Pharmacokinetics and RC Circuit Concepts

Abstract

M ost introductory physics courses include a chapter on RC circuits in which the differential equations for the charging and discharging of a capacitor are derived. A number of papers in this journal^'^ describe lab experiments dealing with the measurement of different parameters in such RC circuits. In this contribution, we report on a lab experiment we developed for students majoring in pharmacy, using RC circuits to simulate a pharmacokinetic process.We see two benefits specific for this project. The first is to lift the RC circuit above its usual electrostatic context by placing it amidst other phenomena, seemingly completely different but described by essentially the same differential equation and so exhibiting similar behavior. The use of a pharmaceutical model to illustrate this adds to the motivation of the students. The second pro is that the lab provides excellent training in the use of exponential functions. This is especially welcome in a curriculum where mathematics is not a first priority but in which exponential behavior will be met frequently and in many domains.Pharmacokinetics^ deals (among other items) with the speed of deactivation or elimination from the body of a medicine after its administration.We present here a setup to mimic a very basic pharmacokinetic model. In this model, a patient is administered a fixed dose íí of a certain drug at regular intervals each of duration ÍQ, e. g., once a day. Administration takes place at the same moment every day. We model the body as one single compartment and we assume that all of the drug is absorbed and spread in the body instantaneously at the moment of administration. After intake, the drug will gradually be eliminated from the body. We do not attempt to explain the (sometimes complicated and maybe multiple) mechanisms of this. Instead, the most simple and most basic model in pharmacokinetics is followed, in which the drug elimination rate is proportional to the amount of drug present in the body. This model can be described by the differential equation dx -= -ax, dt (1) in which x{t) is the amount of agent in the body at time t and Q is a constant. This process can be seen as follows (Fig. 1):• Before the first administration, x = Q;• At time Í = fi, a first dose is administered, so Xj = x{ti) = d;• Between ij and f2, the drug is eliminated (exponentially) and the amount decreases from Xi to Xj; M Xi' Xiamountd ayl day2 time Fig. 1. Schematic time evolution of agent in the body.• At time Í = f2, again a dose is administered, and so X3 = x(f2 + ÍL) =X2 + à, where ÍL is the time required for administration and spreading of the drug. We will assume ij« to.• Again, during the second day (between f2 and Í3), x decays exponentially, with the same time constant, to the value X4 ;• At time t = t2,,a. new dose is injected, so x¡= x^+ d;• This sequence can be repeated as long as is necessary.We want to investigate the long time evolution of the amount of drug in the body, i.e., the amount after "many days." RC circuit simulationTo simulate this process, a...