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 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:Cp=fθPK,X,t Equation 1where θ_PK is a vector of PK parameters determined by model fitting, X represents independent variables associated with the given dose and/or regimen, andt is time. If plasma concentrations are assumed to be proportional to biophase concentrations (C_e), then these expressions, either explicit or differential equations, are often fixed and serve as driving functions in PD models:R=fθPD,Cp or Ce,Z Equation 2where R is the pharmacological response (often abbreviated E for effect or R for response, and are used interchangeably throughout this review) and Zrepresents 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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Figure 1

Basic components of pharmacodynamic models of drug action.

Symbols are defined in text. Adapted from Jusko et al. (1995).

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 (E_max). 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.

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):E=m·log A+e Equation 3where m and e are the slope and intercept terms. Levy (1964, 1966) derived a relationship between monoexponential drug pharmacokinetics (C_p =C₀ · e^{−k · t}) and the time course of in vivo effects from a rearrangement of eq. 3:E=E0−mk2.303t Equation 4where 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-logC_p curve, this simple model essentially introduced the application of linear and log-linear models to in vivo data:E=E0±S·Cp Equation 5 Equation 6where 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 (E_max), and as such, cannot be extrapolated to capture the capacity orE_max 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:dRCdt=kon·RT−RC·C−koff·RC Equation 7where R_T is the maximum receptor density, C is the drug concentration at the site of action,k_on is a second-order association rate constant, and k_off is a first-order dissociation rate constant. If one assumes equilibrium conditions, eq.7 can be rearranged to yieldRC=RT·CKD+C Equation 8where K_D is the equilibrium dissociation constant (k_off/k_on). A further assumption is that the drug effect is directly proportional to the fraction of occupied receptors, such that E = α · RC. Substituting in the relationship of eq. 8, one of the forms of the Hill equation is derived, commonly referred to as theE_max model: Equation 9where E_max is substituted for (α · R_T) and the EC₅₀ is a sensitivity parameter representing the drug concentration producing 50% ofE_max. 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 E_max model, includes an additional parameter, γ, which is a slope term that reflects the steepness of the effect-concentration curve: Equation 10(The actual slope of a plot of E versus logC_p isE_max · γ/4 over the region of 20 to 80% effect). Whereas γ 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: E_maxand EC₅₀ are unchanged as γ is varied). The rationale for the power coefficient is usually uncertain but may be caused by either positive (γ > 1)/negative (γ < 1) cooperativity or homogeneity (γ > 1)/heterogeneity (γ < 1) in receptor/effector functions (Hoffman and Goldberg, 1994). The inhibitory and excitatory sigmoid E_maxmodels 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: Equation 11where K_E is the RC value producing half-maximal 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 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.

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:dCedt=k1e·Cp−ke0·Ce Equation 13where k_1e andk_e0 are first-order distribution rate constants (usually set equal to each other for lack of identifiability). A sigmoid E_max model (eq. 10) is often used to correlate effect-compartment concentrations with the intensity of pharmacological effect (C_e is substituted forC_p). 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. TheC_e profile is thus inferred by the equilibrium delay and is a function of the drug PK andk_e0 parameter. Smallerk_e0 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 peakC_e 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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Figure 2

Biophase distribution model.

Response curves were simulated using eqs. 13 and 9, driven by monoexponential drug concentrations: C_p= C⁰ ·e(−k · t). Increasing doses were achieved by letting C⁰equal 10, 100, and 1000 units. Parameter values werek = 0.3 h⁻¹,E_max = 100 units, EC₅₀= 10 units, and k_1e =k_e0 = 0.1 h⁻¹.

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: 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 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: 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 first-order 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 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:fC=k·C or Kmax·CKC50+C Equation 17a,bwhere k and K_max are second-order cell-kill rate constants and KC₅₀ is the drug concentration producing 50% ofK_max (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 (R_s) and quiescent groups (R_r) (Jusko, 1973). The cell-proliferation and irreversible drug-effect 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 first-order 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 β-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: Equation 19where k_out is a first-order loss rate constant, k_in represents an apparent zero-order production rate of the response (often set equal to the product of k_out 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 =k_in/k_out= R⁰). 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 B₂concentrations 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 5α-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: 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 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) whereH₁(C_p) is subtracted andH₂(C_p) = 0, II (inhibition of dissipation) whereH₁(C_p) = 0 andH₂(C_p) is subtracted, III (stimulation of production) whereH₁(C_p) is added andH₂(C_p) = 0, and IV (stimulation of dissipation) whereH₁(C_p) = 0 andH₂(C_p) 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 (k_in/k_outor R⁰) as drug concentrations decline below the IC₅₀ or SC₅₀values. The time to peak effect is dose-dependent and occurs at later times for larger doses owing to increased duration whenC_p > 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).

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Figure 3

Basic indirect response models.

Response curves were simulated using eq. 20, driven by monoexponential drug concentrations: C_p =C⁰ ·e(−k · t). Increasing doses were achieved by letting C⁰equal 10, 100, and 1000 units. Parameter values werek = 0.3 h⁻¹,E_max = 1 (models I and II) and 5 (models III and IV), EC₅₀ = 10 units,R⁰ = 50 units, andk_out = 0.1 h⁻¹(k_in =k_out ·R⁰).

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: Equation 21a,bwhere k₀ is the apparent zero-order production rate of the precursor (P),k_p is the first-order rate constant of production of the response marker, andk_s 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 ofk_p) whereH₁(C_p) = 0 andH₂(C_p) is subtracted (inhibition) or added (stimulation), and VII and VIII (inhibition or stimulation of k₀) whereH₁(C_p) = is subtracted (inhibition) or added (stimulation) andH₂(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 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-α concentrations by susalimod (VII) (Gozzi et al., 1999).

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Figure 4

Precursor-dependent indirect response models.

Symbols are defined in text.

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 thek_in parameter replaced with a time-dependent 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 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 (k_in), circulate and survive for a specific duration of time (T_R), 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: Equation 23where the baseline condition is constant (R⁰ =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, response-time profiles reveal an apparent zero-order 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 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 [H_n (C_p) 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: 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 EC₅₀) 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).

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Figure 5

Time-dependent transduction model.

Response curves were simulated using eq. 24 (n = 3), driven by monoexponential drug concentrations:C_p = C⁰· e(−k · t). RC was replaced with E* given by eq. 9. Parameter values wereC⁰ = 100 units, k = 0.3 h⁻¹, E_max = 100 units, EC₅₀ = 10 units, and τ = 5 h.

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: Equation 25where 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, 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 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 (R_i) by a first-order process (k_d):dRidt=kd·R−Ri Equation 26This type of equation allows tolerance to evolve in relation to the primary response, but delayed by the operation of thek_d 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: dRC′dt=k3·RC−k4·RC′dRdt=−kon·R·C+koff·RC+k4·RC′ Equation 27a,b,cwhere R represents free receptor concentrations, RC′ is an inactive complex, and k₃ andk₄ 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 (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 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:dRCdt=kon·Cp·R−kt·RC Equation 28where RC is formed via an irreversible second-order rate constant (k_on), andk_t 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-β1a. 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.