P1: PCX Printer: Yet To Come
9780521704632c06 CUFX213A/Peck 9780521618168 December 27, 2007 14:11
Section IBasic principlesByusing a constant of proportionality (k) and observing that as time passes the
concentration falls but more importantly the rate of change also falls, that is, the
constant of proportionality must be negative, we can write:
dC/dt=−kC.
This describes a first-order differential equation, as it is dependent on C raised to
the power one. If we now want an expression for C, we would need to integrate this
expression. We saw earlier that this describes an exponential relationship. It can be
shown that dC/dt=−kC when integrated, gives the expression:
C=C 0 e−kt.
Togive the exact equation we use the condition that when t=0, C=C 0 .Note that
azero-order differential equation is dependent on C raised to the power zero, which
is just one. A zero-order differential equation can therefore be written:
dC/dt=−k.
This tells us that the gradient is constant, which is true only for a straight line.Integration
Integration can be thought of as a way in which we can find the area under a curve
(AUC) if we know the equation that has been used to draw that curve. To find an area
weneed to define the starting and ending points of interest on the x-axis (Figure6.7b).
Inpharmacokinetics the area we are interested in is that under the concentration,
C(equivalent to the y-axis), against time, t (equivalent to the x-axis), curve. Usually
wewant to know the entire area under the curve, which starts at time t=0 and runs
to infinity; occasionally we choose other limits – for example between t=0 and
t=t 1 / 2 (where t 1 / 2 is the half-life). Although knowledge ofhowto integrate functions
is not required, it is useful to know some important integrals related to pharmaco-
kinetics.
Integration is indicated by the symbol∫
with the limiting values written above and
below it. If no limits are given it is assumed that integration is taking place over the
entire range possible. After the symbol for integration we write the function that is
to be integrated together with an indication of the axis along which the limits have
been given. For our concentration/time curve we integrate over time (rather than
concentration) so we are integratingwith respect totime, which is indicated using dt,
just as for differentiation. So for a simple concentration against time curve modelled
using one compartment we can write:AUC=∫
C 0 e−ktdt.This has omitted the limits, which are usually t=0 and t=infinity. Effectively we
find the expression that gives the area under the curve, then evaluate it for the lower
limit (t=0) and subtract this from the value for the upper limit (t=infinity); any
constants will cancel out. We do not need to know how to do this particular integral,