Pharmacology for Anaesthesia and Intensive Care

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6 Mathematics and pharmacokinetics

For any particular model k and Vd are constant, so clearance also must be a constant.

Because clearance is the ratio of volume of distribution and time constant it is pos-

sible for drugs with very different values for Vd andτto have the same clearance as

long as the ratio of these two parameters is the same. So far we have considered the

model in general, but we want to use the model to predict how the plasma concen-

tration changes for a particular dose of drug. If a single bolus dose, X mg, of drug is

given the concentration at time zero is defined as C 0 ;C 0 is X/Vd. If we then follow the

amount of drug remaining in the body (Xt)italso declines in a negative exponential

manner towards zero, as C is Xt/Vd. By substituting for C and C 0 this gives Xt=Xe−kt

so the rate at which drug is eliminated (in mg per minute) is kXt:

dXt/dt=−kXt.

Weknow that clearance is the product of k and Vd, so k is the ratio of clearance and

Vd. If we put this into the expression above we can see that the rate at which drug is

being eliminated is:

kXt=(Cl/Vd)Xt

=Cl(Xt/Vd)

=Cl×C

So,ataplasma concentration C, the drug is eliminated at a rate of C×Cl mg.min−^1.

The time constant is often defined as the time it would have taken plasma concen-

tration to fall to zero if the original rate of elimination had continued. But the time

constantτis the inverse of the rate constant for elimination so it also represents the

time it takes for the plasma concentration to fall by a factor of e (Figure6.11). Since

Concentration

time

C

C/e

C 0

C/2

t

ττ

(^12)
Figure 6.11.Time constant.The dashed line shows the tangent to the concentration–time
curve at C 0 ;ifthis rate of decline had continued then the time it would have taken the
concentration to reach zero is the time constant,τ. The time taken for the plasma
concentration to fall from C to C/e, i.e. to 37% of C, is also one time constant.