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MTH4601 - Research Pro ject 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 ve healthy male volunteers and four patients." This is the data from one of the healthy sub jects. 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 v 1 and v 2 respectively.

De ne variables to be:

x1( t) = concentration of ibuprofen in the gastrointestinal (tract) compartment in g/ml or mg/l; x2( t) = concentration of ibuprofen in the serum/plasma compartment in g/ml or mg/l.

1. Construct a system of linear di erential 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 g/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 0 1 ( t) = v 2x 0 2 ( t) = (1) 2. Then o er a revised system of di erential equations model in which we combine rate constants and volumes of regions of the body, i.e. gastrointestinal tract, x 1( t), and serum/plasma x 2( t). Since we do not know these respective compartment volumes, but have a value of v 2 for a typical human being of 5 liters from the literatures let us assume this data comes from a typical human with v 2 = 5 liters serum/plasma. We then solve the system of di erential equations and identify the functions from solution.

Now v 1x 1( t) = X 1( t) is the actual amount of ibuprofen in mg in the gastrointestinal tract at time tand v 2x 2( t) = X 2( t), using v 2 = 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 di erential equations (1) in terms of the functions X 1( t) and X 2( t), the respective amounts in mg of ibuprofen in the respective compartments, gastrointestinal tract - X 1( t) and serum/plasma - X 2( t).

X 0 1 ( t) = X 0 2 ( t) = (2)Figure 1: Diagram for two compartment model of ibuprofen absorption.

k 1 is called the ab- sorption rate from gastrointestinal tract to serum/plasma while k 2 is called the elimination rate constant. Both have units l/hr.

3. Solve the di erential equations (2) assuming v 2 = 5 for the respective amounts of ibupro- fen. Worried about v 1? 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) k 1 = 0 :91, k 2 = 0 :15 ; (b) k 1 = 0 :65, k 2 = 0 :41 ; (c) k 1 = 0 :85, k 2 = 0 :92 :

GI tract Plasma k 1 k 2 5. Develop a strategy and execute it for determining a truly best estimate of the rate pa- rameters k 1 and k 2.

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.