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6 Mathematics and pharmacokineticsyx1234(^24)
y = 0.5x
y = 0.5x +1
y = 0.5x +2
y = 0.5x +3
y = 0.5x +4
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Figure 6.8.Afamily of straight lines with the same gradient.This family of straight lines,
y=0.5x+c, all have a gradient of 0.5; they differ only in their position with respect to the
y-axis because they have different values for the constant c. Their gradients are the same so
when we differentiate these equations they all differentiate to 0.5, the gradient of the line.
Thus dy/dx=0.5 for this family of curves.
with respect to the y-axis, will have the same differential equation. This is shown in
Figure6.8for a family of simple linear equations where y=mx+c each has the
same gradient at any given time, so each will differentiate to the same expression – a
constant in this case. This is because the value for c in each of these functions does
not change as x changes, so when that part is differentiated it becomes zero. The idea
that a family of curves differentiate to the same expression is important because the
reverse of differentiation is integration – finding the area under the curve. In the case
of integration the area under the curve clearly depends on the position of the graph
with respect to the y-axis. If we have an expression for the rate at which y changes
as x changes then we cannot give a single answer for reversing the process: we do
not know the value for the constant c. It is therefore important to know at least one
point on the curve – often the initial conditions when x=0.
Inpharmacokinetics, this gradient represents the rate at which plasma concen-
tration is changing at a particular point in time. After a bolus dose of drug, we find
that the rate at which plasma concentration changes falls as time passes; the rate of
decline is dependent on the concentration itself. Differentiation defines how con-
centration changes with time and is indicated by:
dC/dt.
This can then be read as ‘the rate of change of concentration with respect to time’.
Inasimple model we know that this is related to concentration itself. This can be
written:
dC/dt∝C.