Pharmacology for Anaesthesia and Intensive Care

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Mathematics and pharmacokinetics

Pharmacokinetics is the study of the way in which the body handles administered

drugs. The use of mathematical models allows us to predict how plasma concentra-

tion changes with time when the dose and interval between doses are changed, or

when infusions of a drug are used. Because there is an association between plasma

concentration of a drug and its pharmacodynamic effect, models allow us to predict

the extent and duration of clinical effects. Mathematical models may therefore be

used to program computers to deliver a variable rate infusion to achieve a predeter-

mined plasma level and hence a desired therapeutic effect.

Itshould be remembered that these pharmacokinetic models make a number of

assumptions. Compartmental models make general assumptions based on virtual

volumes without attempting to model ‘real-life’ volumes such as plasma or extra-

cellular fluid volumes. Therefore, although convenient and useful to associate the

virtual compartments with various tissue groups such as ‘well perfused’ or ‘poorly

perfused’, this remains only an approximation of the physiological state.

Mathematics

Compartmental models are mathematical equations used to predict plasma con-

centrations of drugs based on experimental observations. Mathematical functions

of importance in the understanding of these models are linear, logarithmic and expo-

nential functions. Predicted behaviour and calculation of the parameters that define

the model require manipulation of exponential functions, the logarithmic function

and calculus. The following sections will cover all these concepts, starting with func-

tions, particularly the exponential function, logarithms and finally calculus. In each

section we will relate these to their use in pharmacokinetics, particularly the simple

one-compartment model.

Functions

Afunction defines a unique value for y given a value for x. We write this as:

y=f(x).

Forpharmacokinetics, we are interested in drug concentration (C) as a function of

time (t). We can write:

C=f(t).