Pharmacokinetic-Pharmacodynamic Modeling

Pharmacokinetic-Pharmacodynamic Modeling

Archives of Medical Research 31 (2000) 539–545 REVIEW ARTICLE Pharmacokinetic-Pharmacodynamic Modeling: Why? José Pérez-Urizar,* Vinicio Granados-So...

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Archives of Medical Research 31 (2000) 539–545


Pharmacokinetic-Pharmacodynamic Modeling: Why? José Pérez-Urizar,* Vinicio Granados-Soto,* Francisco J. Flores-Murrieta** and Gilberto Castañeda-Hernández* *Departamento de Farmacología y Toxicología, Centro de Investigación y de Estudios Avanzados del I.P.N. (CINVESTAV), Mexico City, Mexico **Sección de Graduados, Escuela Superior de Medicina del Instituto Politécnico Nacional, Mexico City, Mexico Received for publication February 24, 2000; accepted August 16, 2000 (00/036).

At present, pharmacokinetic-pharmacodynamic (PK-PD) modeling has emerged as a major tool in clinical pharmacology to optimize drug use by designing rational dosage forms and dosage regimes. Quantitative representation of the dose–concentration–response relationship should provide information for prediction of the level of response to a certain level of drug dose. Several mathematical approaches can be used to describe such relationships, depending on the single dose or the steady-state measurements carried out. With concentration and response data on-phase, basic models such as fixed-effect, linear, log-linear, EMAX, and sigmoid EMAX can be sufficient. However, time-variant pharmacodynamic models (effect compartment, acute tolerance, sensitization, and indirect responses) can be required when kinetics and response are out-of-phase. To date, methodologies available for PK-PD analysis barely suppose the use of powerful computing resources. Some of these algorithms are able to generate individual estimates of parameters based on population analysis and Bayesian forecasting. Notwithstanding, attention must be paid to avoid overinterpreted data from mathematical models, so that reliability and clinical significance of estimated parameters will be valuable when underlying physiologic processes (disease, age, gender, etc.) are considered. © 2001 IMSS. Published by Elsevier Science Inc. Key Words: PK-PD modeling, Pharmacokinetics, Pharmacodynamics, Data analysis.

Overview It is well recognized that the rate-factor limiting the advance of most sciences is the ability to measure the variable of relevance for a particular discipline. The sentence recorded by Lord Kelvin is very descriptive: “When you can measure what you are speaking about, and express it in numbers, you know something about it: but when you cannot measure it, when you cannot express it in numbers, your knowledge is of a meagre and unsatisfactory kind: it may be the beginning of knowledge, but you have scarcely, in your thoughts, advanced in the stage of science.” The purpose of this article is to offer some basis for the rationale of state-of-art pharmacokinetic-pharmacodynamic modeling from experi-

Address reprint requests to: José Pérez-Urizar, Departamento de Farmacología y Toxicología, Centro de Investigación y de Estudios Avanzados del I.P.N. (CINVESTAV), Apdo. Postal 22026, 14000 México, D.F., México. Phone and fax: (⫹525) 675-9168; E-mail: [email protected]

mental to clinical settings, focusing on factors that permit adequate data analysis. Introduction Pharmacokinetics (PK) is cited as a science dedicated to the study of rate processes such as absorption, distribution, metabolism, and excretion of a drug and the multiple interrelationships affecting same, such as incomplete absorption, saturability in transport, biotransformation, or binding. A prerequisite for all pharmacokinetic investigation is the availability of appropriate and reliable analytical methodology to measure drug and/or metabolite concentrations in various body fluids. Currently, great advances in elucidating ready properties of drugs have been possible due to the measurement of smaller amounts with more sensitive and selective analytical techniques (1). One consequence of such extensive development in pharmacokinetic research is the recognition of high variability in processes due to genetic, environmental, and pathophysiologic factors. In the

0188-4409/00 $–see front matter. Copyright © 2001 IMSS. Published by Elsevier Science Inc. PII S0188-4409(00)00 2 4 2 - 3


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relative absence of information to the contrary, it has been assumed that pharmacokinetic variability is primarily responsible for quantitative interindividual differences in drug response. On the other hand, the usefulness of a drug does not depend upon the concentration circulating systemically, but on the concentration at the active site on the receptor or enzyme. The same receptors may be involved in mediating the desired therapeutic effect but also in expressing an alteration of their state by toxic manifestations. In most of the drugs used in practice, there is a quantitative difference between the concentration needed for therapeutic activity and the maximal concentration tolerable for reasons of toxicity. Thus, pharmacodynamics (PD) is the term chosen for the quantitative relationship between (observed) plasma and/or tissue concentration(s) of the active moiety and the magnitude of the (observed) pharmacologic; this is in contrast to pharmacokinetics (PK), which describes the quantitative relationship between administered doses and dosing regimens and (observed) plasma/tissues levels of the drug. PK-PD modeling is the mathematical description of the relationships between PK and PD. PK-PD modeling allows the estimation of PK-PD parameters and the prediction of these derived, clinically relevant parameters as well. PK-PD simulations allow the assessment of the descriptive parameters as functions of dose and dose rate. These simulations can provide the dose–response curve for onset, magnitude, and duration of effect. This information can be valuable in optimizing dose and dosing regimens (2). Currently, there is growing recognition of the importance of PK-PD studies in all phases of drug development (3–6). In preclinical studies, PK-PD is used to interpret toxicokinetic data (6), and via physiologic modeling and allometric scaling, it is also used to extrapolate results from animals to humans (7,8). During early clinical testing, PK-PD is used to aid in the interpretation of dose–response and escalation studies. In addition, there are several instances in which PK-PD modeling has been used by regulatory agencies to recommend a dose and/or regimen not originally studied as part of the clinical program (9). As in the case of pharmacokinetics, methods to measure pharmacologic effects and bio-mathematical models had to be developed to characterize and evaluate pharmacodynamic processes. Mathematical models can be considered as simplifications of a phenomenon described in terms of an algebraic or differential equation. In the case of PK-PD modeling, it is expected to not only describe, but also predict distinct situations, such as scaling between preclinical to clinical trials, multiple dosing schemes, or different routes of administration (10). To choose the most appropriate PK-PD model, it is essential to identify the significance of the biological processes involved in eliciting a drug-induced response. Eventually, PK processes, biophase distribution, drug–receptor interaction, signal transduction, and secondary postreceptor

events are factors altering the PD behavior of a drug. If we have that information—although only partially—it is possible to link PK and PD with actual physiologic support instead of only abstract numbers. Then, the model-building process involves fitting the available data and the consideration of possible biological differences that usually are translated into inter- and intravariability. In the face of PD variability, it becomes important to identify the useful predictor (covariates) of PD individuality to facilitate individually optimized pharmacotherapy. It is necessary, therefore, to establish very comprehensive patient profiles during the development of studies. Moreover, the study populations must be representative of the target patient population with respect to age, gender, race, and environmental and pathophysiologic characteristics. If these requirements are absent, the relevance and usefulness of covariates may be questionable (11,12). Because of the multiple factors intervening in a PK-PD study, it then appears adequate to divide the modeling project into the following two basic blocks: 1) that concerning the clinical or experimental design by itself, and 2) the data analysis. Diverse models have been suggested to describe the PK-PD relationship depending upon the nature of drug administration scheme (single doses, multiple doses, long-term infusions, etc.) and the time dependency of PD parameters (described later). Thus, when the system is kinetically at steady state, i.e., the concentrations of the active moiety at the active site are constant (after long-term infusions or multiple doses), relatively simple models are needed to characterize the PK-PD relationship. Otherwise, after single doses (nonsteady-state condition) and when time variant PD parameters are present, more complex models are needed to account for phenomena involved in the PK-PD relationship. Approaches such as disequilibrium between biophase and plasma compartment (13), appearance of active metabolites (14,15), indirect mechanisms of action (16,17), sensitization, and tolerance (18–20) have been proposed to explain the apparent dissociation between time courses of concentration and effect. Recently, the combination of powerful nonlinear, mixedeffect regression models, statistically robust software tools, and the integration of pharmacokinetic-pharmacodynamic knowledge has permitted optimization of the decision process in therapeutic management. By incorporation of previous information into these systems, Bayesian forecasting certainly promises the more adequate individualized therapy for a particular patient.

The PK-PD Study Design Pharmacokinetic issues. The plasma concentration timecourse of a drug is determined by the pharmacokinetic processes of distribution, metabolism, and excretion as well as absorption, when the administration pathway is not sys-

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temic. Pharmacokinetic models developed at present include compartmental, physiologically based, and statistical. Notwithstanding, compartmental models are preferred because they are relatively easily linked to most pharmacodynamic models. Measurements of the active compound, i.e., total, and, when necessary, the free fraction, active metabolite, or active enantiomer must be performed with fully validated analytic methodologies (21–23). When experimental studies are carried out, it is strongly advised that the number of measurements (concentration, but also effect) be as large as possible to obtain the greatest precision in parameter estimation. Additionally, the concentration (and doses) must cover a range sufficiently large and clinically meaningful enough to optimally describe the shape of the PK-PD relationship. This goal can be achieved by studying the concentration and effects after administration of two or three doses in the same individual, or by the administration of a large dose with measurements performed at a very different concentration. The risk of adverse effects and great variability may limit this approach (24). Otherwise, when the measure of concentration (and/or effect) cannot be represented on a regular predefined scheme or is scarce, i.e., in clinical and phase III studies settings, the population approach has rapidly become accepted as a feasible choice. It may explain some of the wide variability observed in drug response, considering that behavior of drugs in population subgroups may be influenced by specific factors such as age, weight, clinical status, etc. (12,25–27). Pharmacodynamic issues. In PK-PD modeling, the availability of realistic measures of pharmacologic response intensity is a basic consideration. This implies that measured effect must be clinically relevant and directly driven by the drug administration. Such measures should ideally meet the following criteria: continuity, objectivity, sensitivity; repeatability, and validity. It is emphasized that pure surrogate end points are useful only if they provide substantial information (28). On the other hand, a drug effect may be considered as any drug-induced change in a physiological parameter when compared to respective predose or baseline value. Pharmacodynamic analysis involves quantifying drug concentration/effect relationships. Ideally, concentrations should be measured at the effect site, the site of action or biophase, where the interaction with the respective biological receptor system takes place, but in most situations, this is not possible. However, frequent sampling is from fluids (often plasma) other than those corresponding to biophase, and in a clearly nonsteady state regimen. Then, the first action during analysis is to explore the shape of time profiles of concentration and effect with special attention to possible delays between them. In the plotting of drug effect vs. concentrations, connecting data in chronological order may confirm the appearance of a hysteresis loop, in which one effect corresponds to more than one concentration. If such a phenomenon were not present, basic pharma-


codynamic models could be used. Otherwise, more complex, time-dependent models must be employed (10). PK-PD Models at Steady State When the concentrations of the active moiety at the site of action are constant and the PD parameters are time-invariant, the system is said to be kinetically at steady state. This is achieved with long-term intravenous infusions or multiple-dose regimens. Several basic PK-PD models have been used to describe the relationship between concentration and effect. Fixed effect model. Also known as the quantal effect model, this is a statistical approach based on a logistic regression analysis. It relates a certain drug concentration with the statistical likelihood of one (or several) effects to be present or absent. For instance, the clinical effects of general anesthetics are quantal (response vs. no response) rather than continuous. In this situation, logistic regression has been used to derive concentration vs. response/no response relationships and estimation of the concentration with 50% probability of no response (29). Also, it can be anticipated that a threshold concentration varies among patients. Therefore, the probability of a certain level of effect at one concentration will be a function of the concentration distribution in the population (30). Major limitations of this model deal with the prediction of complete effect-time profiles. Linear model. This model assumes a direct proportionality between drug concentration and drug effect and can be expressed as Eq. (1): E = S ⋅ C + E0


where S, the slope, and E0, the intercept, are the two parameters of the model. Pharmacodynamically, S represents the effect induced by one unit of C, and E0 represents the value of E when the drug is absent. This model only applies to measured effects with physiological baselines such as blood glucose, blood pressure, etc. Linear model is a particular case of the EMAX model (see later) when the concentration is much lower than EC50; therefore, S ⫽ EMAX/EC50. Friedman et al. (31) found a linear relationship between central activity and diazepam plasma levels. Obviously, the advantage of this model resides in that parameter estimation is easily performed by linear regression. However, this model erroneously assumes that the effect can increase with concentrations without limits. Log-linear model. This model was conceived on the observation that when the concentration-effect is hyperbolic, the log-concentration-effect relationship is roughly linear in the range of 20 to 80% of maximal effect. Therefore, it can also


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be viewed as a particular case of the EMAX model. The loglinear model is givin in Eq. (2): E = S ⋅ log C + E 0


where S and E0 have the same interpretation as in the linear model. Notwithstanding, in this case, S represents the change elicited by one unit of log C. Obvious disadvantages of this model are that it is neither possible to predict the value of E when C ⫽ 0, nor to predict a maximum effect. Nonetheless, this model has been used to successfully predict the pharmacologic activities of beta-blocking agents (32). EMAX model. This model was originally derived from the classical theory of drug-receptor interaction. In this model, effect is related to concentration as in Eq. (3): E MAX ⋅ C E = --------------------- + E0 EC 50 + C


where EMAX is the maximum effect possible, EC50 is the concentration necessary to produce 50% of EMAX, and E0 is the basal value E. The EMAX model describes the concentration-effect relationship over a wide range of concentrations, from zero effect in absence of drug to the maximum effect at concentrations greater than EC50. This is a simple PK-PD model broadly used to characterize a myriad of pharmacologic effects (33–35). Sigmoidal EMAX model. This model is a generalization of the EMAX model. It is based on oxygen-hemoglobin dissociation kinetics (36). This model remarks Eq. (4): γ

E MAX ⋅ C -γ + E 0 E = ------------------------γ EC 50 + C


with EMAX, EC50, and E0 meaning the same as in the EMAX model, and ␥ represents a sigmoidicity factor or steepness of the curve: ␥ ⫽ 1 for the hyperbolic curve (EMAX model); ␥ ⬎1 for a steeper curve, and ␥ ⬍1 for a smoother curve. The additional parameter often allows fitting more conveniently different types of PK-PD data (37–40).

are more sustained than plasma concentrations. Several PD mechanisms have been proposed. If distribution into biophase is rate-limiting, a kinetic biophase compartment can be used to explain the disequilibrium between observed concentrations and effects (13). An indirect mechanism of action, for example, inhibition or stimulation of the synthesis or degradation of endogenous products, can also become rate-limiting, and results in anticlockwise hysteresis (16, 17). Finally, delayed appearance of active metabolite(s) interacting with the same receptors as the parent compound, at steady state or not, also produces anticlockwise hysteresis (14,15). On the other hand, a clockwise hysteresis means that effect decreases quicker than plasma concentration. This reveals a tolerance phenomenon to the drug, either by development of a counterregulatory phenomenon or by desensitization of receptors (41,42). As will be shown, it is possible to develop PK-PD models that can account for both under the nonsteady-state condition or when PD parameters are time-variant to predict the time-course of drug action Effect compartment model. An important conceptual advance in indirectly linking PK and PD models was to realize that the time-course of the effect itself can be used to define the rate of drug movement to the effect site as expressed in the effect compartment model proposed by Sheiner (13). This concept considers a hypothetical effect compartment modeled as an additional compartment of a PK compartment model, and represents the active drug concentration at the effect site. It is linked to the kinetic model by a first-order process, but receives a negligible mass of drug. Therefore, the rate constant for the transfer between the central and effect compartments is also negligible. Consequently, the time-dependent aspects of the equilibrium between plasma concentration and effect are only characterized by the first-order rate constant of ke0. This constant describes the irreversible disappearance of the drug from the effect compartment and is in equilibrium with the first-order rate constant of entrance to effect compartment k1e (thus canceling this). Such events are mathematically defined by the following differential equation [Eq. (5)]: dCe ⁄ dt = k e0 ⋅ ( Cp – Ce )

Nonsteady-State and Time-Dependent Models Nonsteady-state and time-dependent PK-PD models. Under nonsteady-state conditions, i.e., after single doses as well as when time-dependent changes in PD parameters are present, basic PK-PD models are unable to explain concentration– effect relationships. When the data points are connected in chronological order, the manner in which they appear allows defining the presence of a hysteresis loop, i.e., the time-courses of concentration and effect are out of phase. The hysteresis loop may turn in a clockwise or in an anticlockwise sense (30). If an anticlockwise curve is present, i.e., the effect rises slowly, they reach their peak later, and


where Cp and Ce represent the concentrations in the plasma and the effect compartment, respectively. Although semiparametric and nonparametric strategies have been proposed to estimate Ke0 and pharmacodynamic parameters (43–45), the parametric nonlinear regression for simultaneous estimation of Ke0 and PD parameters appears to be the most used approach. In this strategy, a pharmacokinetic model is fitted to Cp vs. time data for obtaining PK parameters, thus used simultaneously with effect-time data by a PD model to obtain ke0 and PD parameters. This approach has been successfully employed to predict the PK-PD relationship of diverse drugs (13,46–49).

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Physiological indirect response models. The observed delay between the kinetics of plasma concentrations and the kinetics of effect may also be explained by an indirect mechanism of drug action (16,17). Indirect response models are models for drugs whose mechanism of action consist of either inhibition or stimulation of a physiologic process involved in the elaboration of the clinical expression of the observed effect. Therefore, if the mechanism is at least partially known and understood, the PK-PD link may evolve from the abstraction of numbers to a physiologic mechanism-oriented tool. Considering the last point, the rate of change of the response over time in the absence of drug can be described by a differential equation [Eq. (6)]: dR ⁄ dt = k in – k out ⋅ R


where kin represents the zero-order constant for the production of the response and kout the first-order rate constant for loss of response. Depending on whether kin or kout are either inhibited or stimulated by the drug, four different submodels have been developed in which the drug effect is mediated by a EMAX-like model. These equations [Eq. (7–10)] are the following: dR ⁄ dt = k in ⋅ I ( t ) – k out ⋅ R, for the inhibition of k in


dR ⁄ dt = k in – k out ⋅ I ( t ) ⋅ R, for the inhibition of k out


dR ⁄ dt = k in ⋅ S ( t ) – k out ⋅ R, for the stimulation of k in


dR ⁄ dt = k in – k out ⋅ S ( t ) ⋅ R, for the stimulation of k out


These consider that I(t) and S(t) are inhibitory and stimulatory functions, respectively, which can be described in terms of Eq. (11–12): C(t ) I ( t ) = 1 – --------------------------IC 50 + C ( t )


E MAX ⋅ C ( t ) S ( t ) = 1 + ---------------------------EC 50 + C ( t )


with IC50 and EC50 being the concentrations eliciting 50% of inhibition or the maximum effect (EMAX), respectively. These physiologic indirect response models have been employed in a variety of studies regarding biological responses such as muscle relaxation, synthesis and secretion, mediator flux, cell trafficking, enzyme induction, or inactivation, among others (17,50,51). Data analysis. A successful PK-PD modeling project is dependent on performing a well-designed study. The selection of doses to be administered and the times that plasma samples and responses should be measured form an intrinsic part of the design step. After data are gathered, it is necessary that an exploratory (graphic) data analysis be performed to confirm


or suggest modifications to the tentative model(s). The next step is the selection of a model and the fitting of that model to experimental data using regression analysis. This is effected to estimate the model parameters and the precision of parameter estimates. Evaluation of goodness-of-fit, correlation between parameters, residual analysis, parameter accuracy, and challenging with new (or reviewed) data will provide validity to the model (10,52,53). On the other hand, PK-PD models are usually complicated by the need to jointly consider the time-course of drug concentration, nonlinear equations that relate effect to concentration, and the usual requirement of two or three dose levels of drug. Thus, the use of computers for fitting experimental data is essential (54). Individual or Population Approach? Once the structure of the model has been chosen, estimation of population mean values of parameters as well as their interindividual variability takes place. There are two ways to confront this: 1) when individual response data are available (limited number of individuals in the same design and number of measurements as large as possible), and 2) when data derive from studies performed on a greater number of individuals in a different design usually with sparse and unbalanced data. Estimation procedure for the first situation is based on the so-called standard two-stage method. This supposes a first stage in which the parameters of each individual are estimated by minimizing an objective function and assuming identical effect variances. Using those estimates, the second stage calculates population mean values and interindividual variability of parameters. Nevertheless, the assumption that estimates obtained during the first stage are equal to true values of the parameters may induce an overestimation of the variability of the parameters in the population. A large number of experiments with minimal error in measurements is required to avoid such limitations in the two-stage method (24). Observational data, often characterized by a lack of control over frequency and timing sampling, require a quite different approach. Nonlinear mixed-effect regression models are the basis of population approach. This approach permits the simultaneous analysis of all data in the studied population, using pharmacokinetic or pharmacodynamic models to describe typical trends (population means) and individual profiles. In these models, the parameters of each individual are modeled in terms of both fixed and random effects. Fixed effects account for large intra- and interindividual differences in the value of parameter and covariates. Random effects are applied to account for unexplained inter- and intraindividual variability. General representation of these models is as follows in Eq. (13): y ij = f ( φ i, D, t ij ) + ε ij


where yij is the pharmacokinetic or pharmacodynamic data at


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jth time in the ith individual, f is the structural model composed of ␾ij, a vector of pharmacokinetic or pharmacodynamic parameters of the ith individual, D is the dose of the drug administered, and ␧ij represents the residual error. On the other hand, ␾i, the vector of individual parameters, may be expressed as ␾i ⫽ g(zi, 0) ⫹ ␩i, where g is a structural-type model, which is a function of fixed effects zi (gender, age, weight, etc.) and fixed effects parameters, 0. ␩i is the interindividual variability affecting the vectors ␾i (55). At present, a variety of software strategies has been proposed to deal with population data but it appears that there is no ideal software. Recently, Aarons (56) pointed out that the numerical and statistical reliability of the methods used, the ease-of-use of the software, and the level of support available may be the most important characteristics. NONMEM® (Non Linear Mixed Modelling, University of California, San Francisco, CA, USA) software is one of the most commonly used programs to build population PK-PD models. Based on this, Mandema et al. (27) suggested an efficient strategy to generate a PK-PD population model using the generalized additive model with functional representation of covariates.






8. 9. 10.


Conclusions and Perspectives A primary objective of PK-PD modeling is to identify key properties of a drug in vivo; this allows characterization and prediction of the time-course of drug effects under physiologic and pathologic conditions. In terms of application, PK-PD modeling has been proposed as relevant in all phases of drug development. In the preclinical phase, during the first-ever-inhuman studies (Phase I), efficacy and safety studies (Phase II and III), and in postmarketing trials (Phase IV), in which the earlier PK-PD approach is incorporated, both a reduction in the number of failed studies and an improvement in the interpretation of data are achieved. Consequently, the application of PK-PD modeling is assuming increasing importance in drug approval processes. On the other hand, in the designing of rational dosage forms and dosage regimes, PK-PD modeling promises to become a routine tool for optimizing drug use. Thus, to accomplish any of the previously mentioned aims, well-carried-out and well-validated PK-PD methodologies are needed. In this review, different PK-PD modeling approaches as descriptive and predictive tools to characterize this timecourse of pharmacologic drug action are presented.










Acknowledgments José Pérez-Urizar is a CONACYT Fellow. 21.

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