Integrated Functions for Four Basic Models of Indirect
Pharmacodynamic Response
WOJCIECH KRZYZANSKI
AND
WILLIAM J. JUSKO*
Contribution from the Department of Pharmaceutics, School of Pharmacy, State University at Buffalo, Buffalo, New York 14260.
Received April 24, 1997.
Final revised manuscript received August 18, 1997.X
Abstract 0 The integrated solutions (ABEC, area between baseline
and effect curve) of four basic models of indirect pharmacodynamic
responses are developed. These models assume that drug can inhibit
or stimulate the production or loss of the response variable. For two
models (I and III) with monoexponential drug disposition, explicit
formulas for the ABEC were obtained, where ABEC is a function of
ln (1 + (D/V)/IC50) or ln (1 + (D/V)/SC50) where D ) dose, V )
volume, and IC50 or SC50 ) 50% effective concentration. Two other
models (II and IV) were treated asymptotically with respect to small
and large doses. Approximate formulas [e.g., ABEC ) constant(1)
‚ ln (1 + (D/V)/IC50) + constant (2)] were derived and the asymptotic
behavior of the ABEC was established. In addition, simulations were
performed to assess the effects of drug absorption rates and
polyexponential disposition on ABEC values. These models show
how pharmacokinetic and pharmacodynamic factors jointly determine
the net response to a single dose of drug.
Introduction
Direct response models (an Emax or sigmoid Emax model)
are usually applied to characterize the relationship between drug concentrations and pharmacological effects
when the pharmacological effects are functions of drug
concentration (i.e. one value of biophase drug concentration
results in exactly one value of drug effect independently
of the time course of concentration). Many drug effects are
indirect in nature. Such drug effects cannot be predicted
knowing the drug concentrations at a moment in time.1-3
Those drug effects depend on the production (kin) and
disposition (kout) of the response factor as well as the time
course of drug concentrations. Four basic models were
proposed to describe the pharmacodynamics of drugs with
mechanisms producing indirect responses.1 A summary
parameter used to characterize the overall effect of drug
is the area between the baseline and the effect curve
(ABEC). This parameter can be considered over a finite
time interval (0, t1)4,5 or as a total net effect (t1 ) ∞).
The purpose of this report is to analyze the dependence
of the integrated pharmacodynamical parameter, ABEC,
for the four basic indirect response models on the drug dose
D in the case of monoexponential drug disposition:
C(t) )
D -kelt
e
V
(1)
where V is the compartment volume and kel is the elimination rate constant. Simulations were also performed to
assess the role of biphasic (drug absorption) and polyexponential disposition on ABEC values.
Theoretical Section
An indirect mechanism produces a measured response
R to a drug (Figure 1). The rate of change of the response
is
dR
) kin(1 + H1(t)) - kout(1 + H2(t))R
dt
Parameter kin represents the zero-order constant for production of the response, and kout defines the first-order rate
constant causing loss of the response. Functions H1 and
H2 specify the type of mechanism, which may be either
stimulation or inhibition of the response. Four basic
models are considered1:
Inhibition of kin: H1(t) ) I(t) and H2(t) ) 0,
Model I (3)
Inhibition of kout: H1(t) ) 0 and H2(t) ) I(t),
Model II (4)
Stimulation of kin: H1(t) ) S(t) and H2(t) ) 0,
Model III (5)
Stimulation of kout: H1(t) ) 0 and H2(t) ) S(t),
Model IV (6)
where I(t) and S(t) are functions responsible for inhibition
and simulation according to:
I(t) ) -
ImaxC(t)
IC50 + C(t)
(7)
and
S(t) )
SmaxC(t)
SC50 + C(t)
(8)
where C(t) is the plasma concentration of the drug, 0 <
Imax e 1 and Smax > 0 are parameters related to maximum
inhibition and stimulation, and IC50 and SC50 are the drug
concentrations which produce 50% of maximum inhibition
and stimulation.
The initial condition is the baseline response:
* To whom correspondence should be addressed. Phone: 716-6452855, ext 225; Fax: 716-645-3693; e-mail: wjjusko@acsu.buffalo.edu.
X Abstract published in Advance ACS Abstracts, November 15, 1997.
© 1998, American Chemical Society and
American Pharmaceutical Association
(2)
S0022-3549(97)00168-8 CCC: $15.00
Published on Web 01/02/1998
R(0) ) R0 )
kin
kout
(9)
Journal of Pharmaceutical Sciences / 67
Vol. 87, No. 1, January 1998
Parameter values of dose (as indicated), V of 90 L, kel of 0.3 h-1,
IC50 or SC50 of 100 ng/mL, Imax or Smax of 1.0 (unless indicated
otherwise), kin of 9 unit/h, and kout of 0.3 h-1 were employed in
examining the effect of dose on ABEC values.
Further simulations were performed to assess the effects of
polyexponential rather then monoexponential disposition on ABEC.
In this case, the pharmacokinetic function employed was:
Cp ) C1e-λ1t + C2e-λ2t
(16)
In eq 16, parameter values were assigned so that C0 and AUC
values were constant with dose. This situation required application of a two-compartment model where central volume (Vc) and
clearance (CL) maintained constant, but tissue volume (VT) was
allowed to vary.
The effect of drug absorption rate (ka) on ABEC values was
examined using the Bateman function:
Cp )
kaD0
(e-kelt - e-kat)
(17)
V(ka - kel)
where ka values were assigned to produce an absorption t1/2
ranging from 0 to 7 h.
Results
For Models I and III, the function H2 ≡ 0. Equation 13
implies that:
Figure 1sFour basic indirect response models representing processes that
inhibit (Models I and II) or stimulate (Models III and IV) the factors controlling
drug response.
A pharmacological effect E is defined as the change of
the response R with respect to the baseline response R0:
E ) |R - R0|
(10)
From eq 2 it follows that:
dE
) kin|H1(t) - H2(t)| - kout(1 + H2(t))E
dt
(11)
and eq 3 implies:
E(0) ) 0
(12)
Because of the nonlinear form of H1(t) and H2(t), the
solution of eq 11 cannot be represented in terms of
elementary functions. A solution of eqs 11 and 12 is of the
form:
E(t) ) kin
∫ |H (τ) - H (τ)|e
t
µ(τ)-µ(t)
1
0
2
dτ
µ(t) ) kout
∫
0
(1 + H2(τ)) dτ
(14)
The area between effect curve (ABEC) parameter is defined
as:
ABEC )
∫
∞
0
E(t) dt
(15)
For Models I and III, an explicit formula for ABEC is
derived. For Models II and IV the asymptotic behavior of
ABEC as D f 0 and D f ∞ is presented.
Methods
Integration of eq 11 was carried out using principles of
asymptotic expansion theory.6 Computer similations were performed by the Runge-Kutta method of numerical integration.7
68 / Journal of Pharmaceutical Sciences
Vol. 87, No. 1, January 1998
∫ |H (t)| dt
∞
0
(18)
1
If the concentration function C(t) is of the form (1), then
the integral in eq 18 can be evaluated and has the following
exact solutions:
{
(
(
)
Imax
D/V
(19)
ln 1 +
, Model I
kel
IC50
ABEC )
Smax
D/V
ln 1 +
, Model III (20)
R0
kel
SC50
R0
)
For Models II and IV the function H1 ≡ 0. Hence, from eq
13:
ABEC ) kin
(13)
where
t
ABEC ) R0
∫ ∫ |H (τ)|e
∞
t
0
µ(τ)-µ(t)
2
dτ dt
(21)
Let parameters Imax, Smax, kel, kin, and kout be fixed. For
Model II, if Imax ) 1, then:
ABEC )
(( ) )
R0 D/V
D/V
+O
kel IC50
IC50
2
as
D/V
f 0 (22)
IC50
where the symbol O(‚) means that the relative error
between the exact and the approximate values is proportional to the expression between the parentheses (for more
detailed definition see Olver6). If 0 < Imax < 1, then:
ABEC )
(
)
R0 Imax
D/V
ln 1 + (1 - Imax)
+
kel 1 - Imax
IC50
D/V
D/V 2
O
as
f 0 (23)
IC50
IC50
(( ) )
For Model IV:
ABEC )
(
)
R0 Smax
D/V
ln 1 + (1 + Smax)
+
kel 1 + Smax
SC50
D/V
D/V 2
O
as
f 0 (24)
SC50
SC50
(( ) )
Proof of eqs 22-24 is presented in Appendix A.
Similar approximate formulas can be obtained for large
doses. For Model II,
{
ABEC )
(
)
( )
Imax
D/V
1
D/V
R0
ln 1 +
+A+
,
kel 1 - Imax
IC50
IC50
if Imax * 1
(25)
D/V
1 kout 2
D/V
R0
ln 1 +
+A+
,
kel 2kel
IC50
IC50
if Imax ) 1
(26)
(
)
( )
where, if Imax * 1, the value of the constant A is:
A)-
2
R0Imax
kout
k2el(1
∫∫
)
∞
0
- Imax
y (k /k )-1
out el
t
0
(1 +
t)-Imax(kout/kel)-1 y-(kout/kel) (1 + y)Imax(kout/kel)-1 dt dy (27)
If Imax ) 1, then:
A)-
R0kout
2k2el
2
∫
∞
0
∞
0
and
(1 + y)-2 ln2 y dy -
( ) ∫∫
R0 kout
kel kel
y (k /k )-1
t out el
0
-(kout/kel)-1
(ln t)(1 + t)
y
-(kout/kel)
The error term is:
( )
{
O
(( ) )
D/V
IC50
D/V
f ∞,
IC50
-1
as
if Imax * 1 and Imax * 1 -
O
(( ) ) ( )
D/V
IC50
-1
ln
D/V
IC50
D/V
f ∞,
IC50
otherwise
kel
(29)
kout
as
(30)
Appendix B provides the rationale for the magnitude of
the error terms. For Model IV:
(
)
( )
Smax
D/V
1
D/V
ABEC ) R0
ln 1 +
+A+
kel 1 + Smax
SC50
SC50
(31)
where
A)
2
kout
R0Smax
k2el(1 + Smax)
∫ ∫
∞
0
y (k /k )-1
out el
0
{
t
(1 +
t)Smax(kout/kel)-1 y-(kout/kel) (1 + y)-Smax(kout/kel)-1 dt dy (32)
(( ) )
D/V
f ∞,
SC50
kel
(33)
if Smax * 1 k
D/V
out
)
D/V
D/V -1
D/V
SC50
O
ln
as
f ∞,
IC50
SC50
SC50
kel
(34)
if Smax ) 1 kout
O
(1 + y)(kout/kel)-1 dt dy (28)
D/V
)
IC50
Figure 2sABEC versus D for the four models. Parameter values were
assigned as described in the Methods Section. For Models I and III, the
curve indicates the exact ABEC based on eqs 19 and 20. For Models II and
IV, the dashed curves show ABECapprox (eqs 25, 26, and 31) and the solid
curves reflect the exact ABEC obtained by numerical integration (eq 10).
( )
D/V
SC50
-1
as
(( ) ( ))
The proof of eqs 25, 26, and 31 is presented in Appendix
B. These equations provide information about behavior of
ABEC for large values of the ratios (D/V)/IC50 and (D/V)/
SC50 for Models II and III. The error term is then small,
so one can use eqs 25, 26, and 31 as approximate values of
ABEC.
For Models I and III, which account for inhibition or
stimulation of kin (Figure 1), the ABEC has exact solutions
as indicated in eqs 19 and 20. These equations show that
a plot of ABEC versus ln (1 + (D/V)/IC50) or ln (1 + (D/V)/
SC50) should be linear with a slope of R0 Imax/kel or R0 Smax/
kel. At high doses, ABEC is simply proportional to ln D2.
Equations 22-26 and 31 served practically as approximations of ABEC. If D/V , IC50 (or D/V , SC50), then
eqs 22 and 23 (or 24 apply; if D/V . IC50 (D/V . SC50),
then eqs 25 and 26 (or 31) are valid. Figure 2 shows the
modest differences between ABECapprox and ABEC for
different dose levels for Models II and IV. The maximum
errors found were 9% for Model II and <1% for Model IV.
Figure 3 shows the pharmacokinetic profiles obtained
by converting to a polyexponential function or having firstorder input to a one-compartment model. Families of
curves were generated for one dose level and scaled to other
doses with an assumption of linear kinetics. The effect of
increasing VT values on the ABEC values for the four
Journal of Pharmaceutical Sciences / 69
Vol. 87, No. 1, January 1998
Figure 5sEffects of varying the drug absorption rate constant (see Figure 3)
on values of ABEC for the four indirect response models.
Figure 3sPharmacokinetic profiles showing polyexponential disposition (upper)
and drug absorption rates (lower) used for simulating changes in ABEC values.
Parameter values were varied as indicated.
increase in ABEC, especially at higher doses. This increase
is because drug concentrations are being held above the
IC50 or SC50 for longer periods, causing an enhancement
in response. In general, however, there remain similar
relationships between ABEC and dose for each set of
pharmacokinetic parameters (Vc, VT, CLd, and CL are
constant). At larger doses ABEC is proportional to ln dose.
Figure 5 shows the effects of absorption rate on ABEC
values for each model. Extending the absorption phase
with smaller ka values produces an increase in ABEC
values at larger doses. Again, this result is due to lengthier
periods of drug concentrations greater than the IC50 or
SC50. A nearly linear relationship is maintained between
ABEC and ln D for larger doses with each set of pharmacokinetic parameters (V, kel, ka).
Discussion
Figure 4sEffects of polyexponential disposition (see Figure 3) on ABEC values
for the four basic indirect response models. The curves corresponding to VT
) 0 show results for monoexponential disposition.
indirect response models is shown in Figure 4. Modest
polyexponential curvature of the plasma concentration
versus time curve produces slight changes in ABEC, but
eventually the prolongation in t1/2 results in a marked
70 / Journal of Pharmaceutical Sciences
Vol. 87, No. 1, January 1998
The ABEC summarizes the influence of primary pharmacokinetic (kel, V) and pharmacodynamic (R0, Imax or Smax,
IC50, or SC50) variables on the net pharmacologic response
for drugs with four basic indirect mechanisms of action.
This parameter is typically calculated from experimental
data and can be related to various doses of administered
drug.2,4
The ABEC parameter depends on the dose D through
the combination (D/V)/IC50 or (D/V)/SC50, which means that
the effective influence on ABEC is the ratio (D/V)/IC50 or
(D/V)/SC50, rather than the dose itself. However, at high
ratios of D/V to IC50 (or SC50), the initial drug concentration
(C0) or dose becomes the major determinant of ABEC.
Because C0 and AUC and D are correlated when clearance
is constant, ABEC will also be proportional to log AUC.
For Models II and IV, which account for inhibition or
stimulation of kout (Figure 1), the ABEC has approximate
solutions. If V and IC50 (or SC50) are fixed and D is varied,
then the error terms in eqs 22 and 23 (or 24) and 25 and
26 (or 31) can be considered as D f 0 and D f ∞,
respectively.
One can observe from eqs 19, 20, 26, and 31 that ABEC
is proportional to ln D as D f ∞, except for Model II when
Imax ) 1:
{
ABEC ∼
Imax
ln D, Model I
kel
Imax
1
ln D, Model II, Imax * 1
R0
kel 1 - Imax
Smax
ln D, Model III
R0
kel
Smax
1
ln D, Model IV
R0
kel 1 + Smax
R0
(35)
(36)
(37)
(38)
For Model II with Imax ) 1, eq 26 implies that ABEC is
proportional to ln2 D:
kout
ABEC ∼ R0 2 ln2 D as D f ∞
2kel
(39)
Numerical simulations were also carried out previously2
to demonstrate the relationships of ABEC to ln (D) for the
four models. The present equations confirm the pattern
found earlier and indicate the general net behavior of
indirect response models in the absence of specific assumptions about pharmacodynamic model parameters.
The ABEC for Model I was derived previously4 over the
time interval 0 to t1, where t1 is a specified time. Our
solution is simpler and reflects the total duration of the
response (time 0 to ∞). The present solution is obtained
from that found previously if t1 f ∞. It is interesting to
note that ABEC for Models I and III are essentially
identical to that for drug providing a direct response
according to: E ) Emax/(EC50 + C(t)), where C(t) is defined
in eq 1. Wagner8 derived a similar formula that contained
Emax in place of R0Imax or R0Smax. Thus, these formulae
have some generality in pharmacodynamics.
The exact solutions for ABEC for Models I and III and
the approximate solutions for Models II and IV could be
obtained in the case of simple monoexponential disposition
(eq 1) of the drug. The derivations for Model II and IV are
complicated because of the presence of the nonlinear H(t)
function attached to the kout parameter in eq 2 producing
the complex integrals shown in eqs B4 and B10. This
permits only approximate solutions under conditions of
small or large doses.
Identification of the exact or approximate values of
ABEC is of value in pharmacokinetic/pharmacodynamic
(PK/PD) modeling for both conceptual and practical reasons. Here-to-fore, the role of pharmacokinetic (V, kel) and
pharmacodynamic (Imax or Smax and IC50 or SC50) values in
controlling indirect responses required exploration by
simulation1,2 or via partially integrated solutions.9 This
requirement remains true in assessing the time-course of
responses, but the determinants of net response can now
be readily found in the ABEC equations. In Models I and
III, ABEC relates to R0, Imax or Smax, and kel in a direct
and linear fashion (eqs 19 and 20). The roles of D, V, and
IC50 and SC50 are slightly obscured by their nonlinear role
in the ABEC equation. Nevertheless, a basic tenet of
pharmacology that the ABEC is proportional to log D holds
true for indirect response models.
Unfortunately, the ABEC equations cannot be generalized to relate clearly to diverse pharmacokinetic models.
Further, such derivations pose difficulties in their even
greater complexity. Thus, the extension of these ABEC
concepts to polyexponential and biphasic plasma concen-
tration versus time profiles was done by simulations.
These simulations (Figures 3-5) show basic similarities
to the simpler pharmacokinetic situation and some interesting differences. All of the ABEC profiles maintain
shapes with a dose threshold and then a nearly proportional increase with log dose. However, at larger doses,
an extended duration of drug exposure by increasing the
terminal t1/2 (even with CL constant) or slowing drug
absorption produces greater net responses. The latter
would also apply in the case of drug infusions because the
biphasic profile is similar. Thus, duration of drug exposure
at values greater than IC50 or SC50 is a fifth determinant
of ABEC.
The present derivations also enhance the value of ABEC
in practical analysis of PK/PD data. For Models I and III,
it becomes possible to use ABEC values at two or more dose
levels to estimate Imax (or Smax) and IC50 (or SC50) by
regression analysis providing that kel and V are supplied
as secondary variables. The ABEC has been used in
simulations to express total drug effects2,10 and as a
comparator in examining changes in responses in studies
of disease effects11 and drug interactions.12,13
Often the ABEC is expressed as a ratio with normalization by baseline responses, (viz., ABEC/R0). The present
derivations validate this practice as a means of removal of
intersubject or treatment variation in R0. Another adjustment is factoring (ABEC/R0) × kel to reflect the role of Imax,
D/V, and IC50 in determining net response:
(
)
ABECkel
D/V
) Imaxln 1 +
R0
IC50
for Model I
(40)
This factoring may be helpful in drug interaction studies
where, if D/V is constant, the pharmacodynamic alterations
in ABEC can be isolated. On the other hand, calculating
ABEC/AUC at low doses does not clearly isolate and reflect
the pharmacodynamic parameters because of the complex
fashion in which D/V and kel control ABEC. At high doses,
however, ABEC/ln D is largely reflects the ratio of R0Imax/
kel or R0Smax/kel (eqs 35-38). These considerations indicate some of the advantages and cautions needed in
examining ABEC values for experimental data.
Appendix A
Proof of Equations 22-24 for Models II and IVsThe
integral ∫∞0 |H2(t)|/[1 + H2(t)] dt can be evaluated by substitution of u ) e-kelt. The results are the leading terms in
eqs 23-25. Because for some positive M > 0: 1 + H2 g
1/M, then
|∫
∞
0
H′2(t)
∫ |H (τ)|e
t
(1 + H2(t))2
µ(τ)-µ(t)
2
0
M2
|
dτ dt e
∫ |H′ (t)| ∫ |H (τ)| dτ dt
∞
0
t
2
2
0
(A1)
The signs of H2 and H′2 are fixed. This allows evaluation
of:
∫ |H′ (t)| ∫
∞
2
0
t
0
|H2(τ)| dτ dt )
∫
∞
0
H2(t)2 dt
(A2)
e-2kelt dt
(A3)
For Model II:
∫
∞
0
H2(t)2 dt e
( )
D/V 2 2
I
IC50 max
∫
∞
0
Journal of Pharmaceutical Sciences / 71
Vol. 87, No. 1, January 1998
An analogous inequality holds for Model IV. Hence, for
Model II:
∫
H′2(t)
∞
∫
(1 + H2(t))2
0
t
0
|H2(τ)|eµ(τ)-µ(t) dτ dt )
(( ) )
D/V
IC50
O
2
as
D/V
f 0 (A4)
IC50
The same statement is true for Model IV, which completes
the proof.
where satisfies the following equations:
{
(R,β,γ) )
1
O
as R f ∞, if β - γ > 0, β - γ * 1 (B8)
R
ln R
(B9)
O
as R f ∞, otherwise
R
()
( )
Let β - γ ) 0. The leading integral in eq B1 evaluated in
terms of R, β, and γ is equal to:
(21)1 +R R ln
2
Appendix B
Proof of Equations 25, 26, and 31 for Models II and
IVsThe proof is carried out for Model II because the case
for Model IV is analogous. Integration by parts transforms
eq 21 into:
ABEC ) R0
∫
R0
∫
1 + H2(t)
0
dt +
H′2(t)
∞
2
0
(1 + H2(t))
∫
t
0
|H2(τ)|eµ(τ)-µ(t) dτ dt (B1)
(21) ln
2
R-
∫
R
y β-1
t
0
(1 + t)-γ-1(1 +
y)γ-1y-β dt dy (B3)
If β - γ ) 0, then:
Imaxkout
-R0
γ
k2el
∫ ∫
R
0
y γ-1
t
(t + 1)-γ-1(ln t)y-γ(1 +
y)γ-1 dt dy (B4)
Let β - γ > 0. Because
∫
y β-1
t
0
(1 + t)-γ-1dt )
{
O(yβ-γ-1) as y f ∞, if β - γ + 1 (B5)
O(ln y) as y f ∞, if β - γ ) 1
(B6)
t
y β-1
R
0
0
∞
0
(1 + t)-γ-1(1 + y)γ-1y-β dt dy )
y β-1
0
(1 + y)-2 ln2 ydy +
(lny y)
(t + 1)-γ-1(ln t) dt ) O
as R f ∞ (B11)
(1 + t)-γ-1(1 + y)γ-1 y-β dt dy + (R,β,γ) (B7)
72 / Journal of Pharmaceutical Sciences
Vol. 87, No. 1, January 1998
as y f ∞
(B12)
one can transform the integral in eq B4 to the following
form:
∫ ∫ t (t + 1) (ln t)y (1 + y) dt dy )
∫ ∫ t (t + 1) (ln t)y (1 + y) dt dy +
R
0
y γ-1
-γ-1
γ-1
-γ
0
∞
y γ-1
-γ-1
γ-1
-γ
0
(lnRR)
O
as R f ∞ (B13)
Thus, the proof of eqs 25, 26, and 31 is completed.
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W. J. Clin. Pharmacol. Ther. 1991, 49, 558-570.
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312-321.
one can transform the integral in eq B3 to the following
form:
∫∫t
∫ ∫t
∞
0
y γ-1
(B2)
Substitute η ) e-kelτ and ζ ) e-koutt into the second integral
in eq B1 and let y ) Rζ1/β and x ) Rη. Then, the integral,
if β - γ > 0, becomes:
0
(21) ∫
(lnRR)
0
kout
kout
D/V
, β)
, γ)
I
IC50
kel
kel max
∫∫
(1 + y)-2 ln2 y dy (B10)
The asymptotic expansion of the expression in eq B10 is:
0
Imaxkout γ
-R0
kel2 β - γ
R
0
Note that ln2 R ) ln2 (1 + R) + O(ln R/R) as R f ∞. Because
The first integral in eq B1 dominates the second as
(D/V)/IC50 f ∞, and it can be evaluated explicitly, yielding
the leading terms in eqs 25 and 26. The remainder terms
come from the second integral in eq B1. To simplify
calculations, it is convenient to introduce nondimensional
parameters:
R)
(21) ∫
O
|H2(t)|
∞
R-
Acknowledgments
This work was supported in part by Grant No. 24211 from the
National Institute of General Medical Science, National Institute
of Health.
JS970168R