Compartment models and drug metabolism Compartment models

Chapter , Problem 53

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Compartment models and drug metabolism Compartment models are used to simulate manufacturing processes, ecological systems, and the assimilation of drugs by the body (pharmacokinetics). The figure shows a two-compartment model for a drug assimilation problem. Compartment 1 corresponds to the blood volume, which has an input of the drug at a specified rate f. Compartment 2 cor- responds to the kidneys and has an outflow of the drug determined by the rate constant k3. In addition, there is exchange of the drug between the compartments determined by the rate constants k1 and k2 (see figure). Let x11t2 and x21t2 be the amount of drug in each compartment t time units after the dosing begins. The equations that describe the amount of drug in each compartment are x1_ = -k1x1 + k2x2 + f 1t2 outflow inflow input from 1 from 2 from outside x2_ = k1x1 - k2x2 - k3x2 inflow outflow outflow from 1 from 2 to outside f(t) k3 k1 2 k2 1 a. Advanced methods can be used to solve two first-order differential equations. Alternatively, we can convert these two equations to an equivalent second-order equation that we can solve. Differentiate the first equation with respect to t and use the second equation to eliminate x2 and x2_. Show that the resulting equation is x_ = -1k1 + k2 + k32x_ - k1k3x + 1k2 + k32f 1t2 + f_1t2, where we have replaced x1 by x. b. Show that the initial conditions x1102 = A, x2102 = 0 become x102 = A, x_102 = -k1x102 + f 102, where again we have replaced x1 by x.