Principles and Practice of Pharmaceutical Medicine

one-compartment model with bolus input of dose
(D) is:

Ae¼

kleD

ke0
K

½e
Kt
e
ke0tŠ

whereKis the elimination rate constant. The con-
centration of drug in the effect compartment,Ce,is
obtained by dividingAeby the effect compartment
volume,Ve:

Ce¼

kleD

Veðke0
KÞ

½e
Kt
eke0tŠ

At equilibrium, the rates of drug transfer between
the central and effect compartments are equal:

k1eA¼ke0Ae

k1eVcC¼ke0VeCe

If the partition coefficient,Kp, equalsCe/Cat equi-
librium (steady state), then we can rearrange the
above equation:

Ve¼

k1eV 1

Kpke0

Substituting for Ve in the above equation
(i.e.kle¼ke0) yields:

Ce¼

ke0DKp

V 1 ðke0
KÞ

½e
Kt
e
ke0tŠ

At equilibrium,Cwill be equal toCe/Kpby defini-
tion, and thus:

Ce¼

ke0D

V 1 ðke0
KÞ

½e
Kt
e
ke0tŠ

This is how the link-model relates the kinetics in
plasma to the kinetics of drug in the effect compart-
ment. When used together with theEmaxmodel for
estimation of the maximal drug-induced effect, the
concentration at half-maximal effect (apparent
EC 50 ) and the rate constant of the disappearance
of the effect (ke0):

E¼

EmaxCen

ECn 50 þCen

Computer fitting of the equations to the effect data

and estimation of the rate constant for the disap-

pearance of the effect,ke0,EC 50 andEmaxfollows,

assuming the sigmoidicity factor (n) to be equal to

unity.

At steady state,Ceis directly proportional to the

plasma concentration (C), asCe¼KpC. Conse-

quently, the potency (EC 50 ) obtained by regressing

the last two equations represents the steady-state

plasma concentration producing 50% ofEmax.

Note that the effect equilibration rate constant

(ke0) may beviewed as a first-order distribution rate

constant. It can also be thought of in terms of the

rate of presentation of a drug to a specific tissue,

determined by, for example, tissue perfusion rate,

apparent volume of the tissue and eventual diffu-

sion into the tissue. The results of the data fitting in

this exercise with the analgesic areEmax4.5; EC 50

0.61 ngml
^1 andke00.07 h
^1.

Effect compartment or link models are limited

by their applicability to situations in which the

equilibrium between plasma and response is due

to distributional phenomena. In reality, there is

often a delay between occurrence of maximum

drug concentration in the effect compartment and

maximum intensity of effect caused by slow devel-

opment of the effect rather than slow distribution to

the site of action. In this situation, indirect or

‘physiological substance’ models are more appro-

priate (Daynekaet al., 1993; Levy, 1994; Sharma

and Jusko, 1997). Warfarin is a good example,

where this drug inhibits the prothrombin complex

activity (PCA) (inhibition of production of effect).

This is illustrated by the following example, which

relates changes inS-warfarin concentration to the

observed PCA. The dose was intravenous. The

change in PCA is shown in Figure 8.6. The plasma

kinetics of (S)-warfarin were described by the fol-

lowing mono-exponential expression:

CwðsÞ¼ 1 :05 e
^0 :0228 t

and the equation for the turnover of clotting factor

[P] was:

dP

dt

¼kd

P 0

1 þ

CwðsÞ

IC50s

n
P

8.3 PHARMACOKINETIC/PHARMACODYNAMIC MODELS 93