### A Mathematical model of drug therapy

#### Abstract

A problem discussed by Burghes, Huntley and McDonald (1, pages 124-129) relating to Drug Therapy is shown to be a very critical case of a generalformulation of the problem. It is also shown that their prescription to give adose of 2S0mg every 4 hours is alright if it is confined to about 24 hours, but,if it is danefor a longer period, the patient will suffer due to an overdose of thedrug. The case is critical because their upper tolerance level of 20 mgjlitreis exactly 4 times their lower tolerance level of 5 mgllitre.

The problem discussed is the following. A patient whose mass is 50kghas to be given a certain drug in fixed dosages at regular intervals. If the drugconcentration in the blood is less than Smgflitre, it is ineffective whereas, if itexceeds 2Omgflitre, it is likely to be toxic. They assume that, in between administrationsof the drug, the rate at any time at which the concentration of thedrug decreases with time is proportional to the concentration of the drug atthat time. Thus, if the initial concentration of the drug is Co, then, after timet before administering the next dose, the concentration c(t)=coe-kt. Theirvalue of k=0.17 hour-I.

Their prescription to give 2SOmgevery 4 hours amounts to an initial concentrationof 1Omg/Iitre, Taking Co = 10, the following table can be constructedfor the maximum and minimum concentrations, corresponding to theconcentrations just after and just before a dose is given.

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