 830 nm and Δλ = 60 nm. Estimate the number of periods within the FWHM of the interference envelope, (b) Rectify the signal by taking the absolute value of the interference signal, (c) Plot the spectral amplitude of the rectified signal. (d) Filter the rectified signal to produce an envelope.
We reformulate Eq. (9.43) to
I AC = exp (-" (τΐ) cos(2A:oA/). (9.55)
10̂
i lo1
10°
10- 101
■ 1 _ 1 . 1 1 1 . 1
01Λ oJU nm 1300 nm
W \0* Bandwidth (nm)
Figure 9.5. Axial resolution at two center wavelengths as a function of bandwidth.
194 OPTICAL COHERENCE TOMOGRAPHY
From Eq. (9.15), we have
41n2 λ^
π Δλ κ
When the interference signal is sampled in MATLAB, we must use a sufficiently high sampling rate to satisfy the Nyquist criterion (more than two data points per period).
Since the FWHM of the interference envelope is the axial resolution AZR and the period of the interference fringes is λο/2, the number of periods within the FWHM is
A representative MATLAB program is listed below:
% Use SI units throughout
lambdaO = 830E-9; % center wavelength dlambda = 60E-9; % bandwidth (delta lambda) c = 3E8; % speed of light
lc = 4*log(2)/pi*lambda(T2/dlambda % coherence length Numberof periods = 0.5*lc/(lambdaO/2) % # of periods in FWHM
figure(1);
N = 2Λ12; % number of sampling points dl = lc*linspace(-2,2, N); % array for Delta_l kO = 2*pi/lambda0; % propagation constant
subplot(4, 1 , 1 ) % interferogram lac = exp(-16*log(2)*(dl/lc).A2) .* cos(2*k0 * dl); plot(dl/lc, lac, 'k') title('(a) Interferogram') xlabel('\Deltal/l_c') ylabel('Signal') axis([-0.6, 0.6, -1, 1])
subplot(4, 1 , 2 ) % rectified interferogram Irec = abs(Iac); plot(dl/lc, Irec, 'k') title('(b) Rectified interferogram') xlabel('\Deltal/l_c') ylabel('Signal') axis([-0.6, 0.6, -1, 1])
subplot(4, 1 , 3 ) % spectrum of the rectified interferogram Fred = fft(Irec)/sqrt(N); % order of frequencies: 0,1...(N/2-1),-N/2,-(N/2-1)...-1
FOURIER-DOMAIN RAPID-SCANNING OPTICAL DELAY LINE 1 9 5
Frec2 = fftshift(Fred); % shifted order of frequencies: -N/2,-(N/2-1)...-1, 0,1...(N/2-1) dfreq = 1/(4*lc); % freq bin size = 1/sampling range freq = dfreq*(-N/2:N/2-1); % frequency array plot(freq*lambdaO, abs(Frec2), 'k') title('(c) Spectrum of the rectified interferogram') xlabel('Frequency (1/\lambda_0)') ylabel('Amplitude') axis([-10, 10, 0, 5])
subplot(4, 1 , 4 ) % envelope freq_cut = 1/lambda0/2; % cut-off frequency for filtering i_cut = round(freq_cut/dfreq); % convert freq_cut to an array index Ffilt = Fred; % initialize array Ffilt(i_cut:N-i_cut+1) = 0; % filter Ifilt = abs(ifft(Ffilt))*sqrt(N); % amplitude of inverse FFT
plot(dl/lc, Ifilt/max(Ifilt), 'k') Iac_en = exp(-16*log(2)*(dl/lc).Λ2); % envelope hold on; plot(dl(1:N/32:N)/lc, Iac_en(1:N/32:N), 'ko') hold off; t i t l e ( ' ( d ) Envelopes') x l a b e l ( ' \ D e l t a l / l _ c * ) y l a b e l ( ' S i g n a l s ' ) a x i s ( [ - 0 . 6 , 0 . 6 , - 1 , 1]) legend( 'Demodulated ' , 'Or ig inal ' )
The graphical output from the MATLAB program is shown in Figure 9.6.
9.5. FOURIER-DOMAIN RAPID-SCANNING OPTICAL DELAY LINE
In addition to the geometric means for varying the reference path length, a frequency-domain approach is also available based on the following inverse Fourier transformation:
I /»+00 E(t — Axg) = — I [£(ü))exp(/ü)A-C£)]exp(—ίωί) άω. (9.57)
^ ^ J—οο
This equation indicates that a linear phase ramp Δτ^ω in the frequency domain leads to a group delay Δτ^ in the time domain. In optics, a grating is a temporal Fourier transformer that can transform a time-domain signal into a temporofrequency-domain signal, whereas a lens is a spatial Fourier transformer that can transform a space-domain signal into a spatiofrequency-domain signal. Both optical elements can serve as inverse Fourier transformers as well.
According to this principle, a Fourier-domain rapid scanning optical delay line was developed using a grating-lens pair, a scanning planar mirror, and a static planar mirror (Figure 9.7). The grating Fourier transforms (disperses) the inci- dent light into temporofrequency (chromatic) components propagating in various
196 OPTICAL COHERENCE TOMOGRAPHY
(a)
Interferogram
Rectified interferogram
ill^iil^^ (b)
(c)
(d)
-10
Spectrum of the rectified interferogram
-5 0 Frequency (IM-Q)
Demodulated O Original
10
-0.4 -0.2 0 MIL
0.2 0.4 0.6
Figure 9.6. Simulated demodulation of OCT signals.
directions; each chromatic component has a spatial frequency. The component for the center wavelength λο is aligned with the optical axis of the lens by adjusting the angle of incidence. The lens focuses each chromatic component into a point on the scanning mirror; hence, it Fourier-transforms the light into a spatiofrequency spectrum along the vertical direction, which represents the temporofrequency spectrum of the original light. The scanning mirror reflects the focused beam with a phase ramp across the temporofrequency spectrum because different tem- porofrequency components accumulate different round-trip path lengths. The lens and the grating then function in their inverse roles to produce a merged beam propagating onto the static mirror, which backreflects the merged beam. Then, the
FOURIER-DOMAIN RAPID-SCANNING OPTICAL DELAY LINE 197
Incident light
Diffraction grating
Double-pass mirror
/ / Figure 9.7. Schematic representation of the Fourier-domain optical delay line.
beam inversely propagates through the entire delay line, which can be analyzed according to the principle of reciprocity. The final beam propagates against the incident beam, but it has experienced a group delay.
The phase shift φ̂ of each temporofrequency component can be expressed by
Snx0Bs 8 π / θ , ( λ - λ ο ) Φ*(λ) = — - — + — — .
λ pg cos θολ (9.58)
Here, λ is the wavelength, XQ is the center-wavelength displacement on the mirror surface relative to the pivot, Θ* is the tilt angle of the scanning mirror, / is the focal length of the lens, pg is the pitch of the grating, and θο is the first- order center-wavelength diffraction angle with respect to the normal vector of the grating. Since θο is zero here, it is excluded in the following derivations. We rewrite Eq. (9.58) as a function of the optical angular frequency ω
Φ*(ω) = 4χοθ5ω 8π/θ 5 (ω —ωο)
c pg(x>o (9.59)
where ωο represents the center angular frequency. The phase delay, defined as the delay at the center frequency, is given by
Axp = φ5(ω0) 4x0Qs
ω0 (9.60)
which can be translated into the following free-space phase-path-length mismatch:
Alp = cAxp = 4χ0θ*. (9.61)
1 9 8 OPTICAL COHERENCE TOMOGRAPHY
The group delay can be derived as follows:
Ax« 9φ5(ω)
3ω
4x0Qs 4 /λ 0 θ 5
ω=ωο cPg
— ΔτΓ 4/λρθ,
cPg (9.62)
which can be translated into the following free-space group-path-length mis- match:
AL CATO = 4xo$s ~ 4/λρθ5
Pg ΔΖ«
4/λρθ,
Pg (9.63)
The phase and the group delays are different, but both are proportional to Qs. Thus, rotating the mirror provides a rapid delay line, which scans several millimeters at a repetition rate of several kilohertz.
9.6. FOURIER-DOMAIN OCT
For any time-domain method, a Fourier-domain equivalent usually exists. A Fourier-domain OCT (FD-OCT) system based on spectral interferometry is shown in Figure 9.8. FD-OCT avoids varying the reference optical path length alto- gether. The recombined beam, however, is dispersed by a spectrometer into spectral components. The corresponding spectral components interfere and form a spectral interferogram. The spectral interferogram is acquired by an optical detector array such as a ID photodiode array. Taking the inverse Fourier trans- formation of the spectrum yields an A-line image in its entirety. As in TD-OCT,
Reference mirror
Low-coherence light source Lens r\|
ΓΤρ> Sample
Imaged reference mirror
] Photodiode array
Figure 9.8. Schematic of an FD-OCT system.
FOURIER-DOMAIN OCT 1 9 9
transverse scanning across the sample provides 2D or 3D imaging. In FD-OCT, all backscatterers on the A-line are measured simultaneously; in TD-OCT, only some of the backscatterers are measured at any one time. Consequently, FD-OCT has a higher frame rate and a greater sensitivity.
Now, we extend the theory for TD-OCT to one for FD-OCT. Multiple backscatterers at various depths on the A-line are considered. Thus, the sample beam consists of multiple partial waves emanating from the backscatterers. The spectral components of the reference and the sample beams can be expressed as
Εκ(ω) = Ε0(ω)Γ* exp(i(2*Α(ω)/Α - ωί)), (9.64)
/
+oo r'sQs) exp(i(2ks(w)ls - ω/)) dls. (9.65)
-00
Here, £Ό(ω) denotes the electric field incident on the reference mirror or the sample surface; r# denotes the amplitude reflectivity of the reference mirror; r's(ls), the object function to be imaged, denotes the apparent amplitude reflec- tivity density (reflectivity per unit depth) of the backscatterers along the A-line in the sample. Since light incident on a scatterer may have been attenuated, r's(ls) represents the apparent—rather than the true—local amplitude reflectivity den- sity. The amplitude reflectivity density r's of a discrete reflector is related to its amplitude reflectivity rs through a delta function
rs(h) = rs(lso)Ws-lso), (9.66)
where /so denotes the location of the reflector. If rfs(l$) is integrated around /so, then rs(/so) is recovered.
If dispersion is neglected, we have
^L = ^=k=™, (9.67) nR ns c
where HR denotes the refractive index of the medium in the reference arm and ns denotes the average refractive index of the sample. For brevity, HR is set to unity here. Apart from a constant scaling factor, the spectral interferogram is given by
I{k) = \ER(kc) + Es(kc)\ 2. (9.68)
Substituting Eqs. (9.64) and (9.65) into Eq. (9.68), we obtain
/(*) = S(k)r\
/
+oo r's(ls)cos(2k(nsls-lR))dls
-00 (9.69) +oo
+ S(k) / r'sds) exp(i2k(nsls)) dls \J~oo
2 0 0 OPTICAL COHERENCE TOMOGRAPHY
where the source power spectral density distribution S(k) = \Eo(kc)\2. The first term on the right-hand side, referred to as the reference-intensity term, can be measured by blocking the sample arm—setting r's(ls) to zero. The second term, referred to as the cross-interference term, encodes rfs(ls) in the integral with a cosine function—that has frequency 2(n$ls — h)—of the wavenum- ber, which is k/(2n) = l/λ. The third term, referred to as the self-interference (sample-intensity) term, originates from power spectrum \Es(kc)\2 and contains the interference among the partial waves from the various sample depths.
The cross-interference term can be decoded to extract r's(ls) by taking the inverse Fourier transformation. The deeper the origin of the backscattered partial wave, the higher the encoding frequency is. The shorter the reference arm, the higher the frequency as well. The Nyquist criterion requires that the frequencies of the spectral interferogram be less than half the spatial sampling frequency of the detector array. Thus, it is advantageous to minimize the frequency of the encoding cosine function. However, if the position of the "imaged reference mirror" is shifted into the sample to minimize the frequency, the two sides around this position will share encoding frequencies, which leads to ambiguity. Therefore, the "imaged reference mirror" must be placed outside the sample unless multiple interferograms with various phase differences between the two arms are measured to resolve the encoding ambiguity.
For brevity in notation, we set IR to zero and shift the reference point for Is to the "imaged reference mirror surface" in the sample arm. If the reference point is outside the sample, a new even function r's(ls) can be crafted as follows so that r's(-ls) = rfs(ls):
\r'sVs) if / s > 0 , P'sds) = , (9.70)
[r's(-ls) if ls<0.
With this new function, Eq. (9.69) can be rewritten as
/(*) - S(k) \r2R+rR ί r's(ls)cxp(i2knsls)dls
9i (9.71) 11 r+o° I 1
+ T / r's(ls)txp(i2knsls)dls\ | . 4 IJ-oo I I
If we change the variable of the integrals by Is = l's/(2ns), Eq. (9.71) can be rewritten as
m-wLj + ̂ ,{fi(A))(t,+ ' |sjfi(i))««f}, (9.72)
FOURIER-DOMAIN OCT 2 0 1
where the following Fourier transformation is employed:
/
+oo f(l's)exp(ikl's)dl's. (9.73)
-00
Using the inverse Fourier transformation
f(l's) = 3-l{F(k)}(lfs) = — / F(k)exp(-ikl's)dk, (9.74)
we rewrite Eq. (9.72) as
$-[{I(k)Ws) = $-l{S(k)}(l's)
\2ns \2ns) 2ns S\2ns) \6n
2 s l 5 V 2 n s / |
(9.75) Here, * denotes convolution, and 6{} denotes the autocorrelation-function oper- ator:
/
+00
f(lsOf«si+ls)<U's\· (9·76) -oo
In this derivation, the following property of the Dirac delta function is used:
δ«>=^5(έ) <9·77 ) The following Wiener-Khinchin theorem is used as well:
e ( M ) } = 7Γ / \F(k)\2exp(-ikl's)dk. (9.78)
With a change of variable by l's = 2^5/5, Eq. (9.75) is converted to
3-l{I(k)}(2nsls)
= $-{{S(k)}(2nsls) * ^-Ws) + ^-?sOs) + 7^-2 < 2 { W ) · (9.79) [2ns 2ns I6nzs J
The second term in the braces is the A-line image r's(ls). The first and last terms, however, represent spurious images. The first term is nonzero only at Is = 0, which is outside the sample; thus, it can be easily removed. Unfortunately, the last term can mingle with the second term; thus, it can be difficult to eliminate. In addition, the convolution with 3~l{S(k)}(2nsls) blurs the image because S(k) functions as a filter.
2 0 2 OPTICAL COHERENCE TOMOGRAPHY
To recover the true image, one may take another interferogram with kl's shifted by π, which causes a sign change in Eq. (9.72):
(9.80) Taking the difference between Eqs. (9.72) and (9.80) yields
Δ/(*) = S(k)^9 lr's (l^-\\(k), (9.81)
where AI(k) = I(k) — h(k). The A-line image can then be recovered by
*(£)- "Vl^K'i). (9-82) rR [ S(k) Changing the variable by l's = 2tish leads to
^ ( / S ) = ^ f - i { ^ > } ( 2 n 5 / s ) . (9.83)
This equation shows that the subtracted-and-deconvolved spectral interferogram in the braces recovers an ideal image; the deconvolution involves simply divid- ing AI(k) by S(k). Although deconvolution can sharpen the image, one should exercise caution in the presence of noise.
An alternative to recovering the true image is to (1) measure the first term (reference-intensity term) in Eq. (9.69) by blocking the sample arm (rfs — 0), (2) measure the third term (self-interference term) in Eq. (9.69) by blocking the reference arm (r/? = 0), and (3) subtract the measured first and third terms from the right-hand side of Eq. (9.69).
In practice, a spectrometer produces a spectrum with uniform wavelength spacing. This wavelength spacing is usually converted to uniform propagation- constant spacing by interpolation as required by the fast Fourier transformation (FFT) algorithm.
By using a single-element photodetector to measure the interference signal while a laser serially sweeps the wavelength, FD-OCT can also construct a spectral interferogram one wavelength at a time. A hardware "k clock" can be installed in the laser to achieve uniform propagation-constant spacing. Once a spectral interferogram is obtained, the theoretical analysis presented above is equally applicable.
Example 9.4. Simulate FD-OCT numerically using MATLAB. Assume a Gaus- sian source spectral density distribution S(k).
FOURIER-DOMAIN OCT 2 0 3
A representative M A T L A B program is shown below:
% Use SI units throughout
lambdaO = 830E-9; % center wavelength of source dlambda = 20E-9; % FWHM wavelength bandwidth of source ns=1.38; % refractive index of sample ls1 = 100E-6; % location of backscatterer 1 ls2 = 150E-6; % location of backscatterer 2 rs1 = 0.5; % reflectivity of backscatterer 1 rs2 = 0.25; % reflectivity of backscatterer 2
k0=2*pi/lambda0; % center propagation constant delta_k=2*pi*dlambda/lambda0A2; % FWHM bandwidth of k sigma_k = delta_k/sqrt(2*log(2)); % standard deviation of k
N=2 10; % number of sampling points nsigma = 5; % number of standard deviations to plot on each side of kO
subplot(4,1,1); % Generate the interferogram k = kO + sigma_k*linspace(-nsigma,nsigma, N); % array for k S_k = exp(-(1/2)*(k-k0). "2/sigmaJT2); % Gaussian source PSD E_s1 = rs1*exp(i*2*k*ns*ls1); % sample electric field from scatter 1 E_s2 = rs2*exp(i*2*k*ns*ls2); % sample electric field from scatter 2 I_k1 = S_k .* abs(1 + E_s1 + E_s2)."2; % interferogram (r_R = 1) plot(k/k0,I_k1/max(I_k1), 'k'); title('Interferogram'); xlabel('Propagation constant k/k_0'); ylabel('Normalized intensity'); axis([0.9 1.1 0 1]);
subplot(4,1,2); % Inverse Fourier transform (IFT) of the interferogram sped =abs(fftshift(ifft(I_k1)))/sqrt(N); dls_prime = 1/(2*nsigma*sigma_k/(2*pi)); % bin = 1/sampling range ls_prime = dls_prime*(-N/2:N/2-1); % frequency array plot(ls_prime/(2*ns),spec1/max(spec1), 'k'); % scale the frequency title('IFT of the interferogram1); xlabel('Depth Is (m)'); ylabel('Relative reflectivity'); axis([-2*ls2 2*ls2 0 1]);
subplot(4,1,3); % IFT of the deconvolved interferogram sped_norm =abs(f f tshi f t ( i f f t ( I_k1. /S_k)) )/sqrt(N); dls_prime = 1/(2*nsigma*sigma_k/(2*pi)); % bin size = 1/sampling range ls_prime = dls_prime*(-N/2:N/2-1); % frequency array plot(ls_prime/(2*ns),spec1_norm/max(spec1_norm), 'k'); title('IFT of the deconvolved interferogram'); xlabel('Depth Is (m)'); ylabel('Relative reflectivity'); axis([-2*ls2 2*ls2 0 1]);
subplot(4,1,4); % IFT of the deconvolved differential interferogram I_k2 = S_k .* abs(-1 + E_s1 + E_s2). 2; % interferogram delta_I_k = I_k1 - I_k2; spec2=abs(fftshift(ifft(delta_I_k./S_k)))/sqrt(N); plot(ls_prime/(2*ns),spec2/max(spec2), 'k'); title('IFT of the deconvolved differential interferogram');
2 0 4 OPTICAL COHERENCE TOMOGRAPHY
x l a b e l ( " D e p t h I s ( m ) ' ) ; y l a b e l ( ' R e l a t i v e r e f l e c t i v i t y ' ) ; a x i s ( [ - 2 * l s 2 2 * l s 2 0 1 ] ) ;
The graphical output of this program is shown in Figure 9.9. The first panel shows the simulated spectral interferogram I(k). The second panel shows the inverse Fourier transform of I(k). Remember to change the independent variable to 2/25/5 after taking the inverse Fourier transformation. The third panel shows the inverse Fourier transform of I(k)/S(k). The fourth panel shows the inverse Fourier transform of AI(k)/S(k) as shown in Eq. (9.82). The locations and
Interferogram
■? 1.5
0.95 1 1.05 Propagation constant k/k0
IFT of the interferogram
Depth ls (m)
IFT of the deconvolved interferogram
Depth ls (m)
IFT of the deconvolved differential interferogram
1.1
2 3
x l O - 4
2 3
xlO"4
2 3 Depth/,(m) χ 1 0 _ 4
Figure 9.9. Simulated signal processing in FD-OCT with two backscatterers.
FOURIER-DOMAIN OCT 205
strengths of the backscatterers are recovered in all the inverse Fourier transforms. In the second panel, however, a DC component and a spurious backscatter at 50 μηι appear; the latter is due to interference between the partial waves from the two backscatterers. The third panel shows that deconvolution sharpens the peaks. The fourth panel shows that the inverse Fourier transformation of the subtracted- and-deconvolved spectral interferogram yields a much cleaner ID image.
If the second backscatterer is obliterated by setting rs2 to zero, we obtain the results shown in Figure 9.10 instead. The DC component still appears in the inverse Fourier transform of I(k). However, no self-interference signal of the sample beam appears because only a single backscatterer exists.
0.9
Interferogram
0.95 1 1.05 Propagation constant k/lc0
IFT of the interferogram
1.1
Depth ls (m)
IFT of the deconvolved interferogram
2 3
xlO"4
> V.D
O
S ! (U
£ 0.5 03
-
I
' - -
I -1 0 1
Depth ls (m)
IFT of the deconvolved differential interferogram
2 3
xlO"4
Depth ls (m) x 1(T
Figure 9.10. Simulated signal processing in FD-OCT with a single backscatterer.
2 0 6 OPTICAL COHERENCE TOMOGRAPHY
9.7. DOPPLER OCT
Like ultrasonography, OCT can image blood flow on the basis of the Doppler effect. Doppler OCT has measured blood flow at a flow rate on the order of 10 pL/s in ΙΟ-μηι-diameter vessels positioned up to 1 mm beneath the tissue surface. The basic principle of Doppler OCT is illustrated by the following analysis in a nondispersive medium. A moving reference mirror varies the arm-length difference by
Δ/(ί) = Δ/ο - vRt, (9.84)
where Δ/ο denotes the arm-length difference at time t = 0 and vR denotes the velocity of the reference mirror. The phase difference between the two arms at the center wavelength λο in Eq. (9.8) is modified to
4π Δφ(ί) = 2&0(Δ/ο ~ vRt) = — (Δ/ο - vRt).
This time-varying phase difference leads to a Doppler shift fR given by
(9.85)
Λ(λ0) = 1
2π
έ/Δφ(ί) dt
2vR λ0 '
(9.86)
The Doppler shift fR is the beat frequency, also referred to as the carrier fre- quency, of the interference fringes. The other spectral components experience similar Doppler shifts given by
Λ(λ) = 2vR (9.87)
which can be used to compute the bandwidth of the interference signal as
Δ/* = 2vR
Δλ. (9.88)
In the presence of flow, an additional Doppler shift leads to the following carrier frequency:
2π
ύ?Δφ(ί) dt λο
(vscosQs - vR) = ΙΛ(λο)-Λ(λο)|. (9.89) Here, vs denotes the velocity of the scatterers, Θ5 denotes the angle between the flow direction and the light-incidence direction, and fs denotes the Doppler shift due to the backscatterers:
Λ(λ0) = 2vs cos ds
(9.90)
GROUP VELOCITY DISPERSION 2 0 7
If fs < /R, we have
/JW(XO) = Λ(λο) ~ Λ(λ 0 ) (9.91)
or Λ(λο) = Λ(λο) - /ÄS(XO). (9.92)
Once the original (predemodulated) interference fringes are acquired, taking the Fourier transformation of the fringes yields fa, which further yields f$ through Eq. (9.92), since //? is known. If θ$ is known, vs can be computed from Eq. (9.90). Because the Fourier transformation is usually performed in sliding short-time windows, a tradeoff between axial and velocity resolutions exists in the flow estimation.
9.8. GROUP VELOCITY DISPERSION
Group velocity dispersion (GVD), which is neglected in previous sections, deteri- orates the axial resolution of OCT. GVD causes polychromatic light to experience nonlinearly frequency-dependent phase delays. As a result, GVD broadens ultra- short laser light. More relevantly, any GVD mismatch between the reference and the sample arms of an OCT system broadens the axial PSF.
To analyze the effect of GVD on OCT signals, we first expand the propaga- tion constant into a Taylor series to the second order around the center angular frequency coo:
*(ω) = *(ωο) + *'(ωο)(ω - ω0) + -*"(ωο)(ω - ω0)2. (9.93)
From the definitions of the phase and the group velocities [Eqs. (9.32) and (9.34)], Eq. (9.93) is reformulated to
*(ω) = — — ω0 + — — (ω - ω0) + -*"(ωο)(ω - ω0)2. (9.94) Vp(co0) ν8(ω0) 2
The frequency-dependent phase mismatch between the reference and the sample beams is
Δφ(ω) = 2ks(<u)ls - 2Μω)//?. (9.95)
If the propagation constants in the reference and the sample paths are equal except for a GVD mismatch in arm length /</, substitution of Eq. (9.94) into Eq. (9.95) yields
Δφ(ω) = — — ωο(2ΔΖ) + — — (ω - ω0)(2Δ/) νρ(ω0) ν8(ω0)
+ ^ΔΑ:,,(ωο)(ω - ω0)2(2/^), (9.96)
2 0 8 OPTICAL COHERENCE TOMOGRAPHY
where Δ/ = Is — IR and
Δ£"(ω0) = *5;(ω0) - *£(ω0).
Substituting Eqs. (9.33) and (9.35) into Eq. (9.96), we obtain
Δφ(ω) = ω0Δτ^ + (ω - ω0)Δτ^ + -Δ*"(ω0)(ω - ω0)2(21α).
Substitution of Eq. (9.98) into Eq. (9.26) leads to
IAC oc Re | βχρ(-ιωοΔχρ) / 5(ω)
x exp I —ι (ω - ω0)Δΐο + —— (ω - ω0) 2(2/^) 1) άω\
(9.97)
(9.98)
(9.99)
If S(co) is Gaussian [Eq. (9.13)], Eq. (9.99) describes an interference signal mod- ulated by a complex Gaussian envelope:
/ A c a R e {rä) C T p (4rW y βχρ(-ι'ωοΔτρ) (9.100)
Here, Γ(2/^) represents the standard deviation of the axial PSF:
r2(2ld) = o 2 T+ix
2 d, (9.101)
where the GVD time constant is defined by
xd = y/Ak"(a>o)(2ld). (9.102)
From Eq. (9.101), we have
* /- "<* r2(2id) σ4 + x4 σ4 + τ;
(9.103)
Substituting Eq. (9.103) into Eq. (9.100), we discover that the real and imaginary components on the right-hand side of Eq. (9.103) cause broadening and chirp- ing, respectively, in the interference signal. The original standard deviation στ is broadened to
*<™<FM- (9.104)
GROUP VELOCITY DISPERSION 2 0 9
Thus, the envelope broadens with increasing τ^ and by a factor of y/l when \d =z οτ. The interference signal chirps with increasing Δ/, as can be seen by differentiating the total phase ΦΑΟ in the two exponents in Eq. (9.100):
dd>Ac 4τ5 /o - ^ = 2*(ωο) - -r^-jk 2(ω0)Δ/. (9.105)
Here, Id is assumed to be independent of Δ/. If Δ/ is uniformly scanned, substi- tuting Eq. (9.84) into Eq. (9.105) leads to
d<\>AC „ f / x 4x2 d //2 2*(ωο)ν* - -rJLÄk Z(O>O)VR(A10 - vRt). (9.106) dt σ4τ + τ«
Thus, the angular frequency of the interference signal varies with time, which is chirping.
Because the amplitude of 1/ Γ(2/^) decreases the peak magnitude of the inter- ference envelope, the system sensitivity —defined as the ratio of the incident light power to the weakest measurable sample light power, usually in dB—degrades. The degradation in the photocurrent amplitude is given by the following multi- plicative factor:
1 '"" (9.107) \Γ(21ά)\ [l + i W a J 4 ] 1 ^
which indicates an inverse proportionality to Λ/Ο^. In practice, the effect of GVD can be reduced by minimizing /</. For example,
the optical fiber lengths in the reference and the sample arms should be matched as closely as possible. In retinal imaging, the clear path in the eye can be matched with one in an optically similar medium—such as water—in the reference arm.
Example 9.5. Estimate the GVD mismatch length Id of a fused-silica fiber rel- ative to air beyond which envelope broadening becomes significant. An SLD is used as the light source, where center wavelength λο = 800 nm and bandwidth Δλ = 20 nm.
From Eqs. (9.39), (9.40), and (9.42), we obtain
V21n2 λ2 στ = ^ - - r f = 2 0 f s . (9.108)
2nc Δλ
In a fused-silica fiber, k" — 350 fs2/ cm at 800-nm wavelength. Therefore, Ak" = 350 fs2/ cm relative to air. When xd = στ, envelope broadening is considered significant. From Eq. (9.102), we obtain
Id = ^ = 0 . 5 1 cm. (9.109)
2 1 0 OPTICAL COHERENCE TOMOGRAPHY
Example 9.6. Derive Eq. (9.105).
The total phase in Eq. (9.100) is
1 τ2, ?
Substituting Eqs. (9.33) and (9.35) into Eq. (9.110), we obtain
1 τ2 ΦΑΓ(Δ/) = 2*(ω0)Δ/ - - —^ Τ [Α: , (ω 0 )2Δ/] 2 ,
which leads to Eq. (9.105) by differentiation with respect to Δ/.
(9.110)
(9.111)
9.9. MONTE CARLO MODELING OF OCT
Although singly backscattered photons are more desirable in OCT, multiple- scattered photons can contribute to OCT signals as well. Here, both single- and multiple-scattered contributions are simulated using the Monte Carlo method. Since only ensemble-averaged quantities are modeled, certain features of OCT—such as speckles—are excluded.
OCT signals are divided into two classes as shown in Figure 9.11. Both classes are based on sample light that is coherent with the reference light. Class I orig- inates from backscattering in a target layer whose central sample arm length zc and thickness Az are determined as follows:
nszc = nRlR,
lr Az =
2ns
(9.112)
(9.113)
Class I
Figure 9.11. Composition of OCT signals.
MONTE CARLO MODELING OF OCT 211
Here, ns and HR denote the refractive indices in the sample and the reference arms, respectively. Class II originates from multiple scattering above the target layer. Class I provides useful imaging information about the target layer, whereas class II does not. Since both classes mingle in the interference signal, class II deteriorates both the contrast and the resolution of OCT.
We assume
/ A C O C T ^ , (9.114)
where
/s = /i + /2. (9.Π5)
Here, I\ and h denote the ensemble-averaged intensity of class I light and class II light, respectively.
Angle-biased sampling, a variance reduction technique, is employed to accel- erate the computation of backscattering. Standard sampling of the scattering angle is inefficient for backscattering because scattering in biological tissue is highly forward-peaked. The angle-biased sampling technique samples an arti- ficially biased scattering phase function in lieu of the true function and then compensates for the bias with a photon-weight correction given by
„ Ρ(θ,φ) w = w. (9.116)
Ρ*(θ,φ)
Here, Θ (0 < θ < π) and φ (0 < φ < 2π) denote the photon deflection polar and azimuthal angles, respectively; P and P* denote the true and biased phase functions, respectively; w and w* denote the photon weights associated with P and P*, respectively.
The Henyey-Greenstein phase function /?(cos0) (see Chapter 3) is adopted here for Ρ(θ, φ):
Pioose) = * " * - 3 ^ , (9.117) 2(1 + gz — 2gcos0)3/2
where g denotes the scattering anisotropy. Further, p(— cosO) is used for Ρ*(θ, φ). Thus, once cos9 is sampled with /?(cos0), — cos0 is actually used to propagate the photon packet. From Eqs. (9.116) and (9.117), the photon-weight correction is given by
i + ^ c o s e y ' V \\+g2-2gCOsBJ <
In the simulation, a photon packet is launched from a pencil beam and then tracked by the conventional Monte Carlo method (see Chapter 3). If the photon packet reaches a scattering site in the target layer, it is first labeled and then
2 1 2 OPTICAL COHERENCE TOMOGRAPHY
scattered with the aforementioned angle-biased sampling. The photon packet is terminated whenever n$ls > nRlR + (/r/2). When reaching the detector, pho- ton packets with nsh < KRIR - (lc/2) are discarded; then, the labeled ones are recorded into class I and the unlabeled ones into class II.
Parameters used in this simulation include lc — 15 μηι, nR = ns = 1.5, absorption coefficient \ia — 1.5 cm"1, scattering coefficient \ks = 60 cm- 1 , and g = 0.9. The detector has a radius of 10 μηι and an acceptance angle of 5°. Figure 9.12 shows that class II is smaller than class I at small probing depths but becomes greater than class I at probing depths beyond ^500 μπι. Figure 9.13 shows that the average number of scattering events in class II is greater than in class I and increases faster with probing depth. Since scattering randomizes polar- ization, class II can be rejected using cross-polarization detection by as much as 50%. If the class I light is completely polarized while the class II light is com- pletely unpolarized, the intersection can be extended from ^500 to ~700 μπτ as indicated in Figure 9.12.
Furthermore, Figure 9.12 shows that class I signal decays at a rate related to the extinction coefficient μ,. The detected signal intensity Is depends on three factors sequentially: (1) the number of photons reaching the target layer, (2) the proportion that is subsequently backscattered, and (3) the proportion that ultimately reaches the detector. Singly backscattered light from different depths has the same angular distribution because the scatterers are assumed to have the same phase function. However, singly backscattered light from a greater depth is broadened over a larger area on the tissue surface and hence is not captured as much by the detector. Consequently, the singly backscattered portion in class I has a decay rate greater than μ,. When the multiple-scattered portion is included, class I ends up with a decay rate that is only slightly different from μ,.
103
102
O
r IO1
c
10°
1 0 - i 0 200 400 600 800 1000
Depth (μηι)
Figure 9.12. Class I and class II signals versus probing depth (zr) .
I I ., I , -
— Θ — Class I — V — Class II: Original polarization — — Class II: Randomized polarization
PROBLEMS 213
ε
0 100 200 300 400 500 600 700 800
Depth (μπι)
Figure 9.13. Average numbers of scattering events in class I and class II signals versus probing depth (zc).
As shown in Figure 9.13, class I also contains contributions from multiple- scattered light; the average number of scattering events increases linearly with the probing depth and reaches ^ 2 at 200 μπι. In principle, only singly backscattered photons can provide exact localized imaging information because they experience no interactions outside the target layer. Conversely, multiple-scattered photons cannot directly furnish localized imaging information because they experience interactions at multiple sites.
PROBLEMS
9.1 Show Eq. (9.15) based on the definition of coherence length. Further, prove lc = 81n2(c/Aoo).
9.2 If direct electronic detection were used to provide a 10-μιτι axial resolution, what would be the required temporal resolution?
9.3 Derive Eq. (9.6). Plot the interference signal versus the arm-length mis- match to show that the AC interference signal rides on a DC background. Under what condition does the contrast (AC amplitude/DC amplitude) reach the maximum? What is the maximum value?
9.4 Prove Eq. (9.41).
9.5 Verify Eq. (9.14).
9.6 (a) Given that if the scatterers are static and the reference mirror is trans- lated at a speed of 40 mm/s toward the incident beam, calculate the beat
214 OPTICAL COHERENCE TOMOGRAPHY
frequency in OCT. The center wavelength of the light source is 830 nm. (b) Assuming that if the scatterers flow toward the incident beam at a speed of 0.5 mm/s with an angle of 75° with respect to the backscattered light, calculate the new beat frequency.
9.7 Implement the simulated TD-OCT shown in Example 9.3 using your own code (a) for Δλ — 100, 50, and 25 nm sequentially and (b) for double backscatterers that are separated by 4/c, 2/c, /c, and lc/2 sequentially. Add a DC background to the interference signal and assume a 5% modulation depth (AC/DC).
9.8 Show analytically that the demodulation demonstrated in Example 9.3 recovers the envelope apart from a constant scaling factor.
9.9 Prove Eq. (9.58) and the subsequent equations in Section 9.5.
9.10 Show that the Fourier-domain rapid scanning optical delay line leads to the following center frequency and bandwidth in the OCT interference signals:
_ 4x0des(t) A A _ 2Δλ (^ 2 A o \ dQs(t) /o = T 7— and Δ / = (*-Ψ) λο dt )ij \ pg ) dt
9.11 Extend Example 9.3 to Doppler OCT. Assume that the reference mirror is translated toward the incident beam at a speed of 40 mm/s and the scatterers flow toward the incident beam at a speed of 0.5 mm/s with an angle of 75° with respect to the backscattered light. In addition, take the Fourier transformation of the interference fringes before rectification and recover the flow velocity of the scatterers when the flow direction and the reference mirror velocity are given.
9.12 Extend the theory in Section 9.4 to one for Doppler OCT.
9.13 Derive Eq. (9.100).
9.14 Derive Eq. (9.103).
9.15 Use MATLAB to demonstrate numerically the broadening and chirping from GVD. Assume a set of realistic parameters.
9.16 Use MATLAB to demonstrate the concepts of phase and group velocities using (a) two copropagating plane waves that have the same magnitude but a small frequency difference and (b) three copropagating plane waves that have the same magnitude but a small frequency difference. Generalize the expressions for the phase and the group velocities.
9.17 Implement a Monte Carlo simulation of OCT to duplicate Figures 9.12 and 9.13.
9.18 Replace the oc sign with an = sign in Eq. (9.25) by adding appropriate parameters.
FURTHER READING 2 1 5
9.19 Relate the interference signal /AC in OCT to the autocorrelation function G i of the electric field.
9.20 Derive the transverse resolution in OCT on the basis of the confocal mech- anism.
READING
Ai J and Wang LHV (2005): Synchronous self-elimination of autocorrelation interference in Fourier-domain optical coherence tomography, Opt. Lett. 30(21): 2939-2941. (See Section 9.6, above.)
Fujimoto JG (2002): Optical coherence tomography: Introduction, in Handbook of Optical Coherence Tomography, Bouma BE and Tearney GJ, eds., Dekker, New York. Marcel (See Section 9.4, above.)
Hee MR (2002): Optical coherence tomography: Theory, in Handbook of Optical Coher- ence Tomography, Bouma BE and Tearney GJ, eds., Marcel Dekker, New York. (See Sections 9.4 and 9.8, above.)
Huang D, Swanson EA, Lin CP, Schuman JS, Stinson WG, Chang W, Hee MR, Rotte T, Gregory K, Puliafito CA, and Fujimoto JG (1991): Optical coherence tomography, Science 254(5035): 1178-1181. (See Sections 9.1 and 9.4, above.)
Lindner MW, Andretzky P, Kiese wetter F, and Hausler G (2002): Spectral radar: Opti- cal coherence tomography in the Fourier domain, in Handbook of Optical Coherence Tomography, Bouma BE and Tearney GJ, eds., Marcel Dekker, New York. (See Section 9.6, above.)
Milner TE, Yazdanfar S, Rollins AM, Izatt JA, Lindmo T, Chen ZP, Nelson JS, and Wang XJ (2002): Doppler optical coherence tomography, in Handbook of Optical Coherence Tomography, Bouma BE and Tearney GJ, eds., Marcel Dekker, New York. (See Section 9.7, above.)
Rollins AM and Izatt JA (2002): Reference optical delay scanning, in Handbook of Optical Coherence Tomography, Bouma BE and Tearney GJ, eds., Marcel Dekker, New York. (See Section 9.5, above.)
Yao G and Wang LHV (1999): Monte Carlo simulation of optical coherence tomography in homogeneous turbid media, Phys. Med. Biol. 44(9): 2307-2320. (See Section 9.9, above.)
FURTHER READING
Bouma BE and Tearney GJ (1999): Power-efficient nonreciprocal interferometer and linear-scanning fiber-optic catheter for optical coherence tomography, Opt. Lett. 24(8): 531-533.
Bouma BE and Tearney GJ (2002): Handbook of Optical Coherence Tomography, Marcel Dekker, New York.
Cense B and Nassif NA (2004): Ultrahigh-resolution high-speed retinal imaging using spectral-domain optical coherence tomography, Opt. Express 12(11): 2435-2447.
Chen ZP, Milner TE, Srinivas S, Wang XJ, Malekafzali A, van Gemert MJC, and Nelson JS (1997): Noninvasive imaging of in vivo blood flow velocity using optical Doppler tomography, Opt. Lett. 22(14): 1119-1121.
2 1 6 OPTICAL COHERENCE TOMOGRAPHY
Chiang HP, Chang WS, and Wang JP (1993): Imaging through random scattering media by using cw-broad-band interferometry, Opt. Lett. 18(7): 546-548.
Chinn SR, Swanson EA, and Fujimoto JG (1997): Optical coherence tomography using a frequency-tunable optical source, Opt. Lett. 22(5): 340-342.
Choma MA, Sarunic MV, Yang CH, and Izatt JA (2003): Sensitivity advantage of swept source and Fourier domain optical coherence tomography, Opt. Express 11(18): 2183-2189.
de Boer JF, Cense B, Park BH, Pierce MC, Tearney GJ, and Bouma BE (2003): Improved signal-to-noise ratio in spectral-domain compared with time-domain optical coherence tomography, Opt. Lett. 28(21): 2067-2069.
Drexler W, Morgner U, Kartner FX, Pitris C, Boppart SA, Li XD, Ippen EP, and Fuji- moto JG (1999): In vivo ultrahigh-resolution optical coherence tomography, Opt. Lett. 24(17): 1221-1223.
Drexler W, Morgner U, Ghanta RK, Kartner FX, Schuman JS, and Fujimoto JG (2001): Ultrahigh-resolution ophthalmic optical coherence tomography, Nature Med. 7(4): 502-507.
Dubois A, Vabre L, Boccara AC, and Beaurepaire E (2002): High-resolution full-field optical coherence tomography with a Linnik microscope, Appl. Opt. 41(4): 805-812.
Fercher AF, Hitzenberger CK, Sticker M, Moreno-Barriuso E, Leitgeb R, Drexler W, and Sattmann H (2000): A thermal light source technique for optical coherence tomography, Opt. Commun. 185(1-3): 57-64.
Fujimoto JG: Optical coherence tomography for ultrahigh resolution in vivo imaging, Nature Biotechnol. 21(11): 1361-1367.
Haiti I, Li XD, Chudoba C, Ghanta RK, Ko TH, Fujimoto JG, Ranka JK, and Windeier RS (2001): Ultrahigh-resolution optical coherence tomography using continuum generation in an air-silica microstructure optical fiber, Opt. Lett. 26(9): 608-610.
Hee MR, Izatt JA, Swanson EA, and Fujimoto JG (1993): Femtosecond transillumination tomography in thick tissues, Opt. Lett. 18(13): 1107-1109.
Hitzenberger CK, Baumgartner A, and Fercher AF (1998): Dispersion induced multi- ple signal peak splitting in partial coherence interferometry, Opt. Commun. 154(4): 179-185.
Hitzenberger CK and Fercher AF (1999): Differential phase contrast in optical coherence tomography, Opt. Lett. 24(9): 622-624.
Izatt JA, Hee MR, Owen GM, Swanson EA, and Fujimoto JG (1994): Optical coherence microscopy in scattering media, Opt. Lett. 19(8): 590-592.
Izatt JA, Kulkami MD, Yazdanfar S, Barton JK, and Welch AJ (1997): In vivo bidirec- tional color Doppler flow imaging of picoliter blood volumes using optical coherence tomography, Opt. Lett. 22(18): 1439-1441.
Knüttel A and Boehlau-Godau M (2000): Spatially confined and temporally resolved refractive index and scattering evaluation in human skin performed with optical coher- ence tomography, J. Biomed. Opt. 5(1): 83-92.
Kowalevicz AM, Ko T, Haiti I, Fujimoto JG, Pollnau M, and Salathe RP (2002): Ultrahigh resolution optical coherence tomography using a superluminescent light source, Opt. Express 10(7): 349-353.
Kulkarni MD, Thomas, CW and Izatt JA (1997): Image enhancement in optical coherence tomography using deconvolution, Electron. Lett. 33(16): 1365-1367.
Kulkarni MD, van Leeuwen TG, Yazdanfar S, and Izatt JA (1998): Velocity-estimation accuracy and frame-rate limitations in color Doppler optical coherence tomography, Opt. Lett. 23(13): 1057-1059.
FURTHER READING 2 1 7
Lee TM, Oldenburg AL, Sitafalwalla S, Marks DL, Luo W, Toublan FJJ, Suslick KS, and Boppart SA (2003): Engineered microsphere contrast agents for optical coherence tomography, Opt. Lett. 28(17): 1546-1548.
Leitgeb R, Hitzenberger CK, and Fercher AF (2003): Performance of Fourier domain vs. time domain optical coherence tomography, Opt. Express 11(8): 889-894.
Leitgeb R, Wojtkowski M, Kowalczyk A, Hitzenberger CK, Sticker M, and Fercher AF (2000): Spectral measurement of absorption by spectroscopic frequency-domain optical coherence tomography, Opt. Lett. 25(11): 820-822.
Li XD, Chudoba C, Ko T, Pitris C, and Fujimoto JG (2000): Imaging needle for optical coherence tomography, Opt. Lett. 25(20): 1520-1522.
Liu HH, Cheng PH, and Wang JP (1993): Spatially coherent white-light interferometer based on a point fluorescent source, Opt. Lett. 18(9): 678-680.
Morgner U, Drexler W, Kartner FX, Li XD, Pitris C, Ippen EP, and Fujimoto JG (2000): Spectroscopic optical coherence tomography, Opt. Lett. 25(2): 111-113.
Pan YT, Birngruber R, and Engelhardt R (1997): Contrast limits of coherence-gated imaging in scattering media, Appl. Opt. 36(13): 2979-2983.
Pan YT, Xie HK, and Fedder GK (2001): Endoscopic optical coherence tomography based on a microelectromechanical mirror, Opt. Lett. 26(24): 1966-1968.
Podoleanu AG, Rogers JA, and Jackson DA (1999): OCT en-face images from the retina with adjustable depth resolution in real time, IEEE J. Select. Topics Quantum Electron. 5(4): 1176-1184.
Povazay B, Bizheva K, Unterhuber A, Hermann B, Sattmann H, Fercher AF, Drexler W, Apolonski A, Wadsworth WJ, Knight JC, Russell PSJ, Vetterlein M, and Scherzer E (2002): Submicrometer axial resolution optical coherence tomography, Opt. Lett. 27(20): 1800-1802.
Rollins AM, Kulkarni MD, Yazdanfar S, Ung-arunyawee R, and Izatt JA (1998): In vivo video rate optical coherence tomography, Opt. Express 3(6): 219-229.
Rollins AM and Izatt JA (1999): Optimal interferometer designs for optical coherence tomography, Opt. Lett. 24(21): 1484-1486.
Schmitt JM, Knüttel A, Yadlowsky M, and Eckhaus MA (1994): Optical-coherence tomog- raphy of a dense tissue—statistics of attenuation, and backscattering, Phys. Med. Biol. 39(10): 1705-1720.
Schmitt JM and Knüttel A (1997): Model of optical coherence tomography of heteroge- neous tissue, J. Opt. Soc. Am. A 14(6): 1231-1242.
Schmitt JM (1999): Optical coherence tomography (OCT): A review, IEEE J. Select. Topics Quantum Electron. 5(4): 1205-1215.
Schmitt JM, Xiang SH, and Yung KM (1999): Speckle in optical coherence tomography, J. Biomed. Opt. 4(1): 95-105.
Swanson EA, Huang D, Hee MR, Fujimoto JG, Lin CP, and Puliafito CA (1992): High- speed optical coherence domain reflectometry, Opt. Lett. 17(2): 151-153.
Tearney GJ, Brezinski ME, Southern JF, Bouma BE, Hee MR, and Fujimoto JG (1995): Determination of the refractive-index of highly scattering human tissue by optical coherence tomography, Opt. Lett. 20(21): 2258-2260.
Tearney GJ, Boppart SA, Bouma BE, Brezinski ME, Weissman NJ, Southern JF, and Fujimoto JG (1996): Scanning single-mode fiber optic catheter-endoscope for optical coherence tomography, Opt. Lett. 21(7): 543-545.
Tearney GJ, Bouma BE, and Fujimoto JG (1997): High-speed phase- and group-delay scanning with a grating-based phase control delay line, Opt. Lett. 22(23): 1811- 1813.
2 1 8 OPTICAL COHERENCE TOMOGRAPHY
Tripathi R, Nassif N, Nelson JS, Park BH, and de Boer JF (2002): Spectral shaping for non-Gaussian source spectra in optical coherence tomography, Opt. Lett. 27(6): 406-408.
Vabre L, Dubois A, and Boccara AC (2002): Thermal-light full-field optical coherence tomography, Opt. Lett. 27(7): 530-532.
Wang RKK, Xu XQ, Tuchin VV, and Elder JB (2001): Concurrent enhancement of imag- ing depth and contrast for optical coherence tomography by hyperosmotic agents, J. Opt. Soc. Am. B 18(7): 948-953.
Wang YM, Zhao YH, Nelson JS, Chen ZP, and Windeler RS (2003): Ultrahigh-resolution optical coherence tomography by broadband continuum generation from a photonic crystal fiber, Opt. Lett. 28(3): 182-184.
Wojtkowski M, Kowalczyk A, Leitgeb R, and Fercher AF (2002): Full range complex spectral optical coherence tomography technique in eye imaging, Opt. Lett. 27(16): 1415-1417.
Wojtkowski M, Bajraszewski T, Targowski P, and Kowalczyk A (2003): Real-time in vivo imaging by high-speed spectral optical coherence tomography, Opt. Lett. 28(19): 1745-1747.
Youngquist RC, Carr S, and Davies DEN (1987): Optical Coherence-domain reflecto- metry—a new optical evaluation technique, Opt. Lett. 12(3): 158-160.
Yun SH, Tearney GJ, de Boer JF, Iftimia N, and Bouma BE (2003): High-speed optical frequency-domain imaging, Opt. Express 11(22): 2953-2963.
CHAPTER 10
Mueller Optical Coherence Tomography
10.1. INTRODUCTION
Mueller optical coherence tomography (Mueller OCT) was conceived to image the polarization properties of biological tissue on the basis of polarization- sensitive detection. Although OCT is analogous to ultrasonography in general, polarization exists in transverse optical waves but not in longitudinal ultrasonic waves. As a result, Mueller OCT has no counterpart in ultrasonography.
10.2. MUELLER CALCULUS VERSUS JONES CALCULUS
The term Mueller calculus refers to the use of the Stokes vector and the Mueller matrix in polarimetry, whereas Jones calculus refers to the use of the Jones vector and matrix. The Stokes vector can quantify the polarization state of any light, whereas the Jones vector can quantify the polarization state of completely polarized light only. The effect of a medium on the polarization state of light can be represented by a Mueller or a Jones matrix; the former operates on a Stokes vector, and the latter operates on a Jones vector. If the medium does not degrade the degree of polarization, the Mueller and the Jones matrices are equivalent, and both are applicable; otherwise, only the Mueller matrix is applicable. Because coherent detection in polarization-sensitive OCT always reports a unity degree of polarization, both matrices apply to OCT. The Mueller matrix, however, clearly separates the two major contrast mechanisms in OCT—backscattering (or back- reflection) and polarization—and hence is preferable for presenting final images. In this book, Mueller OCT refers to polarization-sensitive OCT based on either the Mueller matrix or the Jones matrix.
10.3. POLARIZATION STATE
Polarization of light refers to the orientation of the electric-field vector E(z, t) on the transverse xy plane, where the z axis is aligned with the wave propagation
Biomedical Optics: Principles and Imaging, by Lihong V. Wang and Hsin-i Wu Copyright © 2007 John Wiley & Sons, Inc.
219
2 2 0 MUELLER OPTICAL COHERENCE TOMOGRAPHY
direction and t is time. The Cartesian coordinates (JC, y, z) are right-handed, and the x axis is typically horizontal. An electric field vector of monochromatic light can be decomposed into two orthogonal components along the x and y axes:
E(z,t) = Ex(z,t) + Ey(z,t). (10.1)
The two components can be expressed as
Ex(z, t) = exEXQCO&{kz - ωί + φΛ), (10.2)
Ey(z, t) = eyEyQCOs(kz — ωί + φν). (10.3)
Here, ex and ey denote unit vectors along the x and y axes, respectively; EXQ and φχ denote the amplitude and the phase, respectively, of the horizontal component; Eyo and φν denote the amplitude and the phase, respectively, of the vertical component; ω denotes the angular frequency; and t denotes time. The relative phase between the two components is
Δφ = φ ν - φ χ . (10.4)
We rewrite Eq. (10.3) as
Ey(z, t) = eyEy0cos(kz + φ* - (ωί - Δφ)). (10.5)
For —π < Δφ < 0, Ey(z, t) has a phase lead over Ex(z, t) at a given z\ for 0 < Δφ < π, Ey(z, t) has a phase lag behind Ex(z, t).
In general, the tip of the electric field vector at a given z rotates with time along an ellipse as shown in Figure 10.1a; hence, the light is said to be elliptically polarized. The major axis of the ellipse makes an orientation angle θ() with respect to the JC axis
a ] f 2Ex0Ey0cos(A(\>) % = - arctan2 ' __ 2 , (10.6)
where 0 < θ0 < π. Here, arctan2 denotes four-quadrant inverse tangent, yielding an angle dependent on the quadrant of (E^0 — E^Qy cos Δφ).
An ellipticity angle θ̂ is defined to quantify the shape and the handedness of the ellipse:
b de = :parctan-. (10.7)
a
Here, a and b denote the semimajor and semiminor axes, respectively, of the ellipse; the negative and positive signs represent left- and right-handed polariza- tions, respectively (to be discussed below). Since a > b, —π/4 < θ̂ < π/4. It can be shown that
1 . 2Ex0Ex0sm(A<b) Qe = - - arcsin ^ . (10.8) 2 F 2 4- F2
POLARIZATION STATE 221
(a)
fv ▲
y <
/ \ i --' ^ <^ ^
Ev0
it> /Ί V \ /
\X
ExO
(b)
(c)
Figure 10.1. (a) Elliptical polarization, where the endpoint of the electric field vector traces an ellipse; (b) — π/4 linear polarization; (c) right circular polarization.
An auxiliary angle is introduced as
θ^ = arctan ijcO
(10.9)
where 0 < θ^ < π/2 . From Eq. (10.9), Eqs. (10.6) and (10.8) can be reformu- lated to
30 = -arctan(tan(29</)cos(A(|))),
3e = — arcsin(sin(26<i) sin(A(|>)).
(10.10)
(10.11)
The ellipse can reduce to a line. For instance, if Eyo = 0, the ellipse reduces to a horizontal line; hence, the light is said to be horizontally linearly polarized or simply horizontally polarized. Likewise, if Ex$ = 0, the light is said to be vertically linearly polarized. If EXQ = Eyo and Δφ = 0, the light is + π / 4 linearly polarized. If EXQ = £vo and Δφ = π, the light is —π/4 linearly polarized as shown in Figure 10.1b.
2 2 2 MUELLER OPTICAL COHERENCE TOMOGRAPHY
If Exo = Eyo and Δφ = ± π / 2 , the ellipse reduces to a circle; hence, the light is said to be circularly polarized. When Δφ = —π/2, Eqs. (10.2) and (10.3) become
Ex(z, t) = exEx0cos(kz + φχ - ω/),
Ey(z, t) = eyEyo s'm(kz + φ* — ωί).
(10.12)
(10.13)
Angle — ωί enlarges clockwise with time according to the definition of planar angle. Therefore, the electric field vector at a given z rotates clockwise when an observer looks back at the source (Figure 10.1c). In this case, the light is said to be right circularly polarized because a snapshot of the electric vectors along the z axis resembles a right-handed screw. On the contrary, when Δφ = π/2, Eqs. (10.2) and (10.3) become
Ex(z, t) = exEx0cos((ut - kzo - Φ*),
Ey(z, t) = eyEy0 sin(ooi - kzo - φ*).
(10.14)
(10.15)
Angle +ωί widens counterclockwise with time; hence, the light is said to be left circularly polarized. For any elliptically polarized beam, when — π < Δφ < 0, the polarization is said to be right-handed; when 0 < Δφ < π, the polarization is said to be left-handed.
In an alternative convention, cos(o)i — kz + φ*) and COS(GO/ — kz -f φγ) rep- resent the x and y components of the electric field, respectively; in this case, the interpretation of Δφ must be reversed. Although either convention can be adopted as long as consistency is maintained, the coexistence of the two conven- tions has been a source of confusion in the literature. Here, we use the convention in Eqs. (10.2) and (10.3).
10.4. STOKES VECTOR
The polarization state of any light can be quantified by the Stokes vector, which can be constructed from six intensities measured with different polarization ana- lyzers in front of the detector. Light intensities measured with a horizontal linear analyzer, a vertical linear analyzer, a + π / 4 linear analyzer, a — π/4 linear analyzer, a right circular analyzer, and a left circular analyzer are denoted by ///, Ιγ, /+π/4, / - π /4 , //?, and Ιι, respectively. The Stokes vector S is defined by
S =
(So\ Si
s2 (10.16)
1R ~ h j
All four Stokes parameters are real numbers; parameter So represents the original intensity of the light, and each of the other three parameters (Si, S2, and S3)
STOKES VECTOR 2 2 3
represents the difference between the two intensities that are measured using analyzers with orthogonal polarization states.
The four Stokes parameters are constrained by
O Q ^ Öl " f ^ 2 ' ^λ ' (10.17)
the equality sign applies for completely polarized light and the inequality sign, for partially polarized light; for completely unpolarized light, S\ = 5*2 = S3 = 0. A normalized form of the Stokes vector is
\ So So So/ (10.18)
where the superscript T stands for transposition. From the Stokes vector, the degree of polarization (DOP), the degree of linear
polarization (DOLP), and the degree of circular polarization (DOCP) can be defined as follows:
DOP =
DOLP =
DOCP =
Jsj + S22 + S
So ■
JsJ+sj So
(10.19)
So '
If the DOP of light remains unity after interaction along a path with a medium, the medium is said to be nondepolarizing. Otherwise, the medium is said to be depolarizing.
Because any light can be decomposed into two orthogonally polarized waves, the law of energy conservation requires
IH + Iv = Ι+τι/4 + l-n/4 = 1R + IL = SO- (10.20)
As a result, we can express S with four independent measurements, such as / w , / v , / + π /4 , and IR:
S =
/ IH + IV \ IH-IV
2 / + π / 4 - UH + h) V 2IR-(I„ + IV) )
(10.21)
2 2 4 MUELLER OPTICAL COHERENCE TOMOGRAPHY
For monochromatic light (completely polarized), we have
/ £?n + El, \
s = 2Ex0Ey0cos(A(\)) \ -2Ex0Ey0sm(A(\>) /
(10.22)
which can be related to the polarization ellipse by
S = S0
I l \ cos(26e)cos(20o) cos(20^) sin(20„)
sin(20^)
(10.23)
If a Stokes vector is represented geometrically by vector (S\, 5*2, S3) in Cartesian coordinates, the endpoints of the geometric vectors for all possible polarization states of constant So construct a sphere of radius So, which is referred to as a Poincare sphere. The surface of the sphere represents completely polarized states (DOP = 1), whereas the inside represents partially polarized states (DOP < 1). If DOP = 1, the polar and the azimuthal angles of a vector equal π/2 — 2Qe and 2Θ0, respectively [Eqs. (10.23)]. On the spherical surface, the equator represents linear polarizations; the top and the bottom hemispheres represent right- and left- handed polarizations, respectively; the north and the south poles represent right and left circular polarizations, respectively.
Example 10.1. Derive the normalized Stokes vectors for right and left circularly polarized monochromatic light.
For right and left circularly polarized monochromatic light, we have Exo — Eyo and Δφ = =ρπ/2. From Eqs. (10.22), the Stokes vectors are
s* = / 1 \
0 0
W s, =
/ 1 \ 0 0
V - ' /
(10.24)
where the subscripts R and L denote the right and left circular polarizations, respectively.
10.5. MUELLER MATRIX
The Mueller matrix M can represent the effect of a given medium on a Stokes vector. For an incident Stokes vector Sm, the output Stokes vector Sout is given by
MS:, (10.25)
MUELLER MATRICES FOR A ROTATOR, A POLARIZER, AND A RETARDER 2 2 5
where
M
/ Moo Afoi M02 Mo3 \ M\o Mil Mj 2 Mi 3 M20 M21 M22 M23
\ M30 M31 M32 M33 /
(10.26)
In general, a Mueller matrix has 16 independent elements. The Mueller matrix is determined only by the intrinsic properties of the
medium and the optical path. Conversely, the Mueller matrix can fully char- acterize the optical polarization properties of the medium along a given path. Since element Moo explicitly represents only the intensity-based property of the medium, the Mueller matrix can clearly separate backscattering contrast from polarization contrast in Mueller OCT.
10.6. MUELLER MATRICES FOR A ROTATOR, A POLARIZER, AND A RETARDER
The Mueller matrix for a rotator, which rotates the incident electric field by an angle Θ, can be expressed as
ΜΓ(Θ) -
/ I o 0
U
0 cos(29) sin(29)
0
0 - sin(20)
cos(20) 0
o\ 0 0
1 /
(10.27)
A polarizer, also referred to as a diattenuator, has polarization-dependent attenuation—also termed dichroism. If the two orthogonal eigenpolarization states—states that are unaffected by the polarizing element apart from a con- stant factor—are linear or circular, the diattenuator is referred to as a linear or circular diattenuator.
If the two eigenpolarization axes of a linear polarizer are defined as the x and y axes, the electric field transmittances along the x and y axes can be represented by
px = TpcosQp,
py = TpSinOp,
(10.28)
(10.29)
where Tp denotes the total electric field transmittance. Taking the ratio of these two equations yields tan0/7 = py/px. The Mueller matrix for the linear polarizer is given by
T2
p 2
( 1 cos(20/7)
0 V o
cos(20p) 1 0 0
0 0
sin(29p) 0
0 \ 0 0
sin(29„) /
(10.30)
2 2 6 MUELLER OPTICAL COHERENCE TOMOGRAPHY
If θ/; = 0 (i.e., pY = 0), Mp represents a horizontal polarizer, which can con- vert any—even random—polarization into horizontal polarization. Likewise, if Qp — π /2 (i.e., px = 0), M/7 represents a vertical polarizer.
A phase retarder (also referred to as a wave plate, a phase shifter, or a compensator) has polarization-dependent phase delays. If the two orthogonal eigenpolarization states are linear or circular, the retarder is called a linear or circular retarder.
If the two axes of a linear retarder are defined as the x and y axes, the x and y components of an incident optical beam are phase-shifted differently:
Φ <\>x = Φ * - - ,
Φ Φ ' ν = Φ ν + ? ,
(10.31)
where the primed phases are for the retarded optical beam and φ represents the phase shift between the two orthogonal components. If φ is positive, the x and y axes are referred to as the fast and slow axes, respectively. The Mueller matrix for the linear retarder is given by
Μ Λ =
/ I 0 0 1
V
0 0
0 0
0 0
0 \ 0 x
cos φ sin φ — sin φ cos φ
(10.32)
For a quarter-wave retarder, φ = ± π / 2 . For a half-wave retarder, φ = ±π. The Mueller matrix of a polarizing element whose axes are rotated by Θ in
the xy plane is given by
Μ(θ) = ΜΓ(θ)Μ(0)ΜΓ(-θ), (10.33)
where M(0) and Μ(θ) represent the Mueller matrices of the polarizing element before and after the rotation.
Example 10.2. Show that a quarter-wave retarder converts a ± π / 4 linearly polarized beam into a circularly polarized beam and vice versa.
According to Eq. (10.32) with φ = — π/2, a Mueller matrix operation on the incident Stokes vector leads to
/ 1 0 0 0 1 0 0 0 0
\0 0 1
0 \ 0 -1 0 /
/ * ^ 0
±1 V o )
( x \ 0 0
U1/ (10.34)
MEASUREMENT OF MUELLER MATRIX 2 2 7
The Stokes vectors on the right-hand side represent right and left circular polar- izations, respectively. Conversely, we have
/ l 0 0 0 \ 0 1 0 O i l 0 0 0 -1
\0 0 1 0 / '
( l \ i 0 - o -
{±1 '
(l \ 0
^ 1
I o ) (10.35)
which means that a circular polarization is converted into a linear polarization. This reciprocal conversion between linear and circular polarizations is guaran-
teed by the principle of reciprocity of light. For example, the + π / 4 linear polar- ization is converted into the right circular polarization as shown in Eq. (10.34). When light propagation is time reversed, the right circular polarization becomes a left circular polarization, which is then converted back into the + π / 4 linear polarization as shown in Eq. (10.35).
10.7. MEASUREMENT OF MUELLER MATRIX
A Mueller matrix can be measured with various combinations of source polariz- ers and detection analyzers. One possible measurement scheme is described here. Four incident polarization states—horizontal polarization (//), vertical polar- ization (V), -}-π/4 linear polarization (+π/4) , and right circular polarization (R)—are used sequentially for the incident optical beam. Their normalized Stokes vectors are
S'« = s'„ = ^+π/4 — i \
Ί 0 / S'* = (\
0 0
V (10.36)
where superscript i stands for the incident beams. From Eq. (10.25), the four corresponding output Stokes vectors can be determined as follows:
CO
0+π/4 vo
= MS1,, =Mo + Mi,
= MS l v =Mo-M 1 ,
= M S ^ / ^ M o + IV
= MSfÄ = Mo + M3.
(10.37)
Here, superscript o stands for the output beams; Mo, Mi, M2, and M3 denote the four column vectors of matrix M:
M = (M0 M, M2 M3). (10.38)
2 2 8 MUELLER OPTICAL COHERENCE TOMOGRAPHY
The four column vectors (each has four elements) can be obtained from Eqs. (10.37):
Mo = \(S"H + K),
M, = Us"H-srv), 1 (10.39)
M 2 = ^[2sr+Ji/4-(s 0 H + srv)],
M3 = \[2srR-(srH + s°v)]. At least four independent Stokes vectors must be measured to fully determine a general Mueller matrix, and each Stokes vector requires at least four inde- pendent intensity measurements using different analyzers. Therefore, at least 16 independent intensity measurements must be acquired to completely determine a Mueller matrix. If a Mueller matrix has less than 16 independent elements, fewer intensity measurements are required.
Example 10.3. Given that
/ 1 \ 0 0
V 1 /
GO _ Λ+π/4 —
(\ 0 1
CO _
( · \ -1 0
V o / (10.40)
construct the Mueller matrix M.
We have
Mo = \{&Ή + $ν) = {\ 0 0 0 ) r ,
Mi = X-(S"H-^) = (0 I 0 0)7 ,
M2 = ]-[2S%/4-(S°H + S<{/)] = (0 0 0 -If,
M3 = ^[2S'^ - (S^ + S« )] = (0 0 1 0)T.
Therefore, we construct
(10.41)
M =
(I 0 0
\o
0 1 0 0
0 0 0
- 1
0 0 1 0
(10.42)
which is a quarter-wave plate with φ = π/2.
JONES VECTOR 2 2 9
10.8. JONES VECTOR
The polarization state of fully polarized light can be quantified by the Jones vector E. A Jones vector is a two-element column vector, representing the horizontal (JC) and vertical (y) components of the electric field in phasor expression. From Eqs. (10.2) and (10.3), the Jones vector for monochromatic light is given by
E = / * * o e x p ( i < M \ 1 / Ex0 \ V E y 0 e x p ( l < M / Normalized /F2 , F2 \ ^ θ β Χ ρ Ο ' Δ φ ) ) '
γ ^JCO "*" ^yO
The normalized form is simpler; however, the absolute values of the amplitudes and the phases are forfeited. Unlike the full Jones vector, neither the normalized Jones vector nor the Stokes vector can be used to treat interference between coherent lightbeams.
From Eq. (10.43), the horizontal linear polarization state is expressed as
E /£ ,oexp( /4> , ) \ / l \ \ V J Normalized \ 0 )
Similarly, the vertical linear polarization state is expressed as
EV = ( T7 V * Λ I > ( ? I · (10·45> \ Ey0txp(l(\>y) ) Normalized \ 1 /
The + π / 4 linear polarization state, in which Eyo = EXQ and φ̂ , = φχ, is expressed as
Ε + π / 4 = ( E * ° ^ ί ! ) — ► 4= (! V (io·46)
+ π / 4 \ Εχ0 βΧρ(ΐφ Λ ) / Normalized y / 2 \ l J
The — π /4 linear polarization state is expressed as
E-,/4 = ( Er°eXPii?:\ ) »4= ( \ V 00.47) 1 \ -Εχ0£Χρ(ΐφχ) ) Normalized ^ 2 V _ 1 /
The right circular polarization state is expressed as
E * = ( r EM^') m ) > +=(1.). (10-48) \ £χ0βΧρ(ΐφχ - ΐπ/2) J Normalized ̂ 2 \ ~l /
The left circular polarization state is expressed as
^ = {F ExTftx) n.) - M ' V («0.49)
Y Ex0exp(l<$>x + ΐπ/2) ) Normalized y/2 \ l )
2 3 0 MUELLER OPTICAL COHERENCE TOMOGRAPHY
If the convention of using cos(ooi — kz + φχ) and cos(o)i — kz + φν) for the x and y components of the electric field is adopted, the signs of the second elements in Eqs. (10.48) and (10.49) must be reversed.
A normalized Jones vector satisfies the following inner-product identity:
7-Γ* E = l . (10.50)
The normalized Jones vectors of two orthogonal polarization states satisfy the following orthonormal identity:
Ε [ * · Ε 2 = 0. (10.51)
For example
E^* · Ey = E+* / 4 · Ε_π/4 = Εβ* · EL — 0. (10.52)
10.9. JONES MATRIX
A Jones matrix J can convert an input Jones vector Em into an output Jones vector Eout:
E0Ut = JEin, (10.53)
or
(3) i j \ _ ^ . . ><A(*A ( l054) v / \h\ hi J \Elv I Because both Ejn and Eout can represent fully polarized light only, J is applicable to nondepolarizing media only.
10.10. JONES MATRICES FOR A ROTATOR, A POLARIZER, AND A RETARDER
The Jones matrix for a rotator, which rotates the incident electric field by an angle Θ, is given by
J ^ ) = (C°Sa " S i n f lV 00-55) y sin0 cosO J
The Jones matrix for a linear polarizer that is aligned with the x axis is
3P(°) = (P0 pv)' ( l 0 ' 5 6 )
EIGENVECTORS AND EIGENVALUES OF JONES MATRIX 231
For a linear polarizer oriented at an angle Θ with respect to the x axis, we apply the rotational transformation matrix in Eq. (10.55) to derive its Jones matrix
Jp(O)=Jr(0)J / ?(O)Jr(-9), (10.57)
which leads to
px cos2 Θ + py sin2 Θ (px — py) sin Θcos Θ (Px — Py) s*n Θ cos Θ px sin2 Θ + py cos2 Θ *pw = I ; : _,.=.«„.« _ .:,2al „ _ 2 , 1 · (
|0·58)
The Jones matrix for a linear retarder whose fast axis is aligned with the x axis can be expressed as
/ βχρ(-/φ/2) 0 \ J * < 0 , = ( 0 . χρΟ 'φ /2 ) ) · ( l 0 5 9 )
For a linear retarder oriented at an angle Θ with respect to the x axis, the Jones matrix becomes
Ι φ ( θ ) = Ι Γ ( θ ^ φ ( 0 ) Ι Γ ( - θ ) . (10.60)
10.11. EIGENVECTORS AND EIGENVALUES OF JONES MATRIX
For a linear polarizer, we notice that
Jp(!0)EH = pxEH, (10.61)
Jp(0)Ev = pyEv. (10.62)
Both equations are eigenequations. E// and Ev are the eigenvectors of Jp(0) and are also referred to as eigenpolarizations; px and py are the associated eigenvalues.
If the two eigenvectors are orthogonal, the polarizing element is consid- ered polarization homogeneous. Otherwise, the polarizing element is consid- ered polarization inhomogeneous. Common polarization-homogeneous optical elements include linear polarizers, linear retarders, and circular retarders. A com- mon polarization-inhomogeneous optical element is a circular polarizer that is constructed with a linear polarizer inclined at π /4 with respect to the x axis followed by a λ/4 retarder inclined horizontally.
For a polarization-homogeneous element, the Jones matrix can be constructed from its eigenpolarizations and eigenvalues. We denote the first normalized eigen- vector as
E, = I ' . (10.63)
2 3 2 MUELLER OPTICAL COHERENCE TOMOGRAPHY
On the basis of the orthonormality, the second orthogonal eigenvector is given by
E2 = ( lv ) . (10.64)
The corresponding eigenvalues are λι and λ2:
JE! = λ ι Ε ι , (10.65)
JE2 = X2E2. (10.66)
A new matrix, termed the modal matrix, is constructed from the eigenvectors
/ EXx -E* \ K = (Elf E 2 ) = ( Jy , (10.67)
Eh Eu
which can be easily inverted by
7<\x j K-i = κ^* = ( E*x E*y ) . (10.68)
\ —E\y E]
Another new matrix, termed the diagonal eigenvalue matrix, is constructed from the eigenvalues:
A = ( o x°2)· ( 1 0 · 6 9 )
We rewrite Eqs. (10.65) and (10.66) as
JK = KA. (10.70)
Therefore, we have
J - K A K " 1 . (10.71)
Example 10.4. Show that a circular polarizer that is constructed with a linear polarizer inclined at π/4 with respect to the x axis followed by a λ/4 retarder inclined horizontally is polarization-inhomogeneous.
From Eqs. (10.57) and (10.59), the Jones matrix for the circular polarizer is given by
3pc=h(0)Jp(n/4), (10.72)
EIGENVECTORS AND EIGENVALUES OF JONES MATRIX 2 3 3
where φ = π/2 , ρχ = 1, and py = 0. Matrix operation leads to
i „ - 2 3 ^ ( ; ; ) . 0073)
We can show that
J ,CEL - - ^ E L , (10.74)
JpcE_w/4 = 0Ε_π / 4 . (10.75)
The two eigenvectors, E^ and Ε_π/4, are nonorthogonal because
Ε [ * · Ε _ π / 4 ^ 0 . (10.76)
Therefore, Jpc is polarization inhomogeneous. We can further show that the linear polarizer and the λ/4 retarder are not permutable.
Example 10.5. Construct Jones matrices for homogeneous polarizers J# , Jy, J+TI/4, J-Ti/4» J/?, and J^.
Since the electric field of light can be decomposed into two orthogonal com- ponents, only one of which can be transmitted through an ideal homogeneous polarizer, we use orthogonal eigenvectors to construct the Jones matrices.
For JH and Jv, we may consider the orthogonal pair:
0) - (?)■
U: £)(i)-(J)· (i: £)(?)-(S)·
J„ = (J o ) · <la79>
(£: £)0)-(!)· (2: £)(?)-(?)·
From
we obtain
Likewise, from
2 3 4 MUELLER OPTICAL COHERENCE TOMOGRAPHY
we obtain
For J+Ti/4 and J_
From
we obtain
Likewise, from
we obtain
For a circular pol
From
Jv = (o ij·
π/4, we may consider the orthogonal pair:
7l(0 and 7f(-0·
TiUi ' W w = 7 f w ' J2\J2I J2i)\-l) \0)'
J + H / 4 = 2 ( I i j '
1 (Jn / i 2 \ / l \ / 0 \ V 2 W 2 1 J22j\l)-\0)'
J5\J2i ^ A " 1 / V f l " 1 ) '
J - / 4 = 2 ( - l 1 ) ·
arizer, we may consider the orthogonal pair:
Vl \ Ji\ Jn ) \ ~i ) = V l V -«' j '
V2
(10.82)
(10.83)
(10.84)
(10.85)
(10.86)
(10.87)
(10.88)
(10.89)
(10.90)
CONVERSION FROM JONES CALCULUS TO MUELLER CALCULUS 2 3 5
we obtain
Likewise, from
we obtain
>-& 0 ±_( J\\ Jn\( 1 \ /<Λ V2W21 hijy-i) \o)'
±_(Ju J n \ ( \ \ = l_(\ V2W21 Jn)\i) y/2\i
■KJ v)·
(10.91)
(10.92)
(10.93)
(10.94)
10-12. CONVERSION FROM JONES CALCULUS TO MUELLER CALCULUS
For a nondepolarizing optical element, the Jones matrix and the Mueller matrix are equivalent. Unlike the Mueller matrix, the Jones matrix uses complex ele- ments. Because one phase is typically arbitrary and is set to zero, a Jones matrix has seven independent real parameters. Consequently, a nondepolarizing Mueller matrix has seven independent parameters.
A Jones matrix J can be transformed into an equivalent Mueller matrix M as follows:
- 1 M = U(J(g>J*)U
Here, U is the Jones-Mueller transformation matrix
(10.95)
U V2
( 1 0 0 1 \ 1 0 0 - 1 0 1 1 0
\0 -i i 0 /
(10.96)
and ® represents the Krönecker tensor product. The Krönecker tensor product of A and B is defined as
A ® B =
/ A ( 1 , 1 ) B A(1,2)B . A(2, 1)B A(2,2)B .
\A(m,\)B A(m, 2)B .
. A ( l , n ) B \
. A(2,/i)B
. A(ra, j?)B/
(10.97)
where m and n represent the dimensions of A
2 3 6 MUELLER OPTICAL COHERENCE TOMOGRAPHY
A Jones vector E can be transformed into a Stokes vector S using
S = V2U(E(g)E*), (10.98)
where ensemble averaging is carried out when the light is quasimonochromatic instead of monochromatic.
Example 10.6. Given
A=(au an) and B=(b" *'2 V V a2\ «22 ) \ θ2\ b22 )
calculate the Krönecker tensor product.
A ® B =
I a\\b\\ a\\b\2 «12*11 «12*12 \
«11*21 «11*22 «12*21 «12*22
«21*11 «21*12 «22*11 «22*12
\ «21*21 «21*22 «22*21 « 2 2 * 2 2 /
(10.99)
10.13. DEGREE OF POLARIZATION IN OCT
After completely polarized monochromatic light (DOP = 1) is scattered multi- ple times in a scattering medium, the reemitted light generally becomes partially polarized (DOP < 1) unless the area of the detector is much smaller than the aver- age size of the speckle grains. OCT, however, measures the amplitude—rather than the intensity—of backscattered light. As a result, only the part of the backscattered light that is coherent with the reference beam is detected, which leads to a DOP of unity as explained below.
In OCT, the interference signal /AC received by a detector of finite area can be considered as the sum of the interference contributions from all points on the detector:
/AC oc Err* · E5, + Err* · EJ2 + Err* · Ej 3 + · · ·
= Err* · (E5| + E52 + Ej 3 + · · ·) (10.100)
= Err*-E5.
Here, Er represents the Jones vector of the reference beam, which is assumed to be uniform across the detection plane; ESi(i — 1,2,...) represents the Jones vector of the coherent backscattered wave reaching point / on the detector; Es represents the Jones vector of equivalent total coherent backscattered light; and the dot product represents the interference effect. The projection of each Esi onto Er is summed into /AC· Equivalently, Es/ can be vector summed to E9, whose projection onto Er contributes to /AC· If all Esi components share the same polarization state, Es has the same polarization state. Otherwise, Es has a
PARALLEL MUELLER OCT 2 3 7
net apparent polarization state. In either case, Es has a unique polarization state with a DOP of unity. In an intensity-based detection system, by contrast, the intensities of the backscattered optical fields that reach the various points of the detector are added; thus, unless all Esi components share the same polarization state, the DOP is less than unity.
10.14. SERIAL MUELLER OCT
In serial Mueller OCT discussed in this section, 16 independent OCT measure- ments are acquired to determine the Mueller matrix. Only five measurements, however, are required in Mueller OCT, as is discussed in the following section. From Chapter 9, we know that an interference signal for sample light from a given depth has a peak amplitude /ACO given by
/ACOCX y/hAhVr^ (10.101)
Here, lrA denotes the intensity of the reference beam, which has polarization state Λ; ISA denotes the intensity of the sample beam projected onto polarization state Λ. Thus, we have
/2 h,A oc - P . (10.102)
A serial Mueller OCT system that can measure the Mueller matrix of a scattering medium is shown in Figure 10.2. The light from the source, an SLD (superlumi- nescent diode), has a center wavelength of 850 nm and a FWHM bandwidth of 26 nm. After passing through the polarizer, the lightbeam has a power of 0.4 mW. After passing through the half-wave and the quarter-wave plates, the lightbeam is split by a nonpolarizing beamsplitter. The sample beam is focused into the sample by an objective lens. The reference beam passes through a variable-wave plate and is then reflected back. The reflected beams from the reference and the sample arms are coupled into a single-mode fiber and detected by a photodiode. The spatial resolution is M 0 μπι.
Four different incident polarization states—//, V, + π / 4 , and R—are achieved by rotating the half-wave and the quarter-wave plates in the source arm. For each incident polarization state, the variable-wave plate in the reference arm is adjusted to sequentially achieve polarization states //, V, +π /4 , and R for the round-trip reference beam. Thus, a total of 16 polarization-sensitive OCT images are acquired. Then, four Stokes vectors are computed using Eq. (10.21) and then further processed to construct a Mueller matrix using Eqs. (10.38) and (10.39).
10.15. PARALLEL MUELLER OCT
In parallel Mueller OCT discussed in this section, a Jones matrix is first measured by OCT and then converted into a Mueller matrix. Usually, the coordinates for the Jones vector follow the light propagation direction. For example, a reflection
2 3 8 MUELLER OPTICAL COHERENCE TOMOGRAPHY
Mirror
Zero-order half-wave plate
\
SLD
/ / Polarizer Zero-order
quarter-wave plate
Variable-wave plate
Nonpolarizing beamsplitter
t Lens Sample
Lens
Photodiode
Figure 10.2. Schematic of a serial Mueller OCT system.
Jones matrix converts the Jones vector of the incident light expressed in the forward coordinates (z axis aligned with the direction of incidence) into the Jones vector of the reflected light expressed in the backward coordinates (z axis aligned with the direction of reflection). In this section, however, we use the forward coordinates for the Jones vectors of both the incident and the reflected lightbeams.
An OCT configuration that can measure the Jones matrix of a scattering medium with parallel channels is shown in Figure 10.3. Since at least two inde- pendent incident polarization states are required to fully measure a Jones matrix, two SLD light sources are used to provide horizontal polarization (1 0)T and vertical polarization (0 1 ) r , respectively. The two sources—each of which has a center wavelength of 850 nm and a FWHM bandwidth of 26 nm—are amplitude- modulated at 3 and 3.5 kHz, respectively, for encoded parallel detection. The two source beams are merged by a polarizing beamsplitter, filtered by a spatial filter, and then split into the reference and the sample arms by a nonpolarizing beam- splitter. The sample beam passes through a quarter-wave plate with its fast axis inclined at + π / 4 with respect to the x axis and then is focused into the sam- ple by an objective lens (focal length / == 15 mm and NA — 0.25). Each source delivers about 0.2 mW of power to the sample. At the sample surface, the Jones vectors of the sample beam are (1 i)T and (1 — i)T for the two sources. The reference arm consists of a quarter-wave plate with its fast axis inclined at -f π/8 with respect to the x axis, a lens, and a mirror. After the reference beam passes through the quarter-wave plate twice, the incident horizontal and vertical polarizations are converted into + π / 4 polarization (1 l ) r and —π/4 polarization (1 — l ) r , respectively. Then, the reference and the sample beams
PARALLEL MUELLER OCT 2 3 9
Mirror Q t Lens
Spatial filter
Polarizing ^ 1 ^ beamsplitter ,
Horizontally polarized SLD
Vertically polarized SLD
Polarizing beamsplitter
Lens
J Zeroth-order quarter-wave plates
A Non polarizing beamsplitter
I Lens Sample
Photodiode
Lens
o Photodiode
Figure 10.3. Schematic of a parallel Mueller OCT system.
are combined by the nonpolarizing beamsplitter. The combined light is split into horizontal and vertical components by a polarizing beamsplitter; each component is coupled into a single-mode fiber through an objective lens and then detected by a photodiode. A data-acquisition board, sampling at 50 kHz/channel, digitizes the two signals. The scan speed of the reference arm is 0.5 mm/s, generating a Doppler frequency of about 1.2 kHz. The carrier frequencies—the beat and the sum frequencies between the Doppler frequency and the modulation frequencies of the light source—are 1.8, 2.3, 4.2, and 4.7 kHz.
For singly backscattered light, the incident Jones vector Ejn in the sample arm is converted to the detected Jones vector Eout by
E0ut = JNBsJQB(JsBJMJsi)JQlEin. (10 .103)
Here, JQI and JQB are the Jones matrices of the quarter-wave plate in the incident and the backscattered directions, respectively; Jsi and JSB are the Jones matrices of the sample in the incident and the backscattered directions, respectively; JM is the Jones matrix of the backscattering, which functions as a mirror reflection; JNBS is the Jones matrix of the reflecting surface of the nonpolarizing beamsplit- ter. According to the convention of the coordinates used in this section, both JM and JNBS are equal to the identity matrix:
J M = JNBS -a?) (10.104)
2 4 0 MUELLER OPTICAL COHERENCE TOMOGRAPHY
The combined round-trip Jones matrix of the scattering medium J52 is given by
J S 2 = J S B J A # J S I . (10.105)
The overall round-trip Jones matrix J7 is given by
JT = JNBSJQBJS2JQI· (10.106)
Substituting Eqs. (10.105) and (10.106) into (10.103), we obtain
E0ut = JrE i n , (10.107)
or
(10.108)
The Jones reversibility theorem states that the Jones matrices of an optical element for backward and forward light propagations—JBWD and JFWD> respec- tively—are transposition-symmetric if the same coordinates are used for the Jones vectors:
IBWD = JFWD J?wD· (10.109)
Thus, we have
which together lead to
J S B = J L (10.110)
J Q B = J $ I , (10.111)
J s 2 = J L (10.Π2)
J r = J r · (10.113)
According to these symmetry relations, the number of independent real param- eters in Js2 or JT reduces from seven to five, which means that only five real independent measurements are required to measure a Jones or Mueller matrix in OCT.
As discussed in Chapter 9, multiple-scattered light can contribute to OCT signals. In the presence of multiple-scattering contributions, Eq. (10.112) still holds as long as each photon path is reversible (the probabilities for photons to travel along the same path but in opposite directions are equal). This condition is met, for example, in single-mode optical-fiber-based OCT systems, where light
PARALLEL MUELLER OCT 2 4 1
delivery and detection share the same area and angular distribution. Apart from a constant factor, Js2 is the sum of the Jones matrices for all possible paths:
3s2 = ]T]U>*(JF* + J**). (10.114)
Here, w denotes the weight of a path; subscripts F and R denote the forward and the reversed directions of propagation, respectively; k denotes the index of a path. From the Jones reversibility theorem, we have
IRk iFk^Fk 1 T
JRk' (10.115)
Substituting Eqs. (10.115) into Eq. (10.114), we reach Eq. (10.112) again. For two light sources of different polarization states, Eq. (10.108) leads to
^HX C'H2
^ V l r'V2
JTW hn
h l \ JT22 )(ί: ,«r CjH2t nv2e (10.116) Here, subscripts 1 and 2 for the electric fields denote sources 1 and 2, respectively; ß is a phase difference related to the difference in the spectral characteristics of the two light sources; and ß is zero if the two characteristics are identical. Inverting Eq. (10.116) yields
JTW JT\2
JT2\ JT22
F°
F°
JH\
'VI
^H2e
/ z 'V2 t i (10.117)
The inverse matrix exists and is given by
Fl HI JH2* P*' -1
El Fl 'VI ^ V 2 C /
if the determinant D is nonzero:
1 D
Fl pl
VI
— Fl P*\
Fl Hi (10.118)
D fl fi pi\ ^H\ r'H2e
'VI > V 2 c
'HI
'VI
JH2
'V2 /o. (10.119)
This condition simply means that the two light sources have independent polar- ization states.
Substituting Eq. (10.118) into Eq. (10.117) yields
JT- = - ^Η\ CjH2
Fl ρΦ _ / Γ ' pip
D \ F° F° ^ \ ^ V l CjV2
(10.120) 'VI JH\
2 4 2 MUELLER OPTICAL COHERENCE TOMOGRAPHY
On the basis of the transposition symmetry of J^, ß can be eliminated. Substi- tuting Eq. (10.120) into Eq. (10.113) yields
' i ß(*i#i*if2 + E°v>Ev2) = E°H2eHX + E°V2E^ (10.121)
which can be solved for ß unless
' E°HlE i H2 + E°vlE
i V2=0. (10.122)
If the two incident polarization states are orthogonal and either one happens to be an eigenpolarization state of the sample, Eq. (10.122) holds (see Problem 10.1). This drawback can be overcome by using two nonorthogonal source polarization states, for example, a horizontal polarization state and a + π / 4 polarization state. Once JT is found, J52 can then be determined by solving Eq. (10.106).
A piece of porcine tendon is imaged by this system. The Mueller matrix image is shown in Figure 10.4; each element of the Mueller matrix is a 2D image. Some of the images present periodical stripes presumably due to the birefringence of the collagen fibers in the porcine tendon. Since it is free of polarization effects, image Moo presents no such periodicity but shows backscattering contrast instead.
Figure 10.4. Two-dimensional images, 0.5 x 1 mm in area, of the Mueller matrix of a piece of porcine tendon. Each image except Moo is normalized by Moo pixel-by-pixel.
PROBLEMS 243
PROBLEMS
10.1 Substantiate the interpretation of Eq. (10.122) mathematically.
10.2 Given that
CO _ »v —
I 1 \ -1 0
V o )
CO _ 0 0
V 1 /
C O
I i \ o
- 1 V o /
construct the Mueller matrix M.
10.3 Show that the eigenvalues of the six polarizers given in Example 10.5 are equal to 0 or 1 and that the eigenvectors are orthogonal for each pair.
10.4 Find the Jones matrix for a linear retarder oriented at an angle with respect to the x axis.
10.5 (a) Show that a linear polarizer inclined at - π / 4 with respect to the x axis, followed by a quarter-wave plate, leads to a circular polarizer. (b) Find the eigenvalues and eigenvectors for this matrix, (c) Show that the eigenvectors are nonorthogonal and allow both right and left circularly polarized beams to pass through. Swap the two optical components and derive the new Jones matrix of the system and show whether the two optical components are permutable.
10.6 Construct the Jones matrix for a polarizing element that converts a ± π / 4 linear polarization into a circular polarization but maintains the horizontal and the vertical polarizations.
10.7 Use the Jones representations to show that a quarter-wave plate can be used to convert linear polarization into circular polarization and vice versa.
10.8 Use Jones vectors to prove that the superposition of two equal-amplitude circularly polarized beams with opposite handednesses and a phase dif- ference can be used to form a linear polarization state of an arbitrary orientation. In practice, circular birefringence in either optically active media, such as glucose or magnetooptical media, can be used to produce the phase difference.
10.9 Prove that the Jones or the Mueller matrix of a rotator is independent of the orientation of the rotator.
10.10 Derive Eqs. (10.6), (10.8), (10.10), and (10.11).
10.11 Use MATLAB to produce a movie that shows an electric field vector tracing out the polarization ellipse and its special cases.
2 4 4 MUELLER OPTICAL COHERENCE TOMOGRAPHY
10.12 From the six measurements, ///, Iv, Ι+πμ, / - π /4 , //?, and 1L, assuming that four are acquired to construct the Stokes vector, derive the number of valid choices.
10.13 Prove Eq. (10.22).
10.14 Prove Eq. (10.23).
10.15 When a linear polarizer is rotated a full circle in front of natural light (unpolarized light), draw the locus of the transmitted light polarization on the Poincare sphere in MATLAB in pseudo-3D.
10.16 Prove Eqs. (10.27), (10.30), and (10.32).
10.17 Prove Eq. (10.33).
10.18 Prove Eqs. (10.55) and (10.57)-(10.59).
10.19 Use Eq. (10.71) to form the Jones matrix for a quarter-wave plate.
10.20 Prove conversion Eqs. (10.95) and (10.98).
10.21 Assume that a quarter-wave plate oriented at 45° with respect to the x axis is measured by the system in Figure 10.3. Derive the Jones vector of the lightbeam after each polarizing element, starting from the source. First, use right-handed coordinates whose z axis always follows the propagation direction of the lightbeam. Revise the Jones reversibility theorem in this convention. Then, use right-handed coordinates whose z axis is always in the propagation direction of the lightbeams incident on the reference mirror and the sample as in Section 10.14.
10.22 Extend polarization-difference imaging (see Chapter 8) to birefringent scattering media using the Stokes vector.
READING
Born M and Wolf E (1999): Principles of Optics: Electromagnetic Theory of Propaga- tion, Interference and Diffraction of Light, Cambridge Univ. Press, New York. (See Sections 10.3-10.5, above.)
Chipman RA (1995): Polarimetry, in Handbook of Optics, Bass M. and Optical Society of America, eds., McGraw-Hill, New York. Vol. II, Chapter 22. (See Sections 10.4-10.6, above.)
Collett E (1993): Polarized Light: Fundamentals and Applications, Marcel Dekker, New York. (See Sections 10.3-10.6 and 10.8-10.11, above.)
Jiao SL, Yao G, and Wang LHV (2000): Depth-resolved two-dimensional Stokes vectors of backscattered light and Mueller matrices of biological tissue measured with optical coherence tomography, Appi Opt. 39(34): 6318-6324. (See Sections 10.7 and 10.13, above.)
Jiao SL and Wang LHV (2002): Two-dimensional depth-resolved Mueller matrix of biological tissue measured with double-beam polarization-sensitive optical coherence tomography, Opt Lett 27(2): 101-103. (See Section 10.15, above.)
FURTHER READING 2 4 5
Vansteenkiste N, Vignolo P, and Aspect A (1993): Optical reversibility theorems for polarization—application to remote-control of polarization, J. Opt. Soc. Am. A 10(10): 2240-2245. (See Section 10.12, above.)
Yao G and Wang LHV (1999): Two-dimensional depth-resolved Mueller matrix char- acterization of biological tissue by optical coherence tomography, Opt. Lett. 24(8): 537-539. (See Section 10.14, above.)
FURTHER READING
Brosseau C (1998): Fundamentals of Polarized Light: A Statistical Optics Approach, Wiley, New York.
Bueno JM and Campbell MCW (2002): Confocal scanning laser ophthalmoscopy improve- ment by use of Mueller-matrix polarimetry, Opt. Lett. 27(10): 830-832.
Cameron BD, Rakovic MJ, Mehrubeoglu M, Kattawar GW, Rastegar S, Wang LHV, and Cote GL (1998): Measurement and calculation of the two-dimensional backscattering Mueller matrix of a turbid medium, Opt. Lett. 23(7): 485-487.
Cense B, Chen TC, Park BH, Pierce MC, and de Boer JF (2004): Thickness and birefrin- gence of healthy retinal nerve fiber layer tissue measured with polarization-sensitive optical coherence tomography, Invest. Ophthalmol. Visual Sei. 45(8): 2606-2612.
de Boer JF 1997, Milner TE, vanGemert MJC, and Nelson JS (1997): Two-dimensional birefringence imaging in biological tissue by polarization-sensitive optical coherence tomography, Opt. Lett. 22(12): 934-936.
de Boer JF 1999, Milner TE, and Nelson JS (1999): Determination of the depth-resolved Stokes parameters of light backscattered from turbid media by use of polarization- sensitive optical coherence tomography, Opt. Lett. 24(5): 300-302.
de Boer JF 1999, Srinivas SM, Park BH, Pham TH, Chen ZP, Milner TE, and Nelson JS (1999): Polarization effects in optical coherence tomography of various biological tissues, IEEE J. Select. Topics Quantum Electron. 5(4): 1200-1204.
de Boer JF 2002 and Milner TE (2002): Review of polarization sensitive optical coherence tomography and Stokes vector determination, J. Biomed. Opt. 7(3): 359-371.
Everett MJ, Schoenenberger K, Colston BW, and Da Silva LB (1998): Birefringence characterization of biological tissue by use of optical coherence tomography, Opt. Lett. 23(3): 228-230.
Eyal A and Zadok A (2005): Optical noise induced by Gaussian sources in Stokes param- eter measurements, J. Opt. Soc. Am. A 22(4): 662-671.
Gil JJ and Bernabeu E (1987): Obtainment of the polarizing and retardation parameters of a nondepolarizing optical-system from the polar decomposition of its Mueller matrix, Optik 76(2): 67-71.
Guo SG, Zhang J, Wang L, Nelson JS, and Chen ZP (2004): Depth-resolved birefringence and differential optical axis orientation measurements with fiber-based polarization- sensitive optical coherence tomography, Opt. Lett. 29(17): 2025-2027.
Hecht E (2002): Optics, Addison-Wesley, Reading, MA. Hitzenberger CK, Gotzinger E, Sticker M, Pircher M, and Fercher AF (2001): Mea-
surement and imaging of birefringence and optic axis orientation by phase resolved polarization sensitive optical coherence tomography, Opt. Express 9(13): 780-790.
Jiao SL, Yao G, and Wang LHV (2000): Depth-resolved two-dimensional Stokes vectors of backscattered light and Mueller matrices of biological tissue measured with optical coherence tomography, Appl. Opt. 39(34): 6318-6324.
2 4 6 MUELLER OPTICAL COHERENCE TOMOGRAPHY
Jiao SL and Wang LHV (2002): Jones-matrix imaging of biological tissues with quadruple- channel optical coherence tomography, J. Biomed. Opt. 7(3): 350-358.
Jiao SL and Wang LHV (2002): Two-dimensional depth-resolved Mueller matrix of biological tissue measured with double-beam polarization-sensitive optical coherence tomography, Opt. Lett. 27(2): 101-103.
Jiao SL, Yu WR, Stoica G, and Wang LHV (2003): Contrast mechanisms in polarization- sensitive Mueller-matrix optical coherence tomography and application in burn imag- ing, AppL Opt 42(25): 5191-5197.
Jiao SL, Yu WR, Stoica G, and Wang LHV (2003): Optical-fiber-based Mueller optical coherence tomography, Opt Lett 28(14): 1206-1208.
Jiao SL, Todorovic M, Stoica G, and Wang LHV (2005): Fiber-based polarization-sensitive Mueller matrix optical coherence tomography with continuous source polarization modulation, AppL Opt 44(26): 5463-5467.
Kemp NJ, Zaatari HN, Park J, Rylander HG, and Milner TE (2005): Form-biattenuance in fibrous tissues measured with polarization-sensitive optical coherence tomography (PS-OCT), Opt Express 13(12): 4611-4628.
Li J, Yao G, and Wang LHV (2002): Degree of polarization in laser speckles from turbid media: Implications in tissue optics, J. Biomed. Opt 7(3): 307-312.
Liu B, Harman M, and Brezinski ME (2005): Variables affecting polarization-sensitive optical coherence tomography imaging examined through the modeling of birefringent phantoms, J. Opt Soc. Am. A 22(2): 262-271.
Makita S, Yasuno Y, Endo T, Itoh M, and Yatagai T (2005): Jones matrix imaging of biological samples using parallel-detecting polarization-sensitive Fourier domain optical coherence tomography, Opt Rev. 12(2): 146-148.
Makita S, Yasuno Y, Sutoh Y, Itoh M, and Yatagai T (2003): Polarization-sensitive spectral interferometric optical coherence tomography for human skin imaging, Opt Rev. 10(5): 366-369.
Moreau J, Loriette V, and Boccara AC (2003): Full-field birefringence imaging by thermal-light polarization-sensitive optical coherence tomography. I. Theory, Appl. Opt. 42(19): 3800-3810.
Moreau J, Loriette V, and Boccara AC (2003): Full-field birefringence imaging by thermal-light polarization-sensitive optical coherence tomography. II. Instrument and results, Appl. Opt. 42(19): 3811-3818.
Park BH, Pierce MC, Cense B, and de Boer JF (2004): Jones matrix analysis for a polarization-sensitive optical coherence tomography system using fiber-optic compo- nents, Opt. Lett 29(21): 2512-2514.
Park BH, Pierce MC, Cense B, and de Boer JF (2005): Optic axis determination accuracy for fiber-based polarization-sensitive optical coherence tomography, Opt. Lett. 30(19): 2587-2589.
Pierce MC, Park BH, Cense B, and de Boer JF (2002): Simultaneous intensity, birefrin- gence, and flow measurements with high-speed fiber-based optical coherence tomog- raphy, Opt Lett. 27(17): 1534-1536.
Pircher M, Goetzinger E, Leitgeb R, and Hitzenberger CK (2004): Transversal phase resolved polarization sensitive optical coherence tomography, Phys. Med. Biol. 49(7): 1257-1263.
Ren HW, Ding ZH, Zhao YH, Miao JJ, Nelson JS, and Chen ZP (2002): Phase-resolved functional optical coherence tomography: Simultaneous imaging of insitu tissue struc- ture, blood flow velocity, standard deviation, birefringence, and Stokes vectors in human skin, Opt. Lett. 27(19): 1702-1704.
FURTHER READING 2 4 7
Roth JE, Kozak JA, Yazdanfar S, Rollins AM, and Izatt JA (2001): Simplified method for polarization-sensitive optical coherence tomography, Opt. Lett. 26(14): 1069-1071.
Saxer CE, de Boer JF, Park BH, Zhao YH, Chen ZP, and Nelson JS (2000): High-speed fiber-based polarization-sensitive optical coherence tomography of in vivohuman skin, Opt. Lett. 25(18): 1355-1357.
Schoenenberger K, Colston BW, Maitland DJ, Da Silva LB, and Everett MJ (1998): Mapping of birefringence and thermal damage in tissue by use of polarization-sensitive optical coherence tomography, Appl. Opt. 37(25): 6026-6036.
Todorovic M, Jiao SL, and Wang LHV (2004): Determination of local polarization prop- erties of biological samples in the presence of diattenuation by use of Mueller optical coherence tomography, Opt. Lett. 29(20): 2402-2404.
Vansteenkiste N, Vignolo P, and Aspect A (1993): Optical reversibility theorems for polarization—application to remote-control of polarization, J. Opt. Soc. Am. A 10(10): 2240-2245.
Yang Y, Wu L, Feng YQ, and Wang RK (2003): Observations of birefringence in tis- sues from optic-fibre-based optical coherence tomography, Meas. Sei. Technol. 14(1): 41-46.
Yao G and Wang LHV (2000): Propagation of polarized light in turbid media: Simulated animation sequences, Opt. Express 7(5): 198-203.
Yasuno Y, Sutoh Y, Makita S, Itoh M, and Yatagai T (2003): Polarization sensitive spectral interferometric optical coherence tomography for biological samples, Opt. Rev. 10(5): 498-500.
Yasuno Y, Makita S, Endo T, Itoh M, Yatagai T, Takahashi M, Katada C, and Mutoh M (2004): Polarization-sensitive complex Fourier domain optical coherence tomography for Jones matrix imaging of biological samples, Appl. Phys. Lett. 85(15): 3023-3025.
Zhang J, Guo SG, Jung WG, Nelson JS, and Chen ZP (2003): Determination of birefrin- gence and absolute optic axis orientation using polarization-sensitive optical coherence tomography with PM fibers, Opt. Express 11(24): 3262-3270.
Zhang J, Jung WG, Nelson JS, and Chen ZP (2004): Full range polarization-sensitive Fourier domain optical coherence tomography, Opt. Express 12(24): 6033-6039.
CHAPTER 11
Diffuse Optical Tomography
11.1. INTRODUCTION
The term diffuse optical tomography (DOT) refers to the optical imaging of biological tissue in the diffusive regime. Since it has a l/e penetration depth on the order of 0.5 cm, NIR light around 700-nm wavelength can penetrate several centimeters into biological tissue. As a result, DOT can image the human breast and brain. Image reconstruction in DOT involves both the forward and the inverse problems. The forward problem usually uses the diffusion equation to predict the distribution of reemitted light on the basis of presumed parameters for both the light source and the object. The inverse problem uses the forward problem to reconstruct the distributions of the optical properties of the object from a measured data set. Since the inverse problem is ill-posed, recovering imaging information from diffuse photons remains a challenge. As a rule of thumb, the spatial resolution of DOT is on the order of 20% of the imaging depth; hence, DOT is a low-resolution imaging technology. Nevertheless, DOT provides valuable rapid functional imaging at low cost.
11.2. MODES OF DIFFUSE OPTICAL TOMOGRAPHY
In a DOT system, sources and detectors are placed around the object to be imaged in various geometric configurations. Common geometric configurations, suited for different applications, fall into planar transmission, planar reflection, and cylindrical reemission. Most anatomical sites can be imaged in the planar reflection configuration. Human limbs as well as small animals can be imaged in either the planar transmission or the cylindrical configuration. Human breasts can be imaged in all three configurations. Generally, while one source illuminates the object, all detectors measure the reemitted light. This process is repeated with each source to complete a measurement data set; subsequently, images are reconstructed by computer.
Biomedical Optics: Principles and Imaging, by Lihong V. Wang and Hsin-i Wu Copyright © 2007 John Wiley & Sons, Inc.
249
2 5 0 DIFFUSE OPTICAL TOMOGRAPHY
TABLE 11.1. Modes of DOT with Idealized Parameters0
Mode Source Light Φ,ν^', t') Reemitted Light Om(r, /; r\ t')
Time domain Impulse: h(r')h(t') Frequency Amplitude-modulated:
domain hir')[Ds 4- As cos(oo/' + φ.ν)1
Direct current DC: Dsh(r') (DC)
"Symbol key: Φ—fluence rate; D—DC; A—AC amplitude; ω—angular frequency of AC; φ—phase of AC; r'—source location; t'—source time; r—detection location; t—detection time; subscripts s and m represent source and measurement, respectively
According to the type of signal, DOT is usually classified into three modes (Table 11.1): time domain, frequency domain, and direct current (DC). In all three modes, the reemitted light has the same general form as the source light since the system is linear and time-invariant. In the time-domain mode, the source light is ultra-short-pulsed (typically a few picoseconds wide), and the reemitted light pulses are broadened. In the frequency-domain mode, the source light inten- sity is amplitude-modulated sinusoidally at typically hundreds of MHz, and the reemitted light modulation has reduced modulation depth (AC amplitude/DC). In the DC mode, the source light is usually time-invariant, but it is sometimes modulated at low frequency (e.g., kHz) to improve the SNR or to encode the source. Such low-frequency modulation, however, does not attain the benefits of the frequency-domain mode.
The time-domain and the frequency-domain modes are mathematically related via the Fourier transformation. If measured at many frequencies (including DC) in a sufficiently broad bandwidth, frequency-domain signals can be converted to the time domain using the inverse Fourier transformation. Therefore, the time-domain mode is mathematically equivalent to the combination of the frequency-domain and DC modes. Further, the DC mode is a zero-frequency special case of the frequency-domain mode.
Among the three modes, the time-domain mode is the most information-rich but the slowest in data acquisition and the most expensive. The full extent of the information content, however, is yet to be fully explored. The frequency-domain mode, which typically operates at only a single modulation frequency, contains less information than does the time-domain mode but is faster and less expensive. In addition, it provides better SNR by means of narrowband detection. The DC mode contains no direct information about time of flight and hence has the most limited capability for separating absorption from scattering in a heterogeneous medium; it is the fastest and the least expensive, however.
In the following sections, experimental systems representative of all three modes are introduced. An in-depth description of the frequency-domain mode follows since this mode is the most mature and dominant to date.
Time-resolved: 4>m(r, t\ r', t') Amplitude-modulated:
Dm{r\r') + Am(ry^) cos(cor + $m(r\r'))
DC: Dm{r\r')
TIME-DOMAIN SYSTEM 251
11.3. TIME-DOMAIN SYSTEM
In the time-domain mode, temporal responses to an ultrashort laser pulse are measured around the scattering object to be imaged. Each temporal response represents reemitted light intensity as a function of time; each actually equals the convolution of three functions: the temporal PSF due to light propagation in the medium, the pulse profile of the laser beam, and the impulse response of the detection system. The responses can be temporally resolved using a streak camera (see Chapter 8) or a time-correlated single-photon counting system. The latter has the advantages of larger detection area, better temporal linearity, lower cost, and higher dynamic range over the former; however, it has the disadvantage of slower data acquisition.
A 32-channel imaging system, based on time-correlated single-photon count- ing, was constructed at University College London (UCL) (Figure 11.1). A laser provides picosecond light pulses with tunable wavelength. After going through a beamsplitter, the lightbeam is attenuated to a preset intensity by a neutral-density filter. After further passing through a shutter and a fiber coupler, the lightbeam is coupled to a 1 x 32 fiber switch, which selects one of the 32 source fibers
sl·*-
Reference »
Photodiode Fast fan-out module \
Amplifier-timing Delay discriminator
Nuetral-density V a r i a b l e Longpass filter Picosecond filter optical / time analyzer
attenuator / Preamplifier \
Computer | 1
Pulsed laser
Figure 11.1. Schematic of the UCL time-resolved DOT system. For clarity, only one source fiber and one detection channel are shown.
2 5 2 DIFFUSE OPTICAL TOMOGRAPHY
at a time. The source fibers are routed to distributed locations on the surface of the object.
Reemitted photons are simultaneously detected by 32 detection channels, each with a large-diameter fiber bundle. Since the 32 detection positions are distributed around the object, the received optical signals have vastly different intensities. In other words, some detection fibers—typically those that are close to the source fiber—receive strong optical signals, while other detection fibers—typically the ones that are far away from the source fiber—receive weak optical signals. Since the single-photon counting technique requires that either one photon or none be measured per laser pulse, stronger optical signals are attenuated by computer-controlled variable optical attenuators to extend the dynamic range of the measurement. Then, a longpass filter reduces the ambient light that has shorter wavelengths than the signal light.
In each detection channel, the filtered signal light is delivered to the photocath- ode of a microchannel plate-photomultiplier tube (MCP-PMT), which converts the optical pulse into an analog electronic pulse. After preamplification, the ana- log pulse is shaped by a constant fraction discriminator into a logic pulse with high timing accuracy. In parallel, part of the main beam is split off by a beam- splitter to provide a reference lightbeam. The reference optical pulse is converted by a photodiode into an electronic pulse. The electronic pulse is also preampli- fied and shaped into a logic pulse by a combined amplifier-timing discriminator unit. Each reference logic pulse is then time-delayed and converted through a fast fan-out module into 32 outputs. Each output is connected to a picosecond time analyzer. Each picosecond time analyzer compares the signal logic pulse with the delayed reference logic pulse to measure the time of flight of each indi- vidual photon. The detection cycle is repeated with many laser pulses. At the end, the times of flight of the individual photons detected by each channel build a histogram, which represents the associated temporal spread function.
Once histograms are acquired for a source location, the fiber switch shifts the source light to the next source fiber. This process is repeated through all 32 source fibers to complete the measurement data set. Then, image reconstruction is performed by computer.
11A DIRECT-CURRENT SYSTEM
A DC system based on frequency-division multiplexing was built using 32 lasers and 32 detectors at Massachusetts General Hospital (MGH) (Figure 11.2). Half of the lasers operate at 690-nm wavelength and the other half at 830 nm. The lasers are encoded with 32 frequencies from a master clock; these frequencies are dis- tributed uniformly between 6.4 and 12.6 kHz. The laser outputs are fiberoptically coupled to 16 paired positions around the object to be imaged. Reemitted light is collected through 32 channels of optical fibers and detected by 32 avalanche photodiodes (APDs) in parallel. The electronic output of each APD is bandpass- filtered. The filtered signal is amplified by a programmable-gain stage, which
FREQUENCY-DOMAIN SYSTEM 2 5 3
Laser 1 6.4 kHz
Detector channel (1 of 32) I
(%)—H k ,—, ^ stage
V^y Ί l Object Bandpass A/D convenor . Laser 2, f l , t e r
• 6.6 kHz
Figure 11.2. Schematic representation of the MGH DC imaging system.
matches the signal amplitude to the input range of the analog-to-digital (A/D) converter. The amplified signal is digitized at 45 kHz by the A/D converter and subsequently transferred to a computer. Then, the digitally encoded signal is decoded by the computer on the basis of the modulation frequencies to recover the reemitted optical signal components originating from each source simultaneously. Since all sources and detectors function concurrently, the data-acquisition rate is high. The system is well suited to observing rapid physiological phenomena.
11.5. FREQUENCY-DOMAIN SYSTEM
Before describing a frequency-domain imaging system, we first introduce a single-channel sensing system (Figure 11.3). The output power of a laser diode is modulated by an AC signal at a radiofrequency / (e.g., 200 MHz) from a func- tion generator. The modulated light is delivered to the object through an optical
Optical fiber
Figure 11.3. Schematic representation of a single-channel frequency-domain sensing system.
2 5 4 DIFFUSE OPTICAL TOMOGRAPHY
fiber. Reemitted light is detected by a PMT through another optical fiber. If the PMT has a constant gain, the output electronic signal will have frequency / and can be digitized by an A/D converter. As required by the Nyquist criterion, the A/D sampling frequency must be greater than 2f.
The sampling frequency can be reduced by heterodyne detection, which is applicable to narrowband signals. For heterodyne detection, a local oscillator of frequency / -f Δ/—where Δ / is typically a fraction of / (e.g., tens of kHz)—is produced by another function generator to modulate the gain of the PMT. Because it is proportional to the product of the input light power and the gain, the output of the PMT has multiple frequencies: 0, Δ/, / , / -f Δ / , and 2 / -I- Δ / . The Δ / component is selected by a bandpass filter and then digitized by an A/D converter. The lower frequency Δ / results in less lengthy data.
The amount of data can be further reduced by lock-in detection, which is applicable to single-frequency signals. For lock-in detection, a reliable reference of frequency Δ / is produced by another heterodyne channel. An electronic mixer, which multiplies the two input signals, mixes the outputs from the two function generators to produce a signal of two frequencies: Δ / and 2 / + Δ / . The Δ / component is selected by a bandpass filter to be the reference. The lock-in detector inputs the signal and the reference and outputs the amplitude and the phase of the signal.
The principle of dual-phase lock-in detection (Figure 11.4), also referred to as IQ detection ("Γ for in-phase and "Q" for quadrature), is described as follows.
1
Reference sin (2π/ί)
^
r
90° phase shifter
cos (2nft) ψ
Mixer
1 r
Lowpass filter
Mixer
y r
Lowpass filte r
^ ^
4
^
Signal A sin (2nft + φ)
A cos (φ) / 2
1 r
Amplitude- phase unit
A
A sin (φ)/2
k
Figure 11.4. Block diagram of a dual-phase lock-in detection system.
FREQUENCY-DOMAIN SYSTEM 2 5 5
Both the signal and the reference oscillate at frequency / . The signal is repre- sented by A sin(27r/i -f φ), where A denotes the amplitude, φ denotes the phase, and t denotes time. The reference is represented simply by s'm(2itfl -f- φ) without taking the amplitude into account. The system consists primarily of two mixers, two lowpass filters, and a 90° phase shifter. In the first channel, the mixer mixes the signal and the reference to produce a signal consisting of a DC and a second- harmonic 2 / component. The first lowpass filter passes the DC component and rejects the second-harmonic component, where the DC component can be rep- resented by Si — ^ Α ^ 8 φ apart from a constant factor. In the second channel, the reference signal is first phase shifted by 90° to produce a signal proportional to cos(o)0· Then, the second mixer followed by the second low-pass filter pro- duces a DC component that can be represented by SQ = jA ήηφ. Finally, the amplitude-phase unit outputs the two unknowns—A and φ—based on the two DC components:
A = 2y/sj + S2Q, (11.1)
φ = 3 Γ ^ η - ^ . (11.2) Si
A scanning frequency-domain imaging system was built at Dartmouth College (Figure 11.5). Unlike the system in Figure 11.3, this system automatically scans both the light incidence position and the light detection position around the object using 32 large-core fiberoptic bundles—16 for light delivery and 16 for light
Power Signal generator Signal generator supply 100.000 MHz 100.001 MHz
1 T |»
Fiberoptic bundles
A/D board [
Computer f
Figure 11.5. Schematic of the Dartmouth College frequency-domain imaging system.
2 5 6 DIFFUSE OPTICAL TOMOGRAPHY
detection. Two signal generators—sharing the same timebase—produce signals of radiofrequency fx = 100.000 MHz and f2 = 100.001 MHz, respectively. A bias-T mixes the DC current from a power supply and the radiofrequency current of f\. A laser, whose output light power is modulated by the output of the bias-T, provides 800-nm NIR light. A linear translation stage scans to one of the 16 light delivery fiberoptic bundles to receive the modulated laser light. The selected fiberoptic bundle delivers the light to the object at one of the 16 positions. Another linear translation stage scans to each of the 16 light detection fiberoptic bundles to gather reemitted light from every detection position. A filter wheel—made of neutral-density filters of various optical densities—attenuates stronger optical signals more strongly to compress the range of the received optical signals. A PMT detects the attenuated light and outputs a signal that has various frequencies. A bandpass filter passes the fi — f\ component. An A/D board digitizes the filtered signal and then transfers the digital data to a computer for data processing. This detection process is repeated through all 16 light delivery fiberoptic bundles to complete the data acquisition.
11.6. FREQUENCY-DOMAIN THEORY: BASICS
As shown in Chapter 5, the time-resolved diffusion equation is
3 Φ ( Γ , 0
c3t + μαΦ(Γ, t) - V · [D V4>(r, f)] = 5(r, t). (11.3)
Here, Φ denotes the fluence rate, c denotes the speed of light in the scatter- ing medium, \ia denotes the absorption coefficient, D denotes the diffusion coefficient, and S denotes the source power density. Sometimes, the diffusivity D'—defined by D' = cD—is used instead of D.
For monochromatic light, Eq. (11.3) can be rewritten as
dU(rJ) + c[LaU(r, t) - cV · [D VI/(r, t)] = q(r, f), (11.4) at
where U denotes the photon density and q denotes the photon density source strength. The following relations are used in the conversion:
Φ ( Γ , / ) = hv U(?,t)c, (11.5)
S(r,t) = hvq(r,t), (11.6)
where hv denotes the photon energy. We first examine the spatial impulse response in an infinite homogeneous
scattering medium. A point photon density source is denoted by the following phasor expression:
q(r, t) = [A + Ββχρ(-ιωί)]δ(Γ). (11.7)
FREQUENCY-DOMAIN THEORY: BASICS 2 5 7
Here, ω denotes the angular frequency and A and \B\ denote the DC and the AC source amplitudes, respectively. The ratio \B\/A, termed the modulation depth, ranges between 0 and 1 because q > 0 must hold. From B — \B\ exp(—/φ#), we have
Bexp(-icut) = |2?|exp(—ι'ωί - /φβ). (11.8)
Of course, it is the real part of a phasor expression that represents the actual oscillation:
Re{|^| exp(-/oof - ι'φ*)} = |£|cos(cof+ <M. (11.9)
Because the diffusion equation is linear, we assume the solution to be as follows:
U(r, t) = UDC(?) + UAC(?) βχρ(-ίωί) , (11.10)
where U\c(r) is complex. For the DC part, we have
c\xaUOC(r)-cD V 2UDC(7) = Ah(r). (11.11)
The solution is given by
exp(-^effr) UOC(r) = A— , (Π.12)
4ncDr
where ^eff = Λ/VLJD. For the AC part, we have
-iG)E/Ac(r) +c\xaUAC(?) -cD V2UAC(r) = Bh(r). (11.13)
This equation can be reformulated to a Helmholtz wave equation:
(V2+fc2)i/Ac(?) = - £ ^ , (11.14) cD
where
The solution is given by
t* = - < * · « + '·«>. ( 1 1 . 15 ) cD
ϋΑ^) = Βψφ, (11.16) 4ncDr
which represents a photon-density wave, where k is the propagation constant.
2 5 8 DIFFUSE OPTICAL TOMOGRAPHY
Substituting Eqs. (11.12) and (11.16) into Eq. (11.10), we obtain the overall response:
ii<- ^ Aexp(-^effr) + £exp( / / : r - /Q)0 i / ( r , i ) = : · (Π.17)
AncDr When ω -> 0, the AC solution should approach the DC solution. If ω = 0, Eq. (11.15) becomes
2 _ Z^a __ _ 2 D ~~ f '
As a result, Eq. (11.16) reduces to
ϋ^) = Β^ψ^, (11.18) AncDr
which is identical to Eq. (11.12) except for the amplitudes. Propagation constant k can be separated into real and imaginary parts:
™ ι = (^)',4™(^Β„£),
t, = Imltl = ( ^ f ^ ) "'cos (i arc« £) (11.20)
Substituting k = kr -f iki into Eq. (11.16), we obtain
exp(/A;rr) UAC(r) = flexp(-*fr) / V nr ' . (11.21)
Factor exp(//:rr) represents the phase delay of the photon-density wave, which approximately equals the product of the average time of flight of the photons and the angular frequency ω (see Problems 11.1-11.3); therefore, the phase of UAC(?) provides information about the effective path length that the diffuse photons have taken. The real part kr can be converted to a wavelength by
2π λ = — . (11.22)
kr
Factor exp(—£zr) represents the attenuation of the photon-density wave due to diffusion and absorption in addition to the 1/r geometric decay. Even in the absence of absorption (i.e., \ia = 0), the photon-density wave is still strongly damped as a result of diffusion. Because they follow physical paths of various lengths, photons that reach the observation point arrive at different times. There- fore, photons in the peaks and troughs of a photon-density wave mingle as the wave propagates, which dampens the modulation depth of the overall response.
FREQUENCY-DOMAIN THEORY: BASICS 259
Diffuse photon-density waves can be used to measure optical properties. Once the complex k is measured, \ia and D can be deduced. From Eqs. (11.19) and (11.20), we obtain
ωRe{*2} , n w Va = ΓΊΤΤΓ' (11.23)
c Im{£2} Jc2\u + ω2
c|£|2
Diffuse photon-density waves possess many of the common wave characteristics, such as reflection, refraction, diffraction, and dispersion. For example, SnelFs law is applicable to diffuse photon-density waves:
sin θ; λ; — r L = —. (11.25) sin Θ, \ t
Here, Θ,· and Θ, denote the angles of incidence and transmission, respectively; λ, and \ t denote the wavelengths of the diffuse photon-density waves in the incident and transmitted media, respectively.
Because of its long wavelength (cm scale), a photon-density wave typically provides imaging in the near field, where the spatial resolution is related to the SNR rather than the wavelength. In far-field imaging, spatial resolution based on the Rayleigh criterion is related to ~ λ/2; yet, it can be improved to "superreso- lution" by a factor related to the SNR as well. Ultimately, the resolution in either case is limited by the SNR.
One must distinguish between an optical wave and a photon-density wave. The former is an electromagnetic vector wave, whereas the latter is a photon-density scalar wave. Note also that the latter is based on the former. Furthermore, their wavelengths and attenuation mechanisms are different.
Example 11.1. Illustrate the null plane between two photon-density sources that are 180° out of phase.
A representative MATLAB source code is shown below:
c = 3e8 /1 .37; %m/s mua = 0.1E2; %/m mus = 10E2; %(/m) mus = mus', g = 0 f = 200E6; %Hz
D = 1/(3*(mua+mus)); %m k = sqrt(( -c*mua + i * 2 * p i * f ) / ( c * D ) ) ; %wave vector
disp([ 'Absorpt ion coeff . mua (/cm) = ' , num2str(mua*1E-2)]) disp(['Reduced scatter ing coeff . mus'' (/cm) = ' , num2str(mus*1E-2)]) disp([ 'Frequency f (MHz) = ' , num2str( f*1E-6)] ) disp([ 'Wavelength (cm) = ' , num2st r (2*p i / rea l (k ) *1E2) ] )
2 6 0 DIFFUSE OPTICAL TOMOGRAPHY
disp(['Decay const (cm) = ', num2str(1/imag(k)*1E2)])
xs = 1E-2; %(m) sources at (xs, 0) & (-xs, 0) yd = 3E-2; %(m) detector at (xd, yd) xd = (-3:0.02:3)*1E-2; %m
r1 = sqrt((xd - xs).A2 + yd."2); % distance b/t -src & detector U1 = -exp(i*k*r1)./(4*pi*c*D*r1); % negative src
r2 = sqrt((xd + xs)."2 + yd.A2); % distance b/t +src & detector U2 = exp(i*k*r2)./(4*pi*c*D*r2);
figure(1) subplot(3,1,1) plOt([-XS xs]*1E2, [0, 0], '*', [0], [3], Ό', [0 0], [0 3]) text(-xs*1E2, 0.5, '+Source') text(+xs*1E2, 0.5, '-Source') text(0, 2.5, 'Scanning Detector') axis([-3 3 0 3]) xlabel('Source & Detector Positions (cm)') ylabel('y (cm)') title('Null line')
subplot(3,1,2) plot(xd*1E2, abs(U1+U2)) xlabel('Detector position (cm)') ylabel('Amplitude')
subplot(3,1,3) plot(xd*1E2, unwrap(angle(U1+U2))*180/pi) xlabel('Detector position (cm)') ylabel('Angle (deg)') grid
figure(2) % optional r = (1:1:20)*1E-2; U = exp(i*k*r)./(4*pi*c*D*r); subplot(2,1,1) semilogy(r*1E2, abs(U)) xlabel('Source-detector distance (cm)') ylabel('Amplitude') title('Propagation of PDW)
subplot(2,1,2) plot(r*1E2, unwrap(angle(U))*180/pi) xlabel('Source-detector distance (cm)') ylabel('Angle (deg)') grid
The text output in MATLAB is shown below:
Absorption coeff. mua (/cm) = 0.01
Reduced scattering coeff. mus' (/cm) = 10
Frequency f (MHz) = 200
FREQUENCY-DOMAIN THEORY: LINEAR IMAGE RECONSTRUCTION 2 6 1
Wavelength (cm) = 7.382 Decay const (cm) = 0.9878
The graphical output in MATLAB is shown in Figure 11.6. The detector is scanned along the y = 3 line. Because the two sources are 180° out of phase, the phase difference between any two points that are symmetric about the null plane must be 180°.
-Φ-
Null line
+Source ι -*
Scanning detector
-Source
-1 0 1 Source and detector positions (cm)
x 10-
-1 0 1 Detector position (cm)
- 3 - 2 - 1 0 1 2 3 Detector position (cm)
Figure 11.6. Illustration of the null plane—a null line in 2D space—in the antiphase dual-source response.
11.7. FREQUENCY-DOMAIN THEORY: LINEAR IMAGE RECONSTRUCTION
In this section, we illustrate frequency-domain image reconstruction in an infinite medium using a simple linear inverse theory. The object to be imaged is assumed to have absorption contrast only. The absorption coefficient μα(?) is expressed as
Vatf) = μ*ο(?) + δμβ(Γ), (11.26)
2 6 2 DIFFUSE OPTICAL TOMOGRAPHY
where μαο(/) represents the background absorption coefficient and δμα(?) repre- sents the differential absorption coefficient of the heterogeneities relative to the background. In the forward problem, the perturbation in photon density is calcu- lated from presumed μαο(/) and δμα(?) values. In the inverse problem, δμ0(?) is calculated from the measured data.
For simplicity, we define D as 1/(3μ^) in this section so that D does not depend on μα. We start by rewriting Eq. (11.14) as
i Ί -. - 8(r — rs) (V2 + k2)UAC(r, rs) = -B ", (11.27) cD
where rs denotes the location of the photon-density source. Substituting Eq. (11.26) into Eq. (11.15), we obtain
Jfc2 = fcj + 0( r ) , (11.28)
where
Let
2 = ζ £ μ ^ + ι ω > ( 1 ] 2 9 ) cD
0Cr) = - ^ (H.30)
UAC(r, rs) = U0(r, rs) + f/sc(r, r5), (11.31)
where i/o represents the AC photon density in a homogeneous medium that has the background optical properties and Use represents the differential AC photon density due to the heterogeneities.
Substituting Eq. (11.31) into Eq. (11.27) yields
[V2 + kl + 0(f)] [U0(?, rs) + i/Sc(r, rs)] = - f l 8 ( r " r j ) . (11.32) cD
The diffusion equation for Uo is given by
(V2 + *g)l/o(r,r5) = - f l 8 ( r ^ r f ) > (11-33)
Taking the difference between Eqs. (11.32) and (11.33), we obtain
(V2 + *g)t/Sc(?, r5) = -O(r)[i /o(r, r,) + f/Sc(r, r,)]. (11.34)
If δμα(?) «: μα0(?), we assume that the Born approximation, i/sc(?,?$) <3C £Λ)(/, Γ$)> is valid; hence, Eq. (11.34) becomes
(V2 + k20)Usc(7, rs) = -O(r ) t / 0 ( r , rs), (11.35)
FREQUENCY-DOMAIN THEORY: LINEAR IMAGE RECONSTRUCTION 2 6 3
which can be solved by the following Green function method. For a point source in an infinite medium, Eq. (11.35) becomes
(V2 +k$)G(r - rs) = -8 ( r - rs), (11.36)
where G, referred to as a Green function, is given by
G(r-rs) = ——————. (11.37) 4π|Γ -rs\
The solution to Eq. (11.33) has a similar form:
B txp(ik0\r-rs\) U0(r, rs) = — — - — — — — . (11.38)
cD 4n\r-rs\
In response to the general forcing function on the right-hand side of Eq. (11.35), we have the following solution to the forward problem, based on Green's theorem:
t/sc(r,r ,) = [ U0(r',rs)O(r')G(r-rf) dr'. (11.39)
Because G is shift-invariant, this expression is actually equivalent to a con- volution. The physical meaning of the Green function method is graphically represented in Figure 11.7. As can be seen, Uo(r', rs) represents the light propa- gated from the source to a point inside the object; Uo(r', rs)0(rf) serves as a new source for further light propagation to the detector; the propagation is described by G(r - ?).
To demonstrate the inverse problem for image reconstruction, we discretize Eq. (11.39) in an xy plane:
Use (rj, ?„·) = £ £ i/o (r;„, rsi) O (?'mn) G (?, - ?mn) h\ (11.40)
Here, i is an index of the source, j is an index of the detector, m and n are indices of the coordinates of the perturbations, and h is the size of each grid
Source
U0(r ,rs)
0(r)
Figure 11.7. Illustration of the Green function method.
2 6 4 DIFFUSE OPTICAL TOMOGRAPHY
element (for simplicity, assumed to be cubical). Although the sources, perturba- tions, and detectors are restricted to the plane of interest, photons still propagate in 3D space.
We rewrite Eq. (11.40) as
Use (rj, ?si) = E Σ WV.mn*\La {?'„„) . 0 1 41) m n
where
Wilmn = - i /o {?„„, ?si) G (o - r'mn) h-. (11.42)
To compress each pair of indices to a single index (analogous to counting the squares in a chessboard sequentially), we let
/' = i + jNh (11.43)
m — m + nNm, (11.44)
where Nj is the range of / (the number of values represented by /) and Nm is the range of m. Thus, Eq. (11.41) becomes
t/scO") - Σ Wrtm*[ia(r'm,), (11.45) m'
which can be rewritten in matrix form with separated real and imaginary parts:
"Re{i/scO Jm{i/sc(i /x l | - | τ - ' " ' - -1 i L-f-αν». /J, (11-46)
or
[t/scl = [ΗΊ[δμβ]. (Π.47)
Of course, we can also use complex matrices directly. For matrix [W], the number of rows Nr equals twice the range of /' or twice the number of measurements, and the number of columns Nc equals the range of m' or the number of volume elements (voxels). Matrix [W], which is generally nonsquare, can be inverted to solve for [δμβ]:
[h[La] = [W]- l[Uscl (H.48)
Here, [δμα] represents the image that we are pursuing. Matrix [W] is referred to as the Jacobian matrix; it is also referred to as the sensitivity matrix because it relates the changes in the measurements to the perturbations in the optical properties. The forward problem provides matrix [W], For regular boundaries
FREQUENCY-DOMAIN THEORY: LINEAR IMAGE RECONSTRUCTION 2 6 5
such as semiinfinite, cylindrical, and spherical ones, [W] can be computed by analytical methods. In general, however, [W] must be computed by numerical methods.
Singular-value decomposition (SVD) can be used to invert [W], First, [W] is decomposed into three matrices as follows:
mg(wMNr,Nc[V] T
Nc%Nc. (Π.49)
The middle matrix is diagonal; the subscripts describe the matrix dimensions; [U] and [V] have the following orthogonality:
[Uf[U] = [Il[V]T[V] = Ul (Π.50)
where [/] denotes identity matrix. Then, inversion leads to
[W]~l = {[V] r}-1[diag(u; i ) r I [ l / r 1 - [V] L a g (±) [U]T. (11.51)
To avoid overflow, we set 1/WJ to zero when Wj is less than a preset threshold. Alternatively, we can use the following smoothing algorithm:
w'=Wj + —9 (11.52) wj
where σ is a free parameter. Both methods trade accuracy for stability. Iterative methods, such as the algebraic reconstruction technique (ART) or the
simultaneous iterative reconstruction technique (SIRT) (Appendix 11 A), may be used to solve Eq. (11.47) as well. The advantage of an iterative method over the preceding matrix inversion method is that the former permits hard constraints. For instance, because the reconstructed absorption coefficient should always be nonnegative, a negative value in any voxel can be set to zero at the end of each iteration cycle.
Up until this point, only an infinite medium has been considered. With a finite medium, an extrapolated boundary condition can be used (see Chapter 5). The Green function approach takes the following form instead of Eq. (11.39):
Usc(r) = [ U0(r',rs)O(r')G(r,r')dr'+ -j- Jv 4π
- _, dUsc -., dGl G(r,r')-^-Usc(7)—\dS. (11.53) /.[■
Here, S denotes the extrapolated boundary, V denotes the volume enclosed by 5, and n' denotes the outward surface normal.
Since t/sc is approximately zero on the extrapolated boundary, the surface integral of the second term in the brackets in Eq. (11.53) vanishes. If we choose a Green's function that satisfies the homogeneous Dirichlet boundary condition
2 6 6 DIFFUSE OPTICAL TOMOGRAPHY
(namely, G is zero on the extrapolated boundary), the surface integral of the first term in the brackets in Eq. (11.53) vanishes as well. Therefore, Eq. (11.53) becomes
Usc(r)= I Uo{r\rs)0{r')G{r,r')dr'. (11.54) Jv
It is important to note that the G here is different from the G for an infinite medium. The Green functions that satisfy the homogeneous Dirichlet boundary condition for some regular geometric shapes (e.g., semiinfinite, spherical, cylin- drical spaces) are analytically available. Green's functions for more complex geometric shapes, however, can be computed only using numerical methods. Because no perturbation of optical properties exists outside the real boundary, the volume integration outside the original object is zero and, therefore, V can be reduced to the actual volume of the object.
Example 11.2. Derive Eq. (11.39).
Since the system is linear, a linear operator £ can be used to represent the system as follows:
G(r-rs) = £{h(r-rs)l (11.55)
Usc(r,rs) = £{O(r)U0(r,rs)}. (11.56)
A general forcing function 0(r)Uo(r, rs) can be expanded using 5(r — rs)
O(r)U0(r, rs) = f O(r')U0(r', rs)h(r - ?) dr\ (11.57)
which is based on the sifting property of the delta function (as if a point were sifted out of a function).
Substituting Eq. (11.57) into Eq. (11.56) and operating £ on the r-dependent quantities only, we obtain
f/sc(r, rs) = £{O(r)£/0(r, rs)} = Α [ O(r')U0(r\ rs)h(jr - ΐ') dA
- [ 0(r')Uo(r\rs)£{h(r-r')}dr' (11.58)
= f O(r')U0(r\rs)G(r-r')dr'.
Example 11.3. Implement the linear inverse algorithm in C.
The following data structure is defined (the source code is available on the Web at ftp://ftp.wiley.com/public/sci_tech_med/biomedical_optics):
FREQUENCY-DOMAIN THEORY: GENERAL IMAGE RECONSTRUCTION 2 6 7
typedef struct { double f; /* frequency (Hz). */ double c; /* speed of light in medium (cm/s). */ double h; /* grid size (cm), dx = dy = dz = h. */ int N; /* NxN grid. */ int gap; /* gap between src/det & boundary (grid). */ double muaO; /* mua of background (/cm). */ double musO; /* mus' of background (/cm). */
int obj_x, obj_y, obj_size; /* object location and size in grid. */ double dmua; /* delta mua of object (/cm). */ } ParamStru;
For the following parameters, the reconstructed image is shown in Figure 11.8:
void SetParam(ParamStru *par) { par->f = 500e6; par->c = 3E10 / 1.37; par->h = 0.2; par->N = 20; /* number of voxels in each direction. */ par->gap = 3; par->muaO = 0.1; par->musO = 10;
par->obj_x = 12; par->obj_y = 6; par->obj_size = 2; par->dmua = 0.01;
}
If the frequency is set to zero, the AC photon-density wave reduces to DC. The corresponding reconstructed image is shown in Figure 11.9.
11.8. FREQUENCY-DOMAIN THEORY: GENERAL IMAGE RECONSTRUCTION
In this section, we formulate and solve a general frequency-domain imaging problem in an infinite medium. The goal is to map both the absorption coefficient \\,a{?) and the reduced scattering coefficient \i's{r) from the measured photon- density distribution.
11.8.1. Problem Formulation
The starting point is again the time-dependent diffusion equation for a single fre- quency. From Eq. (11.4), the AC component of the photon-density wave [UAC(?)]
2 6 8 DIFFUSE OPTICAL TOMOGRAPHY
10 jc(x0.2cm)
.M
0.099 0.101 0.103 0.106 0.108 Absorption coefficient
Figure 11.8. Reconstructed image based on simulated frequency-domain data in an infi- nite medium.
ZU 1
18
16
14
12
10
8
6
4
2
(\ \
- "} ?
K
10 x (x0.2 cm)
15 20
r—r™*~~™—f ""f
0.099 0.101 0.103 0.106 0.108
Absorption coefficient
Figure 11.9. Reconstructed image based on simulated DC data in an infinite medium.
FREQUENCY-DOMAIN THEORY: GENERAL IMAGE RECONSTRUCTION 2 6 9
from a point source Bh(r — rs) satisfies
cV · [D(r)Vi/AC(r)] - [c[ia(r) - iü>]t/Ac(r) = -Bh(r - rs). (11.59)
Since [i's(r) is implicit in this equation, we reconstruct D(r) and μα(?), from which [i's(r) can be obtained.
As in the previous section, D(r) and μα(?) are decomposed as
D(r) = D0 + hD(r), (11.60)
μ«(?) = μαο + δμα(?), (11.61)
where Do and μαο represent the constant background optical properties and hμa(r) and hD(r) represent the differential optical properties of the hetero- geneities relative to the background. We duplicate Eq. (11.31) here:
i/Ac(r, rs) = £/0(r, rs) + £/Sc(r, r5). (11.62)
Substituting Eqs. (11.60)—(11.62) into Eq. (11.59), we obtain a differential equa- tion for t/sc> the solution of which is based on the Green function method:
- - f δμα(Γ) _ _ _ ^ t/scfo/, r,) = - / —-—G0(rd , r)UAc(r, rs)dr
3 °° (11.63) + / - ^ ν σ 0 ( Γ ^ Γ ) . ν ΐ / Α € ( Γ , Γ , ) £ / Γ .
Here, r</ denotes the location of the detector, r denotes a position within the object to be imaged, and Go denotes the Green function associated with the diffusion equation for a homogeneous medium that has the background optical properties. The integrals presented above are 3D over the entire object*. The first integral is what we obtained in the previous section from perturbation δμα. The second integral is the contribution from perturbation hD.
Experimentally, we measure a quantity directly related to (/AC- If the back- ground optical properties are known, we can compute U$. Next, we subtract UQ from UAC to yield Use- Then, from Eq. (11.63), we solve for hμa and δ£), which represent two images of different contrasts. Although hμa and hD appear in Eq. (11.63) in apparent linear form, this imaging problem is intrinsically non- linear because i/Ac is an implicit function of hμa and hD.
Example 11.4. Derive Eq. (11.63).
Substituting Eqs. (11.60) and (11.61) into Eq. (11.59), we obtain
cD0V 2UAC + cV · [hD VUAC] - [c([ia0 + hμa) - ico]UAC = -Bh(r - rs).
(11.64)
2 7 0 DIFFUSE OPTICAL TOMOGRAPHY
In a homogeneous medium with the background optical properties, Eq. (11.64) becomes
cD0V 2U0 - (c[ia0 - ίω)ϋ0 = -Bh(r - rs). (11.65)
Subtracting Eq. (11.65) from Eq. (11.64) and using Eq. (11.62), we obtain
V2£/sc + k2Usc = ^ ^ A C - TJ-V . (8DV£/AC), (Π.66) L>0> L>0
where ko is given by Eq. (11.29). Equation (11.66) is a Helmholtz equation and can be solved using the Green function method.
The Green function that satisfies differential equation
V2G0 + k 2 0G0 = -H?d-?) (H.67)
is given by
- _ expO'fcnliv/ — r\) G0(rd, r) = F ; ™d ^ " . (11.68) 4n\rd-r\
Thus, the solution to Eq. (11.66) is
- - f &ΜΆ(?) - -» _ _ _ Usc(rd,rs) = - I — — G 0 ( r d , r)UAC(r, rs)dr
Do! + —] V 'V>D(r)VUAC(r,rs)]G0(rd,r)dr. (11.69)
The second integral on the right-hand side is rewritten as
- ^ ^ V . [hD(?)VUAC(7, rs)]G0(rd, r) dr
= ^ f V5D(r) · Vi/AC(r, rs)G0(rd, r)dr (11.70)
+ -J- [hD(?)V2UAc(?,rs)Go(?d,7)d?.
According to Green's second identity
I (uV2v + Vw · Vv)dr = <b uVv -ndS, (11.71)
FREQUENCY-DOMAIN THEORY: GENERAL IMAGE RECONSTRUCTION 271
where S is an arbitrary surface that encloses the volume to be imaged and n is the surface normal, the second integral on the right-hand side of Eq. (11.70) is expressed as
1 /hD(r)V2OkC(rJs)Go(rdrr)d7
(11.72) = Ϊ Γ / [ S D ( ? ) G o ( ^ ' ^ I ^ A C ^ rs) dr
= J _ I [W(r)G0(rd9 r)]VI/Ac(r, rs) · n dS A) Js
~Έ~0ί V [ 8 D ( ? ) G o ( ^ · ? ) ] ' W A C ( ? ' ?*>rf?·
As S approaches infinity, the surface integral in Eq. (11.72) vanishes, which leads to
^ j hD(r)V2f/AC(?, r JGoir j , r) rfr
= - 7 ^ - [ V[*D<r)Go(jrd,r)].VUAC(r,rs)dr D
{° J (11.73)
= -jj-f *D(r)lVG0(rd, r) · Vt/AC(r, r , ) ]dr
- ^ / G ° f o ' r)[V&D(r) · Vi/AC(r, r,)]rfr.
The second term on the right-hand side of Eq. (11.73) cancels out the first term on the right-hand side of Eq. (11.70); thus, Eq. (11.69) becomes
- - f δμα(Γ) -» - -» - t / s c f o , ^ ) = - / ———G0(rd,r)UAC(r,rs)dr
(11.74) + ^ - f 8D(;)[VGo(^, r) · V£/AC(r, ?,)]</?,
which is Eq. (11.63).
11.8.2. Linearized Problem
Although nonlinear with respect to the optical properties, the solution given by Eq. (11.63) can be linearized when the heterogeneity is weak. In this case, the Born approximation Use ^ ^o is assumed; hence, i/Ac on the right-hand side
2 7 2 DIFFUSE OPTICAL TOMOGRAPHY
of Eq. (11.63) can be replaced by i/o, which can be computed if μ0ο and Do are known. Consequently, Eq. (11.63) can be linearized and discretized to
Nv
Uscfa, rs) = Σ IWa.Maffj) + WsjhD(rj)l (11.75) 7 = 1
Here, the summation is over all Ny voxels within the object to be imaged; Waj and Wsj represent weights
Waj = J J - , (11.76)
Do VGofa, rj) · VUotfj, rs)Ax Ay Az
Dn WsJ = J ^ y? , (11.77)
where AJC, Ay, and Az represent the sizes of the grid elements along the JC, y, and z directions, respectively.
We rewrite Eq. (11.75) in the following matrix form
[WaJi, WsJi) la(rj)'] _ 5(0') J -
^ V | - [ ^ s c ( ^ / , ? , t ) ] (11.78)
or
[W][hx] = [l/scl. (Π.79)
Here, subscript / is the index of the measurement with the source at position rsi and the detector at position r^·; subscript j is the index of the position within the object to be imaged. If Ns source positions and ND detector positions exist in the image acquisition, NM = Ns x ND measurements exist. Measurement column vector [£/sc] n a s NM elements; column vector [hx] has 2Ny elements since vectors [δμα] and [hD] are concatenated; thus, matrix [W^-has dimensions NM x 2Ny. Unknown vector [hx] can be solved from Eq. (11.79) using various mathematical methods.
11.8.3. Nonlinear Problem
When the perturbation is not small, the image reconstruction is nonlinear and is usually solved iteratively with the following steps:
1. Assume the initial optical properties. 2. Solve the forward problem. 3. Calculate the error and check the convergence. If the error is sufficiently
small, terminate the loop. Otherwise, continue to the next step. 4. Set up the inverse problem to update the optical properties.
FREQUENCY-DOMAIN THEORY: GENERAL IMAGE RECONSTRUCTION 2 7 3
5. Solve the inverse problem. 6. Update the optical properties and return to step 2.
In step 1, the initial optical properties are assumed. Usually, a homogeneous distribution of average optical properties is judiciously selected.
In step 2, the forward problem is solved with the current optical properties to calculate the diffuse photon density Uc at all detection locations for each source position. The finite-element or the finite-difference method can provide a solution to the forward problem on the basis of the diffusion equation. For finite objects, boundary conditions must be imposed.
In step 3, the χ2 error is typically calculated as follows:
2 ^[Uc(rdi,rsi) - UM(rdi,rsi)'\ 2
x = ; C [ J . <ll-8°) where UM denotes the measured diffuse photon density and σ, denotes the ith measurement error. If χ2 < ε, where ε is a predefined small quantity, the problem has converged, and the looping is terminated. Otherwise, the looping proceeds to step 4.
In step 4, an inverse problem is set up to update the optical properties. Rather than another random assumption, an optimal update of the optical properties is computed by accounting for the difference between Uc and UM- Since the goal is to reach UM from the current Uc, we expand UM to the first order by a Taylor series in matrix form:
[UM] = [UC] + \^f\ [Δμα] + Γ ^ Ι [AD], (11.81)
where
[dUcl AxAyAz _ ^ _ _ k - M = jr—Go(rdi,rj)Uc(rj,rsi), (11.82) ί 9 μ β ! 7 A)
[JD\ = D VGo(rdi,rj)-VUc(rj,rsi). (11.83)
Here, vectors [UM] and [Uc] have dimension NM\ vectors [Δμα] and [AD] have dimension Ny and denote the differential updates for μα and D, respectively; matrices [3i/c/3|xfl] and [dUc/dD] have dimensions NM X Ny. From Eq. (11.81), the inverse problem can be formulated as
|~Δμα] _ L A D J -[J]\ Λ n = WM ~ Ucl (H.84)
2 7 4 DIFFUSE OPTICAL TOMOGRAPHY
The Jacobian matrix is
m =
'Wc' d\ia dUc dD .
(11.85)
The inverse problem in Eq. (11.84) is a linearized problem, which is anal- ogous to Eq. (11.78).
In step 5, Eq. (11.84) is solved for [Δμα] and [AD] with mathematical tech- niques to be discussed in the following section.
In step 6, the optical properties are updated with [Αμα] and [AD]. The looping then returns to step 2.
Although the entire problem is nonlinear, each iteration cycle is linear. Thus, the nonlinear problem is solved by a series of linear steps. To understand this concept by analogy, draw a parabola with the minimum slightly above the abscissa and the opening facing up. Try to reach the minimum from a higher point on the curve by searching along the local tangent, which is a linear step. Iterate the linear search to approach the neighborhood of the minimum.
11.8.4. Inverse Method
Step 5 is an inverse mathematical problem. The Jacobian matrix can be con- structed explicitly with either the direct or the adjoint method. In the direct method, the forward problem calculates the derivatives of the matrix elements. We write the forward problem in operator form
{A}[UC} = {S},
where [A] denotes the operator and {S} denotes the source. The following equations are solved for derivatives {dUc/d\ia} and{dUc/dD}:
[A]
{A}
dUc 3μα
dUc dD
ds_
dS
JD
dA
dA
d~D
Wd
{Uc
(11.86)
(11.87)
In the adjoint method, the forward problem {A){t/c} = [S] is solved for Uc- Next, the adjoint equation {A'}{i/c) = {S1} is solved for Green's function Go in response to a point source at a detector position. Then, the Jacobian matrix is computed by Eqs. (11.82) and (11.83).
To solve Eq. (11.84), we must invert a nonsquare Jacobian matrix of dimen- sions NM x 2NV. As a result of photon diffusion, the Jacobian matrix is ill-conditioned (nearly singular); hence, direct matrix inversion is unreliable. Usually, the Jacobian matrix is first multiplied by its transpose to form a square
APPENDIX 11 A. ART AND SIRT 2 7 5
matrix; accordingly, Eq. (11.84) becomes
UfU] P ^ l = UY[UM ~ Ud (Π.88)
However, square matrix [ / ] r [7] is still ill-conditioned. A regularization tech- nique is usually used to improve stability at the expense of image quality as follows:
(UfU] + r\r[Crf[Cr]) Γ ^ Ι - [J]T[UM - Ud (H.89)
Here, ηΓ is the regularization parameter, which can be adjusted to control the stability of the inversion; [Cr] is the regularizing operator (sometimes simply the identity matrix). This regularized equation is usually solved using the conjugate- gradient method.
APPENDIX 11 A. ART AND SIRT
If a unique solution exists for a set of linear equations, Ax = b, where A is a matrix with elements a/*,x is an n-tuple vector of unknowns, and b is an n- tuple vector of measurements; then A must be a nonsingular square matrix of dimensions n x n. The goal is to solve for x iteratively.
In the algebraic reconstruction technique (ART), the search typically starts from the origin; the iterative equation for element / of x is
n
Y^ajk(x£)j-i-bj
ixf)j = (x-)j-i ~ ^—η ajh (11.90) Y^ajk<*jk k=\
Here, p is the index of iteration; j and k are the row and column indices of A; and i, j , k — 1, 2 , . . . , n. For brevity, (x£)o denotes the final search point before the current iteration.
The first two movements of the iteration for n — 2 are illustrated in Fig- ure 11.10, where the two lines represent X^ = 1 «i*** = b\ and ^ = 1 a^Xk — ̂ 2· In move 1, the search point moves from the current position (the origin, in this figure) perpendicularly to line 1 and reaches intersection (\p)j=\. In move 2, the search point moves from the current position perpendicularly to line 2 and reaches intersection (xp)7=2· These two moves complete the first iteration cycle for n — 2. The cycle is iterated until convergence is reached.
If there are more equations than unknowns, the problem is overdetermined, and no unique solution exists. In this case, the solution of the ART oscillates in
2 7 6 DIFFUSE OPTICAL TOMOGRAPHY
2nd move
Solution-
Figure 11.10. Illustration of ART.
the vicinity of the "true solution." If there are more unknowns than equations, the solution converges to a subspace, for instance, a line for n — 2.
The simultaneous iterative reconstruction technique (SIRT) is a variant of the ART. In each iteration cycle, n search points are first found by movements toward all the "lines" from the final search point before the current iteration:
«>y - ('Do - —n Y^ajkajk k=\
cijij = 1,2, . . . , n . (11.91)
Then, the final search point for this iteration cycle is the average of all n search points:
*,'=ii>/v (11.92) 7 = 1
The SIRT in general yields better images than the ART does but has a slower convergence speed.
PROBLEMS
11.1 (a) On the basis of the diffusion theory for an infinite medium, derive the mean time of flight between the observation point and the isotropic point source
<0 = 2(D + ry/^D)c 2Dc[\ + (r/δ)]'
PROBLEMS 277
where c is the speed of light in the scattering medium, r is the distance between the observation point and the source, and δ is the penetration depth, (b) Relate the mean path length of flight (p) with (t). (c) Show that (p) = (r2/2D) oc r2 if r « δ and (p) = (rh/2D) oc r given r » δ. //mte: Use fluence rate
c ( r2 \ Φ(Γ , t) = ^ r exp I vinCt 1
and current density R = — Dd<&/dr. Define
I ίΦί/ί / tRdt (ί)Φ = ^γ and (t)R = *j .
/ <$>dt / Rdt Define the differential path length as pd = —d\n R/d\ia, where R is the current density. We have R = — D(9/dr)0(r) , where Φ is the steady- state fluence rate in response to an isotropic point source in an infinite medium: Φ ( Γ ) = exp(—\it^r)/{ATiDr).
(a) Show that
= 1 τ 2 [ 1 + 3 μ α Ρ ] Pd ~ 2 D + yft^Dr '
(b) Relate pd to (p) in Problem 11.1. (c) Show that R a exp (—\v^^r2) when r <̂C δ and R a exp(—\xeffr)
when r ^> δ.
A photon-density wave is generated from an isotropic point source in an infinite medium.
(a) Show that the phase at distance r from the source is
Γ Ar sinCc/2) 1 Ψ ( Γ ) = arctan — — —— - Ar sin(x/2),
[1 + Arcos(t/2)J
where A = [(\iac)2 -f ω 2 ] 1 / 4 / ^ / ^ and τ = arctan[o)/^flc)]. (b) Plot Ψ ( Γ ) and ω(ί) versus ω e [10,100], where (/) is as defined in
Problem 11.1. [Hint: Start with fluence rate Φ(Γ, ω) and current density R = -ϋ(3/3Γ)Φ(Γ,ω).]
Use diffusion theory for an infinite scattering medium to plot the output as a function of time in MATLAB to graphically illustrate the three imaging modalities.
5 Derive Eqs. (11.19) and (11.20).
DIFFUSE OPTICAL TOMOGRAPHY
11.6 Show that
kr =
k> =
and
(11.93)
11.7 (a) Calculate the wavelength of a photon-density wave with the following parameters: frequency / = 200 MHz, \ia = 0 . 1 cm""1, μ̂ = 10 cm- 1 , index of refraction n — 1.37. (b) Plot the wavelength and k[ as a function of / in the range 100-1000 MHz.
11.8 (a) Duplicate the figures presented in Example 11.1. Known parameters include μα =0 .1 cm- 1 , μ̂ = 10 cm- 1 , and frequency / = 200 MHz. (b) Find the location of the maximal amplitude.
11.9 Generate a movie in MATLAB to demonstrate the propagation of a photon-density wave.
11.10 Show Snell's law for photon-density waves.
11.11 Derive the following diffuse photon-density wave image formation for- mula for a spherical refracting surface: Xjn/5in -f Xout/50ut = (λϊη — Xout)/^· Here, \m is the wavelength in the incident medium, λοιη is the wavelength in the transmitted medium, Sm is the distance of object, Soui is the distance of image, and R is the radius of curvature.
11.12 Derive the Rytov approximation counterpart of the Born approximation solution. In the Rytov approximation, the photon-density distribution is expressed as the product—instead of the sum—of the incident (homo- geneous) and scattered (heterogeneous) parts: U(r, rs) = exp[wo(?, rs)+ "sc (^ *s)L where UQ(7, rs) = exp[wo(r, rs)]. Show that the solution is
usc(rd,rs) = Uo(rd
J f δμα(Γ) rd,rs)J D
G(r -rd)U0(f,rs)dr.
11.13 Derive the Green function for a photon-density wave in a semiinfinite scattering medium. The isotropic point source is one transport mean free path below the surface. Use the extrapolated boundary condition.
11.14 Derive the Green function for a photon-density wave in a slab scattering medium. The isotropic point source is one transport mean free path below the surface. Use the extrapolated boundary condition.
FURTHER READING 2 7 9
11.15 Estimate the relative changes in the amplitude and the phase of a photon- density wave when a small absorber within a breast is moved by a small distance. Use realistic parameters for the estimation.
11.16 Rewrite the inverse algorithm in Section 11.7 in MATLAB or C / C + + . Explore whether the real part of Eq. (11.46) is sufficient to provide an image.
11.17 Rewrite the inverse algorithm in Section 11.7 in MATLAB or C / C + + using ART and SIRT, respectively.
READING
Chance B, Kang K, He L, Weng J, and Sevick E (1993): Highly sensitive object location in tissue models with linear in-phase and antiphase multielement optical arrays in one and 2 dimensions, Proc. Natl. Acad. Sei. USA 90(8): 3423-3427. (See Section 11.6, above.)
O'Leary MA (1996): Imaging with Diffuse Photon Density Waves, Ph.D. thesis, Univ. Pennsylvania, Philadelphia. (See Sections 11.5-11.7, above.)
Pogue BW, Testorf M, McBride T, Osterberg U, and Paulsen K (1997): Instrumentation and design of a frequency domain diffuse optical tomography imager for breast cancer detection, Opt. Express 1(13): 391-403. (See Section 11.5, above.)
Schmidt FEW, Fry ME, Hillman EMC, Hebden JC, and Delpy DT (2000): A 32-channel time-resolved instrument for medical optical tomography, Rev. Sei. Instrum. 71(1): 256-265. (See Section 11.3, above.)
Yodh AG and Boas DA (2003): Functional imaging with diffusing light, in Biomedical Photonics Handbook, Vo-Dinh T, ed., CRC Press, Boca Raton, FL, pp. 21.1-21.45. (See Sections 11.4 and 11.8, above.)
FURTHER READING
Aronson R (1995): Boundary-conditions for diffusion of light, J. Opt. Soc. Am. A 12(11): 2532-2539.
Arridge SR (1999): Optical tomography in medical imaging, Inverse Problems 15(2): R41-R93.
Boas DA, Brooks DH, Miller EL, DiMarzio CA, Kilmer M, Gaudette RJ, and Zhang Q (2001): Imaging the body with diffuse optical tomography, IEEE Signal Process. Mag. 18(6): 57-75.
Boas DA, Gaudette T, Strangman G, Cheng XF, Marota JJA, and Mandeville JB (2001): The accuracy of near infrared spectroscopy and imaging during focal changes in cere- bral hemodynamics, Neuroimage 13(1): 76-90.
Boas DA, O'Leary MA, Chance B, and Yodh AG (1997): Detection and characterization of optical inhomogeneities with diffuse photon density waves: A signal-to-noise analysis, Appl. Opt. 36(1): 75-92.
Cerussi AE, Berger AJ, Bevilacqua F, Shah N, Jakubowski D, Butler J, Holcombe RF, and Tromberg BJ (2001): Sources of absorption and scattering contrast for near-infrared optical mammography, Acad. Radiol. 8(3): 211-218.
2 8 0 DIFFUSE OPTICAL TOMOGRAPHY
Cerussi AE, Jakubowski D, Shah N, Bevilacqua F, Lanning R, Berger AJ, Hsiang D, Butler J, Holcombe RF, and Tromberg BJ (2002): Spectroscopy enhances the information content of optical mammography, J. Biomed. Opt. 7(1): 60-71.
Chance B, Anday E, Nioka S, Zhou S, Hong L, Worden K, Li C, Murray T, Ovetsky Y, Pidikiti D, and Thomas R (1998): A novel method for fast imaging of brain function, non-invasively, with light, Opt. Express 2(10): 411-423.
Colak SB, van der Mark MB, Hooft GW, Hoogenraad JH, van der Linden ES, and Kuijpers FA (1999): Clinical optical tomography and NIR spectroscopy for breast cancer detection, IEEE J. Select. Topics Quantum Electron. 5(4): 1143-1158.
Dorn 0 (1998): A transport-backtransport method for optical tomography, Inverse Prob- lems 14(5): 1107-1130.
Durduran T, Choe R, Culver JP, Zubkov L, Holboke MJ, Giammarco J, Chance B, and Yodh AG (2002): Bulk optical properties of healthy female breast tissue, Phys. Med. Biol. 47(16): 2847-2861.
Durian DJ and Rudnick J (1997): Photon migration at short time and distances and in cases of strong absorption, J. Opt. Soc. Am. A 14(1): 235-245.
Fantini S, Franceschini MA, and Gratton E (1994): Semi-infinite-geometry boundary- problem for light migration in highly scattering media—a frequency-domain study in the diffusion-approximation, J. Opt. Soc. Am. B 11(10): 2128-2138.
Fantini S, Franceschini MA, Gaida G, Gratton E, Jess H, Mantulin WW, Moesta KT, Schlag PM, and Kaschke M (1996): Frequency-domain optical mammography: Edge effect corrections, Med. Phys. 23(1): 149-157.
Fantini S, Walker SA, Franceschini MA, Kaschke M, Schlag PM, and Moesta KT (1998): Assessment of the size, position, and optical properties of breast tumors in vivoby noninvasive optical methods, Appl Opt. 37(10): 1982-1989.
Franceschini MA, Moesta KT, Fantini S, Gaida G, Gratton E, Jess H, Mantulin WW, Seeber M, Schlag PM, and Kaschke M (1997): Frequency-domain techniques enhance optical mammography: Initial clinical results, Proc. Natl. Acad. Sei. USA 94(12): 6468-6473.
Franceschini MA, Toronov V, Filiaci ME, Gratton E, and Fantini S (2000): On-line optical imaging of the human brain with 160-ms temporal resolution, Opt. Express 6(3): 49-57.
Gratton G, Fabiani M, Friedman D, Franceschini MA, Fantini Sr Corballis P, and Gratton E (1995): Rapid changes of optical-parameters in the human brain during a tapping task, J. Cogn. Neurosci. 7(4): 446-456.
Graves EE, Ripoll J, Weissleder R, and Ntziachristos V (2003): A submillimeter resolution fluorescence molecular imaging system for small animal imaging, Med. Phys. 30(5): 901-911.
Grosenick D, Moesta KT, Wabnitz H, Mucke J, Stroszczynski C, Macdonald R, Schlag PM, and Rinneberg H (2003): Time-domain optical mammography: Initial clini- cal results on detection and characterization of breast tumors, Appl. Opt. 42(16): 3170-3186.
Grosenick D, Wabnitz H, Rinneberg HH, Moesta KT, and Schlag PM (1999): Develop- ment of a time-domain optical mammography and first in vivo applications, Appl. Opt. 38(13): 2927-2943.
Hawrysz DJ and Sevick-Muraca EM (2000): Developments toward diagnostic breast can- cer imaging using near-infrared optical measurements and fluorescent contrast agents, Neoplasia 2(5): 388-417.
FURTHER READING 281
Hebden JC, Veenstra H, Dehghani H, Hillman EMC, Schweiger M, Arridge SR, and Delpy DT (2001): Three-dimensional time-resolved optical tomography of a conical breast phantom, Appl. Opt. 40(19): 3278-3287.
Hielscher AH, Klose AD, and Hanson KM (1999): Gradient-based iterative image recon- struction scheme for time-resolved optical tomography, IEEE Trans. Med. Imaging 18(3): 262-271.
Klose AD and Hielscher AH (1999): Iterative reconstruction scheme for optical tomog- raphy based on the equation of radiative transfer, Med. Phys. 26(8): 1698-1707.
Li XD, Durduran T, Yodh AG, Chance B, and Pattanayak DN (1997): Diffraction tomog- raphy for biochemical imaging with diffuse-photon density waves, Opt. Lett. 22(8): 573-575.
Licha K, Riefke B, Ntziachristos V, Becker A, Chance B, and Semmler W (2000): Hydrophilic cyanine dyes as contrast agents for near-infrared tumor imaging: Syn- thesis, photophysical properties and spectroscopic in vivocharacterization, Photochem. Photobiol. 72(3): 392-398.
McBride TO, Pogue BW, Gerety ED, Poplack SB, Osterberg UL, and Paulsen KD (1999): Spectroscopic diffuse optical tomography for the quantitative assessment of hemoglobin concentration and oxygen saturation in breast tissue, Appl. Opt. 38(25): 5480-5490.
Ntziachristos V, Ma XH, and Chance B (1998): Time-correlated single photon counting imager for simultaneous magnetic resonance and near-infrared mammography, Rev. Sei. Instrum. 69(12): 4221-4233.
Ntziachristos V, Yodh AG, Schnall M, and Chance B (2000): Concurrent MRI and diffuse optical tomography of breast after indocyanine green enhancement, Proc. Natl. Acad. Sei. USA 97(6): 2767-2772.
O'Leary MA, Boas DA, Chance B, and Yodh AG (1992): Refraction of diffuse photon density waves, Phys. Rev. Lett. 69(18): 2658-2661.
O'Leary MA, Boas DA, Li XD, Chance B, and Yodh AG (1996): Fluorescence lifetime imaging in turbid media, Opt. Lett. 21(2): 158-160.
Pogue BW, Patterson MS, Jiang H, and Paulsen KD (1995): Initial assessment of a simple system for frequency-domain diffuse optical tomography, Phys. Med. Biol. 40(10): 1709-1729.
Pogue BW, Poplack SP, McBride TO, Wells WA, Osterman KS, Osterberg UL, and Paulsen KD (2001): Quantitative hemoglobin tomography with diffuse near-infrared spectroscopy: Pilot results in the breast, Radiology 218(1): 261-266.
Reynolds JS, Troy TL, Mayer RH, Thompson AB, Waters DJ, Cornell KK, Snyder PW, and Sevick-Muraca EM (1999): Imaging of spontaneous canine mammary tumors using fluorescent contrast agents, Photochem. Photobiol. 70(1): 87-94.
Shah N, Cerussi A, Eker C, Espinoza J, Butler J, Fishkin J, Hornung R, and Tromberg B (2001): Noninvasive functional optical spectroscopy of human breast tissue, Proc. Natl. Acad. Sei. USA 98(8): 4420-4425.
Yodh A and Chance B (1995): Spectroscopy and imaging with diffusing light, Phys. Today 48(3): 34-40.
CHAPTER 12
Photoacoustic Tomography
12.1. INTRODUCTION
The term photoacoustic tomography (PAT) refers to imaging that is based on the photoacoustic effect. Although the photoacoustic effect was first reported on by Alexander Graham Bell in 1880, PAT was invented only after the advent of ultrasonic transducers, computers, and lasers. In PAT, the object is usually irradiated by a short-pulsed laser beam. Some of the light is absorbed by the object and partially converted into heat. The heat is then further converted to a pressure rise via thermoelastic expansion. The pressure rise is propagated as an ultrasonic wave, which is referred to as a photoacoustic wave. The photoacoustic wave is detected by ultrasonic transducers and is used by a computer to form an image.
12.2. MOTIVATION FOR PHOTOACOUSTIC TOMOGRAPHY
The motivation for PAT is to combine the contrast of optical absorption with the spatial resolution of ultrasound for deep imaging in the optical quasidiffusive or diffusive regime. Optical absorption is desirable because of its high sensitivity to molecules such as oxygenated and deoxygenated hemoglobin. In Table 12.1, PAT is compared with optical coherence tomography (OCT; see Chapters 9 and 10), diffuse optical tomography (DOT; see Chapter 11), and ultrasonography (US). Because of the strong optical scattering, pure optical imaging in biological tis- sue has either shallow imaging depth or low spatial resolution. Pure ultrasonic imaging can provide better resolution than pure optical imaging in the optical quasidiffusive or diffusive regime because ultrasonic scattering is two to three orders of magnitude weaker than optical scattering. Ultrasonic imaging, however, detects only mechanical properties and has weak contrast in early-stage tumors.
PAT overcomes the limitations of existing pure optical and pure ultrasonic imaging; its contrast is based on optical absorption in the photoacoustic exci- tation phase, whereas its resolution is derived from ultrasonic detection in the
Biomedical Optics: Principles and Imaging, by Lihong V. Wang and Hsin-i Wu Copyright © 2007 John Wiley & Sons, Inc.
283
2 8 4 PHOTOACOUSTIC TOMOGRAPHY
TABLE 12.1. Comparison of Optical Coherence Tomography (OCT), Diffuse Optical Tomography (DOT), Ultrasonography (US) Operating at 5 MHz, and Photoacoustic Tomography (PAT)
Property
Contrast
Imaging depth
Resolution
Speckle artifacts
Scattering coefficient
OCT
Good
Poor (—1 mm)
Excellent (-0.01 mm)
Strong
Strong ( -10 mm-1)
DOT
Excellent
Good (-50 mm)
Poor (—5 mm)
None
Strong (-10 mm"
"Strong for the excitation light and weak for the
■)
US
Poor for early cancers
Excellent and scalable ( -60 mm)
Excellent and scalable (-0.3 mm)
Strong
Weak (-0.03 mm-1)
photoacoustic wave.
PAT
Excellent
Good and scalable
Excellent and scalable
None
Mixed0
photoacoustic emission phase. The image resolution, as well as the maximum imaging depth, is scalable with ultrasonic frequency within the reach of diffuse photons. As ultrasonic center frequency and bandwidth increase, spatial resolu- tion improves at the expense of imaging depth. In addition, PAT provides images that are devoid of speckle artifacts, which are conspicuous in both OCT and US images.
12.3. INITIAL PHOTOACOUSTIC PRESSURE
Two important timescales exist in laser heating:
1. The thermal relaxation time, which characterizes the thermal diffusion, is estimated by
"Cth = otth
(12.1)
where ath is the thermal diffusivity (m2/s) and dc is the characteristic dimen- sion of the heated region (the dimension of the structure of interest or the decay constant of the optical energy deposition, whichever is smaller). The stress relaxation time, which characterizes the pressure propagation, is given by
(12.2)
where vs is the speed of sound (—1480 m/s in water).
INITIAL PHOTOACOUSTIC PRESSURE 2 8 5
If the laser pulsewidth is much shorter than tth, the excitation is said to be in thermal confinement, and heat conduction is negligible during the laser pulse. Likewise, if the laser pulsewidth is much shorter than τν, the excitation is said to be in stress confinement, and stress propagation is negligible during the laser pulse.
On laser excitation, the fractional volume expansion dV / V can be expressed as
dV — = - κ ρ + β7\ (12.3)
Here, κ denotes the isothermal compressibility (~5 x 10~10 Pa- 1 for water or soft tissue); β denotes the thermal coefficient of volume expansion (—4 x 10~4K_1
for muscle); p and T denote the changes in pressure (Pa) and temperature (K), respectively. The isothermal compressibility κ can be expressed as
CP K = - 2 F - <12-4>
Here, p denotes the mass density (~1000 kg/m3 for water and soft tissue); Cp and Cy [~4000 J/(kg K) for muscle] denote the specific heat capacities at constant pressure and volume, respectively.
If the laser excitation is in both the thermal and stress confinements, the frac- tional volume expansion is negligible and the local pressure rise po immediately after the laser pulse can be derived from Eq. (12.3):
Po=—, (12.5) K
which can be rewritten as
β Po = —μ-r)thAe. (12.6)
Here, Ae is the specific optical absorption (J/m3) and η^ is the percentage that is converted into heat. We define the Grueneisen parameter (dimensionless) as
- - ß -M (12.7) KpCV Cp
For water and diluted aqueous solutions, Γ can be estimated by the following empirical formula:
Vw(To) = 0.0043 + 0.0053Γ0, (12.8)
where Γ0 is the temperature in degrees Celsius. At body temperature, rw(31°C) = 0.20. From Eq. (12.7), Eq. (12.6) becomes
Po = rr, thA, (12.9)
2 8 6 PHOTOACOUSTIC TOMOGRAPHY
or
Po = ri\ihVaF. (12.10)
Here, μ^ is the optical absorption coefficient and F is the optical fluence (J/cm2).
Example 12.1. Show that the dimensions of the energy density and the pressure are the same.
J/m3 = N m/m3 = N/m2 = Pa.
Note that 1 bar = 105 Pa.
Example 12.2. Given dc = 0.15 cm and 15 μπι, compute xth and xs in soft tis- sue.
With the typical properties of soft tissue, Eqs. (12,1) and (12.2) predict the fol- lowing values: For dc = 0.15 cm:
(0.15 cm)2
1.3 x 10~J cm2/s 0.15 cm
xs — = 1 μs. 0.15 cm/μs
For dc — 15 μιτι:
(15 x 10-4cm)2 4 xth = , ~— = 17 x 10 s,
th 1.3 x 10-3 cm2/s 15 x 10~4 cm
xs = = 0.01 μβ. 0.15 cm/[i^>
Example 12.3. Estimate the temperature and the pressure rises on a short-pulse laser excitation of soft tissue at body temperature with a fluence of 10 mJ/cm2. Assume μα =0 .1 cm- 1.
Ae = 0.1 cm ' x 10 mJ/cm2 = 1 mJ/cm3
Ae 1 mJ/cm3
~ pC^ ~ 1 g/cm3 x 4 J g - 1 K ' 1
p{) = VAe = 0.20 x 10 mbar = 2 mbar.
= 0.25 mK,
The results also indicate that for each mK (millikelvin) temperature rise, an 8-mbar pressure rise is produced.
GENERAL PHOTOACOUSTIC EQUATION 287
12.4. GENERAL PHOTOACOUSTIC EQUATION
The photoacoustic wave generation and propagation in an inviscid medium is described by the following general photoacoustic equation (see Example 12.4):
p(r,t) = —V- γ }, (12.11) vs
where p(r,t) denotes the acoustic pressure at location r and time t and T denotes the temperature rise. The left-hand side of this equation describes the wave prop- agation, whereas the right-hand side represents the source term.
In thermal confinement, the thermal equation becomes
dT(r,t) 9CV I =H(r,t). (12.12)
at
Here, H(r,t) is the heating function defined as the thermal energy converted per unit volume and per unit time; it is related to the specific optical power deposition Ap by H = y\\^Ap and to the optical fluence rate Φ by H = ηηιμαΦ- Substituting Eq. (12.12) into Eq. (12.11), we obtain the following less general photoacoustic equation:
2 x , & dH p(7,t) = --^ — . (12.13)
The source term is related to the first time derivative of H. Therefore, time- invariant heating does not produce a pressure wave; only time-variant heating does.
Sometimes, velocity potential φ„—which is related to p as follows—is used:
P = -<>'-£. (12-14)
Substituting Eq. (12.14) into Eq. (12.13) yields the following equation, which avoids the time derivative of H:
Example 12.4. Derive the photoacoustic equation shown in Eq. (12.11)
2 λ β φ„ = - £ - / / . (12.15)
The two basic equations responsible for photoacoustic generation are the thermal expansion equation (generalized Hooke's law),
V · ξ(?, t) = -κρ(τ, t) + βΓ(Γ, t) (12.16)
2 8 8 PHOTOACOUSTIC TOMOGRAPHY
and the linear inviscid force equation (the equation of motion)
p-T2%{r,t) = -Vp(r,t), (12.17)
where vector ξ denotes the medium displacement. The left-hand side of Eq. (12.16) represents the fractional volume expansion, while the right-hand side represents the two factors related to the volume expansion. The left-hand side of Eq. (12.17) represents the mass density times the acceleration, and the right-hand side represents the force applied per unit volume. Thus, Eq. (12.17) is an incar- nation of Newton's second law. The reader can reduce the two equations above to their ID counterparts to understand their physical meanings more clearly.
Taking the divergence of Eq. (12.17), we obtain
a2 2 . p — [ V ^ ( r , 0 ] = - V 2 p C - , 0 . (12.18)
Substituting Eq. (12.16) into Eq. (12.18) leads to Eq. (12.11), where vs = 1/^/pic is used. A more detailed derivation of the acoustic wave equation is shown in Appendix 12A.
12.5. GENERAL FORWARD SOLUTION
The general photoacoustic equation shown in Eq. (12.11) can be solved by the Green function approach (see Appendix 12B). Green's function is defined here as the response to a spatial and temporal impulse source term
2 1 G(?, t\ r \ t') = -5( r - r')h(t - r'), (12.19)
where r' and t' denote the source location and time, respectively. In infinite space, where no boundary exists, Green's function is given by
lit-t'-ψ) G(r, t; r', t') = v J \ (12.20)
4n\r -r'\
which represents an impulse diverging spherical wave. The following reciprocity relation holds:
G(r, t- r', t') = G(?\ -t'; r, -t). (12.21)
To see this relationship more clearly, one observes G(f, t\ rf, 0) = G(rf, 0; r, — t) by setting t' — 0.
GENERAL FORWARD SOLUTION 2 8 9
The physical meaning of Green's function should be interpreted carefully. A spatial delta function in the source term of the photoacoustic equation simply represents a point acoustic source. A temporal delta function in the source term, however, is translated into a step heating function or a ramp temperature rise for the following reason; the source term of the photoacoustic equation is proportional to the first time derivative of the heating function in thermal confinement or the second time derivative of the temperature in general. In other words, Green's function represents the response of a point absorber to step heating, rather than impulse heating.
Applying the Green function approach to Eq. (12.11) yields
p(r,t)=[ dt' f ά?0{ΐ,ν,7',ί')^ r Q ° L ' ° , (12.22) J-oo J Kttf dt2
which represents the pressure in response to an arbitrary source. Substituting Eq. (12.20) into Eq. (12.22) leads to
2Trt' t f
(12.23) t'=t-\r-r'\/vs
In thermal confinement, substituting Eq. (12.12) into Eq. (12.23) yields
') p(r, t) = / dr ——- — F 4nCP J \r- r'\ dt'
(12.24) t'=t-\f-r'\/vs
or
FK 4nCP dt J \7-r'\ \ ' vs J
If the heating function can be decomposed as H(r',t') = Hs(r')H,(t'), Eq. (12.25) can be further simplified to
μΚ AitCp dt J I? - r'\ \ vs J
If H,(t') = δ(ί'), Eq. (12.26) provides the delta heating response of an arbitrary absorbing object as
„(?.,) = - * - ! / > f t ^ ( , _ l i Z ^ (12.27) FK 4itCP dt J |r - r'| V Vs )
or
^-hkkhl«^-*-^1)]· <l2'28)
2 9 0 PHOTOACOUSTIC TOMOGRAPHY
where the quantity within the square brackets is the step heating response. From Eq. (12.9), the initial pressure response due to delta heating can be expressed as
p0(r) = rHs(r'). (12.29)
Using Eqs. (12.7) and (12.29), we rewrite Eq. (12.28) as
P^^^ = 7 ^ 2 T \ — f ^ W ^ f i - ^ — ^ ) 1 . (12.30) 4nvjdt lvst J \ vs ) \
Example 12.5. Derive Eq. (12.20) from Eq. (12.19).
The following Fourier transformations are used:
g(jfc, ω ) = II G(r, t\ r', /')exp[-/jfc · (r - r')]exp[i<o(f - t')]drdt, (12.31)
1 = II 8(r - r')h(t - /') expl-ik · (r - r')J
x exp[io)(f -t')]drdt. (12.32)
Taking the Fourier transformation of both sides of Eq. (12.19) yields
g(fc,ü)) = — - L — . (12.33) &z — (joz/vf
Substituting this equation into the following inverse Fourier transformation
G(r, t\ r\ t') = ——- / / g(fc, ω) exp[/fc · (r - r')] exp[-/oo(f - ?')] άω dk, (2π)4 J J
(12.34) we obtain
G(F''; ''''> = ü b / / ^ κ exp[/* ■Cr - r)] x exp[-/oo(i - t')]du> dk. (12.35)
The integral on the right-hand side involves singularities at k = ±ω/υν , but it can be evaluated by Cauchy's contour integration:
ff exp(ifc · ξ - / ω τ ) r
_ /*/· exp(/& · ξ - /ν,τω/υ,) / ω \ -.
GENERAL FORWARD SOLUTION 291
n ■ f / Γ r exp(-/i;,/cx)-exp(/u9/:T) r = 2nivs I βχρ(ικ·ξ) — dk 2k
Ιπνχ I expO/c-ξ) dk. (12.36)
Here, ξ = r — r' and τ = / — t'. Since dfc = Ink2 sin0 t/θ d/c in spherical coor- dinates, we derive
G(r,t;r\tf) = — r̂ / / &exp(/£^cos6) sin(uvÄ:T) sin0 dQ dk 4π2 y0 Jo
= ^— / [exp(—/£ξ) — exp(ik%)]sin(vskx) dk 4πζξ/ Jo
= r - ^ / sin(/^)sin(i;,fcx)</fc
= — Γ - e x p |
1 4π%
ivsk(x I
ivsk I τ H j | d(vsk]
[·κ):·κ): - ^ Κ - ' - ! ^ ) - ( - ' + Ε Ϊ 3 ) :
(12.37)
The second term on the right-hand side of this equation violates causality because the signal detected at distance \r — rf\ from f' can take place only when t > t'\ thus, it must be dropped. As a result, we reach Eq. (12.20).
Example 12.6. From the scalar wave equation, generalize the reciprocity relation to a finite medium.
The Green function for a scalar wave equation satisfies the following differential equation:
\ vjdt2) G(r, t; r', t') = -S(r - r')h(t - t'), (12.38)
where location r' and time t' represent the source point and r and t denote the observation point. Time reversal of Eq. (12.38) leads to
V vjdt2) G(r, -t; r", -t") = -h(r - r)h(t" - t). (12.39)
2 9 2 PHOTOACOUSTIC TOMOGRAPHY
Multiplying Eq. (12.38) by G(?, -t; r", - f " ) , multiplying Eq. (12.39) by G(r,t',r',tr), subtracting the two, and integrating over the volume of investi- gation V and over time t from —oo to /max with tmax > max(i', t"), we obtain
/
'max Γ Γ
<// / rf? C(r, f ; r ' , / ' )V2G(r,- -oo JV L
-t;r , - i )
, · Z" _ , " \ V 7 2 G(r, -t\r", - r ) V z G ( r , i ; F , f ' )
1 32
;G(r, f, r', t')—rG(r, -t; 7", -t") vt dt 2
l d2 + -=G(r, -t; r", - < " ) - r C ( r , f; r', ί')
υί 3ί 2
(12.40)
(r , —χ ; r , —ί ) - G(r , r ; r , f ) .
Note that
G(r, t\ r\ t')V2G{7, - r ; r", - r " ) - G(?, - / ; r", - i")V2G(r , /; r', t')
= V · [G(F, /; r , OVG(r , -f; r", - Γ )
- G(r, -t\ r", -t")VG{r, t\ r\ /')] (12.41)
and
d2 d2
-G(/% f; r', *') — G(r, -t; r", -t") + G(?, -f; r", - f") — G ( r , r; r', f')
8i G(r,t\r,t') — G{r,-t\r\-t")
dt
+ G(r,-f,r",-t") — G(r,f,r',t') at
(12.42)
Substituting Eqs. (12.41) and (12.42) into Eq. (12.41) and applying Green's theorem, we obtain
/
'max f
dt / dS-[G(r,t;r',t')VG(. -oo J S
r, —1\ r ,—t ) -oo J S
- G(r, -t\ 7", -f")VG(r, f; r', i')]
«,2 Λ + — / dr - G ( r , i; r \ t) — G(ry -t; r\ -t") (12.43)
+ G(r, -t\ r \ -t") — G{r, t\ r\ f) dt
G f < J " Jt\ r^ /■~,i Ji ~*i J\ (r ,-t \r , — t ) — G(r ,t ; r , t )
DELTA-PULSE EXCITATION OF A SLAB 2 9 3
where S is an arbitrary surface enclosing V. The first integral on the left-hand side vanishes because both Green functions satisfy the same homogeneous boundary condition (either G or its normal gradient is zero) on S. The second integral also vanishes because both G(r, t = —oo; r', tf) and G(r, —t — — tmax\ r"', —i"), where —oo < /' and — /max < — /", as well as their time derivatives, must be zero as required by causality. Thus, Eq. (12.44) becomes
G(r\ -t'\ r", -t") = G{r", t"\ r', t'), (12.44)
which is the reciprocity relation.
12.6. DELTA-PULSE EXCITATION OF A SLAB
When a delta excitation pulse heats up a slab of thickness d (Figure 12.1), an initial pressure po is first built up within the slab and then propagated out- ward in both the positive and the negative z directions. The pressure distri- bution can be derived by Eq. (12.30), where the integral over the volume is converted into an integral over a solid angle as shown in Figure 12.1. The solid angle is subtended by a section of the spherical shell of radius vst from observation point r = (0, 0, z), where the section is the intersection with the slab.
We first consider an observation point on the positive z axis (z > 0). When the observation point is outside the slab (z > d/2), three cases are considered in terms of the propagation time:
When vst < z — d/2, the spherical shell does not touch the slab; thus, P(z, t) = 0.
When z — d/2 < vst < z 4- d/2, the spherical shell intersects the near edge of the slab at polar angle Θ with respect to the negative z axis. The pressure
d
1st
d/2 z-d/2
Figure 12.1. Diagram for a slab object.
2 9 4 PHOTOACOUSTIC TOMOGRAPHY
distribution is calculated as follows:
Piz,t) = ~—Ύ- — / dr' pohlt-1 ί
P° d P / .-/ *, , , - I dr ö(vst - I 4πυ? 3/ r|)
77 r) Γ 1 Γ Γ Ί = Γ 1 7 " / / (υ,025ΐηθ£/θί/φ
(12.45)
/ (I - COS0)] = 2 dt 2 dt
Po 2 '
When vst > z + d/2, the spherical shell intersects the far edge of the slab at polar angle θ' as well, and we have
p(z,t) = -p-2^\- f f (vstf sine de </φ 4nvjdt It Jo JQ>
2 dt
2 dt
[ i (cos0 '-cos0)]
[/z+d/2 z-d/2\
L V *>■** Vst ) 0. (12.46)
When the observation point is inside the slab (0 < z < d/2), Eq. (12.45) can still be used.
When vst < d/2 — z, the upper limit for the integral over Θ in Eq. (12.45) becomes π because the spherical shell is totally within the slab; as a result, p(z,t) = p0.
When d/2 — z < vst < d/2 + z, the upper limit for the integral over Θ in Eq. (12.45) becomes
= cos z-d/2
Vst
as a result, p(z, 0 = po/2. When vst > d/2 -f z, as in Eq. (12.46), we have /?(z, 0 = 0.
For z < 0, the results are similar owing to symmetry. In summary, the initial pressure distribution can be rewritten as
Po(z) = PoU (ζ + γ \ ν ( - * + γ) ' ( l 2 · 4 7 )
DELTA-PULSE EXCITATION OF A SLAB 2 9 5
where U is the Heaviside function defined as
f 1 for z > 0, ^ ω = | θ for z < 0 . ( 1 2 4 8 )
Consequently, the pressure distribution at any later time can be expressed as
P(z, t) = -p0(z - vst) + -po(z + vst). (12.49)
The first term on the right-hand side represents a right-propagating (along the +z axis) plane wave and the second term, a left-propagating (along the — z axis) plane wave. The physical meaning of Eq. (12.49) is interpreted as follows. Pressure po is generated within the slab on delta heating. Immediately thereafter, po is split into two plane waves, each having a magnitude of ρ$/2 but propagating in opposite directions.
Example 12.7. Plot snapshots of the propagating pressure on the basis of Eq. (12.49).
The following MATLAB code produces Figure 12.2; the partial pressures in Eq. (12.49) are denoted by ppos (/?+) and pneg (/?_), respectively:
% Photoacoustic signal from a homogeneously heated slab % Use SI units
clear all vs = 1500; p0 = 1; d = 1E-3; dhalf = d/2; zmax = 2; z = linspace(-zmax, zmax, 1000)*d;
figure(1) elf
i_axis = 1; for t = [0:1/2:1, 2]*dhalf/vs
ppos = p0/2.*heaviside(z-(-dhalf+vs*t)).*heaviside(dhalf+vs*t-z); pneg = p0/2.*heaviside(z-(-dhalf-vs*t)).*heaviside(dhalf-vs*t-z); p = ppos + pneg;
subplot(4, 2, i_axis, 'align') plot(z/d, ppos/ρθ, 'k-', z/d, pneg/ρθ, 'k--') tick = [.015 .025]; set(0,'DefaultAxesTickLength',tick) title(['\itt\rm = ', num2str(vs*t/d), '\itxd rm/ itv_s']) axis([-zmax, zmax, 0, 1.1]) if (iaxis == 7)
296 PHOTOACOUSTIC TOMOGRAPHY
x l a b e l ( , \ i t z \ r m / \ i t d ' ) end y l a b e l ( ' P a r t i a l p ressu res / \ i t { p } \ rm_0 ' ) i f ( i _ a x i s == 1)
l e g e n d ( ' \ i t p \ r m _ + ' , ' \ i t p \ r m _ - ' ) end i _ax i s = i a x i s + 1 ;
s u b p l o t ( 4 , 2 , i _ a x i s , ' a l i g n ' ) p l o t ( z / d , ρ/ρθ, ' k - ' ) t i ck = [.015 .025] ; set(0, 'DefaultAxesTickLength', t ick) t i t l e ( [ ' \ i t t \ r m = ' , num2str(vs*t/d), ' \ i t xd \ rm / \ i t v_s ' ] ) axis([-zmax, zmax, 0, 1.1]) i f ( i_axis == 8)
x l a b e l ( ' \ i t z \ r m / \ i t d ' ) end y labe l ( 'Tota l pressure/\it{p}\rm_0') i_axis = i_axis + 1;
end
We can also make a movie showing the pressure propagation using the fol- lowing MATLAB script:
% Photoacoust ic s i g n a l from a homogeneously heated s lab % Use SI u n i t s
fig=figure(1); set(fig,'DoubleBuffer','on'); % Flash-free rendering for animations
clear all vs = 1500; PO = 1; d = 1E-3; dhalf = d/2; zmax = 2; z = linspace(-zmax, zmax, 1000)*d;
mov = avifile('Example07_PA_Slab.avi')
for t = [0:0.1:2]*dhalf/vs ppos = p0/2.*heaviside(z-(-dhalf+vs*t)).*heaviside(dhalf+vs*t-z); pneg = p0/2.*heaviside(z-(-dhalf-vs*t)).*heaviside(dhalf-vs*t-z); p = ppos + pneg;
subplot(1, 2, 1) hold off; plot(z/d, ppos/ρθ, 'k-', z/d, pneg/ρθ, 'k--') grid axis([-zmax, zmax, 0, 1.1]) xlabel('\itz/d') ylabel('Partial pressures/\itp\rm_0') legend('\itp\rm_+', ' \itp\rm_-')
DELTA-PULSE EXCITATION OF A SPHERE 2 9 7
subplot(1, 2, 2) hold off; plot(z/d, ρ/ρθ, 'k-') grid axis([-zmax, zmax, 0, 1.1]) xlabelCz/d') ylabel('Total pressure/\itp\rm_0') pause(O.OI)
mov = addframe(mov,getframe(gcf)); end
mov = close(mov);
12.7, DELTA-PULSE EXCITATION OF A SPHERE
When a sphere of radius Rs is heated up with a delta pulse, an initial pressure po is generated inside the sphere. As in the case of a slab, the pressure distribution can be derived from Eq. (12.30). However, the propagation involves spherical waves instead of plane waves. When the observation point is outside the sphere (r > Rs), three cases are considered according to the propagation time. Figure 12.3 shows part of a spherical shell of radius vst centered at the observation point.
When vst < r — Rs, the spherical shell does not touch the heated spherical object; thus, p(r, t) = 0.
When r — Rs < vst < r + Rs, the spherical shell intersects the heated spher- ical object. Thus, the pressure distribution can be derived similarly as in Eq. (12.45):
P<M) = y - [ , ( l -οοβθ)] = - - \t \X — JJ
= ^(r-vst). (12.50)
When vst > r -f Rs, the spherical shell passes the far edge of the heated spherical object and no longer intersects with the heated spherical object; thus, p(r, t) = 0.
When the observation point is inside the sphere (r < Rs), the pressure distri- bution can be derived similarly:
When vst < Rs — r, the spherical shell is entirely enclosed by the heated spherical object, which means that θ = π; as a result, p(r, t) — p$.
When Rs — r < vst < Rs +r, the spherical shell emerges out of the heated spherical object; the pressure distribution can be derived similarly as in Eq. (12.50):
p{rj) = ^(r~vst). (12.51)
2 9 8 PHOTOACOUSTIC TOMOGRAPHY
f = 0x dl\> / = 0x dh\
1
0.5
P- \
-2
C 1 V5
ω 3 C/5
Ö 0.5 ex
"S a a- o
S 0.5
-1
/ = 0.25 x dl\\
-1 0 1
t = 0.5 x i//\\
δ l 8? 3 § 0.5 a. o3 O
1 0 1 t = 0.25 x dlvK
§ 1 c/3
ä 3 C/3
2 °·5 a. aJ o
'
-1 0 1
/ = 0.5 x i//v,
€ I C/5 <U
3 !/5
g 0.5 a, cd O
o ^ 1 C/3 (U
3 (Λ
S 0.5 a-cd c3
t=\xd/\ .V
i ,
1 1
0 zld
€ 1 c/5
3 c/3
ö 0 · 5
Λ O
t- = 1 X i//v,
0
Figure 12.2. Snapshots of propagating pressure from a heated slab.
Figure 12.3. Diagram for a heated spherical object.
DELTA-PULSE EXCITATION OF A SPHERE 2 9 9
When vst > Rs -f r, the spherical shell encloses the heated spherical object and no longer intersects with the heated spherical object; thus, p(r,t) = 0.
The results listed above are summarized using the Heaviside function U as
p(r, t) = p0\ U(RS - vst - r) + ^—^~U(r ~ \Rs ~ vst\)U(R5 + vst-r)\.
(12.52)
If we write the initial pressure as
Po(r) = p0U(r)U(-r + Rs) for 0 < r < Rs,
we have
r + vst p(r, t)
2r r — vst r — vst
p0(r + vst) + ———p0(-r + υ50 + 2r 2r
(12.53)
/7o(̂ ~ vst). (12.54)
The first term on the right-hand side represents a converging spherical wave; the second term represents a diverging spherical wave that originates from the initially converging wave propagating through the center; the third term represents a diverging spherical wave. On delta heating, an initial pressure po—which is constant across the entire heated sphere—is generated. This initial pressure is divided into two equal parts, each initiating a spherical wave. One travels inward
Pressure at r = 2RS
o a.
P V3
a- TD
73 o £
0.2
0.15
0.1
0.05
0
-0.05
-0.1
-0.15
-0.2
-0.25 ( )
1
1
0.5
< I I I
1 1 1 \
1.5 2 2.5 : Normalized time: vst/Rs
1
1
5 3.5
■J
H
^
■\
H
~
A
Figure 12.4. Bipolar (positive followed by negative) pressure profile from a heated sphere versus time.
3 0 0 PHOTOACOUSTIC TOMOGRAPHY
as a converging spherical compression wave (first term), and the other travels outward as a diverging spherical compression wave (third term). When reaching the center of the heated spherical object, the converging spherical wave becomes a diverging spherical rarefaction wave (second term).
Example 12.8. Plot pressure versus time on the basis of Eq. (12.54).
The following MATLAB code produces Figure 12.4; the partial pressures in Eq. (12.54) are denoted by p in l , pinr, and pout, respectively:
% Photoacoustic signal from a homogeneously heated sphere % Use SI units
clear a l l
vs = 1500; PO = 1; Rs = 0.5E-3;
rd = 2*Rs; % Location of detector t = l inspace(0, (rd + 2*Rs)/vs, 1000);
f igure(1) e l f
p inl = p0/2*(1+vs*t . / rd).*heavis ide(rd+vs*t) .*heavis ide(Rs-rd-vs*t) ; pinr = p0/2*(1-vs*t . / rd) .*heavis ide(-rd+vs*t) .*heavis ide(Rs+rd-vs*t) ; pout = p0/2*(1-vs*t . / rd) .*heavis ide(rd-vs*t ) .*heavis ide(Rs-rd+vs*t) ; p = pinl + pinr + pout;
p lo t (vs* t /Rs, p, ' k ' ) t i c k = [.015 .025] ; set(0, 'DefaultAxesTickLength', t ick) xlabel('Normalized time: \ i tv_s t \ rm/ \ i tR_s ' ) ylabel('Normalized pressure: \ i tp \ rm/\ i tp \ rm_0') t i t l e ( 'P ressure at \ i t r \rm= 2\i tR_s')
Example 12.9. Plot snapshots of the propagating pressure on the basis of Eq. (12.54).
The following MATLAB code produces Figure 12.5; the partial pressures in Eq. (12.54) are denoted by p in l , pinr , and pout, respectively.
% Photoacoustic signal from a homogeneously heated sphere % Use SI units
clear a l l vs = 1500; PO = 1; Rs = 0.5E-3; rmax = 4;
DELTA-PULSE EXCITATION OF A SPHERE 3 0 1
rmin = 1E-3*Rs; r = linspace(0, rmax*Rs, 1000) + rmin;
theta = linspace(-pi/2, pi/2);
figure(1) elf
i_axis = 1; for t = [0:1/2:1, 2]*Rs/vs
pin1 = p0/2*(1+vs*t./r).*heaviside(r+vs*t).*heaviside(Rs-r-vs*t); pinr = p0/2*(1-vs*t./r).*heaviside(-r+vs*t).*heaviside(Rs+r-vs*t); pout = p0/2*(1-vs*t./r).*heaviside(r-vs*t).*heaviside(Rs-r+vs*t); p = pinl + pinr + pout;
subplot(4, 2, i_axis, 'align') hold off; plot(r/Rs, pin1/p0, 'k--', ...
r/Rs, pinr/ρθ, 'k-.', ... r/Rs, pout/ρθ, 'k-', ... cos(theta), sin(theta), 'k-')
tick = [.015 .025]; set(0,'DefaultAxesTickLength',tick) title(['t= ', num2str(vs*t/Rs), 'x\itR_s/v_s']) axis equal; axis([0, rmax, -2, 2]) ylabel('Partial pressures/\itp\rm_0') if (i_axis == 1)
legend('p_{in1}', 'pjinr}', 'p_{out}') end if (i_axis == 7)
xlabel('\itr/R_s') end iaxis = iaxis + 1;
subplot(4, 2, iaxis, 'align') hold off; plot(r/Rs, ρ/ρθ, 'k-', cos(theta), sin(theta), 'k-') tick = [.015 .025]; set(0,'DefaultAxesTickLength',tick) title(['t = \ num2str(vs*t/Rs), 'x\itR_s/v_s']) axis equal; axis([0, rmax, -2, 2]) ylabel('Total pressure/\itp\rm_0') if (i_axis == 8)
xlabel('\itr/R_s') end i_axis = iaxis + 1;
3 0 2 PHOTOACOUSTIC TOMOGRAPHY
t = 0xRs/vs
ΊΛ 1
t, -1
2 0 · - H
^ inr Pout
0 2 4 r = 0.5 x/?5/vs
o
c/3
2 ex
"03 O
H
2
1
0
-1
1
f = 0 x Rs/vs
<
0 2 4 r = 0.5 x/?s/vs
Figure 12.5. Snapshots of propagating pressure from a heated sphere.
12.8. FINITE-DURATION PULSE EXCITATION OF A THIN SLAB
Because the photoacoustic response in an infinite medium is both linear to the excitation power and timeshift-invariant, the response to a finite-duration excita- tion pulse R(t) can be computed by convolution:
R(t) /
+oo /·+οο
dt'G(t - t')S(t') = / dt'G(t')S(t - t'), (12.55) -oo J—oo
DARK-FIELD CONFOCAL PHOTOACOUSTIC MICROSCOPY 3 0 3
where G(t) denotes the response to a delta excitation pulse and S(t) denotes the power of the pulse as a function of time.
For an excited slab, G(t) is a top-hat function of time as shown in Figure 12.2. If the slab becomes so thin that the acoustic transit time across the slab is much less than the duration of the excitation pulse, G(t) approaches a delta function apart from a constant factor:
G{t)(xh{vst-z). (12.56)
Substituting Eq. (12.56) into Eq. (12.55) leads to
R(t) a S(vst - z), (12.57)
which means that the photoacoustic pressure is proportional to the excitation pulse. For example, if the pulse is Gaussian, that is,
S(t) = 5 o e x P r - ^ ( r ~ 2 r o ) 1 , (12.58)
the photoacoustic pressure is Gaussian as well. Here, So denotes the peak power, to denotes the center timepoint corresponding to the peak power, and σ denotes the standard deviation.
12.9. FINITE-DURATION PULSE EXCITATION OF A SMALL SPHERE
Outside an excited sphere, G(t) is a bipolar function of time as shown in Figure 12.4. If the sphere becomes so small that the acoustic transit time across the sphere is much less than the width of the excitation pulse, G{t) approaches the derivative of a delta function apart from a constant factor:
G(0oc — h(vst-r). (12.59) at
Substituting Eq. (12.59) into Eq. (12.55) leads to
R(t)<x^-S(vst-r), (12.60) at
which means that for a small spherical object, the photoacoustic pressure is proportional to the time derivative of the excitation pulse.
12.10. DARK-FIELD CONFOCAL PHOTOACOUSTIC MICROSCOPY
Reflection-mode (or backward-mode) confocal photoacoustic microscopy (PAM) has been implemented using both dark-field pulsed-laser illumination and high-NA ultrasonic detection. In conventional dark-field transmission optical
3 0 4 PHOTOACOUSTIC TOMOGRAPHY
microscopy, an opaque disk is placed between the light source and the condenser lens so that ballistic light is rejected; as a result, only nonballistic light—which is scattered by the sample—is detected. In dark-field PAM, the excitation laser beam has a donut-shaped cross section; therefore, the photoacoustic signal from the tissue surface in the field of view is minimized.
In the system shown in Figure 12.6, a Q-switched pulsed Nd:YAG (neodymium: yttrium aluminum garnet) laser, operating at 532-nm wavelength, delivers 0.3 mJ of energy per pulse to the object through a 0.6-mm-diameter optical fiber. The laser beam has a 6.5-ns pulsewidth and a 10-Hz pulse repeti- tion frequency. The Hghtbeam from the fiber output end is coaxially aligned with a focused ultrasonic transducer; both are mounted on a 3D mechanical transla- tion stage. The ultrasonic transducer has a 50-MHz center frequency and a 70% nominal bandwidth. A concave acoustic lens with a 5.5-mm aperture diameter and a 5.6-mm working distance is attached to the ultrasonic transducer. This positive lens provides an NA of 0.44, which is considered large in ultrasonics. Light from the fiber is expanded by a conical lens and then focused onto the object through an optical condenser that has an NA of 1.1. The optical focal region overlaps with the focal spot of the ultrasonic transducer; thus, the optical dark-field illumination and ultrasonic detection are confocal.
Photoacoustic signals are received by the ultrasonic transducer, amplified by a low-noise amplifier, and then recorded digitally. For ultrasonic coupling, the ultrasonic transducer is immersed in water inside a plastic container; the bottom of the container has an opening that is sealed with a thin disposable polyethylene membrane; the object is first coated with ultrasound-coupling gel and then placed below the membrane.
On each laser-pulse excitation, the emitted photoacoustic wave is recorded as a function of time at each location of the ultrasonic transducer. The photoacoustic
Diffuser t Optical
rib fibcr
Conical lens \
Ultrasonic Sample I · transducer
Figure 12.6. Schematic of a PAM system.
DARK-FIELD CONFOCAL PHOTOACOUSTIC MICROSCOPY 3 0 5
(a) 11.3 12.7 14.3 16.0 mm"1
(b) 1 mm
Figure 12.7. (a) Spatial resolution test with a bar chart placed in a tissue phantom (num- bers below the images indicate the spatial modulation frequency); (b) imaging depth test with a black double-stranded cotton thread placed obliquely in the abdominal area of a rat (1 represents the skin surface; 2 represents the thread).
signal is converted into a ID depth-resolved image (A-scan), based on the sound velocity in soft tissue (1.54 mm^xs). Raster scanning of the PAM probing head in a horizontal (xy) plane produces a 3D image.
Four PAM images of a Mylar USAF-1951 target taken through a 4-mm-thick layer of light-scattering tissue phantom made of 2% Intralipid® solution and 1% agar gel are shown in Figure 12.7a; the solid curves show the relative peak-to- peak amplitude of the received photoacoustic pressure across the bars on the target. The reduced scattering coefficient μ5 of the phantom is 15 cm- 1 . The thickness of the phantom equals six transport mean free paths. The modulation transfer function of the system is extracted from Figure 12.7a and extrapolated to its cutoff spatial frequency, producing an estimated lateral resolution of 45 μπι. From another image of a 6^m-diameter carbon fiber, the axial resolution is estimated to be ~15 μπι.
A PAM B-scan image of a black double-stranded cotton thread of 0.2 mm in diameter and 1.25 mm in pitch, which is inserted obliquely into the abdominal area of a sacrificed rat, is shown in Figure 12.7b. The thread is clearly visible in the image up to 3 mm in depth.
PAM images of the vasculature in the dorsal dermis (the upper lumbar area left of the vertebra) of a rat, with the hair removed using commercial hair remover lotion, is shown in Figure 12.8. Four in situ consecutive PAM B-scan images are obtained 0.2 mm apart laterally (Figure 12.8a). Each image is a gray-scale plot of the peak-to-peak amplitudes of the received photoacoustic signals; the vertical and the horizontal axes represent the depth from the skin surface and
3 0 6 PHOTOACOUSTIC TOMOGRAPHY
itrast 1
1 mm
:jl|:
1 mm
Figure 12.8. (a) Four in situ consecutive PAM B-scan images (1 and 2 represent vessels perpendicular to the imaging plane; 3 represents an in-plane vessel); (b) in situ maximum- amplitude projection PAM image taken from epidermal side; (c) photograph taken from dermal side using transmission illumination; (d) in vivo noninvasive maximum-amplitude projection PAM image taken from epidermal side.
the horizontal ultrasonic transducer position, respectively. The focal plane of the ultrasonic transducer is located at a depth of 1.2 mm. The PAM probing head is scanned 100 steps horizontally with a step size of 0.1 mm. The slightly inclined solid line in the upper part of each B-scan delineates the skin surface. The vessels marked by 1 and 2 are nearly perpendicular to the imaging plane, whereas the vessel marked by 3 is nearly parallel to the imaging plane.
An in situ maximum-amplitude projection PAM image on the skin surface (100x100 pixels, 0.1 mm step size), which plots the maximum peak-to-peak amplitude of each received photoacoustic signal (A-scan) within a 0.2-2-mm depth interval from the skin surface versus the ultrasonic transducer position on the tissue surface (JC, y), is shown in Figure 12.8b. For comparison, a photo- graph (obtained using transmission light illumination) of the inner surface of the harvested skin is shown in Figure 12.8c. Good agreement in the vasculature is observed between the PAM image and the photograph. The photograph indicates that the major vessels are M 0 0 μιτι in diameter and the smaller vessels —30 μιη in diameter.
An in vivo maximum-amplitude projection PAM image of a similar area (100 x 100 pixels, 0.05 mm step size, and 0.5-3-mm depth interval) is shown
CO
i*r
(a) 1 mm <b)
1 mm (c)
SYNTHETIC APERTURE IMAGE RECONSTRUCTION 3 0 7
in Figure 12.8d. Blood vessels with a density of up to a few counts per mm are observed.
PAM is capable of imaging optical absorption in biological tissue in the qua- sidiffusive regime, where the spatial resolution is determined primarily by the ultrasonic detection parameters instead of the optical excitation parameters. If the laser pulse is sufficiently short, a high-NA acoustic lens and a high-center- frequency ultrasonic transducer provide high lateral resolution while a wideband ultrasonic transducer provides high axial resolution.
Compared with bright-field illumination, dark-field illumination provides sev- eral advantages: (1) a large illumination area reduces the optical fluence on the tissue surface to less than 1 mJ/cm2, which is well within the safety standards; (2) a large illumination area partially averages out the shadows of superficial het- erogeneities in the image; and (3) dark-field illumination reduces the otherwise strong interference of the extraneous photoacoustic signal from the superficial paraxial area.
12.11. SYNTHETIC APERTURE IMAGE RECONSTRUCTION
Like optical waves, ultrasonic waves can be focused when passing through an acoustic lens, which is the basis of PAM as described in the previous section. Acoustic focusing can also be achieved synthetically by scanning a single-element ultrasonic transducer or using a multielement ultrasonic transducer. According to the principle of reciprocity, ultrasonic transmission and detection of the same acoustic lens are reciprocal.
In Figure 12.9, ultrasonic transmission focusing is illustrated with a five- element array transducer. If all elements are excited at the same time with the same voltage, an approximate plane ultrasonic wave is produced. If the elements are excited at different times, the produced ultrasonic wave can be focused to various points. In this illustration, the excitation of the outermost elements pre- cedes that of the center element by AR/vs, where AR is the difference in radial distance to the desired focal point between the outermost elements and the center element. If similar proper delays are applied to all of the elements accordingly, ultrasonic pulses from these elements arrive at the focal point simultaneously; thus, the produced ultrasonic wave is focused to the desired point. Further, the focal point can be off the ultrasonic axis. If the focal point is set to infinity, an approximate plane wave is produced.
Like an optical grating, an ultrasonic array transducer produces the following far-field amplitude distribution (Figure 12.10):
sin(Nvd) sin ve H(Q) = . - , (12.61)
N sin Vd ve
where
vd = -kde sine (12.62)
3 0 8 PHOTOACOUSTIC TOMOGRAPHY
Figure 12.9. Ultrasonic transmission focusing by a multielement ultrasonic transducer.
Figure 12.10. Acoustic-amplitude pattern produced by an ultrasonic array transducer ver- sus the polar angle in a polar plot.
and
ve = - kwe sinG. (12.63)
Here, Θ denotes the polar angle, N denotes the number of elements, k denotes the magnitude of the wavevector, de denotes the periodic distance between the elements, and we denotes the width of each element. Sidelobes, also termed
GENERAL IMAGE RECONSTRUCTION 3 0 9
grating lobes, appear at polar angles given by
1 ™λα θ^ =s in" 1 — A (12.64)
de where m is a nonzero integer and λα is the acoustic wavelength. The angular width of the mainlobe or each grating lobe is given by
Δθ^ = s i i r l - ^ - . (12.65) Nd( e
Grating lobes deteriorate the lateral resolution in imaging but can be minimized using various designs. For example, the width of each element can be enlarged to reduce the magnitude of the grating lobes relative to the mainlobe, the excitation pulses can be shortened as much as possible, or the spacing between elements can be minimized or randomized.
Synthetic-aperture detection, also referred to as delay-and-sum detection or beamforming, can be applied to imaging. This detection scheme is reciprocal to the aforementioned transmission focusing. For each focal point, the image signal S can be calculated using
5(0 = 5^5 / ( / + Δί/). (12.66)
Here, S/ is the signal from the ith ultrasonic transducer element; Δί,- is the time delay for the ith transducer element, which can be calculated similarly as in Figure 12.9.
12.12. GENERAL IMAGE RECONSTRUCTION
In this section, we consider general image reconstruction for an infinite acousti- cally homogeneous medium. The initial photoacoustic pressure excited by pulse h(t) equals po(r) = T(r)Hs(r) [Eq. (12.29)]. The acoustic pressure p(r,t) at position r and time t, initiated by source po(r), satisfies the following photoa- coustic wave equation [see Eq. (12.13)]:
Po(r)dh(t) p(r, t) = — . (12.67)
vs dt
Three detection configurations are considered. As shown in Figure 12.11, the detection surface is represented by So. For the planar geometry, if another planar surface S'Q parallel to So is added, the combination of S'0 and So encloses the source po(r). For convenience, we write S = So + S'0 for the planar geometry and S = So for the cylindrical or spherical geometry.
3 1 0 PHOTOACOUSTIC TOMOGRAPHY
V
So s
/ (b)
Figure 12.11. (a) During measurement, an ultrasonic point detector at position ro on surface SQ receives photoacoustic signals emitted from source po(r)—during image recon- struction, a quantity related to the measurement at position ro projects backward via a spherical surface centered at ?o; (b) in the planar geometry, another surface Sf0 is combined with SQ to enclose the entire source.
The following Fourier transformation pair is used to convert pressure between the time and frequency domains:
/
+oo F(t)exp(ikJ)dJ, (12.68)
-co
F(t) = — \ F(k)exp(-ikt)dk, (12.69) 2π J_00
where 1 — vst and k — ω/νχ (ω is the angular frequency). According to Green's theorem, the spectrum of the measured pressure P(TQ, 1)
is given by
p(r0,k) = -ik[ d?G ( k out\?, r0)p0(r). (12.70)
Jv
Here, V is a volume enclosing the entire source po(r)\ G^u (f, r0) is a Green function representing a monochromatic diverging spherical wave:
o r ( ^ ) = exp(y-^|) 4n\r -r0\
GENERAL IMAGE RECONSTRUCTION 311
The acoustic pressure p(r,k) inside S can be computed by
p(r, k)= ί dS £*(?<>, k)[2ns0 · V0G<OUV, r0)], (12.72)
where * indicates complex conjugation—equivalent to time reversal (see Prob- lem 12.1), Vo denotes the gradient on ?o, and ns0 denotes the normal vector of S pointing inward. The term in the square brackets indicates dipole radi- ation. Since po(r) — p(r, 1 = 0), taking the inverse Fourier transformation of Eq. (12.72) leads to the following backprojection formula:
1 / *+00 /»
Po(r) = - / dkl dS £*(r0, k)[n50 ■ VoG<out)(r, r0)]. (12.73) π J-oo JS
For the planar geometry, if S is replaced by SQ, the right-hand side yields po(r)/2 instead. Since the reconstructed pressure is real, Eq. (12.73) can be rewritten as
Mr) = - dS dk p(?o, k)[ns0 · V0G<m)(r, r0)], (12.74) ^ Js ./-oo
where G^ (r, ?ο) is a Green function corresponding to a monochromatic con- verging spherical wave:
Min)r - , exp(-ifc|r - r0|) G^ '(r, r0) = T—p——. · (12.75) 4n\r-r0\
A rigorous proof of Eq. (12.74) for the three common detection geometric con- figurations is given in the references for this section.
Employing VoG^in)(r, ro) = — VGJp(r, r0) and inverse-Fourier-transforming p(ro, k)> we can rewrite Eq. (12.74) in the time domain
Po(r) = - ^ ' i flhdSol^] . 02.76) " 0 JSo L * JF=|F-r0|
where Ωο is the solid angle subtended by the entire surface So with respect to the reconstruction point r inside So· We have Ωο = 2π for the planar geometry and Ωο = 4π for the spherical or cylindrical geometry.
Further, we can rewrite Eq. (12.76) in a backprojection form as
= / Po(r)= I 6(ro,i = | r - r o | Ä . (12.77)
3 1 2 PHOTOACOUSTIC TOMOGRAPHY
Figure 12.12. Diagram showing the solid angle dQo subtended by detection element dS{) with respect to point P at r.
Here, b(fo, t) is the backprojection term; dQo is the solid angle subtended by detection element dSo with respect to reconstruction point P at r (Figure 12.12):
b(r0, t) = 2p(r0, t) - It F\- \ (12.78) at
dSo fin- (r — 7Q) dQo = — - j ^ V ° . (12.79)
Factor dQo/ Ωο weighs the contribution from detection element dSo to the recon- struction. The reconstruction simply projects b(?o,l) backward via a spherical surface centered at position rn. The first derivative with respect to time t actually represents a pure ramp filter k, which suppresses low-frequency signals.
The theory described above is based on an ideal infinite bandwidth in both time and space. In practice, if k\r — r0| ^> 1 within the detection bandwidth, we have ldp(ro, 1)1 dl > /?(r0, 7), which means that Eq. (12.78) can be simplified to
b{roJ)^-2t μ \ . \ (12.80) ot
A circular scanning configuration of PAT was implemented to image small- animal brains (Figure 12.13a). A Q-switched Nd:YAG laser provides light pulses (532-nm wavelength, 6.5-ns pulse duration, and 10-Hz pulse repetition fre- quency). The laser beam is expanded and homogenized to provide relatively uniform incident fluence, which is less than 10 mJ/cm2 on the skin surface. Pho- toacoustic waves are coupled through water to an ultrasonic transducer with a center frequency of 3.5 MHz.
Since a circle—rather than a full spherical surface—is scanned, the recon- struction algorithm presented above is only approximately applicable. Neverthe- less, good images are still attainable. Blood vessels in the cortical surface of small animals can be imaged transcranially with the scalp and the skull intact,
APPENDIX 12A. DERIVATION OF ACOUSTIC WAVE EQUATION 3 1 3
Light pulse
Transducer 1 ή
(a)
Sample \ )
Circular scan
Max
(cm)
Optical absorption Min
(b)
Figure 12.13. (a) Diagram of a circular-scanning PAT system for small-animal imaging; (b) a cross-sectional PAT image of a rat brain (RH represents right cerebral hemisphere; LH, left cerebral hemisphere; L, lesion; MCA, middle cerebral artery).
although the hair must be removed (Figure 12.13b). At this optical wavelength, the contrast of hemoglobin is high; the imaging depth is limited to about 1 cm, which is greater than the dimension of the entire brain of a small animal such as a mouse.
APPENDIX 12A. DERIVATION OF ACOUSTIC WAVE EQUATION
A longitudinal small-amplitude acoustic plane wave propagating in a homoge- neous and nondissipative medium in the x direction is considered here. We exam- ine the motion of a differential volume element dV = dxdydz at position x:
1. We derive the material equation. The excess pressure p is a function of the mass density p:
P(P)· (12.81)
3 1 4 PHOTOACOUSTIC TOMOGRAPHY
This equation can be expanded to the first order of the Taylor series around the equilibrium mass density po as
ρ-ρο=(γ)((>-9ο), (12.82)
where po denotes the equilibrium pressure. The condensation parameter s is defined as
S=PZJ*^ (12.83) Po
which can be rewritten as
p = P o ( l + 5 ) . (12.84)
For a small-amplitude acoustic wave, we have s <^ 1. Substituting Eq. (12.84) into Eq. (12.82), we obtain
=p°(l)5· p-Po = Po[-f)s. (12.85) 2. We then derive the force equation. The force F due to pressure p experi-
enced by the differential volume element is given by
m dx dy dz. (12.86) From Newton's second law, we have
dp du — - = P—, (12.87)
dx *dt v
where u denotes the medium velocity and t denotes time. Since s <^1, we replace p by po and yield
dp du - / = Po^-. (12.88)
dx dt This equation, termed the linear inviscid force equation, can be generalized to 3D space as -Vp = po(dü/dt). Substituting Eq. (12.85) into (12.88) yields
ds 1 du = . (12.89)
dx dp/dp dt
3. We derive the continuity equation, based on the conservation of mass:
do d(ou) *L = _Λϋ_Ζ, (12.90)
dt dx
APPENDIX 12A. DERIVATION OF ACOUSTIC WAVE EQUATION 3 1 5
This equation can be generalized to — (dp/dt) = V · (pw). The right-hand side of Eq. (12.90) is expanded to
3(pw) du dp / n n n —— = ΡΤ- + Τ-Μ· (12.91)
ox dx ax Since s <£ 1, we replace p by po in the first term on the right-hand side to yield
d(pu) du dp -ΊΓ1 = Ρο3Γ + 1TU· ( 1 1 9 2 )
ax ax dx We will show later that the second term on the right-hand side is negligible. In this case, Eq. (12.90) becomes
3p du - T 7 = P 0 F - , (12.93)
dt dx which is the linearized continuity equation. Substituting Eq. (12.84) into Eq. (12.93), we obtain
35 du - * - » ; · < i 2 ' 9 4 )
4. Last, differentiating Eq. (12.89) with respect to JC and differentiating Eq. (12.94) with respect to f, and then taking the difference between them, we obtain
d2p _ 1 d2p
'dx2' ~ dp/dp'dt1''
This is a wave equation with
(12.95)
dp „2 vt, (12.96)
(12.97)
ap *'
where vs is the speed of sound in the medium. Thus, we have
d2P _ l d 2 P
'dx1~~^'dt2,
which can be generalized to the 3D case as
9 1 d2p v p = ^ ^ ' < , 2 · 9 8 )
This is the basic acoustic wave equation that describes the propagation of an acoustic wave in a homogeneous nondissipative medium.
As promised, we now show that the second term on the right-hand side of Eq. (12.92) is negligible. The ratio of the second term to the first term on the
3 1 6 PHOTOACOUSTIC TOMOGRAPHY
right-hand side of Eq. (12.92) is
u 3p u (dpjdu\ p09w po \dp/df)J
From Eq. (12.97), we obtain
\8x vs dt J \dx vs dt )
(12.99)
(12.100)
Therefore, we have \(dp/dt)/(dp/dx)\ = vs, which is substituted into Eq. (12.88) to yield
d_p
du POVJ. (12.101)
Substituting Eqs. (12.96) and (12.101) into Eq. (12.99) yields
u 3p p0 du
m (12.102)
If \u\ <£ vs (subsonic medium velocity of a small-amplitude wave), then
dp 1
dx « Po du
Jx
which means that the second term on the right-hand side of Eq. (12.92) is neg- ligible.
APPENDIX 12B. GREEN FUNCTION APPROACH
The general Green function approach is summarized here. The acoustic wave equation with a source term q(r\ tr) is
7'2 ^ ,/x j 32P<r'> ' ') Vlp{r\t') v2 dt'2
= -q(r\t'). (12.103)
On the basis of the reciprocity relation, Green's function G(r, t\ r', t') satisfies the following equation:
V2G(r, t\r\t') 1 d2G(r,t;r',t')
~r2 = -h(r-r')h(t - ί ' ) . (12.104)
Multiplying Eq. (12.103) by G and Eq. (12.104) by p, subtracting them, and then integrating over r' in the volume of interest V and over t' from 0 to i + ,
PROBLEMS 317
we obtain
= - / dt' I dr'G(r, t\ r', t')q(?',t') + p(r, t). Jo Jv
(12.105)
Applying Green's theorem to Eq. (12.105) yields
p(r,t)= f dt' f dr'G(r,t\r',t')q(r\t') Jo Jv
+ [ dt' \ dS' · [GVp(r\ t') - pV'G] Jo Js'
+ — dr' [p G—)\ . vjU Vdt> dt'Jl
Here, S' encloses V". Choosing / + = t + 0+ , we have p(r', t+) = 0 and
dp(r', t>) I
(12.106)
dt' = 0,
t'=l+
due to causality. Thus, Eq. (12.106) becomes
p(r,t)= f dt' [ dr'G(r,t\r',t')q{r',t') Jo Jv
+ [ dt' f dS' ■ [GV'p(?\ t') - pV'G] (12.107) Jo Js' 1 f *( , dG dp
~G | ' , = 0ä^ f'=0 öt
f ' = 0 /
The first integral on the right-hand side depends on the source q(r', t'); the second depends on the boundary condition; the third depends on the initial condition.
PROBLEMS
12.1 Show that complex conjugation of the temporal spectrum is equivalent to time reversal of the temporal function.
12.2 Derive the total acoustic energy transported through an enclosing spher- ical shell that is concentric to a delta heated sphere. Estimate how much optical energy is converted into mechanical energy.
PHOTOACOUSTIC TOMOGRAPHY
12.3 Derive and plot the pressure as a function of time observed outside a sphere excited by (a) a delta pulse and (b) a Gaussian pulse.
12.4 Derive and plot the pressure as a function of time observed outside a thin spherical shell excited by (a) a delta pulse and (b) a Gaussian pulse.
12.5 Derive and plot the pressure as a function of time observed outside a line object first excited by (a) a delta pulse and (b) a Gaussian pulse.
12.6 Derive and plot the pressure as a function of time observed outside a cylindrical object excited by (a) a delta pulse and (b) a Gaussian pulse.
12.7 Derive and plot the pressure as a function of time observed outside an optically absorbing slab object excited by (a) a delta pulse and (b) a Gaussian pulse.
12.8 Derive and plot the pressure as a function of time observed outside a thin disk object excited by (a) a delta pulse and (b) a Gaussian pulse. Set the observation point first in the plane of the disk and then on the axis of the disk.
12.9 Derive and plot the pressure as a function of time observed outside a thin ring object excited by (a) a delta pulse and (b) a Gaussian pulse. Set the observation point first in the plane of the ring and then on the axis of the ring.
12.10 Using velocity potential, derive and plot the pressure as a function of time observed outside a slab object excited by (a) a delta pulse and (b) a Gaussian pulse.
12.11 Make a movie in MATLAB showing the pressure propagation from a sphere in response to a delta-pulse excitation.
12.12 Make a movie in MATLAB showing the pressure propagation from a spherical shell in response to delta-pulse excitation.
12.13 Make a movie in MATLAB showing the pressure propagation from two spheres in response to delta-pulse excitation.
12.14 Fourier-transform the pressure versus time observed outside a slab in response to delta-pulse excitation.
12.15 Fourier-transform the pressure versus time observed outside a sphere in response to delta-pulse excitation.
12.16 Sketch approximately the ideal wavefronts that are produced by a pulsed laser beam incident on a semiinfinite medium when the opti- cal beam diameter is (a) much greater than, (b) much smaller than, and (c) comparable to the optical penetration depth.
12.17 Derive the pressure as a function of time produced from a periodic array of slabs by delta-pulse excitation. Then take its Fourier transformation.
FURTHER READING 3 1 9
12.18 Prove that the focal length of a planoconcave acoustic lens can be approx- imately expressed as / = δ/(1 — \/n), where δ is the radius of curvature, and n = vs\/vS2, where vs\ is the acoustic velocity in the lens and vS2 is the acoustic velocity in the surrounding medium. Assume that the diameter of the lens is small compared with the radius of curvature.
12.19 Derive and plot Eq. (12.61).
12.20 Derive from Eq. (12.74) to Eq. (12.76) and then to Eq. (12.77).
12.21 Implement the delay-and-sum reconstruction algorithm and test it with dummy data generated from the forward solution.
12.22 Implement the general reconstruction algorithm and test it with dummy data generated from the forward solution.
READING
Cho ZH, Jones JP, and Singh M (1993): Foundations of Medical Imaging, Wiley, New York. (See Appendix 12A, above.)
Diebold GJ, Sun T, and Khan MI (1991): Photoacoustic monopole radiation in 1- dimension, 2-dimension, and 3-dimension, Phys. Rev. Lett. 67(24): 3384-3387. (See Sections 12.6 and 12.7, above.)
Feng DZ, Xu Y, Ku G, and Wang LHV (2001): Microwave-induced thermoacoustic tomography: Reconstruction by synthetic aperture, Med. Phys. 28(12): 2427-2431. (See Section 12.11, above.)
Hoelen CG A and de Mul FFM (1999): A new theoretical approach to photoacoustic signal generation, J. Acoust. Soc. Am. 106(2): 695-706. (See Sections 12.6-12.9, above.)
Maslov K, Stoica G, and Wang LHV (2005): In vivo dark-field reflection-mode photoa- coustic microscopy, Opt. Lett. 30(6): 625-627. (See Section 12.10, above.)
Morse PM and Feshbach H (1999): Methods of Theoretical Physics, McGraw-Hill, New York. (See Appendix 12B, above.)
Wang LHV (2003): Ultrasound-mediated biophotonic imaging: A review of acousto- optical tomography and photo-acoustic tomography, Disease Markers 19(2-3): 123-138. (See Section 12.2, above.)
Wang XD, Pang YJ, Ku G, Xie XY, Stoica G, and Wang LHV (2003): Noninvasive laser-induced photoacoustic tomography for structural and functional in vivo imaging of the brain, Nature Biotechnol. 21(7): 803-806. (See Section 12.12, above.)
Xu MH and Wang LHV (2005): Universal back-projection algorithm for photoacous- tic computed tomography, Phys. Rev. E 71(1): 016706. (See Sections 12.3-12.5 and 12.12, above.)
FURTHER READING
Anastasio MA, Zhang J, Pan XC, Zou Y, Ku G, and Wang LHV (2005): Half-time image reconstruction in thermoacoustic tomography, IEEE Trans. Med. Imaging 24(2): 199-210.
Andreev VG, Karabutov AA, and Oraevsky AA (2003): Detection of ultrawide-band ultrasound pulses in optoacoustic tomography, IEEE Trans. Ultrasonics Ferroelectrics Freq. Control 50(10): 1383-1390.
3 2 0 PHOTOACOUSTIC TOMOGRAPHY
Arfken GB and Weber HJ (1995): Mathematical Methods for Physicists, Academic Press, San Diego.
Beard PC, Perennes F, Draguioti E, and Mills TN (1998): Optical fiber photoacoustic- photothermal probe, Opt. Lett. 23(15): 1235-1237.
Bell AG (1880): On the production and reproduction of sound by light, Am. J. Sei. 20: 305-324.
Born M and Wolf E (1999): Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, Cambridge Univ. Press, Cambridge, UK/New York.
Cox BT and Beard PC (2005): Fast calculation of pulsed photoacoustic fields in fluids using k-space methods, /. Acoust. Soc. Am. 117(6): 3616-3627.
Esenaliev RO, Karabutov AA, and Oraevsky AA (1999): Sensitivity of laser opto-acoustic imaging in detection of small deeply embedded tumors, IEEEJ. Select. Topics Quantum Electron. 5(4): 981-988.
Finch D, Patch SK, and Rakesh (2003): Determining a function from its mean values over a family of spheres, SI AM J. Math. Anal. 35(5): 1213-1240.
Gusev VE and Karabutov AA (1993): Laser Optoacoustics, American Institute of Physics, New York.
Haltmeier M, Scherzer O, Burgholzer P, and Paltauf G (2004): Thermoacoustic computed tomography with large planar receivers, Inverse Problems 20(5): 1663-1673.
Hoelen CGA, de Mul FFM, Pongers R, and Dekker A (1998): Three-dimensional pho- toacoustic imaging of blood vessels in tissue, Opt. Lett. 23(8): 648-650.
Karabutov A A, Podymova NB, and Letokhov VS (1996): Time-resolved laser optoacoustic tomography of inhomogeneous media, Appl. Phys. B—Lasers Opt. 63(6): 545-563.
Karabutov AA, Savateeva EV, and Oraevsky AA (2003): Optoacoustic tomography: New modality of laser diagnostic systems, Laser Phys. 13(5): 711-723.
Karabutov A A, Savateeva EV, Podymova NB, and Oraevsky A A (2000): Backward mode detection of laser-induced wide-band ultrasonic transients with optoacoustic transducer, J. Appl. Phys. 87(4): 2003-2014.
Kostii KP, Frenz M, Weber HP, Paltauf G, and Schmidt-Kloiber H (2000): Optoacoustic infrared spectroscopy of soft tissue, J. Appl. Phys. 88(3): 1632-1637.
Kostii KP, Frauchiger D, Niederhauser JJ, Paltauf G, Weber HP, and Frenz M (2001): Optoacoustic imaging using a three-dimensional reconstruction algorithm, IEEE J. Select. Topics Quantum Electron. 7(6): 918-923.
Kostii KP, Frenz M, Weber HP, Paltauf G, and Schmidt-Kloiber H (2001): Optoacoustic tomography: Time-gated measurement of pressure distributions and image reconstruc- tion, Appl. Opt. 40(22): 3800-3809.
Kostii KP and Beard PC (2003): Two-dimensional photoacoustic imaging by use of Fourier-transform image reconstruction and a detector with an anisotropic response, Appl. Opt. 42(10): 1899-1908.
Kruger RA, Reinecke DR, and Kruger GA (1999): Thermoacoustic computed tomography- technical considerations, Med. Phys. 26(9): 1832-1837.
Ku G and Wang LHV (2000): Scanning thermoacoustic tomography in biological tissue, Med. Phys. 27(5): 1195-1202.
Ku G and Wang LHV (2001): Scanning microwave-induced thermoacoustic tomography: Signal, resolution, and contrast, Med. Phys. 28(1): 4-10.
Ku G, Wang XD, Stoica G, and Wang LHV (2004): Multiple-bandwidth photoacoustic tomography, Phys. Med. Biol. 49(7): 1329-1338.
FURTHER READING 321
Ku G and Wang LHV (2005): Deeply penetrating photoacoustic tomography in biological tissues enhanced with an optical contrast agent, Opt. Lett. 30(5): 507-509.
Ku G, Wang XD, Xie XY, Stoica G, and Wang LHV (2005): Imaging of tumor angiogen- esis in rat brains in vivo by photoacoustic tomography, Appl. Opt. 44(5): 770-775.
Oraevsky AA, Jacques SL, and Tittel FK (1997): Measurement of tissue optical properties by time-resolved detection of laser-induced transient stress, Appl. Opt. 36(1): 402-415.
Paltauf G and Schmidt-Kloiber H (2000): Pulsed optoacoustic characterization of layered media, J. Appl Phys. 88(3): 1624-1631.
Paltauf G, Viator JA, Prahl SA, and Jacques SL (2002): Iterative reconstruction algorithm for optoacoustic imaging, J. Acoust. Soc. Am. 112(4): 1536-1544.
Viator JA, Au G, Paltauf G, Jacques SL, Prahl SA, Ren HW, Chen ZP, and Nelson JS (2002): Clinical testing of a photoacoustic probe for port wine stain depth determina- tion, Lasers Surg. Med. 30(2): 141-148.
Wang LHV, Zhao XM, Sun HT, and Ku G (1999): Microwave-induced acoustic imaging of biological tissues, Rev. Sei. Instrum. 70(9): 3744-3748.
Wang XD, Pang YJ, Ku G, Stoica G, and Wang LHV (2003): Three-dimensional laser- induced photoacoustic tomography of mouse brain with the skin and skull intact, Opt. Lett. 28(19): 1739-1741.
Xu MH, Ku G, and Wang LHV (2001): Microwave-induced thermoacoustic tomography using multi-sector scanning, Med. Phys. 28(9): 1958-1963.
Xu MH and Wang LHV (2002): Time-domain reconstruction for thermoacoustic tomog- raphy in a spherical geometry, IEEE Trans. Med. Imaging 21(7): 814-822.
Xu MH and Wang LHV (2003): Analytic explanation of spatial resolution related to bandwidth and detector aperture size in thermoacoustic or photoacoustic reconstruction, Phys. Rev. E 67(5): 056605.
Xu Y and Wang LHV (2001): Signal processing in scanning thermoacoustic tomography in biological tissues, Med. Phys. 28(7): 1519-1524.
Xu Y, Feng DZ, and Wang LHV (2002): Exact frequency-domain reconstruction for thermoacoustic tomography—I: Planar geometry, IEEE Trans. Med. Imaging 21(7): 823-828.
Xu Y, Xu MH, and Wang LHV (2002): Exact frequency-domain reconstruction for ther- moacoustic tomography—II: Cylindrical geometry, IEEE Trans. Med. Imaging 21(7): 829-833.
Xu Y and Wang LHV (2004): Time reversal and its application to tomography with diffracting sources, Phys. Rev. Lett. 92(3): 033902.
Zhang HF, Maslov K, Stoica G, and Wang LHV (2006): Functional photoacoustic microscopy for high-resolution and noninvasive in vivo imaging, Nature Biotechnol. 24(7): 848-851.
Zhang J, Anastasio MA, Pan XC, and Wang LHV (2005): Weighted expectation maxi- mization reconstruction algorithms for thermoacoustic tomography, IEEE Trans. Med. Imaging 24(6): 817-820.
CHAPTER 13
Ultrasound-Modulated Optical Tomography
13.1. INTRODUCTION
Ultrasound-modulated optical tomography (UOT), first demonstrated in the 1990s, is another hybrid method that combines optical contrast and ultrasonic resolution as does photoacoustic tomography. UOT is based on the ultrasonic modulation of coherent laser light in a scattering medium. The medium is irradiated by both a laser beam and a focused ultrasonic wave. The ultrasound- modulated component of the reemitted light, which carries information about the local optical and acoustic properties, is used to provide tomographic imaging. Consequently, the image contrast is related to the optical and acoustic prop- erties, whereas the spatial resolution is determined primarily by the ultrasonic wave. Because all the ultrasound-modulated light—regardless of the number of scattering events experienced—contributes to the imaging, UOT is capable of imaging deeply into the optical quasidiffusive or diffusive regime.
13.2. MECHANISMS OF ULTRASONIC MODULATION OF COHERENT LIGHT
Three mechanisms have been identified to account for ultrasonic modulation of light in a scattering medium:
1. Incoherent Modulation of Light Due to Ultrasound-Induced Variations in Optical Properties of Medium. As an ultrasonic wave propagates in a scat- tering medium, the medium is compressed or rarefied depending on the location and time, which causes the mass density to vary. The variations in the mass density further modulate the optical properties—including the absorption coefficient, scattering coefficient, and index of refraction—of the medium. Consequently, the reemitted light intensity varies with the
Biomedical Optics: Principles and Imaging, by Lihong V. Wang and Hsin-i Wu Copyright © 2007 John Wiley & Sons, Inc.
323
324 ULTRASOUND-MODULATED OPTICAL TOMOGRAPHY
ultrasonic wave. Although this mechanism does not require light coher- ence, ultrasonic modulation of low-coherence light is much weaker than that of high-coherence light.
2. Variations in Optical Phase in Response to Ultrasound-Induced Displace- ments of Scatterers. The displacements of scatterers modulate the free-path lengths (hence the phases) of light traversing the ultrasonic field. The mod- ulated optical phases in the free-path lengths are accumulated along each complete path. Consequently, the reemitted light, which forms a speckle pattern, fluctuates with the ultrasonic wave.
3. Variations in Optical Phase in Response to Ultrasonic Modulation of Index of Refraction of Background Medium. The modulated index of refraction modulates the free-path phases of light traversing the ultrasonic field. The modulated optical phases contribute to speckle fluctuations as in the second mechanism.
Mechanisms 2 and 3 require light coherence. Both analytical and Monte Carlo models for the two coherent mechanisms have been developed. Only the former, however, is introduced here. In theory, the models for the coherent mechanisms can be unified by using the dielectric constant and further extended to include the incoherent mechanism.
In the analytical model, a plane ultrasonic wave irradiates a homogenous isotropic scattering medium. We assume that (1) the optical wavelength is much shorter than the mean free path (weak-scattering approximation) and (2) the ultrasound-induced change in the optical path length is much less than the opti- cal wavelength (weak-modulation approximation). Owing to the weak-scattering approximation, ensemble-averaged correlations between electric fields from dif- ferent paths are negligible compared with those from the same paths.
The autocorrelation function G \ (τ) of the scalar electric field of the scattered light can be expressed as
poo
d ( x ) = / p(s)(E5(t)E*{t + x))ds. (13.1) Jo
Here, () denotes ensemble and time averaging, Es denotes the unit-amplitude electric field of the scattered light of a path of length s, and p(s) denotes the prob- ability density function of s. The contributions to G\(x) from Brownian motion and from the ultrasonic field are independent and hence are treated separately. For brevity, only the latter is considered here.
In the following model, a coherent optical plane wave is incident normally on a slab of thickness J, and a point detector detects the transmitted light. The diffusion theory with a zero-boundary condition provides a solution to p(s). From Eq. (13.1), we obtain
C (τ) = W > s i n h « e n - c o s ^ t ) ] } 1 / 2 ) K sinh((d//,'){e[l -οο*(ωαχ)]}1'2)'
MECHANISMS OF ULTRASONIC MODULATION OF COHERENT LIGHT 3 2 5
where ωα is the acoustic angular frequency, and the other parameters are
ε = 6 (δ„+δ ί / ) (ηοΜ) 2 , (13.3)
hn = (απ, +α„2)η2 , (13.4)
«ni = -/cfl/,'arctan(/;fl/,'), (13.5)
a»2 = Tir,—, n{n ,/v—7' ( 1 3 · 6 ) fcfl/j/arctanifcfl/,) — 1
8</ = i (13.7) o
Here, no is the background index of refraction; ko is the magnitude of the optical wave vector in vacuo; A is the acoustic amplitude, which is proportional to the acoustic pressure; ka is the magnitude of the acoustic wavevector; l't is the optical transport mean free path; η is the elastooptical coefficient, related to the adiabatic piezooptical coefficient of the material dn/dp (derivative of refractive index n with respect to pressure /?), the mass density p, and the speed of sound vs: η = (dn/dp)pv]:; δ„ and 5</ are related to the average contributions per free path (or per scattering event) to the ultrasonic modulation of light via index of refraction and displacement, respectively.
Whereas δη increases with kaVr hj remains constant at £; thus, the ratio of δη to hd increases with kal'r The correlation between the two modulation mechanisms is neglected here for simplicity.
According to the Wiener-Khinchin theorem, the power spectral density £(ω) of the modulated speckle is related to G\(x) through the following Fourier trans- formation:
Gi(x)exp(/ooT)dT. (13.8) -oo
Frequency ω is relative to the angular frequency of the unmodulated light (ωο) because exp(—ιωοτ), which is dropped for convenience, is implicit in G\(x). Therefore, ω = 0 in Ξ(ω) corresponds to absolute angular frequency ωο·
Since G\(x) is an even periodic function of τ, the spectral intensity at fre- quency ηωα can be calculated by
Ta cos(A20)aT)Gi(T)<iT:, (13.9)
where n = 0, ± 1 , ±2..., and Ta is the acoustic period. The frequency spectrum /„ is symmetric about ωο· We define the one-sided modulation depth as
M{ = !±. (13.10)
3 2 6 ULTRASOUND-MODULATED OPTICAL TOMOGRAPHY
In the weak-modulation approximation, (d//,)e1/2 <^ 1; thus, Eq. (13.2) can be simplified to
GiW = l - \ ( f ) e[l-cos(a>flx)]. (13.11)
Thus, we have
Ml=zh(?)εαΛ2, (1312)
which indicates a quadratic relationship between M\ and A. If the modulated light is measured experimentally using a Fabry-Perot interferometer, this quadratic dependence is observed. Otherwise, the detected apparent modulation depth (Μ')—defined as the ratio of the observed AC signal to the observed DC signal—can have a different dependence on A. In some cases, the AC signal originates from the beat between the electric field components at the fundamental frequency of the modulated light (ωο ± ωα) and the electric field component at the unmodulated optical angular frequency (ωο). As a result, we have approximately M' a (/i//o)1//2 = M / a A, which indicates that M' is proportional to A.
13.3. TIME-RESOLVED FREQUENCY-SWEPT UOT
If a single-frequency ultrasonic wave is used in UOT, the axial resolution along the ultrasonic axis is much worse than the lateral resolution because of the elon- gated ultrasonic focal zone. Ultrasonic frequency sweeping (chirping), however, can improve the axial resolution. A time-resolved frequency-swept UOT system is shown in Figure 13.1. A frequency-swept signal is produced from a function generator and then amplified by a power amplifier followed by a transformer. The instantaneous frequency of the signal is
fs(t)=as+bt, (13.13)
where as denotes the starting frequency, b denotes the sweep rate, and t denotes time. Here, the frequency sweeps from 7.0 to 10.0 MHz at a rate of 297 MHz/s. The amplified signal is applied to an ultrasonic transducer, which transmits a focused ultrasonic wave vertically into a scattering medium in a glass tank. An ultrasound absorber is placed at the bottom of the tank to minimize reflection from the water-glass interface.
After being broadened to 15 mm in diameter, a laser beam illuminates the scattering medium perpendicularly to the ultrasonic beam. The ultrasonic beam modulates the laser light with the following instantaneous frequency distribution along the ultrasonic axis:
fs(t,z) = as + b(t-?—Q\ for ί>^—^, (13.14) V vs ) vs
TIME-RESOLVED FREQUENCY-SWEPT UOT 3 2 7
Frequency sweep
Scattering medium
Function generator
T Power
amplifier
^ ψ
Function generator
Modulation signal
Transformer
Laser
Ultrasonic transducer
ψ~*J Ultrasonic
I wave
Ultrasound absorber
Computer
ZiZ W Oscilloscope
Modulated PMT
Amplifier
Bandpass filter
Figure 13.1. Schematic of an experimental setup for frequency-swept UOT. The z axis is along the acoustic axis; the y axis is along the optical axis; the x axis points into the paper.
Here, z denotes the ultrasonic axis and zo denotes a reference point along the ultrasonic axis at time zero. A PMT converts the received transmitted light into an electric signal. The gain of the PMT is modulated for heterodyne detection by a reference signal produced by another function generator. The reference modulation signal, also frequency-swept, has an instantaneous frequency given by
fr{t) =ar+bt,
where ar denotes the starting frequency. The heterodyned signal has the following frequency distribution:
Mz) = \fsit,z)-fr(t)\ as - ar - b(z - zo)
(13.15)
(13.16)
which is independent of time t. The heterodyned signal at the output of the PMT is bandpass-filtered and then amplified. The bandwidth of the filter Afh is determined by the desired range on the z axis to be imaged (region of interest) Δζ as follows:
Afh = - Δ ζ . (13.17)
The signal from the amplifier is digitized by an oscilloscope and then transferred to a computer for postprocessing.
An object made of rubber is placed in the middle plane of the tank. The thickness of the tank along the laser beam is 17 cm. The scattering coefficient and the anisotropy of the scattering medium are 0.16 cm"1 and 0.73, respectively.
3 2 8 ULTRASOUND-MODULATED OPTICAL TOMOGRAPHY
The object is translated in the tank along the x axis with a step size of 1 mm. A time-domain signal is recorded at each stop and then Fourier-transformed into a spectrum by a computer. Each spectrum is further converted into a ID raw image of the scattering medium along the ultrasonic axis (z axis) on the basis of Eq. (13.16).
Two sample frequency spectra are depicted in Figure 13.2. Figure 13.2a shows a spectrum when the object is far from the ultrasonic axis. Figure 13.2b shows a spectrum when the object blocks part of the ultrasonic axis. As can be seen, the frequency components corresponding to the location of the object disappear.
3000 r
2500
< 2000
2 1500
δ 1000 ex CO
(a)
500
0 1
3000
2500
< 2000
£ 1500
UAJW^KJ/OJUKJ LκψL·kAXL·l*hλAιλ^
18 18.5 19 19.5 20 20.5 21 21.5 22 Frequency (kHz)
1000
500
JJUM
Blocked frequency components
L^AvLMÜJUJAi-Uv lJ4v>^4*-ly*H*J<W
18.5 19 (b)
19.5 20 20.5 Frequency (kHz)
21 21.5 22
Figure 13.2. Frequency spectra of the heterodyned frequency-swept ultrasound- modulated optical signal when the object is (a) far from and (b) on the ultrasonic axis (AU = arbitrary units).
FREQUENCY-SWEPT (JOT WITH PARALLEL-SPECKLE DETECTION 3 2 9
—
1 j
1
1
1
" 1 — | - - l 2 m m H 6 mm
1
1 j
i |
(a) 1 0.5 0
(b) Relative spectral intensity
- 0
S 5
«10
0 5 10 15 Horizontal axis x (mm)
(c) (d)
0 5 10 15 Horizontal axis x (mm)
0.5 1.0 1.5 Relative spectral intensity
Figure 13.3. (a) Schematic cross-sectional view of the buried object in a scattering medium; (b) signals along the vertical dashed line in panel (a); (c) signals along the horizontal dashed line in panel (a); (d) 2D image of the scattering medium.
This figure demonstrates the one-to-one correspondence between the heterodyne frequency and the position along the ultrasonic axis. The image contrast reflects the spatial variation in the optical and acoustic properties.
Combining all the ID spectra yields a 2D image (Figure 13.3). The first spec- trum, which is taken when the object is far from the ultrasonic axis, is used as a reference. All spectra are divided by the reference spectrum point-by-point to yield relative spectra, which are ID images. All ID images are pieced together to form a 2D image. Signals along the dashed lines in Figure 13.3a are plotted in Figures 13.3b and 13.3c, respectively. As can be seen, the edge resolution in both directions is approximately 0.5 mm. The z-axis resolution is determined by the ultrasonic sweep parameters, whereas the jc-axis resolution is determined by the ultrasonic focal diameter.
In summary, a frequency-swept (chirped) ultrasonic wave can encode laser light traversing the acoustic axis with various frequencies. Decoding the trans- mitted light provides resolution along the acoustic axis. This scheme is analogous to MRI.
13.4. FREQUENCY-SWEPT UOT WITH PARALLEL-SPECKLE DETECTION
The frequency-swept UOT based on a single-element photodetector in the previ- ous section is demonstrated only in the quasiballistic regime. By improving the
3 3 0 ULTRASOUND-MODULATED OPTICAL TOMOGRAPHY
Laser
ΐ 1 Function
generator 1 i V
Delay generator
i k
T Lens ^n s f\_
-ih-^ir Water-,
—►
Power amplifier
▲
Function generator 2
1 Sample
1 t
Function generator 3
CCD camera
▲ Ultrasonic
T transducer
z J ▲
x T
U— Computer
Figure 13.4. Schematic of a multiple-speckle UOT system. The z axis is along the acous- tic axis; the y axis is along the optical axis; the x axis points out of the paper.
SNR with a CCD camera that detects multiple optical speckle grains in parallel, one can image in the diffusive regime. In this detection scheme, both the laser and the ultrasonic transducer are modulated with chirp functions. Imaging along the ultrasonic axis can be accomplished by electronically scanning the time delay between these two chirp functions.
An experimental setup is shown in Figure 13.4. A focused ultrasonic trans- ducer with a 2.54-cm focal length in water, and a 1-MHz center frequency generates an ultrasonic wave with a peak pressure of ~2 x 105 Pa at the focus. A laser emits a lightbeam modulated at 1 MHz with 690-nm wavelength, 12- mW average power, and 7-cm coherence length. The laser beam is expanded to 1.6 x 0.3 cm in cross section and then projected onto the sample, which is par- tially immersed in water for acoustic coupling. The light transmitted through the sample generates a speckle pattern, which is detected by a high-speed 12-bit CCD camera. The average speckle grain size is adjusted to match the CCD pixel size. Three function generators share the same timebase to ensure synchronization. Function generators 1 and 2 produce chirp functions to modulate the diode laser and to excite the ultrasonic transducer, respectively. A delay generator controls the time delay between the trigger signals to these two function generators.
If the chirp signal from function generator 2 were not amplitude-modulated, the frequency of the heterodyned signal along the ultrasonic axis (z axis) would be
Λ(ζ ,τ ) -Ή)· (13.18) where b is the frequency sweep rate and t is the time delay between the two chirps from function generators 2 and 1.
To implement the source-synchronized lock-in technique, the chirp signal from function generator 2 is amplitude-modulated by a reference sinusoidal wave of frequency fh(z,x) from generator 3. After lowpass filtering by the CCD, the
ULTRASONICALLY MODULATED VIRTUAL OPTICAL SOURCE 3 3 1
signal from each CCD pixel can be expressed as
/ί(φΓ) <xlb + 1m c o s ^ + φΓ). (13.19)
Here, h denotes the background signal, lm denotes the signal related to the ultrasound-modulated light component, <\>s denotes the initial phase of the speckle grain, and φΓ denotes the initial phase of the reference sinusoidal wave. The apparent modulation depth Mf — Im/Ib, which is related to the local optical and acoustic properties, is recovered for imaging. The initial phase φΓ is set sequentially to 0°, 90°, 180°, and 270°. The corresponding four frames of CCD images are acquired to calculate M' by
M' = — >/[/«(90°) - //(2700)]2 + [7,(0°) - /,(180°)]2. (13.20) lib
Each pixel of the CCD camera produces an M'\ the average of the M' values from all 256 x 256 CCD pixels represents a single point (pixel) in the final image.
From Eq. (13.18), the spatial location being imaged is given by
z = vslx — J . (13.21)
At the same time, the ultrasound-modulated light from other spatial locations results in AC signals in the CCD pixels and hence is rejected by the CCD camera. One-dimensional images are obtained along the z axis by electroni- cally varying τ. Further, 2D images are obtained by mechanically scanning the ultrasonic transducer along the x axis.
The spatial resolution ZR along the z axis is determined by
ZR « ~ , (13.22)
where the speed of sound vs is M500 m/s in most soft tissues and Δ / denotes the frequency span of the chirp. Therefore, ZR is inversely proportional to Δ / .
13.5. ULTRASONICALLY MODULATED VIRTUAL OPTICAL SOURCE
The original ultrasound-modulated optical signal can be considered a virtual light source. The virtual source is initially localized but is broadened with light prop- agation. If imaged near the acoustic axis, the virtual source can be seen clearly. Figure 13.5 shows a series of images at various z coordinates associated with different Λ ί ζ , τ ) values [Eq. (13.18)]. Ultrasonic modulation of light locally improves the spatial resolution of imaging because scanning a virtual small light source inside a highly scattering medium can produce a better image of the scanned cross section than scanning an actual small light source outside the medium.
3 3 2 ULTRASOUND-MODULATED OPTICAL TOMOGRAPHY
0.6 1 1.4 Transverse axis (cm)
Figure 13.5. Demonstration of ultrasound-modulated light as a virtual source. The left frame represents the entire virtual source acquired without chirping. The following frames represent the virtual sources at various z values acquired with chirping Courtesy of Atlan et al. (2003).
13.6. RECONSTRUCTION-BASED UOT
Axial resolution along the ultrasonic axis can be achieved by reconstruction as in X-ray CT. In X-ray CT, a cross-sectional image of a sample is reconstructed from the transmitted X-ray intensities, which are acquired from multiple linear and angular scans around the object. In UOT, ultrasound-modulated optical signals are acquired while the ultrasonic beam is scanned linearly and angularly around the object. Subsequently, a filtered backprojection algorithm reconstructs an image of the cross section formed by the scanned ultrasonic axis.
A reconstruction-based UOT system that can operate in either reflection or transmission configuration is shown in Figure 13.6a. The CCD camera and the incident laser beam are on the same side of the sample for the reflection con- figuration and on opposite sides for the transmission configuration. After being expanded to 20 mm in diameter, light from a diode laser (690-nm wavelength, 11-mW power) illuminates the sample. The power density used is much lower than the ^200-mW/cm2 safety limit. An ultrasonic wave from a focused ultra- sonic transducer is coupled into the sample through water in which the sample is partially immersed. Since the ultrasonic transducer has a 38-mm focal length in water and a 1-MHz center frequency, the focal zone is close to 2.8 mm in diameter and -^20 mm in length; the peak pressure at the focus is —105 Pa. The speckle pattern generated by the reemitted light is detected by a 12-bit CCD camera with 256 x 256 pixels. Ultrasound-modulated optical signals are extracted using parallel-speckle detection without chirping.
Linear and angular scans are required for the data acquisition. For experimental convenience, the buried object is translated horizontally and rotated about the optical axis while the imaging system is held stationary. A coordinate system is affixed to the buried object: the y axis is along the optical axis, and the z axis is initially along the ultrasonic axis. As shown in Figure 13.6b, the x.z coordinates rotate with the buried object, whereas the measurement coordinates (x\ y', z!) are stationary with the z axis parallel to the ultrasonic beam.
RECONSTRUCTION-BASED UOT 3 3 3
(Reflection detection)
(a)
(Transmission detection)
Laser
Sample
Ultrasonic beam
ev Rotation \ \ l Linear scan
\
Ultrasonic transducer
Linear
Ultrasonic beam
(b)
Figure 13.6. (a) Schematic of a reconstruction-based UOT system; (b) coordinate systems.
The detected ultrasound-modulated optical signal can be expressed as an inte- gration of the signal originating from the z! axis:
(13.23) 5(φ, x) = / s^y(z)dz\
which is a Radon transform (see Chapter 8). The integrand can be expressed as
*M'(z') = Cx Q^Az')M^Azf)G^Az'). (13.24)
Here, C\ denotes a constant; οφ,χ'(ζ') denotes the optical fluence rate; Μψ^'(ζ') denotes the ultrasonic modulation depth, which is related to the optical and the ultrasonic properties; and G^x'(z') denotes the Green function that describes the transport of the original ultrasound-modulated light to the detector. In the diffusive regime, Q^x'(z') and G^y(z') have a weak dependence on z!.
3 3 4 ULTRASOUND-MODULATED OPTICAL TOMOGRAPHY
From projection data ς(φ,χ'), a filtered backprojection algorithm is used to reconstruct an image of the sample as follows:
f(x,z)= I / S(<\>,k)\k\exp(ikx')dkd<\>. (13.25)
Here, k is the spatial ίΓεςμεΜ^γ in the x' direction; ^(φ, k) is the spatial Fourier transform of 5*(φ, x')\ \k\ is referred to as the Ram-Lak filter.
A tissue sample is imaged using the transmission configuration. Here, the step size of the linear scan along the x' axis is 1.2 mm, and the step size of the angular scan is 5°. By placing the CCD camera at an angle of 25° with respect to the optical axis, contributions from ballistic photons can be ruled out and the capability of nonballistic imaging can be demonstrated. Figure 13.7a shows a photograph of the middle cross section (xz plane) of a 14-mm-thick chicken breast tissue sample containing a chicken blood vessel, which measures — 8 x 3 mm2 on the xz plane and —2 mm along the y axis. Figure 13.7b shows the reconstructed image, which clearly reveals the buried object.
(b) 0.5 0.7 0.9
Figure 13.7. (a) Photograph of the middle xz cross section of a 14-mm-thick chicken breast tissue sample in which a blood vessel is buried; (b) reconstructed 2D image using the transmission configuration.
UOT WITH FABRY-PEROT INTERFEROMETRY 3 3 5
13.7. UOT WITH FABRY-PEROT INTERFEROMETRY
UOT can also be implemented with a long-cavity scanning confocal Fabry-Perot interferometer (CFPI), which provides a large etendue (the product of the detec- tion area and the acceptance solid angle) and a short response time (Figure 13.8a). As shown in Figure 13.8b, the sample is gently pressed to a semicylindrical shape through a slit along the x axis; the orthogonal ultrasonic and optical beams are confocal below the sample surface. Reemitted light is collected on the opposite side of the ultrasound beam from the incident lightbeam. This configuration min- imizes the effect of unmodulated light from the shallow region and enhances the ultrasonic modulation of some of the quasiballistic light that still exists at small imaging depths.
A focused ultrasonic transducer (15-MHz center frequency, 15-MHz band- width, 4.7-mm lens diameter, and 4.7-mm focal length) is driven by a pulser. The peak ultrasonic pressure at the focal spot measures 3.9 MPa, which is within the ultrasound safety limit at this frequency for biological tissue with no well-defined gas bodies. The laser light (532-nm wavelength, 100-mW power) has a 0.1-mm focal diameter in a clear medium. The sample is mounted on a three-axis (x\ yr, z') translation stage. The ultrasonic transducer and the sample are immersed in
Coupling Confocal Fabry-Perot Beam splitter Avalanche optics interferometer photodiode
Collecting fibei Focusing ]
°Pt i c s ' Laser
Ultrasonic transducer
Pulser-receiver Trigger (a) generator
Shutter
(b)
»A Collected light Incident
lightbeam
Ultrasound beam
Figure 13.8. (a) Schematic of a CFPI-based UOT system (PZT represents a piezoelectric transducer made of lead zirconate titanate); (b) top view of the sample.
3 3 6 ULTRASOUND-MODULATED OPTICAL TOMOGRAPHY
water for acoustic coupling. The light-focusing optics and the collection optical fiber (0.6-mm core diameter) are also immersed in the same water tank. The optical-fiber output is coupled to the CFPI (50-cm cavity length, 0.1-mm2 sr etendue and >20 finesse), which operates in transmission mode.
A beamsplitter splits off some light for cavity tuning. Initially, one of the CFPI mirrors is scanned by more than one free spectral range of the cavity to search for a cavity length that matches the center frequency of the unmodulated light. Then, the mirror is displaced by a calibrated amount so that the cavity length matches the frequency of the positive sideband of the ultrasound-modulated light (15 MHz greater than the center frequency of the unmodulated light).
Next, an avalanche photodiode (APD) detects the light filtered through the CFPI; the output signal is sampled at 100 MHz by a data acquisition board. The entire system is controlled by computer. The APD signal is acquired during ultrasound propagation through the sample as a function of time. Converting the propagation time into z via the acoustic speed vs & 1500 m/s yields the distribution of the ultrasound-modulated optical intensity along the ultrasonic axis I\(z), which provides a ID image.
In each operational cycle, the resonant frequency of the CFPI is tuned first; then, both the ultrasonic transducer and the data-acquisition board are triggered by a trigger generator, and 4000 APD signals are acquired in one second. Aver- aging over 10 cycles produces a ID image of satisfactory SNR. Two-dimensional images are obtained by further scanning of the sample along the x direction.
A typical profile of I\(z)—which peaks at the intersection between the optical and the ultrasonic axes because ultrasonic modulation is related to both optical fluence and ultrasound intensity—is shown in Figure 13.9. Chicken breast tissue is pressed through a 4-mm-wide slit to form a cylindrical tissue bump with a
16
14
12
< 10
1 8 3 c ~ 6
4
2
0 2 2.5 3 3.5 4 4.5 5
Ultrasonic axis z (mm)
Crossing point . between optical and ultrasonic .
Figure 13.9. Ultrasound-modulated light intensity along the ultrasonic axis (AU = arbitrary units).
UOT WITH FABRY-PEROT INTERFEROMETRY 3 3 7
2-mm radius. A long black latex rod with a 60-μπι diameter—which is transpar- ent for ultrasound but absorptive for light—is placed along the x axis below the sample surface. When the ultrasound pulse passes through the object, the optical contrast produces a dip in I\(z).
The axial and lateral resolutions are investigated by imaging two chicken breast tissue samples (Figure 13.10). The samples are in semi-cylindrical shapes with 3.2- and 3-mm radii of curvature, respectively; each contains a 0.1-mm-thick black latex object (Figures 13.10b and 13.10d) at the axis of the semicylinder.
15.0
15.5 H
16.0
(a)
3.8 3,6 3.4 3,2 3.0 2,8
z (mm) (b)
(c)
3.8 3.6 3,4 3,2 3.0
z (mm) (d)
(e)
— Cut 1 "°--Cut2
z (mm) (0
π—r—r 16.6 17.0
X (mm) 17.4
Figure 13.10. (a) Image and (b) photograph of an object; (c) image and (d) photograph of another object; (e) ID axial profiles of intensity from the data in part (a); (f) ID lateral profile of intensity from the data in part (c) (AU represents arbitrary units).
3 3 8 ULTRASOUND-MODULATED OPTICAL TOMOGRAPHY
A typical ID image of the background tissue is used as a reference /,ref(z). Each ID image of the object 7° J(z) is converted into a relative profile 7[e,(z) by
7, (z)
These relative profiles form 2D images are shown in gray scale with five equally spaced gray levels in Figures 13.10a and 13.10c. Figure 13.10e presents two I\d(z) profiles taken from Figure 13.10a along two cut lines. Along cut line 1, the actual gap is 70 μηι and is resolved with a 55% contrast; along cut line 2, the actual gap reduces to 50 μιτι, and the contrast decreases to 40%. Similarly, Figure 13.1 Of represents the ID lateral profile of intensity versus x taken from Figure 13.10c along the cut line. The actual gap along the cut line is 120 μΐΏ and is resolved with a 50% contrast. If a gap resolvable with a 50% contrast is defined as the resolution, the estimated axial and lateral resolutions are 70 and 120 μιη, respectively.
PROBLEMS
13.1 Show that given (d/l't)zx/1 «; 1, Eq. (13.2) can be simplified to Eq. (13.11) and the modulation depth can be approximated by Eq. (13.12).
13.2 Plot the ratio of hn to δ^ as a function of kal't using the following parame- ters: dn/dp = 1.466 x 10~10 m2/N, p = 1000 kg/m3, and vs = 1488 m/s.
13.3 Plot the modulation depth as a function of l't using the following param- eters: dn/dp = 1.466 x 10~10 m2/N, p = 1000 kg/m3, υ, = 1488 m/s, ultrasonic frequency fa — 1 MHz, no = 1.33, λο = 500 nm, d = 5 cm, and A = 0.01 nm. Convert A into pressure.
13.4 Plot h/h versus Λ using the following parameters: dn/dp = 1.466 x 10~10 m2/N, p = 1000 kg/m3, vs = 1488 m/s, ultrasonic frequency fa = 5 MHz, no = 1.33, λο = 500 nm, d = 5 cm, and kal't = 1.
13.5 In frequency-swept ultrasound-modulated optical tomography, the source that is applied to the ultrasonic transducer sweeps from 7 to 10 MHz in 10 ms. The reference signal that is applied to the PMT sweeps from 7.01 to 10.01 MHz in 10 ms. Derive the formula that converts the frequency of the heterodyned signal to the position along the ultrasonic axis.
13.6 Derive the phase of the chirped sinusoidal function that provides frequency sweep fs(t) = as + bt.
13.7 (a) Derive Eq. (13.20). (b) Extend it to a three-phase method, where the three phases are 90° apart, (c) Extend it to a two-phase method, where the two phases are 180° apart.
FURTHER READING 3 3 9
13.8 A mechanical index—which is used to estimate the likelihood of mechan- ical bioeffects—is defined as MI = P / ( C M I \ Z J ) . Here, P is the local peak-rarefactional pressure in MPa, / is the acoustic frequency in MHz, and CMI is a coefficient equal to 1 M P a / V M H z . Calculate the MI for the ultrasonic parameters used in this chapter and compare with the current safety standards.
13.9 Given μα = 0.03 c m - 1 , μ5 = 100 cm" 1 , and g = 0.9, compute μ6ίί in both c m - 1 and dB/cm. Compare μ,^ with the ultrasonic attenuation coeffi- cient at frequency / = 3 MHz in biological tissue, which is approximately 0 . 5 / dB/(cm MHz).
13.10 Implement the Radon transformation in C/C++.
13.11 Implement the inverse Radon transformation in C/C++.
READING
Atlan M, Forget BC, Ramaz F, and Boccara AC (2003): Private communication. (See Section 13.5, above.)
Li J and Wang LHV (2004): Ultrasound-modulated optical computed tomography of biological tissues, Appl. Phys. Lett. 84(9): 1597-1599. (See Section 13.6, above.)
Sakadzic S and Wang LHV (2004): High-resolution ultrasound-modulated optical tomog- raphy in biological tissues, Opt. Lett. 29(23): 2770-2772. (See Section 13.7, above.)
Wang LHV, Jacques SL, and Zhao XM (1995): Continuous-wave ultrasonic modulation of scattered laser-light to image objects in turbid media, Opt. Lett. 20(6): 629-631. (See Section 13.1, above.)
Wang LHV (2001): Mechanisms of ultrasonic modulation of multiply scattered coherent light: An analytic model, Phys. Rev. Lett. 8704(4). (See Section 13.2, above.)
Wang LHV (2003): Ultrasound-mediated biophotonic imaging: A review of acousto- optical tomography and photo-acoustic tomography, Disease Markers 19(2-3): 123-138. (See Section 13.2, above.)
Wang LHV and Ku G (1998): Frequency-swept ultrasound-modulated optical tomography of scattering media, Opt. Lett. 23(12): 975-977. (See Section 13.3, above.)
Yao G, Jiao S, and Wang LHV (2000): Frequency-swept ultrasound-modulated opti- cal tomography in biological tissue by use of parallel detection, Opt. Lett. 25(10): 734-736. (See Section 13.4, above.)
FURTHER READING
Abbott JG (1999): Rationale and derivation of MI and TI—a review, Ultrasound Med. Biol. 25(3): 431-441.
Atlan M, Forget BC, Ramaz F, Boccara AC, and Gross M (2005): Pulsed acousto-optic imaging in dynamic scattering media with heterodyne parallel speckle detection, Opt. Lett. 30(11): 1360-1362.
Blonigen FJ, Nieva A, DiMarzio CA, Manneville S, Sui L, Maguluri G, Murray TW, and Roy RA (2005): Computations of the acoustically induced phase shifts of optical paths
3 4 0 ULTRASOUND-MODULATED OPTICAL TOMOGRAPHY
in acoustophotonic imaging with photorefractive-based detection, Appl. Opt. 44(18): 3735-3746.
Bossy E, Sui L, Murray TW, and Roy RA (2005): Fusion of conventional ultrasound imaging and acousto-optic sensing by use of a standard pulsed-ultrasound scanner, Opt. Lett. 30(7): 744-746.
Granot E, Lev A, Kotier Z, Sfez BG, and Taitelbaum H (2001): Detection of inhomo- geneities with ultrasound tagging of light, J. Opt. Soc. Am. A 18(8): 1962-1967.
Gross M, Goy P, and Al-Koussa M (2003): Shot-noise detection of ultrasound-tagged photons in ultrasound-modulated optical imaging, Opt. Lett. 28(24): 2482-2484.
Gross M, Ramaz F, Forget BC, Atlan M, Boccara C, Delaye P, and Roosen G (2005): Theoretical description of the photorefractive detection of the ultrasound modulated photons in scattering media, Opt. Express 13(18): 7097-7112.
Hisaka M, Sugiura T, and Kawata S (2001): Optical cross-sectional imaging with pulse ultrasound wave assistance, J. Opt. Soc. Am. A 18(7): 1531-1534.
Hisaka M (2005): Ultrasound-modulated optical speckle measurement for scattering medium in a coaxial transmission system, Appl. Phys. Lett. 87(6): 063504.
Kempe M, Larionov M, Zaslavsky D, and Genack AZ (1997): Acousto-optic tomography with multiply scattered light, J. Opt. Soc. Am. A 14(5): 1151-1158.
Leutz W and Maret G (1995): Ultrasonic modulation of multiply scattered-light, Physica B 204(1-4): 14-19.
Lev A, Kotier Z, and Sfez BG (2000): Ultrasound tagged light imaging in turbid media in a reflectance geometry, Opt. Lett. 25(6): 378-380.
Lev A and Sfez BG (2002): Direct, noninvasive detection of photon density in turbid media, Opt. Lett. 27(7): 473-475.
Lev A and Sfez BG (2003): Pulsed ultrasound-modulated light tomography, Opt. Lett. 28(17): 1549-1551.
Lev A, Rubanov E, Sfez B, Shany S, and Foldes AJ (2005): Ultrasound-modulated light tomography assessment of osteoporosis, Opt. Lett. 30(13): 1692-1694.
Leveque S, Boccara AC, Lebec M, and Saint-Jalmes H (1999): Ultrasonic tagging of photon paths in scattering media: Parallel speckle modulation processing, Opt. Lett. 24(3): 181-183.
Leveque-Fort S (2001): Three-dimensional acousto-optic imaging in biological tissues with parallel signal processing, Appl. Opt. 40(7): 1029-1036.
Li H and Wang LHV (2002): Autocorrelation of scattered laser light for ultrasound- modulated optical tomography in dense turbid media, Appl. Opt. 41(22): 4739-4742.
Li J and Wang LHV (2002): Methods for parallel-detection-based ultrasound-modulated optical tomography, Appl. Opt. 41(10): 2079-2084.
Li J, Ku G, and Wang LHV (2002): Ultrasound-modulated optical tomography of biolog- ical tissue by use of contrast of laser speckles, Appl. Opt. 41(28): 6030-6035.
Li J, Sakadzic S, Ku G, and Wang LHV (2003): Transmission- and side-detection config- urations in ultrasound-modulated optical tomography of thick biological tissues, Appl. Opt. 42(19): 4088-4094.
Li J and Wang LHV (2004): Ultrasound-modulated optical computed tomography of biological tissues, Appl. Phys. Lett. 84(9): 1597-1599.
Mahan GD, Engler WE, Tiemann JJ, and Uzgiris E (1998): Ultrasonic tagging of light: Theory, Proce. Nad. Acad. Sei. USA 95(24): 14015-14019.
Marks FA, Tomlinson HW, and Brooksby GW (1993): A comprehensive approach to breast cancer detection using light: Photon localization by ultrasound modulation and
FURTHER READING 341
tissue characterization by spectral discrimination. Proc. Soc. Photo-opt. Instrum. Eng. 1888: 500-510.
Murray TW, Sui L, Maguluri G, Roy RA, Nieva A, Blonigen F, and DiMarzio CA (2004): Detection of ultrasound-modulated photons in diffuse media using the photorefractive effect, Opt. Lett. 29(21): 2509-2511.
Sakadzic S and Wang LHV (2002): Ultrasonic modulation of multiply scattered coherent light: An analytical model for anisotropically scattering media, Phys. Rev. E 66(2): 026603.
Sakadzic S and Wang LHV (2004): High-resolution ultrasound-modulated optical tomog- raphy in biological tissues, Opt. Lett. 29(23): 2770-2772.
Schenk JO and Brezinski ME (2002): Ultrasound induced improvement in optical coher- ence tomography (OCT) resolution, Proc. Natl. Acad. USA 99(15): 9761-9764.
Selb J, Pottier L, and Boccara AC (2002): Nonlinear effects in acousto-optic imaging, Opt. Lett. 27(11): 918-920.
Solov'ev AP, Sinichkin YP, and Zyuryukina OV (2002): Acoustooptic visualization of scattering media, Opt. Spectrosc. 92(2): 214-220.
Sui L, Roy RA, DiMarzio CA, and Murray TW (2005): Imaging in diffuse media with pulsed-ultrasound-modulated light and the photorefractive effect, Appl. Opt. 44(19): 4041-4048.
Wang LHV and Zhao XM (1997): Ultrasound-modulated optical tomography of absorbing objects buried in dense tissue-simulating turbid media, Appl. Opt. 36(28): 7277-7282.
Wang LHV (1998): Ultrasonic modulation of scattered light in turbid media and a potential novel tomography in biomedicine, Photochem. Photobiol. 67(1): 41-49.
Wang LHV and Ku G (1998): Frequency-swept ultrasound-modulated optical tomography of scattering media, Opt. Lett. 23(12): 975-977.
Wang LHV (2001): Mechanisms of ultrasonic modulation of multiply scattered coherent light: A Monte Carlo model, Opt. Lett. 26(15): 1191-1193.
Wang LHV (2003): Ultrasound-mediated biophotonic imaging: A review of acousto- optical tomography and photo-acoustic tomography, Disease Markers 19(2-3): 123-138.
Yao G and Wang LHV (2000): Theoretical and experimental studies of ultrasound- modulated optical tomography in biological tissue, Appl. Opt. 39(4): 659-664.
Yao G, Jiao SL, and Wang LHV (2000): Frequency-swept ultrasound-modulated opti- cal tomography in biological tissue by use of parallel detection, Opt. Lett. 25(10): 734-736.
Yao G and Wang LHV (2004): Signal dependence and noise source in ultrasound- modulated optical tomography, Appl. Opt. 43(6): 1320-1326.
Yao Y, Xing D, and He YH (2001): AM ultrasound-modulated optical tomography with real-time FFT, Chinese Sei. Bulle. 46(22): 1869-1872.
Yao Y, Xing D, He YH, and Ueda K (2001): Acousto-optic tomography using amplitude- modulated focused ultrasound and a near-IR laser, Quantum Electron. 31(11): 1023-1026.
Zhu Q, Durduran T, Ntziachristos V, Holboke M, and Yodh AG (1999): Imager that combines near-infrared diffusive light and ultrasound, Opt. Lett. 24(15): 1050-1052.
APPENDIX A
Definitions of Optical Properties
TABLE A.l. Basic Optical Properties
Parameter
Absorption coefficient
Scattering coefficient
Anisotropy
Index of refraction
Definition
Probability of photon absorption by a medium per unit (infinitesimal) path length
Probability of photon scattering by a medium per unit (infinitesimal) path length
Average of cosine of scattering polar angle by single scattering
Ratio of speed of light in vacuum to phase velocity in medium; only the real part is considered here
Symbol Unit Typical Value
μ0
cm - i
0.1
100
0.9
1.38
TABLE A.2. Derived Optical Properties
Parameter
Albedo Diffusion
coefficient
Definition
a = μ.9/(μα 4- μ5) 0 = 1 / [ 3 ( μ β + μ;)];ΰιβ
coefficient linking the current to the gradient of fluence in Fick's law
Symbol
a D
Unit
— cm
Typical Value
0.999 0.033
Biomedical Optics: Principles and Imaging, by Lihong V. Wang and Hsin-i Copyright © 2007 John Wiley & Sons, Inc.
Wu
343
344 DEFINITIONS OF OPTICAL PROPERTIES
TABLE A.2. (Continued)
Parameter Definition Symbol Unit Typical Value
Effective attenuation coefficient
Extinction coefficient
μβίτ = V iW#; exponential decay rate of fluence far from the source
μ, = μ0 4- μ ν; probability of photon interaction with a medium per unit (infinitesimal) path length, where interaction includes both absorption and scattering (also referred to as total interaction coefficient)
^ef f .74
100.1
Mean free path
Penetration depth
Reduced scattering coefficient
Transport albedo
Transport interaction coefficient
Transport mean free path
lt = Ι/μ,; mean free path length between interactions
5 = l/^eff; exponential decay constant of fluence far from the source
μ.ν = μ.ν(ΐ -g); probability of equivalent isotropic photon scattering by a medium per unit (infinitesimal) path length in diffusive regime (also referred to as transport scattering coefficient)
a' = \i'J(\ia + \i's)
μ; = μ.α + μ;
/; = ι/μ;
It
δ
μ;
a'
μί
/;
cm 0.01
0.575
cm 10
cm
cm
0.990
10.1
0.099
APPENDIX B
List of Acronyms
ID 2D 3D AC APD ART CCD CDF CFPI CNR CT CW dB DC DOP DOT DRR EEM ESF FFT FOV FWHM GVD IDM IFT IR LSF MCML
one dimension, or one-dimensional two dimensions, or two-dimensional three dimensions, or three-dimensional alternating current avalanche photodiode algebraic reconstruction technique charge-coupled device cumulative distribution function confocal Fabry-Perot interferometer contrast-to-noise ratio computed tomography continuous wave decibel direct current degree of polarization diffuse optical tomography depth-to-resolution ratio excitation-emission matrix edge spread function Fast Fourier transform field of view full width at half maximum group velocity dispersion inverse distribution method inverse Fourier transform infrared line spread function Monte Carlo modeling of light transport in
multilayered scattering media
Biomedical Optics: Principles and imaging, by Lihong V. Wang and Hsin-i Copyright © 2007 John Wiley & Sons, Inc.
Wu
345
3 4 6 LIST OF ACRONYMS
MCP MRI MTF NA NIR OCT OD PAM PAT PDF PDI PMT PSF PZT rms RTE SIRT SLD SNR STF SVD TPM UOT US UV
microchannel plate magnetic resonance imaging modulation transfer function numerical aperture near-infrared optical coherence tomography optical density photoacoustic microscopy photoacoustic tomography probability density function polarization-difference imaging photomultiplier tube point spread function lead (Pb) zirconate titanate (piezoelectric transducer) root-mean-squared radiative transfer equation simultaneous iterative reconstruction technique superluminescent diode signal-to-noise ratio system transfer function singular-value decomposition two-photon microscopy ultrasound-modulated optical tomography ultrasonography ultraviolet
APPENDIX B
List of Acronyms
ID 2D 3D AC APD ART CCD CDF CFPI CNR CT CW dB DC DOP DOT DRR EEM ESF FFT FOV FWHM GVD IDM IFT IR LSF MCML
one dimension, or one-dimensional two dimensions, or two-dimensional three dimensions, or three-dimensional alternating current avalanche photodiode algebraic reconstruction technique charge-coupled device cumulative distribution function confocal Fabry-Perot interferometer contrast-to-noise ratio computed tomography continuous wave decibel direct current degree of polarization diffuse optical tomography depth-to-resolution ratio excitation-emission matrix edge spread function Fast Fourier transform field of view full width at half maximum group velocity dispersion inverse distribution method inverse Fourier transform infrared line spread function Monte Carlo modeling of light transport in
multilayered scattering media
Biomedical Optics: Principles and imaging, by Lihong V. Wang and Hsin-i Copyright © 2007 John Wiley & Sons, Inc.
Wu
345
3 4 6 LIST OF ACRONYMS
MCP MRI MTF NA NIR OCT OD PAM PAT PDF PDI PMT PSF PZT rms RTE SIRT SLD SNR STF SVD TPM UOT US UV
microchannel plate magnetic resonance imaging modulation transfer function numerical aperture near-infrared optical coherence tomography optical density photoacoustic microscopy photoacoustic tomography probability density function polarization-difference imaging photomultiplier tube point spread function lead (Pb) zirconate titanate (piezoelectric transducer) root-mean-squared radiative transfer equation simultaneous iterative reconstruction technique superluminescent diode signal-to-noise ratio system transfer function singular-value decomposition two-photon microscopy ultrasound-modulated optical tomography ultrasonography ultraviolet
INDEX
Absorbance, 139, 164 Absorbers, primary, 6, 7 Absorption
coefficient. See Absorption coefficient cross section, 5 cross section (two photon), 171 efficiency, 5 illustration, 4 origins, 5 spectrum, 1 spectra of primary absorbers (plot), 7
Absorption coefficient conversion from fluence to specific
absorption, 85 conversion from specific absorption to
fluence, 54 conversion to pressure, 286 definition, 5, 343 hemoglobin, 6 map, 267 sensing, 140, 145, 146 spectra of primary absorbers (plot), 7
Acceptance angle antenna theorem, 161, 162 collimated transmission method,
136, 137 OCT, 212 solid, 161, 335
Acoustic focusing, 307 Acoustic lens, 304, 307, 319 Acoustic pressure, 285 Acoustic transit time, 303 Acoustic wave equation, 313, 316 Acoustooptic modulator, 161 Activatable retarder, 155 Adjoint method, 274 Agargel, 135, 305 Airy disc, 176 Albedo, 88, 123
definition, 343
Biomedical Optics: Principles and Imaging, by Copyright © 2007 John Wiley & Sons, Inc.
Algebraic reconstruction technique (ART), 265, 275
illustration, 276 A-line. See also A-scan
definition, 186 Fourier-domain OCT, 198, 199, 201, 202 time-domain OCT, 187
Amino acid, 9 Amplitude reflectivity, 167, 187, 199
density, 199 Analyzer, polarization. See Polarization
analyzer Angiogenesis, 1, 7 Angle-biased sampling, 211, 212 Angular wavenumber. See Propagation
constant Anisotropy
definition, 47, 343 formula, 87 Mie theory, 20, 33 plot, 24 similarity relation, 111 structural, 38
ANSI Standard C, 40, 67 Antenna theorem, 161, 162
illustration, 161 Anti-Stokes transition, 5 APD. See Avalanche photodiode Arm-length difference, 183, 186, 206 Arm-length mismatch. See Arm-length
difference ART. See Algebraic reconstruction technique A-scan. See also A-line
definition, 186 PAM, 305, 306 time-domain OCT, 186
Attenuation coefficient, ultrasonic, 339 Autocorrelation function, 184, 201, 324 Auxiliary angle, 221 Avalanche photodiode (APD), 252, 336
V. Wang and Hsin-i Wu
347
348 INDEX
Axial resolution OCT, 181, 186, 189, 193, 207 PAM, 305, 307 UOT, 326, 329, 332, 337
Azimuthal angle (illustration), 18
Backprojection, 311-334 Ballistic imaging, 2, 153 Ballistic light. See Ballistic photon Ballistic photon, 153, 161, 334
arrival time, 154, 158 polarization, 157 spatial frequency, 156, 160
Ballistic regime, 115 definition, 114 OCT, 186
Ballistic transmittance, 8, 135. See also Unscattered transmittance
Bandwidth axial resolution, 193 chirp, 327 coherence length, 184 coherence-gated holographic
imaging, 160 ideal, 312 interference signal, 206 SLD, 209, 237, 238 spatial frequency, 161 ultrasound, 284, 304, 335
Bar chart, 305 Basis set, 88 Beam splitter
confocal microscope, 165 DOT, 252 Michelson interferometer, 181 nonpolarizing, 237-239 OCT, 182 optical heterodyne imaging, 161 polarizing, 238, 239 UOT, 336
Beamforming, 309 Beat frequency, 160, 161, 206 Beer's law
absorption, 5 ballistic imaging, 153 depth, 99 generalized, 164 primary beam, 147 probability, 45 scattering, 8 thickness, 135, 136 time, 98, 115
Bessel equation, transformed, 100
Bessel function, 21, 166 modified, 70, 74 spherical, 21, 28
Binary tree, 74 Biochemical information, 1, 146 Bioluminescence, 2 Biomarker, 2 Bipolar pressure profile, 299 Birefringence, 154, 158, 242. See also
Polarization circular, 9 dextrorotatory, 9 levorotatory, 9 linear, 9 negative, 9 positive, 9
Blood flow, 2, 206 Blue sky, 18 Body temperature, 285, 286 Boltzmann equation. See Radiative transfer
equation Born approximation, 147, 262, 271 Boundary
refractive-index-matched. See Refractive-index-matched boundary
refractive-index-mismatched. See Refractive-index-mismatched boundary
Boundary condition Cauchy, 102 Dirichlet, 103, 265,266 DOT, 273 extrapolated (illustration), 102 Green's function approach, 317 homogeneous, 293 Mie theory, 26, 32, 33 RTE, 97, 101 semiinfinite medium, 106, 107 slab, 120
Brain, 249, 312, 313 Breast, 249 Broad beam, 67, 156, 199, 326 Brownian motion, 324 B-scan, 186, 305, 306
Carbon fiber, 305 Carrier, 189 Carrier frequency, 206, 239 Cauchy boundary condition, 102 Cauchy's contour integration, 290 Causality, 98, 117, 291, 293, 317 CCD
holography, 158 reflectometry, 140, 145 speckle imaging, 330-332, 334
INDEX 349
CDF. See Cumulative distribution function Cell nuclei, 1, 8 Center frequency
OCT, 187, 197 PAM, 304 PAT, 284, 312 UOT, 330, 332, 335, 336
Central limit theorem, 55 Chirping, 208, 209, 326, 329, 332 Circular polarization, 9, 222, 224, 229
Mueller matrix, 227 Circular polarizer, 231, 232, 234 CNR. See Contrast-to-noise ratio Coefficient
absorption. See Absorption coefficient extinction. See Extinction coefficient molar extinction. See Molar extinction
coefficient scattering. See Scattering coefficient total interaction. See Extinction coefficient
Coherence gating, 153, 158, 160, 186 Coherence length, 184-186, 190, 330 Coherence time, 184 Coherence-gated holographic imaging, 153, 158
illustration, 159 Collagen, 1, 9, 10, 38, 242 Collimated transmission method, 135, 139
illustration, 136 Comparison of imaging modalities, 2, 284 Compensator. See Retarder Complex conjugation, 89, 311 Complex expression. See Phasor representation Compressibility, 285 Computed tomography (CT), 1, 163, 332 Concentration of hemoglobin. See Hemoglobin
concentration Condensation parameter, 314 Condenser lens, 164, 304 Confocal microscopy
fluorescence (illustration), 166 optical, 154, 164 photoacoustic. See Photoacoustic microscopy system (illustration), 165
Conical lens, 304 Conjugate-gradient method, 275 Conservation of energy, 85, 88, 223 Conservation of mass, 314 Constant fraction discriminator, 252 Continuity equation, 314, 315 Contrast
comparison, 2 definition, 12 DOT, 269
functional imaging, 1 illustration, 13 molecular imaging, 2 OCT, 181, 219 PAT, 313 UOT, 329, 338
Contrast-to-noise ratio (CNR), 13 CONV program, 67, 77 Converging spherical wave, 299, 300, 311 Conversion
between optical wavelength and photon energy, 14
from pencil beam to isotropic source, 106
from temperature to pressure, 286 Convolution
arbitrary source, 98 broad beam, 67 coherent, 168 CONV program, 77 DOT, 251, 263 fluorescence source, 147 Fourier-domain OCT, 201 Gaussian beam, 69 incoherent, 167-170 infinitely wide beam, 54 LSF, 14 numerical example, 77 numerical solution, 72 object function, 11 PAT, 302 top-hat beam, 71 truncation error, 76
Correlation, 324, 325 Critical angle, 49, 104 Critical depth, 122, 128, 129 Cross section
absorption, 5 absorption (two photon), 171 scattering. See Scattering cross section
Cross-correlation theorem, 184 Cross-interference term, 200 C-scan, 190 CT. See Computed tomography Cumulative distribution function (CDF), 41, 42,
45, 60 Current density
conservation of energy, 117 definition, 84 direction, 85, 97 effect on radiance (illustration), 91 fractional change, 96, 97, 117 projection, 108
Cytoplasm, 8
350 INDEX
Dark-field confocal photoacoustic microscopy (PAM), 303
Dark-field illumination, 307 Data-acquisition board, 239 dB. See Decibel Decibel (dB), 139, 209 Deconvolution, 202, 205 Defocus distance, 166 Degeneracy, 97 Degree of circular polarization (DOCP), 223 Degree of linear polarization (DOLP), 223 Degree of polarization (DOP), 219, 223, 224,
236, 237 Delay-and-sum detection, 309 Delta heating, 289, 290, 295, 299
slab, 293 sphere, 297
Demodulation, 191, 193, 196 Depolarizing medium, 223 Depth of focus. See Focal zone Depth-priority scanning, 190 Depth-to-resolution ratio (DRR), 13, 181 Dermis, 305 Determinant, 241 Diattenuator, 225 Dichroic mirror, 165 Dichroism, 225 Differential path length, 277 Diffraction limit, 156, 165, 170 Diffraction theory, 165 Diffuse optical tomography (DOT), 2, 249, 283
DC, 250, 252 DC (illustration), 253 frequency domain, 250, 253 frequency domain (illustration), 253, 255 reconstructed image (plot), 268 time domain, 250, 251 time domain (illustration), 251
Diffuse reemittance relative, 39
Diffuse reflectance angularly resolved, 56 angularly resolved (plot), 57 approximations (illustration), 107 diffusion step, 123, 124 diffusion theory, 106 diffusion theory (plot), 109 experimental data (plot), 142 far, 140, 143, 144 hybrid, 124 hybrid (plot), 130, 131 image source (plot), 112 isotropic source in slab (plot), 126, 128 Monte Carlo data (plot), 142
Monte Carlo step, 123, 124 oblique incidence, 143 optical fibers, 144 pencil beam and isotropic source (plot), 111 pencil beam on slab (plot), 127 projection of current density, 108 relative, 39, 106 representation, 51 similarity relation (plot), 110, 114 slab, 121, 125 source depth (plot), 113 time-resolved, 145 total, 52, 55, 145 weight recording, 49
Diffuse transmittance angularly resolved, 56 angularly resolved (plot), 57 isotropic source in slab (plot), 126 Monte Carlo step, 123 pencil beam on slab (plot), 127 relative, 39 representation, 51 slab, 121, 125 total, 52, 55 weight recording, 49
Diffusion approximation boundary condition, 102 directional and temporal broadening, 97 expansion of radiance, 88, 105 high albedo, 88 P\ approximation, 89 similarity relation, 97
Diffusion coefficient, 120, 145, 256 definition, 97, 343 oblique incidence, 143
Diffusion equation approximated RTE, 83 background, 262, 269 derivation, 96 DOT, 249, 267 excitation, 147 expression, 97 fluorescence, 147 impulse response, 98 linearity, 257 numerical methods, 273 photon density, 256
Diffusion expansion of radiance, 88, 105 Diffusion theory
accuracy, 122 accuracy and speed, 106 breakdown, 99 derivation, 88 diffuse reflectance (plot), 109
INDEX 351
fluence, 56 fluorescence, 147 hybrid, 119 oblique incidence, 143 validation, 110 zero-boundary condition, 145, 324
Diffusive regime boundary condition, 106 definition, 114, 115 DOT, 249 effective reflection coefficient, 105 OCT, 186 PAT, 283 penetration depth, 56 UOT, 323, 330, 333
Diffusivity optical, 256 thermal, 284
Digital holography, 158, 175 Dimensionless step size, 40, 43, 45, 46, 48 Dipole moment, 23 Dipole radiation, 24, 311 Dirac delta function, 2, 11, 38, 201 Direct method, 274 Direction cosines, 39, 43, 47, 49 Dirichlet boundary condition, 103, 265, 266 Dispersion compensation, 209 Divergence, 86, 154, 288 Diverging spherical wave, 29, 288, 299, 310 DOCP. See Degree of circular polarization DOLP. See Degree of linear polarization DOP. See Degree of polarization Doppler
effect, 2, 206 frequency. See Doppler shift OCT, 206 shift, 186, 206, 239
DOT. See Diffuse optical tomography Dot product, 89 Double refraction, 9 Dynamic focusing, 190 Dynamic range, 251, 252
Early-photon imaging, 154 Edge spread function (ESF), 11
illustration, 11 EEM. See Excitation-emission matrix Effective attenuation coefficient, 98, 120, 143
definition, 344 Effective path length, 258 Effective reflection coefficient, 104, 105, 120 Eigenequation, 231 Eigenpolarization, 225, 226, 231, 242 Eigenvalue, 231, 232
Eigenvector, 231-233 Elastic scattering, 5, 85, 165 Elastooptical coefficient, 325 Elliptical polarization, 220, 222
illustration, 221 Ellipticity angle, 220 Empirical formula
center shift of diffuse reflectance, 143 effective reflection coefficient, 120 Grueneisen parameter, 285
Encoding ambiguity, 200 Energy density, 85, 286 Energy flow (illustration), 84 Ensemble averaging, 37, 210, 211, 236, 324 Envelope of interference fringes, 186
axial resolution, 189 demodulation, 191 expression, 189 GVD, 208, 209 number of periods, 193, 194
Equation acoustic wave. See Acoustic wave equation Bessel (transformed). See Bessel equation,
transformed Boltzmann. See Radiative transfer equation continuity. See Continuity equation diffusion. See Diffusion equation eigen. See Eigenequation force. See Force equation Helmholtz. See Helmholtz equation inviscid force. See Inviscid force equation Maxwell. See Maxwell equations motion. See Equation of motion photoacoustic. See Photoacoustic equation radiative transfer. See Radiative transfer
equation telegraphy. See Telegraphy equation thermal. See Thermal equation thermal expansion. See Thermal expansion
equation Equation of motion, 288 ESF. See Edge spread function Etendue, 335, 336 Excitation
definition, 3 illustration, 4 nonlinear optical, 169 one-photon, 170 one-photon (illustration), 170 time, 4 two-photon. See Two-photon excitation two-photon (illustration), 170
Excitation-emission matrix (EEM), 146
352 INDEX
Expansion of radiance, diffusion. See Diffusi« expansion of radiance
Extended trapezoidal rule, 72 integrand evaluation (illustration), 73
Extinction coefficient ballistic imaging, 153 collimated transmission method, 135 definition, 6, 344 formula, 8, 44 OCT, 212
Extracellular fluid, 8 Extraordinary ray, 9 Extrapolated boundary
DOT, 265 illustration, 102 image point, 106 oblique incidence, 143 refractive-index-matched, 102 refractive-index-mismatched, 105 slab, 120
Extrapolation (illustration), 73
Fabry-Perot interferometer, 326, 335 Far field, 24, 259, 307 Fast axis, 155, 226, 231, 238 Fast Fourier transformation (FFT), 202 FFT. See Fast Fourier transformation Fick's law, 102, 106, 343
diffuse reflectance, 108 formula, 97
Field of view (FOV), 13, 304 Filter wheel, 256 Finite-difference method, 273 Finite-element method, 273 First photon-tissue interaction, 76 First-order diffraction term, 160 Flowchart for tracking photons, 40 Fluence, 53
conversion to pressure, 286 definition, 84 depth resolved, 56 distribution (plot), 58 plot, 78, 79 relative, 39
Fluence rate boundary value, 145 conversion to pressure, 287 definition, 84 diffusion theory, 119 primary beam, 147
Fluorescence characteristics, 9 confocal imaging (illustration), 166 definition, 3
emission spectrum, 146 excitation spectrum, 146 illustration, 4 incoherent, 9 lifetime, 3, 4, 10, 146, 147 modeling, 147 origins, 9 quantum yield. See Quantum yield red shift, 9 spectroscopy, 146 spectrum, 1, 9 Stokes shift, 9 time scales, 4
Fluorophore, endogenous, 10 Flux, 56
energy, 85 photon, 171
Focal plane, 156, 157, 306 Focal zone, 190, 326, 332 Focused ultrasonic transducer, 304, 330, 332,
335 Force equation, 288, 314 Forcing function, 263, 266 Forward problem, 249, 272-274
perturbation, 262-264 Fourier optics, 156 Fourier space-gated imaging, 156 Fourier transformation
autocorrelation function, 325 differential equation, 99, 100, 290 Doppler OCT, 207 Fourier-domain OCT, 201 PAT, 310 spatial, 12, 156, 160 temporal, 184, 191, 250 UOT, 328
Fourier transformer inverse, 195 spatial, 157, 195 temporal, 195
Fourier-domain OCT, 198 signal processing (illustration), 204, 205 system (illustration), 198
Fourier-domain optical delay line illustration, 197
FOV. See Field of view Frame rate, 13, 199 Frequency sweeping. See Chirping Frequency-division multiplexing, 252 Frequency-swept gating, 160, 202, 326, 329 Fresnel reflection, 43, 103, 105, 122 Full-field image, 164 Functional imaging, 1, 7, 249 Fused-silica fiber, 209
INDEX 353
Gabor holography, 173, 174 illustration, 173
Gaussian beam, 69, 190 fluence (plot), 79
Gaussian envelope, 189, 208 Gaussian line shape, 184, 189 Gaussian quadratures, 125 Gene expression, 2 Glucose, 9 Grating, 140, 145, 195, 307 Grating lobe, 309 Grating-lens pair, 195 Green's function, 316. See also Impulse
response DOT, 263, 265, 269, 270, 274 PAT, 288, 289, 291, 293, 310, 311 pencil beam, 38, 67, 68 point source, 98, 99 PSF, 10 UOT, 333
Green's function approach DOT, 263, 265, 269, 270 illustration, 263 PAT, 288, 289 summary, 316
Green's second identity, 270 Green's theorem
diffusion theory, 98 DOT, 263 PAT, 292, 310, 317
Group delay, 188, 195, 197, 198 Group velocity, 188 Group velocity dispersion, 207 Group-path-length mismatch, 198 Grueneisen parameter, 285
Hankel function, 21, 29, 33 Hankel transform, 166 Heated slab
illustration, 293 snapshots of pressure (plot), 298
Heated sphere illustration, 298 pressure versus time (plot), 299 snapshots of pressure (plot), 302
Heating function, 287, 289 Heaviside function, 295, 299 Helmholtz equation, 257, 270
scalar, 27, 29 vector, 30
Hemoglobin concentration, 1, 6, 7, 145 deoxygenated, 6, 283 oxygen saturation, 1, 6, 7 oxygenated, 6, 283 PAT, 313 primary absorber, 6 spectrum of molar extinction coefficient
(plot), 6 two forms, 6
Henyey-Greenstein phase function hybrid model, 119, 123 Monte Carlo method, 46, 47 OCT, 211 sensing, 137
Heterodyne detection, 160, 161, 186, 254, 327
Heterodyne frequency, 161, 329, 330 Heterodyne imaging, optical. See Optical
heterodyne imaging High resolution, 2, 13, 164, 181, 284 Highpass filter, 191 High-speed shutter, 154 Histogram, 252 Hologram, 158, 159, 171-173
reconstruction (illustration), 172, 173, 175 recording (illustration), 172, 173
Holographic imaging, 153, 158 Holography, 158, 171 Homogeneous boundary condition, 293 Hooke's law, 287 Hybrid model, 119, 122
illustration, 123 Hypermetabolism, I, 7
IDM. See Inverse distribution method Image function, 11 Image plane, 164 Image reconstruction
CT, 163 DOT, 249, 252, 261, 263, 267 PAT, 309 UOT, 332
Image source, 106, 108, 120, 143 illustration, 107 slab (illustration), 121
Imaging modalities, comparison of, 2, 284 Implicit function, 269 Implicit photon capture, 42 Impulse function. See Dirac delta function Impulse heating. See Delta heating Impulse response, 68, 76, 251, 256 See also
Green's function pencil beam, 38, 67
3 5 4 INDEX
Impulse response, (Continued) point source, 98 PSF, 10
Index of refraction. See Refractive index definition, 343
Inelastic scattering, 4 Infinitely narrow photon beam. See Pencil
beam Initial condition, 317 Initial photoacoustic pressure, 284, 309
slab, 293 sphere, 297
Inner product, 89 Instantaneous frequency, 326 Intensity, 84 Interference fringes
antenna theorem, 161 Doppier OCT, 206, 207 Michelson interferometry, 183 time-domain OCT, 186, 194
Interferogram, 171, 183. See also Interference fringes
spectral, 198-200, 202, 204, 205 Interferometer, Fabry-Perot. See Fabry-Perot
interferometer Interferometry, Michelson. See Michelson
interferometry Internal conversion, 4 Interpolation (illustration), 73 Intralipid® solution, 135, 305 Inverse distribution method (IDM)
definition, 41 illustration, 41 proof, 42 scattering angles, 46-48 step size, 44, 45
Inverse Fourier transformation differential equation, 100, 290 Fourier-domain OCT, 198, 200, 201, 204,
205 PAT, 311 spatial, 160 temporal, 186, 189, 195, 250
Inverse method, 274 Inverse problem, 249, 262, 263, 272 Inverse Radon transformation, 163 Inviscid force equation, 288, 314 IQ detection, 254 Isosbestic point, 6 Isothermal compressibility, 285 Isotropie scattering
definition, 47 primary beam, 147
similarity relation, 123 UOT, 324
Isotropie source, 92 array, 121 normal and oblique incidence (illustration),
141 plane, 99 point, 108 slab (plot), 126, 128 volume, 122
Iterative method, 265, 272
Jablonski energy diagram, 4 illustration, 4, 170
Jacobian matrix, 264, 274 Jones calculus, 219, 235 Jones matrix, 219, 240
conversion to Mueller matrix, 235 definition, 230 OCT, 237 requisite independent measurements, 240 rotator, polarizer, and retarder, 230
Jones reversibility theorem, 240, 241 Jones vector, 219, 229, 238, 239
conversion to Stokes vector, 236 Jones-Mueller transformation, 235
k clock, 202 Keratin, 9 Kerr effect, 154 Kerr gate (illustration), 155 Krönecker delta function, 89 Krönecker tensor product, 235, 236
Lateral resolution. See Transverse resolution Law
Beer's. See Beer's law Fick's. See Fick's law Hooke's. See Hooke's law Newton's second. See Newton's second law Snell's. See Snell's law
Left circular polarization, 222, 224, 227, 229 Legendre polynomials, 89
associated, 29, 89 Leith-Upatnieks holography (illustration), 174,
175 Lifetime, fluorescence. See Fluorescence
lifetime Line spread function (LSF), 11
illustration, 11 Linear inverse algorithm, 261, 266 Linear polarization
decomposition, 49 definition, 221
INDEX 355
Jones vector, 229 Mueller matrix, 227 Poincare sphere, 224 principal axes, 9
Linear polarizer eigenpolarization, 225 Jones matrix, 230, 231 part of a circular polarizer, 233 polarization-difference imaging, 157 time gate, 155
Linearity, 67, 251 Local oscillator, 161, 254 Lock-in, 254, 330
illustration, 254 LSF. See Line spread function
Magnetic resonance imaging (MRI), 1, 2, 329 Markov chain, 50 MATLAB
conventional and confocal microscopes, 169 heated slab, 295, 296 heated sphere, 300 Mie theory, 22 null plane, 259 OCT, Fourier-domain, 202 OCT, time-domain, 194 Rayleigh theory, 20
Maximum imaging depth ballistic imaging, 153 comparison, 2 definition, 13 DOT, 249 OCT, 181 PAM (plot), 305 PAT, 284, 313
Maximum-amplitude projection definition, 306 plot, 306
Maxwell equations, 17, 26 MCML, 40, 58, 67 MCP. See Microchannel plate Mean absorption length, 2, 5 Mean free path, 8, 115, 154, 324
definition, 344 typical value, 2
Mean path length of flight, 277 Mean time of flight, 276 Mechanisms of ultrasonic modulation, 323 Medical imaging modalities, 1 Medium displacement, 288 Medium velocity, 314, 316 Melanin, 6, 8
primary absorber, 6
Michelson interferometry, 181, 185 illustration, 182
Microchannel plate (MCP), 155, 252 Mie theory, 7, 17
derivation, 26 particle sizing, 145 phase function, 47
Mitochondria, 8, 10 Mixer, 161, 254, 255 Modal matrix, 232 Modes of DOT, 249 Modulation depth
DOT, 250, 257, 258 UOT, 325, 326, 331, 333
Modulation transfer function (MTF), 12, 305 Molar concentration, 6, 7 Molar extinction coefficient, 6, 10
spectrum (plot), 6 Molecular conformation, 1 Molecular imaging, 2 Monochromator, 140 Monte Carlo method, 37
accuracy and speed, 106 benchmark, 110, 138 broad beam, 67 equivalence to RTE, 83 flowchart, 40 hybrid, 119 oblique incidence reflectometry, 141 OCT, 210 UOT, 324
MRI. See Magnetic resonance imaging - MTF. See Modulation transfer function Mueller calculus, 219, 235 Mueller matrix, 219
conversion from Jones matrix, 235 definition, 224 image, 242 measurement, 227 OCT, 237 requisite independent measurements, 240 rotator, polarizer, and retarder, 225
Mueller OCT, 219 images of tendon (plot), 242 system (illustration), 239
Multimode fiber, 186 Muscle fiber, 1, 9, 38 Myelin, 9
NA. See Numerical aperture NAD(P)H, 10 NADH, 10 Nd:YAG, 304, 312 Near field, 259
356 INDEX
Nepers, 139 Neutral-density filter, 139, 251, 256 Newton's second law, 288, 314 Nonballistic light. See Nonballistic photon Nonballistic photon, 115, 153
arrival time, 154, 158 polarization, 157 spatial frequency, 156, 160
Nondepolarizing medium, 223, 230 Nonionizing radiation, 2 Nonlinear inverse algorithm, 267, 272 Nonlinear optical excitation, 169 Nonlinear problem, 272 Nonlinear reconstruction, 272 Nonradiative relaxation, 3 Null line (plot), 261 Null plane, 259, 261
plot, 261 Numerical aperture (NA)
definition, 166 OCT, 190 PAM, 303, 304 two-photon microscopy, 171
Nyquist criterion, 194, 200, 254
Object function, 11 Objective lens
confocal microscopy, 165 conventional microscopy, 164 OCT, 190, 237-239 two-photon microscopy, 171
Oblique-incidence reflectometry, 140 illustration, 141 white-light (illustration), 144
OCT. See Optical coherence tomography Offset-reference holography. See
Leith-Upatnieks holography Operator, 274
autocorrelation, 201 linear, 159, 266 real-part, 159
Optical absorption, 1, 2, 6, 283, 307 Optical attenuator, variable, 252 Optical coherence tomography (OCT), 181, 283
axial resolution versus bandwidth (plot), 193 class I signal, 210 class II signal, 211 classes of signals (illustration), 210 degree of polarization, 236 demodulation (illustration), 192, 196 Doppier. See Doppier OCT Fourier-domain. See Fourier-domain OCT Monte Carlo modeling, 210 Mueller. See Mueller OCT
polarization-sensitive, 219 scattering versus depth (plot), 213 signal versus depth (plot), 212 system (illustration), 185, 191 time-domain, 185
Optical coordinates, 166 Optical delay line, 195 Optical density, 139, 256 Optical fiber
fused-silica, 209 multimode, 186 single-mode, 185, 237, 239
Optical heterodyne imaging, 154, 160 illustration, 162
Optical imaging, motivation for, 1 Optical microscopy, conventional, 164
axial PSF, 167 dark-field, 303 illustration, 165 lateral PSF, 167 PSF, 166 PSF (plot), 168
Optical properties, 2, 85, 259 Optical sectioning, 164, 169 Optically thick medium, 154 Optically thin medium, 154 Optimal coordinates, 51, 53 Ordinary ray, 9 Orientation angle of major axis, 220 Orthogonality, 34, 89, 265. See also
Orthonormality Orthonormality, 89, 230, 232. See also
Orthogonality Outgoing spherical wave. See Diverging
spherical wave Oximetry, 7 Oxygen saturation of hemoglobin. See
Hemoglobin oxygen saturation
PAM. See Photoacoustic microscopy Parallel Mueller OCT, 237 Parallel-speckle detection, 329 Paraxial approximation, 166 PAT. See Photoacoustic tomography Path-length difference, 183, 186 Path-length distribution, 186 Path-length mismatch. See Path-length
difference PDF. See Probability density function Pencil beam, 2
conversion to isotropic source, 106, 107, 112, 122
conversion to isotropic source (plot), 129 conversion to photon cloud, 115
INDEX 357
cylindrical symmetry, 39 definition, 38 diffuse reflectance, 106 diffuse reflectance (plot), 109, 111 fluence (plot), 78, 79 Green's function for broad beam, 67, 68 hybrid model, 119 normal and oblique incidence, 140 normal and oblique incidence (illustration),
141 oblique-incidence reflectometry, 143 OCT, 211 slab (plot), 127, 130, 131
Penetration depth, 56, 249 definition, 99, 344 scale for grid, 51 water, 6
Perturbation, 262-264, 269, 272 Phase
delay in GVD, 207 delay in OCT, 188, 197 delay in photon-density wave, 258 delay in retarder, 226 lag, 220 lead, 220 shift, 197 shifter. See Retarder velocity, 188, 343
Phase function Henyey-Greenstein. See Henyey-
Greenstein phase function Mie theory, 47
Phasor representation, 159 holography, 158, 172 Jones vector, 229 Michelson interferometry, 182 Mie theory, 29 photon density, 256, 257
Phosphor screen, 155 Phosphorescence, 3, 4
illustration, 4 Photoacoustic effect, 283 Photoacoustic equation, 287-289, 309 Photoacoustic microscopy (PAM), 303
image (plot), 306 imaging depth (plot), 305 spatial resolution (plot), 305 system (illustration), 304
Photoacoustic tomography (PAT), 2, 283 brain image (plot), 313 reconstruction (illustration), 310, 312
Photoacoustic wave, 283, 284, 287, 304, 312 Photobleaching, 170
Photocathode, 155, 252 Photocurrent, 163, 182, 183, 209 Photoelectric effect, 155 Photoelectron, 155 Photomultiplier tube (PMT)
frequency-domain DOT, 254, 256 frequency-swept UOT, 327 optical heterodyne imaging, 161 time-domain DOT, 252
Photon cloud, 115 Photon current, 97 Photon density, 256, 273
definition, 85 perturbation, 262 snapshot (plot), 3 wave, 257-259, 267
Photon energy, 4, 10, 171, 183, 256 relation with wavelength, 14
Photon packet absorption, 46 boundary crossing, 48 launching, 43 moving, 46 representation, 42 scattering, 46 step size, 44 termination, 50
Photon propagation regimes, 114 ballistic. See Ballistic regime diffusive. See Diffusive regime quasiballistic. See Quasiballistic regime quasidiffusive. See Quasidiffusive regime
Physical depth distribution, 186 Physical quantities, 50, 72, 83
interpolation and extrapolation (illustration), 73
Picosecond time analyzer, 252 Piezooptical coefficient, 325 Pinhole
coherence-gated holographic imaging, 158 confocal microscopy, 165 spatiofrequency filtered imaging, 156, 157 two-photon microscopy, 169, 170
Planck constant, 14, 85 PMT. See Photomultiplier tube Poincare sphere, 224 Point spread function (PSF)
confocal and conventional microscopy (plot), 168
confocal and two-photon microscopy, 170 confocal microscopy, 165, 167 conversion to LSF, 11 convolution, 11, 168 definition, 10
358 INDEX
Point spread function (PSF) (Continued) lens, 166 OCT, 189, 207, 208
Poisson distribution, 136 Polar angle
illustration, 18 sampling, 46
Polarimetry, 219 Polarizability, 17, 18, 23 Polarization, 157, 219, 236 Polarization analyzer
measurement of Mueller matrix, 227 measurement of Stokes vector, 222, 223, 228 polarization-difference imaging, 157, 158 polarization-difference imaging (illustration),
157 Polarization ellipse, 220, 222, 224
illustration, 221 Polarization homogeneous medium, 231 Polarization inhomogeneous medium, 231 Polarization origins, 9 Polarization state, 219 Polarization-difference imaging, 154, 157
illustration, 157 Polarization-sensitive OCT, 219 Polarizer, 225, 230 Polarizing element, 225, 226, 231 Polystyrene sphere, 135 Positive lens, 304 Power spectral density, 184, 200, 325 Primary absorbers, 6, 7 Primary beam, 147 Primary scatterers, 8 Probability density function (PDF)
definition, 60 free path length, 44, 45 path length, 324 phase function, 87, 137 sampling, 41
Projection data, 163, 334 Projection image, 154, 156, 163. See also
Shadowgram Propagation constant
GVD, 207 Michelson interferometry, 182 Mie theory, 27 OCT, 187, 189 photon density wave, 257, 258 Rayleigh theory, 17
Pseudorandom number, 41 azimuthal angle, 47 polar angle, 47 Russian roulette, 50
specular reflection, 49 step size, 122
PSF. See Point spread function Pulse oximetry, 7 Pupil, 166
Quantum efficiency of detector, 182 Quantum yield, 146, 147, 171
definition, 3 fluorophores, 10
Quasiballistic photon, 153 Quasiballistic regime
definition, 114, 115 OCT, 186 UOT, 329
Quasidiffusive regime, 115 definition, 114, 115 OCT, 186 PAM, 307 PAT, 283 UOT, 323
Radiance, 83, 88 diffusion expansion (illustration), 91 illustration, 84
Radiative transfer equation (RTE), 83, 88 derivation (illustration), 86
Radon transformation, 163, 333 illustration, 164
Raman scattering, 4, 5 illustration, 4 spectrum, 1
Ram-Lak filter, 334 Ramp filter, 312 Random walk, 14, 37 Raster scanning, 154, 305 Rayleigh criterion, 259 Rayleigh range, 190 Rayleigh theory, 7, 17, 23
scattering efficiency (plot), 24 Real image, 160, 173-175 Reciprocity
acoustic wave, 288, 291, 307, 316 confocal microscopy, 167 grating-lens pair, 197 optical fluence rate, 98 polarization, 227
Reconstruction of a hologram, 158, 173, 174 Red shift, 9 Reduced interaction coefficient. See Transport
interaction coefficient Reduced scattering coefficient
definition, 94, 344 fluorescence excitation, 147 map, 267
INDEX 359
Mie theory, 21 oblique-incidence reflectometry, 140 PAM, 305 particle sizing, 145 similarity relation, 123 wavelength dependence, 170
Reemission, 39, 52, 106 imaging configuration, 249 modeling, 50 photon termination, 50
Reemittance, 49, 50 Reference-intensity term, 200 Reflectance, 51 Reflection mode, 158, 165 Reflectometry, 140 Refractive index
glass, 44 tissue, 8, 44 water, 8, 44
Refractive index, relative, 18, 104 Refractive-index-matched boundary, 106
boundary condition, 101 definition, 55 effect on fluence, 56 effect on penetration depth, 57 fluence distribution (plot), 58 photon entry, 122
Refractive-index-mismatched boundary, 55, 119 boundary condition, 103 computation time, 132 effect on fluence, 56 effect on penetration depth, 57 fluence distribution (plot), 58
Regularization, 275 Relative refractive index, 18, 104 Relaxation (illustration), 4 Resonant frequency, 336 Retarder, 226, 231
activatable, 155 half-wave, 154, 155, 226, 237 part of a circular polarizer, 233 quarter-wave, 226, 228, 237-239 variable, 237
Retina, 2, 9, 181, 209 Riccati-Bessel function, 21 Right circular polarization, 222, 224, 227
illustration, 221 Rotator, 225, 230 RTE. See Radiative transfer equation Russian roulette, 43, 50, 123, 132 Rytov approximation, 278
Sample-intensity term, 200 Sampling of a random variable, 41
Scalar wave, 259 Scatterers, primary, 8 Scattering
biological structures (illustration), 8 coefficient, 8, 21, 343 cross section, 8, 18, 25, 87 efficiency, 8, 19, 20, 33 efficiency (plot), 24 mean free path, 8 media, 2 medium, layers (illustration), 38 medium, slab (illustration), 121 optical depth, 136 origins, 7
Secondary electron, 155 Self-interference term, 200 Sensitivity, 199, 209 Sensitivity matrix, 264 Serial Mueller OCT, 237
system (illustration), 238 Shadowgram, 154, 156. See also Projection
image Shift invariance. See Translation invariance Shutter speed, 155 Side lobe, 308 Signal-to-noise ratio (SNR), 13, 250
definition, 12 effect on spatial resolution, 259 UOT, 330, 336
Similarity relation, 97, 106, 114, 123, 147 diffuse reflectance (plot), 110
Simultaneous iterative reconstruction technique (SIRT), 265, 275
sine function, 163 Single-mode fiber, 185, 237, 239 Single-photon counting, 145, 155, 251, 252 Singly backscattered light, 153, 210, 212, 213,
239 Singularity, 29, 290 Singular-value decomposition (SVD), 265 Sinogram. See Projection data SIRT. See Simultaneous iterative reconstruction
technique Size parameter, 18, 21 Slab, heated, 293, 302 SLD. See Superluminescent diode Slow axis, 226 Snell's law, 49, 104, 137, 140
photon-density wave, 259 SNR. See Signal-to-noise ratio Soft tissue contrast, 2 Sound velocity. See Speed of sound Source density function, relative, 123
plot, 129
360 INDEX
Spatial filter, 157, 158, 238 Spatial frequency, 12
ballistic imaging, 156, 161 grating-lens pair, 196 holography, 160, 175 PAM, 305 spectrum, 156 UOT, 334
Spatial modulation frequency (plot), 305 Spatial resolution
ballistic imaging, 153 comparison, 2 confocal microscopy, 165 definition, 11 DOT, 249, 259 OCT, 181, 237 PAM, 307 PAM (plot), 305 PAT, 284 streak camera, 155 time gate, 156 tradeoff with FOV, 13 two-photon microscopy, 169 UOT, 323, 331
Spatiofrequency filtered imaging, 154, 156 illustration, 156
Specific absorption, 53, 72, 285 definition, 85 relative, 39
Specific absorption rate, 287 definition, 85
Specific energy deposition. See Specific absorption
Specific heat capacity, 285 Specific power deposition. See Specific
absorption rate Speckle
coherence-gated holographic imaging, 158
grain, 236 OCT, 210, 236, 284 UOT, 324, 325, 329, 332
Spectral intensity, 325 Spectral interferometry, 198 Spectral radiance, 83 Spectral resolution, streak camera, 155 Spectrograph, 145 Spectrometer, 155, 198, 202 Spectrophotometer, 139, 140 Spectrophotometry, 139 Spectroscopy, 2, 135
fluorescence, 146 white-light, 144
Specular reflectance example, 44, 56 formula, 43 incident beam, 143, 147 photon packet, 48 unscattered reflectance, 52
Specular reflection. See Fresnel reflection Speed of sound
comparison with speed of light, 181 formula, 315 soft tissue, 305, 331 UOT, 325, 336 water, 284
Sphere, heated, 297, 303 Spherical harmonics, 88 Spherical polar coordinates (illustration), 18 Standard error, 55 Step heating, 289 Step size, 45, 122 STF. See System transfer function Stokes parameters, 222, 223 Stokes shift, 9 Stokes transition, 5 Stokes vector, 219, 222
construction of Mueller matrix, 237 conversion from Jones vector, 236 polarization-difference imaging, 158
Streak camera, 145, 155, 251 illustration, 155
Stress confinement, 285 Stress relaxation time, 284 Subsonic medium velocity, 316 Superluminescent diode (SLD), 186, 209, 237,
238 Surface normal, 108, 265, 271 SVD. See Singular-value decomposition Synthetic aperture, 307, 309 System transfer function (STF), 12
Targeted contrast agent, 2 Taylor expansion, 137
fluence rate at boundary, 102 matrix form, 273 pressure, 314 propagation constant, 187, 207
Telegraphy equation, 117 Temporal PSF, 251 Temporal resolution, 155 Temporal spread function, 252 Tendon, 242 Thermal coefficient of volume expansion,
285 Thermal confinement, 285, 287, 289
INDEX 361
Thermal diffusion, 284 Thermal diffusivity, 284 Thermal equation, 287 Thermal expansion equation, 287 Thermal relaxation time, 284 Thermoelastic expansion, 283 Time gate, 154 Time of flight, 250, 252, 258 Time reversal, 227, 291, 311 Time-domain OCT, 185 Time-gated imaging, 153, 154
illustration, 154 Time-resolved measurement, 145 Tomography, 1 Top-hat beam, 71
fluence (plot), 78 Top-hat function, 303 Total interaction coefficient. See Extinction
coefficient Total transmittance, 55 Translation invariance, 67
spatial, 11, 98, 263 temporal, 68, 302
Translation stage, 256, 304, 335 Transmission mode, 165, 336 Transmittance, 5, 51, 158 Transport albedo, 106, 108, 123, 143
definition, 344 Transport interaction coefficient, 94
definition, 344 Transport mean free path
conversion to isotropic source, 123 definition, 344 diffusion scale, 56, 97, 115, 132 far diffuse reflectance, 140 formula, 94 PAM, 305 scale for grid, 51 UOT, 325
Transport mean free time, 96, 97 Transport scattering coefficient. See Reduced
scattering coefficient Transposition symmetry, 240, 242 Transverse resolution, 181, 309
OCT, 190 PAM, 305, 307 UOT, 326, 329, 331, 337
Transverse scanning, 156, 161 Transverse-priority scanning, 190 Turbid media, 2 Two-photon excitation, 169, 170 Two-photon microscopy, 154, 169
illustration, 169
Ultrafast laser, 154 Ultrasonic array. See Ultrasonic transducer
array Ultrasonic attenuation coefficient, 339 Ultrasonic coupling, 304, 330, 336 Ultrasonic pressure, 335. See also Acoustic
pressure Ultrasonic scattering, 283 Ultrasonic transducer, 283
array, 307 array (illustration), 308 PAM, 304, 306, 307 PAT, 312 UOT, 326, 330, 332, 335
Ultrasonography, 1 analog of OCT, 181 comparison, 2, 284 Doppler flow, 206 lack of polarization, 219
Ultrasound absorber, 326 Ultrasound imaging. See Ultrasonography Ultrasound-coupling gel, 304 Ultrasound-modulated optical tomography
(UOT), 323 Unscattered absorption, 39, 46, 53 Unscattered reflectance, 51, 52 Unscattered transmittance, 51, 52, 55 Unscattered transmitted photons, 136 UOT. See Ultrasound-modulated optical
tomography UOT, Fabry-Perot interferometry
image (plot), 336, 337 system (illustration), 335
UOT, frequency-swept image (plot), 329 spectrum (plot), 328 system (illustration), 327 virtual source (plot), 332
UOT, multiple-speckle system (illustration), 330
UOT, reconstruction-based image (plot), 334 system (illustration), 333
USAF-1951 target, 305 User times, 130, 132
Variable optical attenuator, 252 Variance reduction technique, 42, 211 Vasculature, 305, 306 Vector wave, 259 Velocity potential, 287 Velocity resolution, 207 Vibrational relaxation, 4, 9, 170
362 INDEX
Video rate, 13 Waveplate. See Retarder Virtual image, 160, 173-175 Weak-modulation approximation, 324, 326 Virtual optical source, 331 Weak-scattering approximation, 324
Weight of a photon packet, 43 Water White-light spectroscopy, 144
absorption, 6 Wideband light source, 160 compressibility, 285 Wiener-Khinchin theorem, 184, 189, dispersion compensation, 209 201, 325 Grueneisen parameter, 285 primary absorber, 6 X-ray refractive index, 8 CT, 163, 332 refractive-index-matching, 101. 135 projection imaging, 156 speed of sound, 284 radiography, I ultrasonic coupling, 304, 312, 330, 332, 336
Wavelength of photon-density wave, 258 Zero-boundary condition, 145, 324