Modeling ibuprofen

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MTH4601 - Research Project

Name

MODELING IBUPROFEN

STATEMENT

Ibuprofen, an analgesic pain reliever, is ingested into the gastrointestinal tract by swallowing

a pill of the substance or drinking a solution. The drug then moves to the serum or plasma

(blood) where it travels to sites to do its work of pain relief. The following data (see Table

1) comes from an experiment in which “Following oral dosing with 400 mg ibuprofen, serial

blood samples were taken from five healthy male volunteers and four patients.” This is the data

from one of the healthy subjects. It is worth noting that a standard dose in each pill of Advil

Liqui-Gels is 200 mg of ibuprofen, but often patients take two pills for relief, resulting in a dose

of 400 mg ibuprofen.

Model 1

This is an example of a two compartment model. We show a diagram of the situation in

Figure 1. For our data we do not know the volume of the gastrointestinal tract nor the volume

of the plasma for our patients, so we shall call these v1 and v2 respectively.

Define variables to be:

x1(t) = concentration of ibuprofen in the gastrointestinal (tract) compartment in µg/ml or

mg/l;

x2(t) = concentration of ibuprofen in the serum/plasma compartment in µg/ml or mg/l.

1. Construct a system of linear differential equations to model the absorption of ibuprofen as

depicted in Figure 1. You might consider modeling the change in the amount of ibuprofen

in the two compartments:

Time

(hr) 0 0.65 1.03 1.26 1.63 1.73 2.10 3.00 3.97 5.08 6.02 7.00

Ibuprofen

Conc. in

µg/ml

0 25.81 34.22 33.47 32.91 28.42 27.16 16.64 9.91 7.48 5.24 4.86

Table 1: serum/plasma concentration of Ibuprofen at time intervals after an initial oral dose

of 400 mg of ibuprofen was administered to a healthy patient.

v1x ′ 1(t) =

v2x ′ 2(t) = (1)

2. Then offer a revised system of differential equations model in which we combine rate

constants and volumes of regions of the body, i.e. gastrointestinal tract, x1(t), and

serum/plasma x2(t). Since we do not know these respective compartment volumes, but

have a value of v2 for a typical human being of 5 liters from the literatures let us assume

this data comes from a typical human with v2 = 5 liters serum/plasma. We then solve

the system of differential equations and identify the functions from solution.

Now v1x1(t) = X1(t) is the actual amount of ibuprofen in mg in the gastrointestinal tract

at time t and v2x2(t) = X2(t), using v2 = 5, is the actual amount of ibuprofen in mg in

the serum/plasma in the body at time t.

Let us rewrite the system of differential equations (1) in terms of the functions X1(t)

and X2(t), the respective amounts in mg of ibuprofen in the respective compartments,

gastrointestinal tract - X1(t) and serum/plasma - X2(t).

X′1(t) =

X′2(t) = (2)

GI tract Plasma k1 k2

Figure 1: Diagram for two compartment model of ibuprofen absorption. k1 is called the ab-

sorption rate from gastrointestinal tract to serum/plasma while k2 is called the elimination rate

constant. Both have units l/hr.

3. Solve the differential equations (2) assuming v2 = 5 for the respective amounts of ibupro-

fen. Worried about v1? Be patient and watch and see as your work progresses.

4. Here are three sets of estimates of the parameters for this model. Which one is best (of

the three presented) for predicting the data?

(a) k1 = 0.91, k2 = 0.15,

(b) k1 = 0.65, k2 = 0.41,

(c) k1 = 0.85, k2 = 0.92.

5. Develop a strategy and execute it for determining a truly best estimate of the rate pa-

rameters k1 and k2.

6. Compute the sum of square errors between the data and your best model for the concen-

tration of ibuprofen in the serum/plasma.

7. Plot the resulting model over your data and comment on its ability to predict the drug

behavior.