Abstract
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 mechanismbased 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.
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 mid1960s (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 ExposureResponse 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 mechanismbased 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 MichaelisMenten 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 Table1, 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, C_{p}) are typically represented by a mathematical function:
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 drugtarget interaction, most pharmacodynamic models require description using differential equations. Minimally, it is desirable to resolve drugdependent parameters, such as capacity and sensitivity terms in receptoroccupancy 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 (EC_{50}) and capacity constants (E_{max}). Resolution of timedependent 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 ratelimiting steps (turnover, transduction, and tolerance) in the biology of the system.
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):
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% (loglinear) of the maximum effect (E_{max}), and as such, cannot be extrapolated to capture the capacity orE_{max} parameter. Furthermore, the loglinear 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 concentrationresponse 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 drugreceptor complex (RC) is given by the following equation:
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 drugreceptor complex: Equation 11where K_{E} is the RC value producing halfmaximal effect. Thus, combining eqs. 8 and 11: Equation 12where E_{m} is a system maximum and τ represents a transducer or efficacy function (R_{T}/K_{E}). A power parameter, analogous to γ 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 abovementioned 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.
Biophase Distribution Model
Drug distribution to the site of action may represent a ratelimiting step for drugs in producing their biological effect.Furchgott (1955) coined the term “biophase” and provided the first diffusiontype 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 effectcompartment 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:
Slow ReceptorBinding 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. 810). In contrast, an ionchannel 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 drugreceptor complex, and in a direct parallelism to eq. 7, can be defined as follows: Equation 14The 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 K_{D} 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.
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 phasenonspecific 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 phasenonspecific cell killing is as follows:
Equation 15where the response (R) represents cell number (e.g., malignant cells, bacteria, parasites, or viral load) and Cis either C_{p} orC_{e}. 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:
Equation 16a,bwhere the apparent firstorder growth rate constant (k_{g}) 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/R_{ss}) is added to reflect an upper limit (R_{ss}) in cell number, depending on the data available and the organism or cell type (Mouton et al., 1997). Plasma or effectcompartment drug concentrations may be involved in an irreversible interaction with target cells through the f(C) function, which is often defined as follows:
Cell Proliferation Model with CycleSpecific Inactivation.
Some chemotherapeutic agents exert antiproliferative effects only during specific phases of the cell cycle. This property has been characterized using a twocompartment model, which separates the total cell population into proliferating (R_{s}) and quiescent groups (R_{r}) (Jusko, 1973). The cellproliferation and irreversible drugeffect functions are operable only in the former, and the system of equations is as follows: Equation 18a,bThe interconversion of cells between these populations is governed by the firstorder transformation rate constantsk_{sr} andk_{rs}. 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(R_{s}) into eq. 18a, and successfully applied the model to in vitro bactericidal kinetics data of several βlactam 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: Equation 19where k_{out} is a firstorder loss rate constant, k_{in} represents an apparent zeroorder production rate of the response (often set equal to the product of k_{out} and the initial response value, R^{0}, 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 =k_{in}/k_{out}= R^{0}). Such a model was used byYamamoto 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 B_{2}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. 19have resulted in PD models used to capture in vivo dynamics of 5αreductase 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: Equation 20where the rate constants k_{in} andk_{out} are as previously defined (eq.19), and theH_{n} (C_{p}) functions (n = 1 or 2) are given by theE_{max} model (eq. 9). In this context,E_{max} is redefined as maximum factors of either fractional inhibition (0 <I_{max} ≤ 1) or stimulation (S_{max} > 0). The EC_{50} parameter retains its common definition, but is often symbolically replaced with IC_{50} or SC_{50} for inhibition or stimulation. Individual models are as follows: I (inhibition of production) whereH_{1}(C_{p}) is subtracted andH_{2}(C_{p}) = 0, II (inhibition of dissipation) whereH_{1}(C_{p}) = 0 andH_{2}(C_{p}) is subtracted, III (stimulation of production) whereH_{1}(C_{p}) is added andH_{2}(C_{p}) = 0, and IV (stimulation of dissipation) whereH_{1}(C_{p}) = 0 andH_{2}(C_{p}) is added. A schematic of the basic indirect response models and typical responsetime profiles for increasing drug doses are shown in Fig.3. Responsetime 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 (k_{in}/k_{out}or R^{0}) as drug concentrations decline below the IC_{50} or SC_{50}values. The time to peak effect is dosedependent and occurs at later times for larger doses owing to increased duration whenC_{p} > IC_{50} or SC_{50}. 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 precursorPK/PD model (see model VI below) was used to capture the release of prolactin by the administration of the antipsychotic drug remoxipride (MovinOsswald and HammarlundUdenaes, 1995). Further development and characterization of such models was subsequently reported (Sharma et al., 1998). A more general set of precursordependent indirect response models can be defined by the following system of equations: Equation 21a,bwhere k_{0} is the apparent zeroorder production rate of the precursor (P),k_{p} is the firstorder rate constant of production of the response marker, andk_{s} is an optional firstorder rate constant of precursor elimination, the need for which may be tested using traditional modelfitting criteria. Specific models can be defined as follows: V and VI (inhibition or stimulation ofk_{p}) whereH_{1}(C_{p}) = 0 andH_{2}(C_{p}) is subtracted (inhibition) or added (stimulation), and VII and VIII (inhibition or stimulation of k_{0}) whereH_{1}(C_{p}) = is subtracted (inhibition) or added (stimulation) andH_{2}(C_{p}) = 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 Tcell 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 factorα concentrations by susalimod (VII) (Gozzi et al., 1999).
Whereas the basic indirect response models assume a constant steadystate baseline value in the absence of drug (R^{0}), some biomarkers may exhibit nonstationarity or a timedependent 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 thek_{in} parameter replaced with a timedependent function. One of the simplest examples involves the use of a single cosine function: Equation 22where R_{m} is the mean input rate,R_{b} is the amplitude of the input rate, t_{z} is the peak time (baseline acrophase), and 2π/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 lifespan 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 (k_{in}), circulate and survive for a specific duration of time (T_{R}), and are then eliminated from the system not by a firstorder 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 onecompartment model, the operative equation is as follows: Equation 23where the baseline condition is constant (R^{0} =k_{in} ·T_{R}) and drug is assumed to inhibit [1 − H(C)] or stimulate [1 +H(C)] cell production via the Hill orE_{max} model (eqs. 9 and 10). For stimulation of cell production, responsetime profiles reveal an apparent zeroorder rise in cell density until a peak is reached at time T_{R}, 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 precursorstyle 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 lifespan 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 leastsquares 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 [H_{n} (C_{p}) in eq. 20] was applied to model the antilipolytic effects of adenosine A_{1} 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 setpoint 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 modelbuilding 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 timedependent 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 postreceptor events are ratelimiting, 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: Equation 24where M_{n} are the nth secondary messengers, RC is given by eq. 7, and τ represents the mean transit time. To avoid the specific receptor dynamics required by eq. 7, RC may be replaced with the E_{max} 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 (E_{max} and EC_{50}) and system parameters (τ) 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: counterregulation, desensitization, up or downregulation, and precursor pool depletion.
Counterregulation 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: Equation 25where the production of M is driven by the primary response (R) as governed by firstorder rate constants of production (k_{1}) and loss (k_{2}). In turn, the net response will reflect the difference, R_{net} =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. k_{out} · (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 proteincoupled 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 (R_{i}) by a firstorder process (k_{d}):
The final two mechanisms of up or downregulation 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 drugreceptorDNA 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 offsetting 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 mechanismbased 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 mechanismbased concepts that capture primary ratelimiting 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 drugtarget interactions are emerging with increasing frequency. Such a model was shown to be useful for characterizing the pharmacodynamics of humanized antiFactor IX monoclonal antibody in monkeys (Benincosa et al., 2000). The inclusion of drugtarget microconstants (k_{on} andk_{off}) provides a flexible means of bridging the pharmacokinetics of drugs and the time course of effects, thereby inferring the temporal pattern of drugtarget 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 targetmediated drug disposition and dynamics, where the drugtarget 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 drugreceptor complex was defined as follows:
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 mechanismbased pharmacodynamic models are reflected in the recognition of the nonlinear drugtarget 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.
Footnotes

↵1 Current address: Gerontology Research Center, 5600 Nathan Shock Dr., Baltimore, MD 21224.

↵2 Current address: Department of Pharmacokinetics and Physical Pharmacy, Jagiellonian University, 9 Medyczna St., 30688 Krakow, Poland.

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
 Received September 26, 2002.
 Accepted February 3, 2003.
 The American Society for Pharmacology and Experimental Therapeutics
References
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 postdoctoral 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 interferonβ and corticosteroids as model compounds. He has been a fellow of the American Foundation for Pharmaceutical Education (2000–2002), was recognized in the Eli Lilly Graduate Symposium in Pharmacokinetics, Pharmacodynamics, and Clinical Sciences at the AAPS annual meeting in Toronto (2002), and was given the Buffalo Pharmaceutics Scholar Award in 2001. Dr. Mager currently is an Adjunct Assistant Professor of Pharmaceutical Sciences at UB and is pursuing postdoctoral training at the National Institute on Aging of the National Institutes of Health (Baltimore, MD) in the laboratory of Dr. Darrell R. Abernethy.
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 RawlsPalmer 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 immunosuppressants. He has over 420 publications.