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 concentration 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 the 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 predetermined plasma level and hence a desired therapeutic effect.
It should 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 a plasma or extracellular 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.
Compartmental models are mathematical equations used to predict plasma concentrations of drugs based on experimental observations. Mathematical functions of importance in the understanding of these models are linear, logarithmic, and exponential functions. Predicted behavior 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 functions, 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.
The linear function
The equation for a straight line with a gradient of m is given by: y = MX + c. The constant, c, tells us the intercept on the y-axis and allows us to position the straight line in relation to the axes. If we knew only the gradient, we couldn’t draw the line; we need at least one point to fix the exact place to draw it. Thus we need two pieces of information to draw a particular straight line: its gradient and its intercept on the y-axis.
If m is negative the slope of the line is downward, if m is positive then the slope is upward (Figure 6.1). In a later section, we meet differentiation; for a straight line the differential equation simply gives a constant, the value of the gradient, m. We meet a straight line in pharmacokinetics when taking a semi-logarithmic plot of the concentration-time curve for a simple one-compartmental model.
The expression may look more complicated than the one above ln(C) = ln(C0) − kt. In this case, we think of the y-axis as being ln(C) and the x-axis as being t. If we then compare this expression with y = MX + c it should be clear that −k is like m and represents the gradient and ln(C0) represents the intercept on the y-axis
The exponential function
An exponential function takes the form: y = Anax. In this relationship n is the base and x the exponent; A and a are constants. Although it is possible to use any base for our exponential function, the natural number e is chosen for its mathematical properties. The exponential function, y = ex, is the only function that integrates and differentiates to itself, making manipulation of relationships involving exponentials much easier than if another base were chosen.
The number e is irrational, it cannot be expressed as a fraction, and takes the value 2.716 . . . where there is an infinite number of digits following the decimal point. Exponentials are positive if the rate of change of y increases or negative if the rate of change decreases as x increases; in the example above ‘a’ is positive for a positive exponential and negative for a negative exponential.
Bacterial cell growth is an example of a positive exponential relationship between the number of bacteria and time; compound interest is a further example relating to the growth of an investment with time. For a negative exponential, we write y = Be−bx.