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Population Pharmacokinetics and Pharmacokinetic-Pharmacodynamics in Clinical Pharmacology

  • Daniel F. B. WrightEmail author
  • Chihiro Hasegawa
  • Hesham S. Al-Sallami
Living reference work entry

Abstract

Clinical pharmacology is a broad professional and scientific discipline concerned with all aspects of drug use in humans. One of the primary goals of this field is to improve health outcomes by supporting the development, rational use, and safety of medicines. Clinical pharmacology and pharmacometrics are closely related and share common goals and research themes. Notable amongst these are pharmacokinetics and pharmacodynamics. In the pharmaceutical industry, population pharmacokinetic and pharmacokinetic-pharmacodynamic studies aid dose selection, assess links between drug exposure and efficacy and safety metrics, and inform the dosing information that will be presented on the drug label. In the clinical environment, population pharmacokinetic and pharmacokinetic-pharmacodynamic studies are conducted to aid dose optimization for an individual patient. The aim of this chapter is to present an overview of population pharmacokinetic and pharmacokinetic-pharmacodynamic concepts and methodology as they apply in the industrial and clinical setting. The chapter is divided into four parts: Part 1 will provide a board overview of general concepts and definitions related to population pharmacokinetic and pharmacokinetic-pharmacodynamic analyses, Part 2 will look at commonly used models, Part 3 will explore methodology, primarily nonlinear mixed effects modeling, and Part 4 will present examples of pharmacokinetic and pharmackinetic-pharmacodynamic analyses, presented in the style that is typical for a regulatory submission involving phase I data.

In the pharmaceutical industry, population pharmacokinetic (PK) and pharmacokinetic-pharmacodynamic (PKPD) studies play an important role in the analysis of Phase 1, 2, and 3 data, and to a lesser extent, data that arises from preclinical and post-marketing trials. The goals of PK and PKPD analyses align with those of the broader clinical pharmacology team, i.e., to aid decisions about dose selection for the prospective drug, to assess efficacy and safety data, and ultimately, to inform the dosing information that will be presented on the drug label. Post-marketing, in the clinical setting, PK and PKPD are often viewed as an applied field within the medical discipline of clinical pharmacology, focused primarily on optimizing drug therapy for individual patients. In both cases, the types of analyses conducted may extend beyond population PK and PKPD to include exposure-response (ER) analyses, quantitative systems pharmacology modeling (QSP), physiological-pharmacokinetic (PBPK) modeling, and optimal design work, amongst others. These approaches are bound together by the application of mathematical and statistical methodologies, referred to collectively as “pharmacometrics.”

In this chapter, we assume that pharmacometric analyses performed in the industrial or clinical setting will share broadly similar goals, particularly with regards to optimizing drug doses. The only assumed difference is that the research conducted in the drug industry will tend to focus on the optimization of dose to the level of regulatory requirement (e.g., dosing guidance for the drug label) while in the clinical environment, the focus is usually dose optimization for an individual patient. The aim of this chapter is to present an overview of population pharmacokinetic and pharmacokinetic-pharmacodynamic concepts and methodology as they apply in the industrial or clinical setting. The chapter is divided into four parts: Part 1 will provide a board overview of general concepts and definitions related to population PK and PKPD analyses, Part 2 will look at commonly used PK and PKPD models, Part 3 will explore methodology, primarily nonlinear mixed effects modeling, and Part 4 will present an example of PK and PKPD analyses, presented in the style that might be expected for a regulatory submission involving phase I data.

Part 1: General Concepts and Definitions

Clinical pharmacology is a broad professional and scientific discipline concerned with all aspects of drug use in humans. One of the primary goals of this field is to improve health outcomes by supporting the development, efficacious use, and safety of medicines. While a diverse discipline, this chapter will focus primarily on two central themes in clinical pharmacology research, pharmacokinetics and pharmacodynamics. Here pharmacokinetics is defined as the time course of drug concentrations in the body, a science focused on the relationship between the drug dose and exposure (Fig. 1a). The generic term “drug exposure” will be used to refer to drug concentration at any time point, i.e., C(t). Note that it is convenient and common to summarize C(t) as a time-invariant measure such as the maximum, minimum, or steady-state average plasma concentration post dose (\( {C}_{p_{max}} \), \( {C}_{p_{min}}, \) and \( {C}_{p_{ss, ave}} \), respectively), as well as the area under the plasma concentration time curve (AUC). Pharmacodynamics is concerned with the relationship between drug exposure and response (Fig. 1b) and is independent of time. Pharmacokinetics-pharmacodynamics (PKPD) is a combination of PK and PD that allows us to explore the time course and magnitude of drug response (Fig. 1c).
Fig. 1

Conceptual framework for PK, PD, and PKPD data (Adapted from Wright et al. 2011, used with permission from John Wiley and Sons licence number 4277280083455)

In this chapter, we will consider pharmacodynamic responses that are quantified by measuring a biomarker or surrogate (e.g., blood pressure, prothrombin time to measure anticoagulation), rather than dichotomous measures of drug effect (e.g., seizure or no seizure) or other clinical endpoints (e.g., death, hospitalization, etc). A definition of a pharmacodynamic biomarker will be presented below.

Pharmacometrics

Pharmacometrics is concerned with the analysis and interpretation of data that arises from drug studies. The discipline can be traced to the seminal work of Louis Sheiner (Sheiner 1977) and, in particular, his collaboration with Stuart Beal to develop nonlinear mixed effects modeling methodology for drug studies (Sheiner and Beal 1980). Pharmacometrics involves the use of mathematical and statistical models to predict drug exposure, physiological response, and clinical outcomes, and to describe the variability in these measures between (and within) individuals. As such, pharmacometric analyses play an important role in drug development as well as clinical practice. The terms “population analysis,” “population approach,” and “population modeling” are also commonly used to refer to pharmacometric analyses.

Optimizing the Dose

All things are poison and not without poison; only the dose makes a thing not a poison

Paracelsus (physician and botanist, 1493–1541)

The notion that a relationship exists between the amount of drug given to an individual and the intensity of the resulting response (desired or adverse) is intuitive and has likely been understood since antiquity. It was certainly recognized by Paracelsus in the sixteenth century who noted, in the above quote, that a poison and a therapeutic agent differ only in the dose that is administered. However, it was not until the last century that it was possible to quantify the relationship between drug dose and physiological response.

Research intended to inform dose selection or optimization, whether to support drug labeling decisions for regulatory submission or to individualize therapy in the clinic, requires a quantitative understanding of drug exposure and physiological response, and how these vary between and within individuals. The optimal dose is expected to have the highest probability of achieving a desired physiological response in an individual patient or a population of patients while carrying a minimal risk of adverse effects. The general concept is that the drug will achieve the desired physiological response in most people once a threshold exposure (or steady-state plasma concentration) has been reached. Increasing the exposure may increase the magnitude of the response in some cases, depending on the shape of the exposure-response curve (see Fig. 3), but may also increase the risk of adverse effects and toxicity. This concept underpins the clinical practice of therapeutic drug monitoring, where plasma concentrations are measured and doses adjusted to achieve a specified target range. TDM is particularly useful for drugs with a narrow therapeutic range, i.e., where the desired physiological response and toxicity can occur within a narrow exposure range, and where the physiological response is difficult to measure, i.e., cases where the clinical endpoint is to prevent an adverse event (e.g., seizure).

Biomarkers for Physiological Response

The ideal measure of drug effectiveness is the consistent achievement of the clinical outcome of interest in the intended patient population. However, when the outcome is the long-term prevention of an adverse clinical event (e.g., stroke with anticoagulant therapy or cardiovascular disease with lipid-lowering drugs) or requires long-term observation (e.g., the cessation of gouty attacks with urate-lowering therapy), a biomarker for the physiological response is required.

A biomarker is an indicator of the biological, pathological, or pharmacological response to drug therapy (Biomarkers Definitions Working Group 2001). The biomarker may be on the causal pathway between the disease and the clinical outcome or may be correlated in some way with the outcome. The drug may alter the underlying disease which in turn will alter the biomarker or the drug may act directly on the biomarker itself which, in turn, will be correlated with the clinical outcome (e.g., symptomatic treatment).

The term biomarker can be distinguished from “biomeasure” and “surrogate.” A biomeasure is any physiological measurement that can be obtained clinically (e.g., blood pressure). A biomarker is therefore a special case of a biomeasure. A surrogate is a biomarker that is intended to substitute for a clinical outcome. A surrogate is usually approved by a regulatory agency and can be used in the drug approval process.

Part 2: Pharmacokinetic and Pharmacokinetic-Pharmacodynamic Models

What Is a Model?

Essentially, all models are wrong, but some are useful

George E. P. Box (statistician, 1918–2013)

A model is a construct that allow us to simplify reality. A toy car is a trite example. It is small enough to fit into a child’s hand yet, to a child, it retains all of the essential features of the real thing such as the right body shape, wheels, and perhaps a flashy paint job. Similarly, a pharmacometric model is a mathematical representation of the complex interaction between a drug and a biological system. It is used to describe the relationship between input variables, such as drug dose and time, and output variables including plasma drug concentration and drug response.

For a pharmacometric model to be useful, it must be simple enough for practical use yet retain the essential mathematical features that allow us to understand the relationship between input (e.g., drug dose, time) and output (e.g., plasma drug concentrations, drug response). This concept is embodied in the often-used quote above from George Box. All models are wrong because a model cannot recreate reality; however, a model can serve a useful purpose if it retains a close relationship with important aspects of reality.

The term “population model” is commonly used in pharmacometrics and may seem misleading. In the context of a pharmacometric analysis, “population” refers to the group (population) of individuals who are the intended recipients of the medicine. This may also include healthy volunteers who may be recruited, for example, in a Phase I pharmacokinetic study. A population model will, therefore, provide typical drug exposure and response information, estimates of random between subject variance, predictable variance in drug exposure or response associated with measurable patient factors (e.g., weight), and random within subject variance. Hence, the information contributed by an individual patient is retained and contributes to the overall understanding of drug behavior.

Pharmacokinetic Models

Pharmacokinetics is the science that relates the dose administered to the time course of measured drug concentrations in the body (usually the plasma) and therefore provides information about drug exposure. A PK model can predict the typical time course of drug concentration C(t) as a function of the administered dose (D) over time (t) and is dependent on unknown pharmacokinetic parameters (θpk). The primary PK parameters of interest are clearance (CL), the apparent volume of distribution (V), the absorption rate constant (ka) (for extravascular administration), systemic availability (F) (for extravascular administration), and the secondary parameter elimination rate constant (ke), which is given by CL/V. CL is a constant that relates the rate of elimination to the measured drug concentration and is related to the functional capacity of the body. V is the apparent volume into which the drug distributes and is related to body composition. F is the fraction of drug that reaches the systemic circulation.

A PK model can be constructed using a compartmental structure with an input model and a disposition model (Fig. 2). The input model describes the time course of drug movement from the site of administration (e.g., the gut) to the site of drug measurement (e.g., the plasma). The disposition model describes the time course of drug distribution, metabolism, and elimination from the body and can be depicted as single or multiple compartments. Input, distribution, and elimination occur simultaneously and the relative time course of each will determine the pharmacokinetic behavior of the drug.
Fig. 2

Schematic of a one compartment PK model (a) and a two compartment PK model (b) for an orally administered drug with first-order absorption and elimination

Figure 2 depicts one and two compartment PK models for an orally administered drug. The body is represented as a series of discrete units into which the drug distributes and from which drug elimination occurs. The compartments do not represent true physiological spaces, and it is assumed that drug behavior is similar within each compartment. In a one compartment model (Fig. 2a), the drug is assumed to distribute evenly throughout a single compartment. In a two compartment model (Fig. 2b), the drug distributes into an additional peripheral compartment.

For a one compartment extravascular administration model, the concentration of drug in the plasma at any time (after a single dose) is given by

Equation 1

The equation for a one compartment PK model with first-order absorption and elimination
$$ C(t)=\frac{D\times F\times {k}_a}{V\times \left({k}_a-{k}_e\right)}\times \left({\mathit{\exp}}^{\left(-{k}_e\times t\right)}-{\mathit{\exp}}^{\left(-{k}_a\times t\right)}\right) $$

By convention the number of compartments in a PK model is defined by the number of exponential terms needed to describe the disposition of the data. Therefore, the model above, while having two exponential terms, has only one term related to disposition and hence is considered a one compartment model.

Pharmacodynamic Models

Pharmacodynamics relates drug concentrations to the observed pharmacological response. Since in vitro pharmacodynamic experiments are often conducted in equilibrium conditions, the relationship between concentration and response is typically independent of time. A PD model can predict the typical response (denoted E for effect) of a drug as a function of the drug concentration (C) and unknown pharmacodynamic parameters (θpd). Note that we use the terms 'drug effect' and 'drug response' interchangebly in this chapter.

The relationship between drug concentration and response is usually characterized by a hyperbolic function, which yields a nonlinear relationship between effect and concentration. Hence, doubling the concentration (or dose) will not necessarily result in a doubling of drug response and the drug response will asymptote to a maximum despite increased concentrations.

Much of the underpinning theory, and the models used to describe the exposure-response relationship, is based on receptor-binding theory. However, in pharmacometrics, we are primarily concerned with the magnitude of drug response that will result from a given dose or exposure. . Therefore, a model that will predict drug response, not drug binding is required. The relationship between receptor occupancy and drug effect can be described using a proportionality constant known as ‘intrinsic activity’, which can range from 1 for a full agonist to 0 for an antagonist. This enables the relationship between concentration and effect to be expressed using an empirical version of the Emax model:

Equation 2

The E max model
$$ E={E}_{max}\times \frac{C}{C_{50}+C} $$

where Emax is the maximum effect of the drug, C is the drug concentration, and C50 is the drug concentration which results in 50% of maximum response.

The Emax model has been widely applied in pharmacodynamics and is the basis for many models in pharmacometrics. There are two parameters of interest: Emax and C50. A generalization of the Emax model, called the sigmoidal Emax, includes an empirical exponent termed the Hill coefficient (λ) which changes the shape of the exposure-response curve (Fig. 3):
Fig. 3

The Emax model showing the change in shape with different Hill coefficients, which will dictate the shape of the curve. In this model Emax = 1, C50 = 0.025 units/L

Equation 3

The sigmoidal E max model
$$ E={E}_{max}\times \frac{C^{\lambda }}{{C_{50}}^{\lambda }+{C}^{\lambda }} $$

Values of λ greater than 1 produce a steep exposure-response curve and predict that relatively small changes in drug concentration will produce a rapid change in effect which, at the extreme, can be observed as an “on-off” phenomenon (e.g., anti-arrhythmic agents). Values of λ less than 1 produce a shallow concentration-response curve where response increases rapidly at low concentrations but approaches the Emax asymptote slowly at high concentrations (Fig. 3).

The Emax model can be further modified to include baseline physiological status (S0) in the absence of drug. This model can be used to predict the change in this status (e.g., blood pressure) after drug administration.

Equation 4

The Emaxmodel with a constant baseline status (S0)
$$ E={S}_0+{E}_{max}\times \frac{C}{C_{50}+C} $$

Pharmacokinetic-Pharmacodynamic Models

A PKPD model can predict the typical response (E) of a drug as a function of the dose (D) and time (t) dependant on pharmacokinetic (θpk) and pharmacodynamic parameters (θpd).

PKPD models conventionally include two basic subtypes:
  • Immediate effects model

  • Delayed effects model

Immediate Effects PKPD Model

Under nonequilibrium conditions, an immediate effects PKPD model predicts that drug effects will reach a maximum at approximately the same time as the maximum plasma concentration of the drug (see Fig. 4). The PK model provides the time-dependent plasma concentration (C(t)) so that the immediate drugs effects model can be described by the Emax model (equation 5).
Fig. 4

The time course of drug concentrations (blue line) and drug effects (green line) for an immediate effects model. In this model Emax = 1, C50 = 0.1 units/L. The concentrations were generated using a one compartment model with first-order input. Dose = 1 unit, CL = 1 L/hour, V = 1 L, and ka = 0.5 h

Equation 5

Immediate effects PKPD model
$$ E(t)={E}_{max}\times \frac{C(t)}{C_{50}+C(t)} $$

By contrast, under equilibrium conditions, it is assumed that the steady state average concentration (Css, ave) is sufficient to describe the important pharmacokinetic characteristics of the drug. This is given by

Equation 6

A model for C ss,ave under equilibrium conditions
$$ {C}_{ss, ave}=\frac{dose\ rate}{CL} $$

where dose rate is the maintenance dose (e.g., mg/day). This greatly reduces the complexity of the PK model, but it will only predict the magnitude of drug effects not the onset or duration. The PK model can be substituted into a PD model to provide a steady-state PKPD model:

Equation 7

Immediate effects PKPD model under equilibrium conditions
$$ E={E}_{max}\times \frac{C_{ss, ave}}{C_{50}+{C}_{ss, ave}} $$

Delayed Effect PKPD Models

An important limitation of many PKPD models is that the data used to develop the model will usually lack detail concerning underlying PK and PD mechanisms, i.e., they are empirical in nature. In the absence of this mechanistic information it will only be possible to model the rate-limiting step in the time course of drug response. Three mechanisms for delayed effects will be considered as possible rate-limiting processes in the time course of drug effects: drug-receptor binding, distribution to the biophase, and delay related to biological systems (e.g., secondary messengers).

Delay in drug-receptor binding. If the rate-limiting step in the time course of drug effects is driven by the drug-receptor interaction, the onset of drug effects will be related to the dissociation constant, koff (Wright et al. 2011). If the koff value is large, providing a short equilibration half-life, then the drug will be seen to behave as if it has an immediate effect (also termed an “immediate effect” or “direct action”).

Delay in distribution to the biophase. For most drugs the site of action is distal to the venous compartment. A model can therefore be constructed to account for the delay in drug distribution to the so-called “effect compartment” or “biophase.” The distinction between the terms “biophase” and “effect compartment” is one of semantics. The biophase is the supposed true site of action. The effect compartment is a theoretical space that appears to have the same distributional properties as the biophase. The effect compartment is therefore not a true model for the biophase but rather a model for the delay in effect due to drug distribution to the biophase. This distribution is essentially a pharmacokinetic phenomenon, but it is described entirely and empirically by the observed delay in pharmacodynamic behavior.

An effect compartment model includes a link function between the PK model and PD model (Holford and Sheiner 1982). If the distribution to the biophase is a first-order process, the delay between peak plasma concentration and peak drug effect will be independent of dose and clearance but dependent only on the rate constant of elimination from the effect compartment. The model assumes that only a small amount of drug distributes into the biophase so that the overall impact on mass-balance is negligible. While the volume of the effects compartment cannot be determined, the rate constant for the loss of drug from the effect compartment can be estimated during the modeling analysis. The full effect compartment model therefore includes a PK model, a link model for the effect compartment, and a simple Emax model driven by the effect compartment concentration (Ce):

Equation 8

Effect compartment PKPD model. k e0 is the equilibrium rate constant for the effect compartment
$$ \frac{dC_e}{dt}={k}_{e0}\times \left(C(t)-{C}_e\right);{C}_e\left(t=0\right)=0 $$
$$ E(t)={E}_{max}\times \frac{C_e(t)}{C_{50}+{C}_e(t)} $$
Delay in system response. The mechanism of action for many drugs involves the activation or blockade of a receptor which, in turn, initiates a physiological response mediated by a series of biological processes (e.g., second messengers) (Fig. 5) (Dayneka et al. 1993; Jusko and Ho 1994; Sharma and Jusko 1998). These processes have a time course of their own and often constitute the rate-limiting step in the time course of drug effects. Like effect compartment models, these models are characterized by a delay in the observed effect with respect to the measured plasma concentration and are commonly referred to as “turnover models” (and sometimes “indirect-response models”) (Jusko and Ho 1994). An important difference between models based on physiological intermediaries and those based on an effect compartment is that the time of maximum effect for the turnover model will be dose dependent (Fig. 5).
Fig. 5

Schematic of four turnover models including inhibition or stimulation of the intermediary production or inhibition or stimulation of the intermediary elimination (Adapted from Wright et al. 2011, used with permission from John Wiley and Sons licence number 4277311151819)

There are four proposed mechanisms for drugs acting on a single biological intermediary (I) resulting in four turnover PKPD models (Dayneka et al. 1993; Jusko and Ho 1994; Sharma and Jusko 1998). These are presented in Table 1, while the typical concentration-effect profiles for each model are shown in Fig. 6.
Table 1

Models for the turnover of a single biological intermediate (I). Rin is the zero-order production rate of I, kout is the elimination rate constant for I

 

Mechanism

Model

A

Inhibition of input

\( \frac{dI}{dt}={R_{in}}^{\ast}\left[1-{E_{max}}^{\ast}\frac{C(t)}{C_{50}+C(t)}\right]-{k_{out}}^{\ast }I \)

B

Stimulation of input

\( \frac{dI}{dt}={R_{in}}^{\ast}\left[1+{E_{max}}^{\ast}\frac{C(t)}{C_{50}+C(t)}\right]-{k_{out}}^{\ast }I \)

C

Inhibition of loss

\( \frac{dI}{dt}={R}_{in}-{k_{out}}^{\ast}\left[1-{E_{max}}^{\ast}\frac{C(t)}{C_{50}+C(t)}\right]\ast I \)

D

Stimulation of loss

\( \frac{dI}{dt}={R}_{in}-{k_{out}}^{\ast}\left[1+{E_{max}}^{\ast}\frac{C(t)}{C_{50}+C(t)}\right]\ast I \)

Fig. 6

Concentration-effect profiles using the four turnover PKPD models and three dose levels: 50 mg (dotted curve), 500 mg (dashed curve), and 2000 mg (solid curve). CL = 10, V = 20, Emax = 1,C50 = 1, E0 = 1

Selection of the most appropriate PKPD model should be guided by prior knowledge of the drug pharmacology. Data from more than one dose level greatly helps to distinguish between models (immediate or delayed action and, if delayed, an effect compartment or a turnover model, see Fig. 6). For a direct effect model, the time of the peak concentration is also the time of peak effect for all dose levels. The effect compartment model introduces a delay between the peak concentration and peak effect; this could be due to drug transport to the site of action. If this transport is a first-order process (e.g., diffusion), the time delay is independent of dose. All four turnover models exhibit a dose-dependent delay between peak concentration and peak effects.

Statistical Models for Uncertainty in Drug Response Measurement

A population model includes a statistical model to describe the variability between the model predictions and the observations. This is termed residual unexplained variability (RUV) or “uncertainty.” It is assumed that uncertainty arises typically from four sources:
  1. 1.

    Process error – where the dose or timing of dose or timing of blood samples are not conducted at the times that they are recorded

     
  2. 2.

    Measurement error – where the response is not measured exactly due to assay error

     
  3. 3.

    Model misspecification

     
  4. 4.

    Moment to moment variability within a patient

     

In mathematical terms, a model for one individual can be described as:

Equation 9

The mathematical form of a model with residual error
$$ {y}_j=f\left(D,{x}_j,\theta \right)+{\varepsilon}_j $$

where the jth observation (e.g., drug concentration) for the individual yj is a function of the administered dose (D) and time, and θ is an np-by-1 vector of unknown mean parameters for the individual. In this model, the jth observation deviates from the model prediction by an error, εj, which is assumed to be normally distributed with a mean of zero and a variance of σ2:

Equation 10

The distribution of ε j
$$ {\varepsilon}_j\sim N\left(0,{\upsigma}^2\right) $$

Statistical Models for Variability Between Individuals (Heterogeneity)

An expansion of a model for a single subject to a population of subjects includes an additional consideration of the variability between people. This yields the well-known population approach model. An important goal of population modeling is to establish the relationship between pharmacokinetic and pharmacodynamics parameters of interest and covariates (i.e., observable patient characteristics such as sex, age, weight, height, organ function indices, and concomitant drugs). Genetic covariates may also predict differences in PKPD parameters for some drugs. Covariates explain some of the variability between individuals and therefore provide the basis for decisions about individualized dosing in clinical practice. In population analyses, variability between individuals is also called heterogeneity, between subject variability (BSV), intra-individual variability (IIV), and population parameter variability (PPV).

In mathematical terms, a population model for repeated measures in a series of individuals can be generalized as:

Equation 11

The mathematical form of a population model with between-subject variability and residual error
$$ {y}_{ij}=f\left({D}_i,{x}_{ij},\theta; {\eta}_i\right)+{\varepsilon}_{ij} $$

where the jth observed concentration for the ith individual (yij) is a function of the administered dose (Di) and time(xij) and ηi is an np-by-1 vector of the difference between the parameter estimates for the ith individual and the typical values (geometric mean values) for the population. The distribution of ηi for all subjects in the study population is often assumed to be normally distributed (although transformations can be applied to account for other distributions) with a mean of zero and a variance-covariance given by Ω:

Equation 12

The distribution of η i
$$ {\eta}_i\sim N\left(0,\Omega \right) $$
BSV can be described by two distinct models. Firstly, a model is developed to describe predictable reasons why individuals are different (BSVP, where “P” means predictable) and then a model is developed to quantify random variability (BSVR, where “R” means random) (Holford and Buclin 2012). Simply put, BSVP is the variability that can be explained by covariates while BSVR is the remaining random component. By quantifying both BSVP and BSVR in a population it is possible to predict the likely range of PKPD responses that may occur. BSVP can be reduced by accounting for influential covariates (e.g., size, organ function, disease state, or genetics) on parameter estimates. However, random variability in PK parameters across a patient population still remains even after accounting for patient covariates. A recent review showed that the average CV% for clearance (based on 181 population PK studies) was about 40% (IQR 26–48) (Al-Sallami et al. 2014). This corresponds to a fivefold variability in steady state average concentration (Css, ave) which clinically necessities a fivefold difference in dose-requirements to achieve a target Css across the population (Fig. 7). The authors suggest a recalibration of current perception of what constitutes normal PK variability. Traditionally, between-subject variability in PK parameters was considered “low” for CV% ≤ 10%, “medium” for CV% of around 25%, and “high” for CV% > 40% (Rowland and Tozer 2011). We propose that a CV of 25–50% for clearance should be considered normal variability for most drugs.
Fig. 7

Illustration of variability and fold-difference in Css as a result of BSV in CL of a hypothetical drug with PK described by a one compartment model with a CL of 1% and 100% bioavailability. Css calculated for a dose of 1 unit

Part 3: Methodology (Nonlinear Mixed Effects Modeling)

The following is a brief overview of nonlinear mixed effects modeling. It is not intended to be an exhaustive review of technical details or of the statistical methodology. For further details, please refer to Bonate 2011, Davidian and Giltinan 1995, and Vonesh and ChinChilli 1997.

Population analyses have traditionally employed three approaches: the naïve pooled approach; the two stage analysis; and the population nonlinear mixed effects modeling. The naïve pooled method, in essence, assumes that all observations arise from a single individual and hence differences between individuals cannot be quantitated (Sheiner and Beal 1980). For the two stage approach, the parameters of interest are estimated for each individual in the data set separately using ordinary least squares or a similar estimation method. The population parameters are then determined by calculating the arithmetic or geometric mean of the parameter values across all of the subjects. While a simple method for estimating population parameter values and BSV, the “two-stage” method requires rich data from each individual and may result in inflated and/or biased estimates of BSV.

In nonlinear mixed effects modeling, the population parameters and variance terms are estimated simultaneously for all individuals. Sparse data and unbalanced sampling designs can potentially be accommodated (although see Siripuram et al. 2017 for limitations in this regard), and both population and individual parameter values can be estimated. The term “mixed effects” refers to the combination of fixed effect parameters (e.g., mean drug CL in a patient population) and random effect parameters (e.g., between-subject variance for CL in a patient population).

Nonlinear mixed effects modeling provides a means of assessing the probability of the data arising from particular structural and variance models, given the parameters θ, Ω, and σ2. In other words, the parameter estimation will often involve the computation of the likelihood of the observed data arising from the given model. It is usual to search for the best set of parameter values (often maximum likelihood estimates) via iterative algorithms.

There are several nonlinear mixed effects modeling software packages, but NONMEM® is commonly used in pharmacometrics. It was originally developed by Lewis Sheiner and Stuart Beal (Beal et al. 2011) and was the first software developed specifically for analyzing pharmacokinetic and pharmacodynamic data. Because nonlinear mixed effects models are nonlinear with respect to the random effects parameters, no closed-form solutions are available to solve the integrals for the expectation and variance of the likelihood. This problem has been addressed by using a linearization process during parameter estimation. The first-order (FO) method for approximating the likelihood in NONMEM® was first proposed by Sheiner and Beal and uses a first-order Taylor series approximation around the random effects (evaluated at ηi = 0). The first-order conditional estimation (FOCE) method uses a similar principle, but the expansion is evaluated at each iteration based on a conditional estimate of ηi, i.e., the empirical Bayes estimates (EBEs) of the BSV. An interaction term can be added (FOCE-I) for heteroscedastic error models to account for interaction between η and ε. The Laplacian approximation method uses a second-order Taylor series around the conditional estimates of ηi (Davidian and Giltinan 1995). For NONMEM the objective function value (OFV) is proportional to minus twice the log likelihood (−2LL).

Once a population model is fitted to the data, the model performance is evaluated based on statistical significance, predictive performance, and biological plausibility. To show statistically whether one model performs better than another, the likelihood ratio test is usually used. As the OFV is proportional to –2LL, and the likelihood ratio is asymptotically and approximately chi-squared distributed, a decrease in OFV between two nested models of ≥3.84 points denotes a p-value <0.05 with one degree of freedom. The precision of parameter estimates is computed using either the asymptotic standard errors obtained via maximum likelihood estimation or through nonparametric bootstraps. The predictive performance of the model can be assessed through the use of diagnostic plots and visual predictive checks which can show whether model predictions are biased. The predictive performance can also be assessed by using the model to predict into a new data set. Additionally, parameter values and covariate effects in the final model are checked for biological and/or mechanistic plausibility.

Part 4: Example of Population PK and PKPD Analyses

The following example presents a population PKPD analysis that is similar to the type of work conducted as part of a drug development process in the pharmaceutical industry. The drug in this case is a fictitious example of a lipid-lowering drug simply termed “Drug P.” No similarity to any marketed drug is intended. The structure of Part 4 is based on the style that is often presented in population PKPD analysis reports which may be included in regulatory submission documents.

Introduction

Drug P is a lipid-lowering drug that inhibits 3-hydroxy-3-methyglutaryl coenzyme A reductase (HMG-coA reductase) in the liver. HMG reductase is the rate-limiting enzyme in the biosynthesis of low-density lipoprotein (LDL)-cholesterol. Drug-P is inactive but is rapidly converted to an active metabolite, Metabolite-M. The conversion of Drug-P to Metabolite-M is approximately 30%.

This document provides a description of the population pharmacokinetics and pharmacokinetic-pharmacodynamic analyses of Drug-P in healthy volunteers conducted as part of the Phase I clinical pharmacology program.

Objectives

The primary objectives of the population analyses were:
  1. (a)

    To characterize the plasma pharmacokinetics and pharmacokinetic-pharmacodynamics of Drug-P and Metabolite-M in healthy volunteers after oral administration

     
  2. (b)

    To quantify the PK and PKPD variability between subjects

     
  3. (c)

    To predict dosing regimens that will achieve a long-term reduction in LDL of 50–90%. This information will be used to inform dose selection for phase II studies

     

Methods

Study Design and Data Description

Data for the population PKPD analyses was sourced from a phase I dose-ascending study. The study included 50 healthy volunteers who received either 1 mg, 3 mg, 10 mg, 30 mg, or 100 mg daily of Drug-P for 14 days. An additional 10 healthy volunteers were also included in a placebo group in each dosing arm (i.e., an extra two subjects were given placebo in each arm). Samples for Drug P and Metabolite-M plasma concentrations were collected pre-dose, and 0.5, 1, 1.5, 2, 3, 3.5, 4, 5, 6, 8, 10, 12, and, 24 h after the dose on study days 1, 7, and 14. LDL-cholesterol plasma concentrations were collected daily just prior to the dose of Drug-P. All analyses were conducted using untransformed data. Data from placebo groups are excluded from the analyses.

Handling Missing Data

Drug concentration data below the limit of quantification (BLQ) was analyzed using a likelihood based censoring method if >10% of the data were BLQ (Beal 2001). If <10% of the data were BLQ, the BLQ samples are omitted from the analysis.

Data with missing independent variables (e.g., dose, sampling times) were to be excluded.

Handling Data Outliers

Outlying data points were identified using the conditional weighted residuals (CWRES) where observations that were more than six standard deviations away from the null (based on CWRES) were excluded from the analysis. A sensitivity analysis was performed where outlying data were included, and model fits with and without outlying data were to be compared.

Computer Hardware and Software Platforms Used in the Analysis

The population analysis was performed using NONMEM (version 7.3) and the Intel visual fortran compiler. Runs were performed using first-order conditional estimation (FOCE) with INTERACTION. Pre- and post-processing was performed using SAS v.9.3. The computer processors used were Intel Xeon® CPU, 2.53 GHz.

Model Building Process

Model building was based on a two stage process: (1) development of the best base population model (the best model in the absence of covariates), including structural components and statistical models for random residual variability and between subject variability, and (2) final model refinement and evaluation. No covariate modeling was conducted.

Structural Models Tested

One, two, and three compartment structural models were considered for the disposition of Drug-P. Various absorption models, including transit compartment models, were also considered. A parent-metabolite model was constructed with a one compartment model for the metabolite, Metabolite-M. The conversion of Drug-P to Metabolite-M was fixed to 30%.

An inhibitory turn-over model was considered for the concentration-response relationship of Metabolite-M on plasma LDL-cholesterol concentrations as follows:
$$ \frac{dLDL}{dt}={R}_{in}\times \left(1-{E}_{max}\times \frac{C{(t)}^{\gamma }}{C_{50}^{\gamma }+C{(t)}^{\gamma }}\right)-{k}_{out}\times LDL $$
where Rin is the rate of cholesterol synthesis, kout is the rate of cholesterol loss, Emax is the maximum value of synthesis inhibition, C(t) is the concentration of Metabolite-M, γ is the sigmoidicity parameter (also termed the Hill coefficient), and LDL is the plasma concentration of LDL-cholesterol. The pharmacokinetic and pharmacodynamic (PKPD) data were analyzed using a sequential modeling method (IPP, individual pharmacokinetic parameters) (Zhang et al. 2003a, b). Metabolite-M plasma concentrations were introduced into the PD models using linear, Emax, and sigmoid Emax models.

Variability and Error Models Tested

Between-subject variability was described using an exponential model in the following format:
$$ {\theta}_i={\theta}_{pop}{e}^{\eta_i} $$
where θi is the ith individual’s value of parameter θ, θpop is the typical value of θ in the population, and ηi is the difference between the ith individual’s value and the typical value. ηi is assumed to be independent and identically distributed with variance Ω.
Residual variability was described using both an additive and proportional error model in the following format:
$$ {y}_{ij}={\widehat{y}}_{ij}\ \left(1+{\varepsilon}_{1 ij}\right)+{\varepsilon}_{2 ij} $$
where yij is the jth observation for the ith individual, \( {\widehat{y}}_{ij} \) is the model predicted observation and ε1 and ε2 are the proportional and additive residual errors, respectively.

Covariate Models to Be Tested

No covariates were tested as part of this analysis.

Model Selection

Model selection was guided by: (1) the likelihood ratio test where a decrease in the objective function value (OFV) of 3.84 units (χ2, p < 0.05) with one degree of freedom for nested models was considered significant, (2) graphical goodness of fit plots, (3) visual predictive checks (VPCs), (4) parameter precision, and (5) the biological plausibility of parameter estimates.

Procedures Used for Model Evaluation

Model evaluation was performed by assessing the visual predictive check (VPC) and prediction-corrected VPC (pcVPC) plots. In both cases, 100 data sets were simulated under the final model. For pcVPC, the 95% confidence interval (CI) of 5th, 50th, and 95th percentiles were shown against the observed data.

Model Simulations to Inform Dose Selection

The final PKPD model was implemented in NONMEM. Drug-P, Metabolite-M, and LDL plasma concentrations were stochastically simulated for 1, 3, 10, 20, 30, and 100 mg daily dosing regimens and for 0.5, 1.5, 5, 10, 15, and 50 mg twice daily dosing. Simulations were conducted assuming 50 subjects in each dosing arm and for 12-week duration of Drug-P. Each dosing regimen was assessed by determining the percent reduction in LDL plasma concentrations for each simulation at week 12 of Drug-P. The dosing regimen was selected if the 95% CI for LDL reduction was within 50–90% of baseline.

Results

Data Analyzed

The dataset included 2100 Drug-P and Metabolite-M plasma concentrations and 700 LDL-cholesterol plasma concentrations from 50 subjects. All 50 subjects who took Drug-P completed the study and contributed measureable PK and PD samples. All Drug P and M plasma concentrations collected prior to the first dose of Drug Y were BLQ and were excluded from the analysis. No other BLQ samples were present. The final dataset therefore included 2050 Drug P and Metabolite-M plasma concentrations and 700 LDL-cholesterol plasma concentrations from five different dosing regimens. Spaghetti plots of the raw data are presented in Figs. 8 and 9.
Fig. 8

Spaghetti plots of the raw data for the Parent (Drug P) and metabolite (Drug M) for each dose group. Blue line is the median

Fig. 9

Spaghetti plots of the percent change in LDL-cholesterol from baseline for each dose group

Population Pharmacokinetic Model

The best fit to the data for Drug-P and Metabolite-M was a two compartment parent-metabolite model (one compartment for each compound) with first-order absorption and elimination. A schematic of the final PK model is shown in Fig. 10. The final model parameters are summarized in Table 2. Standard goodness-of-fit plots are presented in Fig. 11. The conditional-weighted residual plots (bottom row) show no obvious bias across the model predictions and time. A scatter of under-predicted plasma concentrations for Metabolite B is noted for both the population predictions and individual predictions. These are few in number and are all at higher plasma concentrations. In addition, the predicted versus observed plots presented in Fig. 11 are of limited diagnostic value in the setting of repeated measures data. In this case, the under-predicted values predictions are from two individuals and therefore can be seen to clump together in the plots. Overall, there is no obvious model misspecification evident from the goodness of fit plots.
Fig. 10

Schematic of the final PK model for Drug-P and Metabolite-M

Table 2

Parameter estimates for the final PK model for Drug-P and Metabolite-M

Parameter

Population estimate (RSE%)

BSV as CV% (RSE%)

CLP (L/hr)

2040 (5.6)

39.0 (8.7)

VP (L)

9210 (4.9)

34.3 (10.1)

CLM (L/hr)

535 (5.7)

40.1 (9.7)

VM (L)

948 (6.8)

45.3 (9.0)

ka (hr−1)

2.81 (6.9)

38.8 (11.2)

LAG (hr)

0.203 (4.4)

0 FIX

σprop (Drug-P) %

30.7 (2.0)

 

σprop (Drug-M) %

21.0 (1.5)

 

CLP Drug-P clearance, VP Drug-P volume, CLM Metabolite-M clearance, VM Metabolite-M volume, ka absorption rate constant, LAG absorption time lag, σprop proportional residual error, σadd additive residual error

Fig. 11

Goodness of fit plots for Drug-P (parent) and Metabolite-M (metabolite)

Visual predictive checks (VPC) and prediction-corrected visual predictive checks (pcVPC) are presented in Figs. 12 and 13 for Drug P and Metabolite M. In Fig. 12, the median model-prediction and the 90% prediction interval suggest an acceptable model fit. Note that the confidence intervals are not shown in Fig. 12 due to the small number of subjects. In Fig. 13, the median and 5th and 95th percentiles of the observed data were included within the confidence interval of each percentile for model predicted plasma concentrations, suggesting an acceptable model fit in both Drug-P and Metabolite-M.
Fig. 12

Visual predictive check (VPC) for the final PK model for Drug-P and Metabolite-M. The median model-prediction (solid line) and the 90% prediction interval (shaded area) are displayed against the observed data (black dots)

Fig. 13

Prediction-corrected visual predictive check (pcVPC) for the final PK model for Drug-P and Metabolite-M. The median model-predictions (solid line) and the 5th and 95th percentiles (dashed lines), as well as the 95% CI around the percentiles (shaded area) are displayed against the observed data (black dots)

Population Pharmacokinetic-Pharmacodynamic Model

The effect of Metabolite-M on LDL-cholesterol was modeled using an inhibitory turnover model. A schematic of the model is presented in Fig. 14. An IPP model was used so the individual PK parameters were fixed to the EBE (empirical Bayes estimate) values obtained from the final PK model. Metabolite-M plasma concentrations entered into the turnover model using an Emax model. The final model parameters are summarized in Table 3.
Fig. 14

Schematic of the final PKPD model

Table 3

Parameter estimates for the final PD parameters

Parameter

Population estimate (RSE%)

BSV as CV% (RSE%)

Rin (mg/dL/hr)

1.04 (3.3)

53.1 (20.5)

Emax

1 Fix

C50 (ng/mL)

0.101 (3.1)

115.0 (11.8)

Baseline LDL (mg/dL)

102 (2.6)

17.9 (18.7)

Kout (hr−1)

Rin/baseline

σprop (LDL) %

7.7 (7.0)

 

Rin production rate of LDL, C50 the plasma concentration at ½ maximum effect, Kout elimination rate constant for LDL, σprop proportional residual error, σadd additive residual error

Standard goodness-of-fit plots are presented in Fig. 15. The conditional-weighted residual plots (bottom row) show no obvious bias for the model predictions. Overall, there is no obvious model misspecification evident from the goodness of fit plots.
Fig. 15

Goodness of fit plots for the LDL data

Visual predictive checks (VPC) and prediction-corrected visual predictive checks (pcVPC) are presented in Figs. 16 and 17. In Fig. 16, the median model-prediction and the 90% prediction interval suggest an acceptable model fit. Note that the confidence intervals are not shown in Fig. 12 due to the small number of subjects. In Fig. 17, the median and 5th and 95th percentiles of the observed data were included within the confidence interval of each percentile for model predicted LDL concentrations, suggesting an acceptable model fit in both cases.
Fig. 16

Visual predictive check (VPC) for the final PKPD model (LDL). The median model-prediction (solid line) and the 90% prediction interval (shaded area) are displayed against the observed data (black dots)

Fig. 17

Prediction-corrected visual predictive check (pcVPC) for the final PKPD model (LDL). The median model-predictions (solid line) and the 5th and 95th percentiles (dashed lines), as well as the 95% CI around the percentiles (shaded area) are displayed against the observed data (black dots)

Model Simulations to Inform Dose Selection

Model-predicted LDL plasma concentrations under once daily Drug-P dosing regimens are presented in Fig. 18. Doses of 10, 20, 30, and 100 mg daily resulted in median LDL concentrations that met the target of a 50% reduction in LDL concentration from baseline. A dose of 100 mg daily resulted in a median LDL reduction from baseline of about 85%; however, the 95% CI of the model-predicted LDL exceeded the 90% target (95% CI of approximately 75–95). Doses of 20 mg and 30 mg produced median LDL reductions of about 65% and 70% (95% CI of approximately 55–75 and 60–80), respectively.
Fig. 18

Model-predicted percent change from baseline for LDL concentrations after once daily dosing for 12 weeks. The black line is the median prediction and the shaped blue areas are the 95% CI

Model-predicted LDL plasma concentrations under twice daily Drug-P dosing regimens are presented in Fig. 19. Doses of 5, 10, 15, and 50 mg twice daily resulted in LDL concentrations that met the target of a 50% reduction in LDL concentration from baseline. Dosing at 50 mg and 15 mg twice daily resulted in LDL reductions that exceeded the predefined upper limit of 90% (95% CI for each dose includes 90%). Doses of 5 mg and 10 mg produced median LDL reductions of about 63% and 77% (95% CI of approximately 55–74, and 66–86), respectively.
Fig. 19

Model-predicted percent change from baseline for LDL concentrations after twice daily dosing for 12 weeks. The black line is the median prediction and the shaped blue areas are the 95% CI

Dose Selection for Phase II Studies

A dose of 20 mg once daily was selected as the dosing regimen to progress to Phase II trials based on the pharmacometric analyses. The median and 95% CI of the model-predicted LDL reduction were within the predefined target of 50–90% reduction from baseline (65%, 95% CI of approximately 55–75). Although the 5 mg twice daily produced LDL concentrations that met the LDL reduction target with lower daily dose compared to 20 mg once daily, the once daily dosing was preferred due to ease of administration in the clinic leading to higher compliance.

Conclusion to Part 4

Population pharmacokinetic and pharmacokinetic-pharmacodynamic models were developed for Drug P, Metabolite M, and the impact on LDL-cholesterol reduction. The best fit to the data for Drug P and Metabolite-M was a two compartment parent-metabolite model (one compartment for each compound) with first-order absorption and elimination. The effect of Metabolite M on LDL-cholesterol was modeled using an inhibitory turnover model. A dose of 20 mg once daily was selected as the dosing regimen to progress to Phase II trials based on the model-predicted LDL reduction of 65% (95% CI of approximately 55–75) from baseline.

Conclusion

Clinical pharmacology and pharmacometrics share common goals and research themes, including pharmacokinetics and pharmacodynamics. A primary purpose of population pharmacokinetic and pharmacokinetic-pharmacodynamic analyses is to aid dosing decisions, whether in the industrial or clinical setting. The goal is to ensure that medicines are safe and effective and that dosing guidance is underpinned by a scientific understanding of drug behavior and pharmacological response. In addition, variability in drug response between and within individuals can be quantified, and factors which predict this variability, such as body weight, can be accounted for. The resulting models have utility for predicting drug response into new settings, such as the simulation of Phase II studies using a model developed from Phase I data presented in this chapter. By optimizing dose selection, population pharmacokinetic and pharmacokinetic-pharmacodynamic analyses help to improve the chances of success in confirmatory Phase III trials for new drugs and to aid dose individualization in the clinical setting.

References and Further Reading

  1. Al-Sallami HS, Cheah SL, Han SY et al (2014) Between-subject variability: should high be the new normal? Eur J Clin Pharmacol 70:1403–1404CrossRefGoogle Scholar
  2. Ariens EJ, Van Rossum JM, Simonis AM (1957) Affinity, intrinsic activity and drug interactions. Pharmacol Rev 9:218–236PubMedGoogle Scholar
  3. Beal SL (2001) Ways to fit a PK model with some data below the quantification limit. J Pharmacokinet Pharmacodyn 28:481–504CrossRefGoogle Scholar
  4. Beal S, Sheiner LB, Boeckmann A, Bauer RJ (2011) NONMEM user’s guides. (1989–2011). Icon Development Solutions, Ellicott CityGoogle Scholar
  5. Biomarkers Definitions Working Group (2001) Biomarkers and surrogate endpoints: preferred definitions and conceptual framework. Clin Pharmacol Ther 69:89–95CrossRefGoogle Scholar
  6. Bonate PL (2011) Pharmacokinetic-pharmacodynamic modeling and simulation, 2nd edn. Springer, New YorkCrossRefGoogle Scholar
  7. Davidian M, Giltinan DM (1995) Nonlinear models for repeated measurement data. Chapman & Hill, LondonGoogle Scholar
  8. Dayneka NL, Garg V, Jusko WJ (1993) Comparison of four basic models of indirect pharmacodynamic responses. J Pharmacokinet Biopharm 21:457–478CrossRefGoogle Scholar
  9. Duffull SB (2012) Is the ideal anticoagulant a myth? Expert Rev Clin Pharmacol 5:231–236CrossRefGoogle Scholar
  10. Holford NH, Buclin T (2012) Safe and effective variability – a criterion for dose individualization. Ther Drug Monit 34:565–568CrossRefGoogle Scholar
  11. Holford NH, Sheiner LB (1982) Kinetics of pharmacologic response. Pharmacol Ther 16:143–166CrossRefGoogle Scholar
  12. Jusko WJ, Ko HC (1994) Physiologic indirect response models characterize diverse types of pharmacodynamic effects. Clin Pharmacol Ther 56:406–419CrossRefGoogle Scholar
  13. Leake CD (1961) The scientific status of pharmacology. Science 134:2069–2079CrossRefGoogle Scholar
  14. Rowland M, Tozer TN (2011) Variability. Clinical pharmacokinetics and pharmacodynamics: concepts and applications, 4th edn. Lippincott Williams & Wilkins, PhiladelphiaGoogle Scholar
  15. Sharma A, Jusko WJ (1998) Characteristics of indirect pharmacodynamic models and applications to clinical drug responses. Br J Clin Pharmacol 45:229–239CrossRefGoogle Scholar
  16. Sheiner LB, Beal SL (1980) Evaluation of methods for estimating population pharmacokinetic parameters. I. Michaelis-Menten model: routine clinical pharmacokinetic data. J Pharmacokinet Biopharm 8:553–571CrossRefGoogle Scholar
  17. Sheiner LB, Rosenberg B, Marathe VV (1977) Estimation of population characteristics of pharmacokinetic parameters from routine clinical data. J Pharmacokinet Biopharm 5:445–479CrossRefGoogle Scholar
  18. Siripuram VK, Wright DFB, Barclay ML et al (2017) Deterministic identifiability of population pharmacokinetic and pharmacodynamic models. J Pharmacokinet Pharmacodyn 44:415–423CrossRefGoogle Scholar
  19. Vonesh EF, ChinChilli VM (1997) Linear and nonlinear models for the analysis of repeated measurements. Marcel Dekker, New YorkGoogle Scholar
  20. Wright DFB, Winter HR, Duffull SB (2011) Understanding the time course of pharmacological effect: a PKPD approach. Br J Clin Pharmacol 71:815–823CrossRefGoogle Scholar
  21. Zhang L, Beal SL, Sheiner LB (2003a) Simultaneous vs. sequential analysis for population PK/PD data I: best-case performance. J Pharmacokinet Pharmacodyn 30:387–404CrossRefGoogle Scholar
  22. Zhang L, Beal SL, Sheinerz LB (2003b) Simultaneous vs. sequential analysis for population PK/PD data II: robustness of methods. J Pharmacokinet Pharmacodyn 30:405–416CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  • Daniel F. B. Wright
    • 1
    Email author
  • Chihiro Hasegawa
    • 1
    • 2
  • Hesham S. Al-Sallami
    • 1
  1. 1.School of PharmacyUniversity of OtagoDunedinNew Zealand
  2. 2.Translational Medicine CenterOno Pharmaceutical Co., Ltd.OsakaJapan

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