Department of Pharmaceutical Sciences, School of Pharmacy and
Pharmaceutical Sciences, University at Buffalo, State University
of New York, Buffalo, New York
Pharmacodynamics is the study of the time course of pharmacological
effects of drugs. The field of pharmacodynamic modeling has made many
advances, due in part to the relatively recent development of basic and
extended mechanism-based models. The purpose of this article is to
describe the classic as well as contemporary approaches, with an
emphasis on pertinent equations and salient model features. In
addition, current methods of integrating various system complexities into these models are discussed. Future pharmacodynamic models will
most likely reflect an assembly of the basic components outlined in
this review.
 |
Introduction |
Pharmacodynamics
(PD)3 has evolved from an empirical to a quantitative
scientific endeavor that seeks to characterize the time course of drug
effects through the application of mathematical modeling to such data.
This shift has primarily resulted from improved analytical
methodologies, which has enhanced the ability to measure various
biomarkers of drug effects, advances in computer hardware and software,
increased regulatory and academic interest, and the continued
construction and refinement of pharmacodynamic models based on
underlying physiological mechanisms. Linking the pharmacokinetics (PK)
of a drug with the subsequent temporal pattern of in vivo
pharmacological response can be traced to the pioneering work of
Gerhard Levy in the mid-1960s (Levy, 1964
, 1966). Since then, PK/PD
modeling has emerged as a firmly established scientific discipline,
some of the major goals of which are to codify current facts and data
sets, test competing hypotheses regarding processes altered by the
drug, make predictions of system responses under new conditions, and
estimate inaccessible system variables (Yates, 1975). In addition to
providing a systematic framework for studying and understanding in vivo
pharmacology and systems biology, the implications of PK/PD modeling
are far reaching. At a recent meeting related to NIGMS Pharmacologic
Sciences Training, it was indicated "there was remarkable consensus
that the core subject matter of pharmacology remains the principles of
pharmacokinetics and pharmacodynamics" (Preusch, 2002). Applications
of PK/PD have been extended to virtually all phases of drug development
(Peck et al., 1994), which has resulted in the current Guidance for
Industry on Exposure-Response Relationships: Study Design, Data
Analysis, and Regulatory Applications from the Food and Drug
Administration (http://www.fda.gov/cder/guidance/index.htm).
The main objective of this report is to review the major
mechanism-based pharmacodynamic models in use today, highlighting operable equations, simple signature profiles, and important model features. Additionally, techniques for incorporating various system complexities into these models are discussed. For purposes of clarity, efforts were made to keep equations as general as possible and
the number of symbols to a minimum.
| |
General Perspectives |
The analysis of PK data is often considered routine and
straightforward, but major physiological insights have derived from basic principles embodied in Fick's Law of Diffusion, Fick's Law of
Perfusion, and the Michaelis-Menten equation. The vast array of
pharmacological mechanisms and physiological processes controlling drug
responses complicate PD modeling. The major types of PK/PD models used
to conceptualize these mechanisms of action are listed in Table
1, and a general scheme depicting the
basic processes of such models is shown in Fig.
1 (Jusko et al., 1995). The time course
of drug concentrations in a relevant biological fluid (e.g., plasma, Cp) are typically represented
by a mathematical function:
|
(1)
|
where 
PK is a vector of PK parameters
determined by model fitting, X represents independent
variables associated with the given dose and/or regimen, and
t is time. If plasma concentrations are assumed to be
proportional to biophase concentrations
(Ce), then these expressions, either
explicit or differential equations, are often fixed and serve as
driving functions in PD models:
|
(2)
|
where R is the pharmacological response (often
abbreviated E for effect or R for response, and
are used interchangeably throughout this review) and Z
represents a vector of drug-independent system parameters. Analogous to
PK models, eq. 2 may be either explicit, as for some simple systems, or
more commonly as differential equations requiring numerical methods for
solution. When biophase distribution represents a rate-limiting step
for drugs in producing their effects, a link-compartment can be
incorporated to accommodate this delay. Direct measurements at or near
the site of action are preferable, however. The biosensor process
involves the interaction between the drug and the pharmacological
target and may be described using various receptor-occupancy models,
may require equations that consider the kinetics of the drug-receptor
complex formation and dissociation, or may encompass irreversible
drug-target interactions. In addition to rapid direct responses, many
drugs act via indirect mechanisms and the biosensor process may serve
to stimulate or inhibit the production or loss of endogenous mediators
(biosignal flux). These altered mediators may not represent the final
observed response and further transduction processes may occur, thus
accounting for further time delays and requiring additional modeling
components. Finally, a host of system complexities such as drug
interactions, functional adaptation, changes with pathophysiology, and
other factors may have a significant role in controlling drug effects after acute and long-term drug exposure.
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Fig. 1.
Basic components of pharmacodynamic models
of drug action.
Symbols are defined in text. Adapted from Jusko et al. (1995).
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The final PK/PD model chosen for a particular data set should be based,
as much as possible, on the pharmacology of the drug and system (Levy,
1994a). Once a model is defined, unknown parameter values are typically
estimated using nonlinear regression techniques contained within
computer programs such as WinNonlin (Pharsight, Mountain View, CA),
Kinetica (Innaphase, Philadelphia, PA), and ADAPT II (Biomedical
Simulations Resource, Los Angeles, CA). Owing to the nonlinear
drug-target interaction, most pharmacodynamic models require
description using differential equations. Minimally, it is desirable to
resolve drug-dependent parameters, such as capacity and sensitivity
terms in receptor-occupancy models, as well as system parameters often
in the form of rate constants for biophase distribution, biosignal
turnover, or signal transduction processes.
| |
PK/PD Modeling Requirements |
The resolution of PK/PD models and parameters is best achieved by
having relevant pharmacokinetics (preferably at the biophase), an
understanding of the mechanism of action of the drug, appreciating the
determinants of any time dependence in responses, and collecting a
suitable array of experimental measurements as a function of dose and
time. When possible, such measurements should be sensitive, gradual,
quantitative, reproducible, and meaningful. Owing to the nonlinearity
in most biosensor processes, a sufficiently wide range of drug
concentrations and doses are needed to extract the sensitivity
(EC50) and capacity constants
(Emax). Resolution of time-dependent
steps requires careful assessment of the stationarity of the
baseline (or placebo response), response profiles at two or more dose
levels (signature profiles), and how the system returns to the baseline
in the face of possible functional adaptation processes exhibited as
tolerance and/or rebound. Models with greater complexity require
multiple and more comprehensive sets of experimental data, more astute
design, and greater skill in data analysis because of possible multiple
nonlinear components. Drug therapy seeks to ameliorate alterations in
biochemical or physiological processes caused by disease that
necessitates studies of pathophysiology to assess aberrations in the
system. In the end it should be recognized that drugs can serve as
probes that perturb the normal (or abnormal) homeostasis of the body,
and a suitable model should reveal not only the pharmacological
properties of the drug but also the major rate-limiting steps
(turnover, transduction, and tolerance) in the biology of the system.
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Simple Direct Effects. |
In the early days of pharmacodynamics, it was recognized that the
intensity of many pharmacological effects is linearly related to the
logarithm of dose (A) (Levy, 1964):
|
(3)
|
where m and e are the slope and intercept
terms. Levy (1964, 1966) derived a relationship between monoexponential
drug pharmacokinetics (Cp = C0 · e
k · t)
and the time course of in vivo effects from a rearrangement of eq. 3:
|
(4)
|
where E0 is a theoretical
intercept, m is the linear slope describing the response-log
concentration relationship, and k is the first-order
elimination rate constant of the drug. The Levy equation was supported
by clinical data for drugs like tubocurarine that showed plasma
concentrations decreasing exponentially after intramuscular injection
and the degree of muscle relaxation decreasing linearly with time
(Levy, 1966). Concepts such as duration of effect and role of serial
dosing also evolved. Although originally limited to monoexponential
kinetics and rapidly reversible dynamics with a linear effect-log
Cp curve, this simple model
essentially introduced the application of linear and log-linear models
to in vivo data:
|
(5)
|
|
(6)
|
where E0 is the baseline effect
and S and m are the slopes of the respective relationships.
The simple models described by eqs. 4 to 6 provided early
pharmacodynamic parameters (slope values) that could be easily
calculated (simple linear regression) and compared for different drugs
or drug combinations. However, these models are only valid when the effect is either less than 20% (linear) or within 20 to 80%
(log-linear) of the maximum effect
(Emax), and as such, cannot be
extrapolated to capture the capacity or
Emax parameter. Furthermore, the
log-linear model breaks down when drug concentrations are less than the
apparent intercept. Owing to these limitations, Wagner (1968) proposed the use of the Hill equation to describe the in vivo
concentration-response relationship. The rationale for this approach
was based on the law of mass action and classical receptor occupancy
theory (Ariens, 1954). The rate of change of the drug-receptor complex
(RC) is given by the following equation:
|
(7)
|
where RT is the maximum receptor
density, C is the drug concentration at the site of action,
kon is a second-order association rate
constant, and koff is a first-order
dissociation rate constant. If one assumes equilibrium conditions, eq.
7 can be rearranged to yield
|
(8)
|
where KD is the equilibrium
dissociation constant
(koff/kon).
A further assumption is that the drug effect is directly proportional to the fraction of occupied receptors, such that E = 
· RC. Substituting in the relationship of eq. 8, one of the
forms of the Hill equation is derived, commonly referred to as the
Emax model:
|
(9)
|
where Emax is substituted for
( · RT) and the
EC50 is a sensitivity parameter representing the
drug concentration producing 50% of
Emax. The typical effect-log
concentration relationship is thus curvilinear and avoids the
disadvantages of the previous models. Furthermore, initial estimates of
the PD parameters to be used in nonlinear regression analysis are
readily identifiable from the effect-concentration curve. The full Hill
equation, or the sigmoidal Emax model,
includes an additional parameter, 
, which is a slope term that
reflects the steepness of the effect-concentration curve:
|
(10)
|
(The actual slope of a plot of E versus log
Cp is
Emax · /4 over the region of
20 to 80% effect). Whereas values of less than unity will produce
broad slopes, higher values (greater than 4) are reflective of an
all-or-none type of response (NB: Emax and EC50 are unchanged as is varied). The
rationale for the power coefficient is usually uncertain but may be
caused by either positive ( > 1)/negative ( < 1)
cooperativity or homogeneity ( > 1)/heterogeneity ( < 1) in receptor/effector functions (Hoffman and Goldberg, 1994). The
inhibitory and excitatory sigmoid Emax models are extensions of eq. 10 where the right-hand side of the equation is either subtracted from (inhibitory) or added to
(excitatory) a baseline value (E0).
These models are still commonly used today for drugs that satisfy the
condition that a rapid equilibrium is obtained between plasma and
biophase concentrations (e.g., many central nervous system and
cardiovascular agents) and where agonism is the relevant action of the drug.
The operational model of agonism (Black and Leff, 1983) couples the
receptor occupancy concept to effector processes that control in vivo
drug responses. More realistically, the effect is assumed to be
nonlinearly related to the drug-receptor complex:
|
(11)
|
where KE is the RC value
producing half-maximal effect. Thus, combining eqs. 8 and 11:
|
(12)
|
where Em is a system maximum and
represents a transducer or efficacy function
(RT/KE).
A power parameter, analogous to in eq. 10, can also be added to the
concentration terms of this function to improve model fitting. This
model requires the determination or estimation of receptor affinity and
capacity to unravel the intrinsic efficacy properties from in vivo
data. Consequently, it may be possible to predict the time course of in
vivo effects of relevant drugs from in vitro measurements. Such
correlations were shown by Visser et al. (2003) in a comprehensive
assessment of the effects of GABA receptor modulators on
electroencephalogram effects in rats based on the Black and Leff principles.
The above-mentioned equations fundamentally describe simple activities
of drugs that are agonists. Equations for drug antagonists are more
complex and textbooks (Kenakin, 1997) should be consulted to deal with
the diversity of less common but more complicated mechanisms of drug
action that can occur.
A feature common to all models discussed in this section is the
assumption that a rapid equilibrium is obtained between plasma and
biophase concentrations. Accordingly, maximum or peak effects are
predicted to occur simultaneously with peak drug concentrations. However, most in vivo responses lag behind drug concentrations, a
phenomenon resulting in hysteresis in plots of response versus concentration. This temporal displacement may result from various physiological and/or pharmacological causes, and several models attempt
to capture such delays in terms of the mechanism of action of drugs and
the affected biological systems.
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Biophase Distribution Model |
Drug distribution to the site of action may represent a
rate-limiting step for drugs in producing their biological effect. Furchgott (1955) coined the term "biophase" and provided the first diffusion-type equations to describe drug permeation to receptors in
such sites. Sheiner et al. (1979) subsequently described a modeling
approach for drugs exhibiting response delays using a hypothetical
effect-compartment as a mathematical link between the time course of
plasma concentrations and drug effects. The amount of drug entering
this compartment is considered to be negligible and therefore is not
reflected in the PK of the drug. Plasma concentrations are typically
fixed functions (eq. 1) and the rate of change of biophase drug
concentrations can be defined as follows:
|
(13)
|
where k1e and
ke0 are first-order distribution rate
constants (usually set equal to each other for lack of
identifiability). A sigmoid Emax model
(eq. 10) is often used to correlate effect-compartment concentrations
with the intensity of pharmacological effect
(Ce is substituted for
Cp). A model diagram and typical
response-time profiles are shown in Fig.
2. The kinetics of the drug is assumed to
be monoexponential for this and later simulations. The
Ce profile is thus inferred by the
equilibrium delay and is a function of the drug PK and
ke0 parameter. Smaller
ke0 values may result in flip-flop
kinetics and will produce later peaks and prolongation of the response
in accordance with slower distribution to and from the site of action.
Although peak effects will be delayed relative to plasma
concentrations, the times at which the peak effect occurs will be dose
independent. Therefore, larger drug doses will yield identical peak
Ce and effect times and the recession slopes (return to baseline conditions in the effect-time curve) are
parallel and linear over the region of 80 to 20% of maximum effect.
This approach was the first to allow mathematical modeling of
nonsteady-state in vivo PK/PD data and is highly useful for delayed
drug effects owing to distributional processes. Because the biophase
model was the first used to account for time delays in drug effects, it
has often been applied inappropriately before it was recognized that
various other processes are more likely to cause such delays.
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Slow Receptor-Binding Model |
The majority of pharmacodynamic models assume that the binding of
the drug with its pharmacological target occurs rapidly, is reversible,
and can be described using equations derived under equilibrium
conditions (e.g., eqs. 8-10). In contrast, an ion-channel binding
model has been developed by Shimada et al. (1996) on the basis of in
vitro binding data of calcium channel antagonists, which demonstrate
relatively slow rates of association and dissociation. The
pharmacological effect is still assumed to be proportional to the
concentration of the drug-receptor complex, and in a direct parallelism
to eq. 7, can be defined as follows:
|
(14)
|
The inclusion of the binding parameters was sufficient to account
for the temporal discrepancy between the PK and antihypertensive effect
of eight calcium channel antagonists in Japanese patients. Additionally, the estimated KD values
were shown to be significantly correlated with those obtained from in
vitro experiments. These results suggest that the model could be used
to predict the pharmacodynamic profile of future drugs in this class
from PK and in vitro binding data, a goal that is often sought in drug
development. Although attractive in principle, the model was developed
using single doses and, to our knowledge, a rigorous evaluation of this
model over a wide dosing range has yet to be performed.
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Irreversible Effects |
Select chemotherapeutic agents (including numerous anticancer and
antimicrobial compounds) and enzyme inhibitors exert their biological
effects through irreversible bimolecular interactions with cells and/or
proteins. Jusko (1971) described a basic pharmacodynamic modeling
approach for phase-nonspecific chemotherapeutic drugs. The original
model and its various modified forms continue to be used to
characterize drugs that elicit irreversible effects.
Cell Proliferation Model with Irreversible Inactivation.
A general equation for cellular proliferation and phase-nonspecific
cell killing is as follows:
|
(15)
|
where the response (R) represents cell number (e.g.,
malignant cells, bacteria, parasites, or viral load) and C
is either Cp or
Ce. The natural proliferation of
cells, in the absence of drug, is described by the g(R)
function. In the simplest models, the density of viable cells grows
exponentially:
|
(16a,b)
|
where the apparent first-order growth rate constant
(kg) is the difference between the
true natural rates of growth and degradation. A number of additional
growth or population models may be incorporated, such as the
"logistic" model where a term (1 
R/Rss) is added to reflect
an upper limit (Rss) in cell number,
depending on the data available and the organism or cell type (Mouton
et al., 1997). Plasma or effect-compartment drug concentrations may be
involved in an irreversible interaction with target cells through the
f(C) function, which is often defined as follows:
|
(17a,b)
|
where k and Kmax are
second-order cell-kill rate constants and KC50 is
the drug concentration producing 50% of
Kmax (Jusko, 1971; Zhi et al., 1988).
As a result of the joint effects of cell-killing and natural growth
dynamics, effect-time profiles or survival curves are often biphasic,
with an initial phase of cell-killing followed by regrowth after drug
concentrations decline below a minimum effective value (e.g.,
KC50).
Cell Proliferation Model with Cycle-Specific Inactivation.
Some chemotherapeutic agents exert antiproliferative effects only
during specific phases of the cell cycle. This property has been
characterized using a two-compartment model, which separates the total
cell population into proliferating
(Rs) and quiescent groups
(Rr) (Jusko, 1973). The
cell-proliferation and irreversible drug-effect functions are operable
only in the former, and the system of equations is as follows:
|
(18a,b)
|
The interconversion of cells between these populations is governed
by the first-order transformation rate constants
ksr and krs. The effects of vincristine and
vinblastine on hematopoietic and lymphoma cells in the mouse femur were
well characterized in the original derivation of the model (Jusko,
1973). More recently, Yano et al. (1998) substituted eq. 17b and the
logistic growth function for g(Rs)
into eq. 18a, and successfully applied the model to in vitro
bactericidal kinetics data of several 
-lactam antibiotics.
Turnover Model.
Irreversible inactivation also extends to the interaction between some
drugs and endogenous enzymes, which can be modeled with an indirect
turnover model:
|
(19)
|
where kout is a first-order loss
rate constant, kin represents an
apparent zero-order production rate of the response (often set equal to
the product of kout and the initial
response value, R0, for purposes of
stationarity), and f(C) is as previously defined (eq.
17, a and b). In the absence of drug, eq. 19 reduces to the baseline
condition where the rate of change is zero and the response variable is
a constant value (i.e., R = kin/kout = R0). Such a model was used by
Yamamoto et al. (1996) to explain the long duration of antiplatelet
effect of aspirin in humans. The response was driven by plasma drug
concentrations. Plasma thromboxane B2
concentrations and the percentage of prostacyclin production served as
biomarkers of cyclooxygenase activity in platelets and vessel wall
endothelium after oral aspirin administration. Modifications to eq. 19
have resulted in PD models used to capture in vivo dynamics of
5-reductase inhibition (Gisleskog et al., 1998; Katashima et al.,
1998b) and H+,K+-ATPase
inactivation by several proton pump inhibitors (Katashima et al.,
1998a; Abelo et al., 2000).
| |
Indirect Effects |
The earliest description of drugs acting through indirect
mechanisms came from Ariens (1964). He explained how drugs might induce their effects not by direct interactions with receptors, but
rather on the ability of this interaction to affect the fate of
endogenous compounds and the subsequent effects that are mediated by
those substances. Nagashima et al. (1969) were the first to report the
PK/PD modeling of indirect PD data, capturing prothrombin complex
activity in the blood of normal volunteers who received oral doses of
warfarin. However, a systematic modeling approach for characterizing
diverse types of indirect responses was not described until relatively
recently. The four basic models of Dayneka et al. (1993) initiated the
formal PK/PD modeling of responses generated by indirect mechanisms of
action and were shown to characterize numerous clinical pharmacodynamic
effects (Jusko and Ko, 1994). Subsequent efforts have yielded a series
of extended indirect response models that serve to capture additional
complexities related to specific drugs and biological systems.
Basic Indirect Response Models.
The four original indirect response models are based on drug effects
that either stimulate or inhibit the production or loss of a mediator
or response variable. A general equation for the rate of change of the
response variable can be written as follows:
|
(20)
|
where the rate constants kin and
kout are as previously defined (eq.
19), and the
Hn(Cp)
functions (n = 1 or 2) are given by the
Emax model (eq. 9). In this context,
Emax is redefined as maximum factors
of either fractional inhibition (0 < Imax 
1) or stimulation
(Smax > 0). The
EC50 parameter retains its common definition, but
is often symbolically replaced with IC50 or
SC50 for inhibition or stimulation. Individual
models are as follows: I (inhibition of production) where
H1(Cp)
is subtracted and
H2(Cp) = 0, II (inhibition of dissipation) where
H1(Cp) = 0 and
H2(Cp) is subtracted, III (stimulation of production) where
H1(Cp)
is added and
H2(Cp) = 0, and IV (stimulation of dissipation) where H1(Cp) = 0 and
H2(Cp)
is added. A schematic of the basic indirect response models and typical
response-time profiles for increasing drug doses are shown in Fig.
3. Response-time profiles of these models
typically show a slow decline or rise in the biomarker to some maximum
level, followed by a gradual return to baseline conditions
(kin/kout
or R0) as drug concentrations decline
below the IC50 or SC50
values. The time to peak effect is dose-dependent and occurs at later times for larger doses owing to increased duration when
Cp > IC50 or
SC50. More complete reviews of the basic
properties of these models are available and provide useful information
as to appropriate model selection, model sensitivity to parameter
values and dose levels, and methods of obtaining initial parameter
estimates from experimental data (Sharma and Jusko, 1996; Krzyzanski
and Jusko, 1998).
Extended Indirect Response Models.
In addition to describing indirect effects, Ariens (1964) also noted
that certain drugs may cause a liberation of certain endogenous
compounds, causing a subsequent depletion that may require time to
replenish. When that substance produces the pharmacological effect, a
form of tolerance may be observed after continued drug exposure. One
form of an integrated precursor-PK/PD model (see model VI below) was
used to capture the release of prolactin by the administration of the
antipsychotic drug remoxipride (Movin-Osswald and Hammarlund-Udenaes,
1995). Further development and characterization of such models was
subsequently reported (Sharma et al., 1998). A more general set of
precursor-dependent indirect response models can be defined by the
following system of equations:
|
(21a,b)
|
where k0 is the apparent
zero-order production rate of the precursor (P),
kp is the first-order rate constant of
production of the response marker, and
ks is an optional first-order rate constant of precursor elimination, the need for which may be tested using traditional model-fitting criteria. Specific models can be
defined as follows: V and VI (inhibition or stimulation of kp) where
H1(Cp) = 0 and
H2(Cp)
is subtracted (inhibition) or added (stimulation), and VII and VIII
(inhibition or stimulation of k0)
where
H1(Cp) = is subtracted (inhibition) or added (stimulation) and
H2(Cp) = 0 (model diagram shown in Fig. 4).
Models V and VI possess the unique ability to characterize both
tolerance and rebound phenomena (Sharma et al., 1998). Some examples in
the literature can be found for T-cell lymphocyte trafficking by
prednisolone (model V) (Magee et al., 2001), inhibition of leukocyte
survival by paclitaxel (model VII) (Minami et al., 1998), and
inhibition of experimentally induced tumor necrosis factor-
concentrations by susalimod (VII) (Gozzi et al., 1999).
Whereas the basic indirect response models assume a constant
steady-state baseline value in the absence of drug
(R0), some biomarkers may exhibit
nonstationarity or a time-dependent baseline. A classic example
is the time course of endogenous cortisol concentrations, which follow
a circadian rhythm and can be suppressed by the administration of
exogenous corticosteroids. This physiological response has been well
characterized by indirect response model I with the
kin parameter replaced with a
time-dependent function. One of the simplest examples involves the use
of a single cosine function:
|
(22)
|
where Rm is the mean input rate,
Rb is the amplitude of the input rate,
tz is the peak time (baseline acrophase), and
2
/24 converts time into radians (Lew et al., 1993). However, more
robust mathematical functions have been developed for cortisol
secretion, including a form of Fourier analysis and are the subject of
comparison (Chakraborty et al., 1999). These techniques are not
specific for cortisol and may be applied to other irregular biorhythmic baselines.
A cell life-span concept has been integrated into the indirect response
models for drugs that alter the generation of natural cells (e.g.,
erythro- and thrombopoietin) (Krzyzanski et al., 1999). Cells are
assumed to be produced at a constant rate
(kin), circulate and survive for a
specific duration of time (TR), and are then eliminated from the system not by a first-order process, but
at the same rate as the input, delayed by the cell life span (senescence or conversion to another cell type). For a simple one-compartment model, the operative equation is as follows:
|
(23)
|
where the baseline condition is constant
(R0 = kin · TR) and drug is assumed to inhibit
[1 H(C)] or stimulate [1 + H(C)] cell production via the Hill or
Emax model (eqs. 9 and 10). For stimulation of cell production, response-time profiles reveal an
apparent zero-order rise in cell density until a peak is reached at
time TR, followed by a gradual return
to baseline levels. The shape of the peak response will be sharper for
larger doses and broader with longer life spans. Additional
compartments may be added in a precursor-style format, which may
represent other cell types or various levels of cell maturation. Unlike
cumbersome physiological models that often contain many compartments
and parameters that are not amenable to routine clinical
pharmacodynamic modeling (Pantel et al., 1990), the cell life-span
indirect response model introduces a relevant approach with few
equations and pharmacologically meaningful parameters. However, they
require use of delay differential equations that are difficult to
operate in nonlinear least-squares fitting of data.
Finally, several other modifications to eq. 20 have shown to enhance
model performance under certain conditions. For example, the biophase
distribution concept was combined with indirect response model I to
describe the central nervous system effects of tiagabine in rats
(Cleton et al., 1999). Also, the operational model of agonism (eq. 12)
with a slope parameter used in the place of the Hill function
[Hn(Cp)
in eq. 20] was applied to model the antilipolytic effects of adenosine
A1 receptor agonists in rats (Van der Graaf et
al., 1999). Zuideveld et al. (2001) used a heat production/heat loss
indirect response model with drug effect on a set-point temperature as
the pharmacological mechanism. The four basic models have also been
extended to include a peripheral response pool (Krzyzanski and Jusko,
2001), which was recently applied to the cell trafficking dynamics of a
novel immunosuppressant (Li et al., 2002). Indirect response modeling
can account for the dynamics of numerous drugs that alter the
production or loss of response variables, and a systematic
model-building process is required for ascertaining the need of
additional components to accommodate drug or system complexities.
| |
Signal Transduction Models |
The pharmacological effect of compounds may be mediated by
time-dependent transduction, whereby the final drug response is a
result of a signaling cascade controlled by secondary messengers (e.g.,
intracellular calcium ions, cyclic AMP). When these post-receptor events are rate-limiting, drug effects can lag considerably behind plasma concentrations. Although empirical time lags may be added to
previously described models, this approach rarely captures the gradual
onset typical of transduction cascades. On the other hand, elaborate
physiological models of biological signaling pathways (Bhalla and
Iyengar, 1999) do not lend themselves to PD modeling of typical data.
Furthermore, many of the individual steps in the cascade are unknown or
cannot be readily measured experimentally. A simple transit compartment
model was suggested to describe delayed responses owing to transduction
processes (Sun and Jusko, 1998). It requires a series of differential
equations:
|
(24)
|
where Mn are the nth secondary
messengers, RC is given by eq. 7, and represents the mean transit
time. To avoid the specific receptor dynamics required by eq. 7, RC may
be replaced with the Emax or Hill
equation (eqs. 9 and 10), assuming rapid receptor binding (model
diagram and simulations are shown in Fig.
5). Thus, a model is derived that
contains a minimal number of drug
(Emax and EC50)
and system parameters () that may be applied to human clinical data
(Mager and Jusko, 2001b) and may provide a simple structure on which
future knowledge of specific processes may be integrated. Recent
applications of this approach include the PK/PD modeling of the
parasympathomimetic activity of scopolamine and atropine in rats
(Perlstein et al., 2002) and the chemotherapeutic effects of
methotrexate (Lobo and Balthasar, 2002).
| |
Tolerance Models |
Drug tolerance can be broadly defined as a diminution of the
expected pharmacological response after repeated or continuous drug
exposure. The mechanisms of tolerance are complex and not always
completely understood. However, the frequency with which tolerance is
observed and its clinical implications warrant discussion. The primary
tolerance mechanisms are: counter-regulation, desensitization, up- or
down-regulation, and precursor pool depletion.
Counter-regulation models typically use an opposing effect or signal
that attenuates the response to a drug. The structure of this model can
take on many forms; however, the rate of change of a mediator or
opposing response (M) can be defined as follows:
|
(25)
|
where the production of M is driven by the primary response
(R) as governed by first-order rate constants of production
(k1) and loss
(k2). In turn, the net response will
reflect the difference, Rnet = R M, perhaps with an intermediary
transduction step (Bauer and Fung, 1994). Alternatively, negative
feedback can be achieved by integrating mediator values into
appropriate pharmacodynamic models, such as stimulation of loss of a
response variable, viz. kout · (1 + M) (Gabrielsson and Weiner, 1997).
Receptors may undergo desensitization, reflecting either
internalization or an apparent decrease in drug affinity, and represent another source of the lessening of drug effects on prolonged exposure. A classic example is the desensitization of G protein-coupled receptors
by protein kinases in response to stimulation by select agonists
(Foreman and Johansen, 1996). One modeling technique is to allow the
receptors or response to be temporarily "lost" to an inactive pool
(Ri) by a first-order process
(kd):
|
(26)
|
This type of equation allows tolerance to evolve in relation to
the primary response, but delayed by the operation of the kd constant. A more mechanistic
approach to desensitization may be reflected in receptor-inactivation
theory (Kenakin, 1997). This model can encompass both
receptor-occupancy and the rate theory of drug action and is defined by
the following equations:
|
(27a,b,c)
|
where R represents free receptor concentrations, RC' is an
inactive complex, and k3 and
k4 are first-order rate constants of
RC' production and dissociation. The RC complex is formed in the usual
manner (eq. 27a, see also eq. 7); however, this complex drives the
production of an inactive receptor form, which can further dissociate
back into the free receptor. The effect is assumed to be proportional
to the rate of receptor inactivation (k3 · RC). Interestingly, given
the proper rate constants, simulations reveal a transient peak response
followed by a fade to a new steady state (Kenakin, 1997). Furthermore,
the magnitude of the fade is dose-dependent, increasing with larger doses.
The final two mechanisms of up- or down-regulation and precursor pool
depletion may be modeled using previously described pharmacodynamic
models. In the case of altered receptor density, the basic indirect
response models can be used where the drug-receptor-DNA complex serves
to inhibit or stimulate the production or loss of either receptor
messenger RNA or receptor density (eq. 20). This pharmacogenomic
approach has been used in part to capture the complex receptor dynamics
of the glucocorticoid receptor in rat liver after the acute and
continuous exposure to methylprednisolone (Ramakrishnan et al., 2002),
and along with mass law depletion of free receptors, can capture the
observed apparent tolerance phenomenon. The precursor pool depletion
model also has been discussed (model VI, eq. 21, a and b).
Other tolerance models have been proposed which are driven by drug
concentrations and have often been found to work interchangeably (Gardmark et al., 1999). Those described here are more mechanistic in
having some component of the response system producing loss of the
primary effect via four physiologically relevant processes. The present
tolerance models largely couple the primary PD response model with an
off-setting process, which seeks to return the system to its normal
homeostasis and often resulting in a temporary rebound. Experimental
designs involving repeated or lengthy drug administration along with
capturing the full return to baseline are needed to examine and model
functional adaptation processes.
| |
More Complex Models |
The major mechanism-based pharmacodynamic models have been
presented in terms of basic theory, operable equations, essential model
features, and selected examples. Methods of incorporating drug and
system complexities have been discussed, particularly those associated
with indirect effects and tolerance phenomena. Many other factors, such
as, drug interactions, the presence of active metabolites and
enantiomers, opposing drug effects, binding to multiple receptor sites,
and disease progression may complicate the analysis of PD data and
require additional components to be integrated into the basic models
outlined above.
The focus of this review was on fundamental mechanism-based concepts
that capture primary rate-limiting steps in drug responses with
simplicity and parsimony. In considering future pharmacodynamic models,
however, two additional approaches merit discussion. First, models that
incorporate the binding kinetics of drug-target interactions are
emerging with increasing frequency. Such a model was shown to be useful
for characterizing the pharmacodynamics of humanized anti-Factor IX
monoclonal antibody in monkeys (Benincosa et al., 2000). The inclusion
of drug-target microconstants (kon and
koff) provides a flexible means of
bridging the pharmacokinetics of drugs and the time course of effects,
thereby inferring the temporal pattern of drug-target concentrations
and potentially binding capacity, both of which are not typically
measurable for in vivo systems. This concept is of considerable
importance for drugs exhibiting target-mediated drug disposition and
dynamics, where the drug-target interaction not only drives the
pharmacological effect but also is reflected also in the
pharmacokinetics of the drug (Levy, 1994b; Mager and Jusko, 2001a).
Second, complex PD models will most likely represent a compilation of
several of the basic components described in this review, of which the
recent 5th generation model of corticosteroid pharmacodynamics is an example (Ramakrishnan et al., 2002). The overall model is constructed with a series of differential equations in a piecewise manner. Separate
modeling was carried out for methylprednisolone PK, glucocorticoid receptor and receptor mRNA dynamics, and hepatic tyrosine
aminotransferase (TAT) mRNA and activity in rats. The rate of change of
the drug-receptor complex was defined as follows:
|
(28)
|
where RC is formed via an irreversible second-order rate constant
(kon), and
kt is a first-order rate constant for
translocation of the drug-receptor complex into the nucleus. A
transduction compartment model was used to describe the time course of
the active drug-receptor complex in the nuclei of cells. These
concentrations were used to stimulate the production of TAT mRNA, which
is translated into TAT activity levels, in a manner similar to
precursor-dependent indirect response models. Indirect response model I
is used to describe the down-regulation of the production of the
receptor mRNA, also driven by the nuclear drug-receptor complex
density. This imparts a form of tolerance because fewer receptors are
produced and available to interact with the drug. A third example of a complex PK/PD model is for the target-mediated PK/PD of
interferon-1a. Components of receptor binding, a precursor indirect
response model, and two mechanisms of tolerance were needed to explain the effects of this agent on neopterin dynamics in humans and monkeys
(Mager and Jusko, 2002). Complex PD models of the general form depicted
in Fig. 1 may become commonplace as more components of drug action are
determined experimentally.
| |
Conclusions |
Drugs interact with receptors, enzymes, transporters, and/or other
biological macromolecules in specific as well as multiple ways to block
or trigger an incredible array of molecular, biochemical, and
physiological events. This represents a rich tableau of possible biomarkers, functions, and models to describe the time course of
ensuing drug responses at various levels of biological organization. The essential components of many mechanism-based pharmacodynamic models
are reflected in the recognition of the nonlinear drug-target interaction and the key steps that are subsequently altered in the
normal and pathological biological cascades that control the homeostasis of affected physiological systems. The field of PK/PD modeling has clearly emerged from using empirical functions to characterize data to the employment of a diverse array of basic to
complex models, which allow entire data sets and systems to be captured
using equations and models that reflect the essential underlying rules
of pharmacology and physiology.
This work was supported by Grant GM57980 from the National
Institutes of General Medicine (National Institutes of Health) and a
Predoctoral Fellowship to D.E.M. from the American Foundation for
Pharmaceutical Education.
Abbreviations used are:
PD, pharmacodynamics;
PK, pharmacokinetics;
TAT, tyrosine aminotransferase.
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Abstract
Introduction
Elzbieta Wyska
received a Master's degree in pharmacy (1989) and a Ph.D. degree in
pharmaceutical sciences (1998) from the Jagiellonian University,
Kraków, Poland. Her doctoral research concerned the role of free
antidepressant drug concentration monitoring in the treatment of major
depression. In years 2000 to 2002, Dr. Wyska was trained as a
post-doctoral fellow at the Department of Pharmaceutical Sciences,
State University of New York at Buffalo in the field of PK/PD modeling
with emphasis on performing model comparisons and seeking applications
to literature data under the direction of Professor William J. Jusko.
At present, she works as a Research Assistant Professor at the
Department of Pharmacokinetics and Physical Pharmacy, Faculty of
Pharmacy, Jagiellonian University.
Donald E. Mager received a
Bachelors of Science degree in Pharmacy from the University at Buffalo,
State University of New York (UB) in 1991 followed by the Pharm.D.
(2000) and Ph.D. (2002) degrees. The focus of his Ph.D. dissertation
was on the application of mechanistic modeling and biomathematical
techniques to experimental data from human and preclinical models to
delineate the controlling factors in the PK/PD profiles of selected
immunomodulatory agents using interferon-
Dr. Jusko is Professor of
Pharmaceutical Sciences and received his BS in Pharmacy (1965) and
Ph.D. (1970) degrees from the State University of New York at Buffalo.
He then joined the Clinical Pharmacology Section of the Boston Veterans
Administration Hospital and was Assistant Professor of Pharmacology at
Boston University School of Medicine. He returned to Buffalo in 1972 as
Director of the Clinical Pharmacokinetics Laboratory and Assistant
Professor. He was a Fulbright Scholar at The Mario Negri Institute for
Pharmacology in Italy in 1978/79, received the Rawls-Palmer Award in
1987 from ASCPT, the Doctor Honoris Causae from Jagellonian University
of Cracow in 1987, the Russell R. Miller Award from ACCP in 1988, the
Distinguished Service Award from the Am. College of Clinical Pharmacology in 1989, and the Research Achievement Award in PPDM from
AAPS in 1998. He is a Fellow of AAPS, ACCP, and AAAS and serves on the
editorial boards of seven journals. His research covers clinical,
basic, and theoretical pharmacokinetics and pharmcodynamics of diverse
drugs, particularly corticosteroids and immuno-suppressants. He has
over 420 publications.
