Pharmacology for Anaesthesia and Intensive Care

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6 Mathematics and pharmacokinetics

although we can note that it simplifies to:

AUC=C 0 /k.

Wementioned in the previous section that integration can be thought of as the

opposite of differentiation, but we need a little more information before integrating

adifferential equation back to the original relationship between variables. The infor-

mation required is usually the initial conditions. For example, taking the differential

equation we met previously:

dC/dt=−k·C.

Wecan rearrange this and write:

(1/C)·dC=−k·dt.

Ifwethen integrate both sides, knowing that the integral of 1/x=ln(x) and the

integral of the constant k is kx+c, where c is a different constant, we end up with an

expression:

ln(C)=−kt+c.

When t=0, C=C 0 so putting in these initial conditions we find the value for c:

ln(C 0 )=c.

Wecan now substitute for c and write:

ln(C)=−kt+ln(C 0 ).

This is the linear relationship we have met before. We can now simplify this to get a

relationship that does not involve logarithms; first re-arrange:

ln(C)−ln(C 0 )=−kt.

Wesaw in a previous section that subtracting the logarithm of two numbers is the

same as dividing one by the other, so we can write:

ln(C/C 0 )=−kt.

Wecan now convert back to the exponent form:

C/C 0 =e−kt.

Re-arranging this now gives the familiar equation for the one-compartmental

model:

C=C 0 e−kt.

Pharmacokinetic models

Modelling involves fitting a mathematical equation to experimental observations

of plasma concentration following drug administration to a group of volunteers or

patients. These models can then be used to predict plasma concentration under

avariety of conditions and because for many drugs there is a close relationship