MTH4601 – Research ProjectNameMODELING IBUPROFENSTATEMENTIbuprofen, an analgesic pain reliever, is ingested into the gastrointestinal tract by swallowinga 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 Table1) comes from an experiment in which “Following oral dosing with 400 mg ibuprofen, serialblood samples were taken from five healthy male volunteers and four patients.” This is the datafrom one of the healthy subjects. It is worth noting that a standard dose in each pill of AdvilLiqui-Gels is 200 mg of ibuprofen, but often patients take two pills for relief, resulting in a doseof 400 mg ibuprofen.Model 1This is an example of a two compartment model. We show a diagram of the situation inFigure 1. For our data we do not know the volume of the gastrointestinal tract nor the volumeof the plasma for our patients, so we shall call thesev1andv2respectively.Define variables to be:x1(t) = concentration of ibuprofen in the gastrointestinal (tract) compartment inμg/ml ormg/l;x2(t) = concentration of ibuprofen in the serum/plasma compartment inμg/ml or mg/l.1. Construct a system of linear differential equations to model the absorption of ibuprofen asdepicted in Figure 1. You might consider modeling the change in the amount of ibuprofenin the two compartments:Time(hr)00.651.031.261.631.732.103.003.975.086.027.00IbuprofenConc. inμg/ml025.8134.2233.4732.9128.4227.1616.649.917.485.244.86Table 1:serum/plasma concentration of Ibuprofen at time intervals after an initial oral doseof 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 rateconstants and volumes of regions of the body, i.e. gastrointestinal tract,x1(t), andserum/plasmax2(t). Since we do not know these respective compartment volumes, buthave a value ofv2for a typical human being of 5 liters from the literatures let us assumethis data comes from a typical human withv2= 5 liters serum/plasma. We then solvethe system of differential equations and identify the functions from solution.Nowv1x1(t) =X1(t) is the actual amount of ibuprofen in mg in the gastrointestinal tractat timetandv2x2(t) =X2(t), usingv2= 5, is the actual amount of ibuprofen in mg inthe serum/plasma in the body at timet.Let us rewrite the system of differential equations (1) in terms of the functionsX1(t)andX2(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 Plasmak1 k2Figure 1: Diagram for two compartment model of ibuprofen absorption.k1is called theab-sorption ratefrom gastrointestinal tract to serum/plasma whilek2is called theelimination rateconstant. Both have units l/hr.3. Solve the differential equations (2) assumingv2= 5 for the respective amounts of ibupro-fen. Worried aboutv1? 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 (ofthe 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.
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