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6 Mathematics and pharmacokineticscompartment. The rate constant for elimination from the central compartment is
now referred to as k 10 .Drugisgiven into the central compartment and there is a
rapid initial decline in concentration due to distribution into the peripheral com-
partment together with a slower decline, the terminal elimination, which is due both
to elimination from the body and re-distribution of drug to plasma from the second
compartment.
Consider a dose X mg of drug given into the central compartment with X 1 and
X 2 representing the amount of drug in the central and peripheral compartments
respectively after time t. We know that movement of drug is an exponential process
depending on the rate constant and the amount of drug present so the rate at which
the amount of drug in the central compartment changes with time depends on
three processes: (1) removal of drug from the central compartment by metabolism
and excretion; (2) drug distribution to the second compartment, both of which are
dependent on X 1 ; and (3) re-distribution from the second compartment, which is
dependent on X 2 and the rate constant for transfer, k 21 .Wecan therefore write a
differential equation for the rate of change of the amount of drug in the central
compartment:
dX 1 /dt=−k 10 X 1 −k 12 X 1 +k 21 X 2.
This is much more complicated than the simple single compartment model and
requires a special form of integral calculus to solve (Laplace transforms). It can be
shown that using the result from integral calculus for the amount of drug, and divid-
ing X 1 by V 1 to give the concentration in the central compartment, C, we get the
equation:
C=A·e−αt+B·e−βt.
The semi-logarithmic plot of ln(C) against time is the sum of two straight lines rep-
resenting the two exponential processes in the relationship above (Figure6.12). The
intercepts of these two straight lines on the ln(C) axis (i.e. when t=0) allow the
constants A and B to be found and C 0 is the sum of A and B. The rate constants,
αandβare found from the gradients of these lines and the reciprocals of these
rate constants give the time constantsταandτβ, which are related to the half-lives
t1/2αand t1/2β,respectively (see above for relationship between time constant and
half-life). Neither of these rate constants equates to any specific rate constant in the
model, but each is a complex combination of all three. The steepness of the initial
decline is determined by the ratio k 12 /k 21 .Ifthe ratio k 12 /k 21 is high then the initial
phase will be very steep, for a lower ratio the initial phase is much less steep. For
example, after a bolus dose of fentanyl, where the ratio of k 12 /k 21 is about 4:1; the
plasma concentration falls very rapidly. For propofol the ratio is close to 2:1 and the
initial phase is less steep than for fentanyl. The absolute values for k 12 and k 21 deter-
mine the relative contribution of distribution to the plasma-concentration curve;
the higher their value, the faster distribution occurs and the smaller the contribu-
tion so that for very high distribution rates the closer the model approximates a single