Pharmacology for Anaesthesia and Intensive Care

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6 Mathematics and pharmacokinetics

Concentration

C 0

time

A

Css

B

0

Colony

size

time

(a)

(b)

Figure 6.2.The exponential function.(a) Negative exponential function. Curve A is a simple

wash-out curve for drug elimination, for example after a single bolus dose; the equation is C=

C 0 e−kt,the asymptote is zero and the starting point on the concentration-axis is defined as C 0.

Curve B is a simple wash-in curve such as is seen when a constant rate infusion is used. The

starting point on both axes is now zero and the asymptote is the concentration at steady-state

Css.The equation is Css(1−e−kt). In both cases the rate constant for elimination is k. (b)

Positive exponential function. This represents the exponential growth in a bacterial colony

starting from a single organism. This organism divided; the two resultant organisms both

divide and so on. There is a regular doubling of the number of bacteria so the equation is N=

2 t/d,where N is the number of organisms at time t and d is the time between consecutive cell

divisions.

achieve steady-state. Because the time constant is longer than the half-life, it will

take fewer time constants than half-lives to reach an approximate steady-state. After

one time constant it will have fallen by 63.21%, and an approximate steady-state

value is reached in three time constants.

The characteristic of an exponential relationship is that therateat which the

dependent variable changes is dependent on thevalueof that variable. We shall

see later that this implies that when an exponential function is differentiated, the