Introduction to Pharmacometrics and Quantitative Pharmacology with an Emphasis on Physiologically Based Pharmacokinetics

Abstract

Pharmacometrics, enabled by advanced pharmacostatistical modeling and simulation technique, has gained significant importance as a tool for model-based drug development and for pharmacotherapy. This introductory chapter provides an overview of the models and mathematical framework that are commonly used in the field. Both pharmacokinetic and pharmacodynamic models are discussed; the physiologically based pharmacokinetic (PBPK) approach is emphasized through illustrations and examples. We have shown how systemic and tissue clearances are related through mathematical proofs using the PBPK approach. As for the pharmacodynamic section, models for continuous and noncontinuous responses were explained, as well as a disease progression model. We also provided select examples of systems pharmacology models that investigate physiological and biochemical processes. This chapter also discusses the software tools and their specific application. As pharmacometrics and quantitative pharmacology has evolved as a discipline, more and more different types of statistical and mathematical models have been utilized. The advancement of computing software and systems facilitate these complex computations.

1.1 Introduction

Pharmacometrics has become a term that encompasses modeling and simulation for pharmacokinetics (PK), exposure–response relationship, and disease progression . Mechanistic models that describe the biochemical processes involved in a physiological system have become more utilized in drug development . The models of complex systems are generally classified as systems pharmacology . A quote from William Jusko describes the role of pharmacometrics in drug development: “Pharmacometrics lies at the heart of what drug companies do: collecting data from animals, normal volunteers, and patients; quantifying it, and then being able to determine what that data mean for optimizing drug efficacy and minimizing toxicity” (Nielsen and Friberg 2013). Pharmaceutical and biotech companies have invested heavily in establishing pharmacometrics expertise to utilize the preclinical, clinical, as well as human genomic data to understand the disease progression, the drug behavior, and its effect on individual patients and to personalize medicine to specific groups of patient population. The purpose of this chapter is to provide an overview of different approaches that were used in pharmacometrics in the context of pharmaceutical drug development.

1.2 Classical PK Analysis

There are two primary approaches in the classical PK analysis: the compartmental modeling and the noncompartmental analysis. Compartmental modeling is based on the mass balance equations on the compartment and the noncompartmental modeling is based on the statistical moments derived from the time course of the drug concentration data.

Compartmental PK models are widely used to characterize the disposition of a drug using its concentration–time profiles sampled from body fluid such as plasma, serum, or whole blood following an administered dose. The general expression of the compartmental model is given in Eq. 1.1, where a series of exponential terms are used to fit to the drug concentration–time profile:

$$ C(t)=\sum{}_{i}^{n}{{{A}_{i}}}{{e}^{-{{\alpha }_{i}}t}}$$

(1.1)

where $i$ indicates each compartment, $n$ is the total number of compartment, and ${{A}_{i}}$ and ${{\alpha }_{i}}$ are called macroconstants reflecting the amount of the drug administered, the mass transfer between the compartments, and the elimination of the drug from the body. The number of compartments ($n$) determined by curve fitting is a rather abstract mathematical construct. The interpretation of Eq. 1.1 is that the body is a series of compartments; the drug is distributed between compartments, and is eliminated from the body. It was recognized that Eq. 1.1 was the solution of a series of ordinary differential equations derived by mass balance of each compartment.

The simplest compartmental model has one compartment with a bolus injection. The differential equation for the one-compartment model can be derived from mass balance; that is, the rate of change of the drug amount in the compartment equals the rate of the input minus the rate of output:

$$ \frac{dVC}{dt}=rate\,of\,input-{{k}_{e}}VC,\quad at\,t=0,C={{C}_{0}}$$

(1.2)

where $V$ is the volume of the compartment, $C$ is the drug concentration of the compartment, and ${{k}_{e}}$ is the first-order elimination rate constant. ${{C}_{0}}$ is the initial condition of the differential equation, which is the drug concentration prior to drug administration. For a bolus injection, when using the delta function to represent the rate of input, the solution to the above equation, assuming V is a constant, is:

$$ C(t)=\frac{Dose}{V}{{e}^{-{{k}_{e}}t}}$$

(1.3)

where $Dose$ is the input amount of the drug. Comparing Eq. 1.1 with Eq. 1.3, it is obvious that $A=Dose/V$, $\alpha ={{k}_{e}}$, and $n=1$.

Because Eq. 1.1 is first-order kinetics, the half-life (t _1/2) can be estimated as ${{t}_{1/2}}=\frac{\ln \,(2)}{{{k}_{e}}}$. Keep in mind that only for first-order kinetics, the half-life is a constant. Equation 3 is often reparameterized to

$$ C(t)=\frac{Dose}{V}{{e}^{-\frac{CL}{V}t}}$$

(1.4)

where ${{k}_{e}}=CL/V$ and $CL$ refers to clearance. Clearance is one of the most important concepts introduced in PK. Additional information on clearance will be discussed in the section on the clearance definition.

In many cases, the disposition of a drug in the body follows a multi-exponential decline, which shows as multi-linear phases in the log concentration versus time profile shown in Fig. 1.1. This type of drug concentration profile is often characterized by two or more compartments. For a two-compartmental model with a bolus injection, Eq. 1.1 becomes

Fig. 1.1

Log concentration–time profile that follows a bi-exponential decline

$$ C(t)=A{{e}^{-\propto t}}+B{{e}^{-\propto t}}$$

(1.5)

The half-life of the α phase and the β phase (or called the terminal phase) of the drug can be estimated as

$$ {{t}_{1/2,\alpha }}=\frac{\ln \,(2)}{\alpha },\quad {{t}_{1/2,\beta }}=\frac{\ln \,(2)}{\beta } $$

(1.6)

To estimate the overall half-life of a drug in the body following multi-exponential decline, the concept of “effective half-life” was introduced and the calculation is given in Eq. 1.7, e.g. for a two-compartment model:

$$ effective\text{ }{{t}_{1/2}}=\frac{1}{AUC}\left(\frac{A}{\alpha }{{t}_{\frac{1}{2},\alpha }}+\frac{B}{\beta }{{t}_{\frac{1}{2},\beta }}\right), $$

(1.7)

where $AUC$ is the area under the concentration–time profile.

The differential equations for a two-compartmental model can be derived through mass balance on each compartment:

$$ \begin{aligned}& \frac{d{{A}_{c}}}{dt}=Input rate-{{k}_{12}}{{A}_{c}}-{{k}_{e}}{{A}_{c}}+{{k}_{21}}{{A}_{p}}\\& \frac{d{{A}_{p}}}{dt}={{k}_{12}}{{A}_{c}}-{{k}_{21}}{{A}_{p}}{{A}_{p}}\end{aligned} $$

(1.8)

where A _c and A _p refer to the drug amounts in the central and peripheral compartments, respectively; k ₁₂ and k ₂₁ are the mass transfer rate constants between the central and the peripheral compartments, also called microconstants. The analytical solution to Eq. 1.8 for a bolus injection can be expressed in the same manner as Eq. 1.5, with the following micro- and macroconstant conversion:

$$ \alpha +\beta ={{k}_{e}}+{{k}_{12}}+{{k}_{21}}$$

$$ \alpha \times \beta ={{k}_{e}}\cdot {{k}_{21}}$$

$$ A=\frac{Dose(\propto -{{k}_{21}})}{{{V}_{c}}(\propto -\beta)} $$

$$ B=\frac{Dose({{k}_{21}}-\beta)}{{{V}_{c}}(\propto -\beta)}. $$

Re-parameterization for a two-compartment model in terms of $CL$, intercompartmental clearance ($Q$), ${{V}_{c}}$, and ${{V}_{p}}$ can be expressed as ${{k}_{e}}=\frac{CL}{{{V}_{c}}},\ {{k}_{12}}=\frac{Q}{{{V}_{c}}}$, ${{k}_{21}}=\frac{Q}{{{V}_{p}}}$. For more detailed discussions of commonly used PK models including intravenous infusion and extravascular routes, the reader may refer to the textbooks on PK and pharmacodynamic (PD) analysis (Derendorf and Hochhaus 1995; Gabrielsson and Weiner 2000; Gibaldi and Perrier 1999; Rowland and Tozer 1989).

The compartmental models are often used to simulate concentration profiles from one dosing regimen to another, or from a single dose to a steady-state concentration profile. The compartmental model has its limitation, however. First, the number of compartments and the property of the compartments are rather abstract mathematical constructs. The underlying physiology of the model and the resulting model representation is subject to the analyst’s interpretation. Second, the parameters do not have a clear physiological meaning, and so the source of the variability of the parameters cannot be clearly identified and be correlated to physiological reality.

A noncompartmental model is based on statistical moments of the concentration–time data (Dunne 1993; Yamaoka et al. 1978). The nth-order statistical moment has the following mathematical form:

$$ \int_{0}^{\infty }{{{t}^{n}}C(t)dt}, $$

(1.9)

where $t$ is time, $n$ is the order of moment, and $C(t)$ is the drug concentration as a function of time. The area under the concentration–time curve (AUC), the moment curve (AUMC), and subsequently the mean residence time (MRT) can then be computed through integrating the concentration–time profile:

$$ AUC=\int_{0}^{\infty }{{{t}^{0}}C(t)dt=}\int_{0}^{\infty }{C(t)dt} $$

(1.10)

$$ AUMC=\int_{0}^{\infty }{{{t}^{1}}C(t)dt=}\int_{0}^{\infty }{tC(t)dt} $$

(1.11)

$$ MRT=\frac{\mathop{\int }_{0}^{\infty }tC(t)dt}{\mathop{\int }_{0}^{\infty }C(t)dt}=\frac{AUMC}{AUC}. $$

(1.12)

In practice, the computations of the above parameters are carried out using numerical integrators such as the linear or log-linear trapezoidal rule on the discrete concentration–time data. PK parameters such as $CL$, ${{V}_{\text{ss}}}$, and ${{t}_{1/2}}$ can be derived from those statistical moments:

$$ CL=\frac{Dose}{AUC} $$

(1.13)

$$ {{V}_{\text{SS}}}=CL*MRT. $$

(1.14)

The terminal half-life can be calculated using the slope, λz, of the log concentration curve, ${{t}_{1/2}}=\frac{\ln{}(2)}{{{\lambda }_{z}}}$. If the concentration profile shows mono-exponential decline, the terminal half-life can also be calculated using the ${{t}_{1/2}}=\text{ln(2)}*MRT$. If the concentration profile shows multi-exponential decline, the half-life calculated using $\text{ln(2)}*MRT$ will be the “effective half-life,” the same as the solution using Eq. 1.7 in the compartmental modeling approach. The underlying assumption of the noncompartmental modeling is that the PK of a drug is linear (Gibaldi and Perrier 1999). A special case is that the noncompartmental model is equivalent to a one-compartment PK model, where the PK parameters derived through noncompartmental analysis can also be obtained from a one-compartment model through integration of Eq. 1.2.

The advantage of a noncompartmental method compared to the compartmental model is that the results from the moment approach are less subjective on the analysts’ bias of their model of choice (Yamaoka et al. 1978). From a numerical analysis point of view, noncompartmental analysis is using numerical integration over the time course of drug concentration to derive PK parameters rather than optimization on either algebraic or differential equations. Thus, the “noise” in the drug concentration–time profile has less impact on the PK parameters than that of compartmental modeling. For example, when calculating the effective half-life using Eq. 1.7, an unrealistically long effective half-life could be generated when the terminal phase slope cannot be accurately estimated. In that situation, the effective half-life estimated using $\ln \,(2)*MRT$ is more reliable. The noncompartmental analysis is often the choice for computing PK parameters of a drug for regulatory submission.

1.3 Physiologically Based PK Modeling

The physiologically based pharmacokinetic (PBPK) model was developed to overcome the limitations of the compartmental modeling. The structure of the model, the property of the compartments, and the parameters are based on the underlying physiological and biological processes that are responsible for drug disposition.

1.3.1 History and Methodology of PBPK Approach

The concept of predicting the effect of a xenobiotic on a living organism based on mathematical models that incorporate real physiological parameters such as organ functions and flow rates was initially proposed by Teorell in (1937a, b). No progress was made in PBPK since Teorell’s postulation of using mathematical models to describe xenobiotic disposition until the late 1950s, possibly due to the limitation in computational power. The most comprehensive development was made by Bellman and colleagues in the early 1960s (Bellman et al. 1963). The depiction of a PBPK model proposed by Bellman and his colleagues is shown in Fig. 1.2 with modifications. In the model, the tissue or lumped tissue was connected through blood flow. Blood flow through the main arteries and veins was assumed to be similar to a plug flow, that is, the drug concentration during the circulation was changing with time and longitude. The interstitial fluid and intracellular region were treated as perfectly mixed compartments. Based on those assumptions, the mathematical expression for the model was a set of difference-differential equations. The assumption of the plug flow leads to computational difficulties, as the entire past history of the drug concentration at each region of the body needs to be retained for the calculation of the successive time interval. Their work was discussed in detail and summarized by Bischoff and Brown in 1966 (Bischoff 1966). In the same publication, Bischoff and Brown discussed the application of mass transfer concept at great length at the levels of capillary, interstitial, and intracellular region. They also discussed the time needed for “mixed” drug concentration in the blood circulation versus the transient time of a typical human (~ 1 min). Based on the physiological reality and transport phenomena, the compartment property including the capillary, interstitial, and cellular region in Fig. 1.2 was characterized without accounting for every detail. They turned a set of difference-differential equations into differential equations such as Eqs. 1.15 and 1.16. Using a delta function to represent a bolus injection, the drug concentration profiles in different regions of the body were simulated.

Fig. 1.2

An illustration of a PBPK model for a mammalian circulation system

To illustrate the mathematical expression and the parameters of a general PBPK model, for simplicity, if we assume that the interstitial and cellular regions are at equilibrium, a compartment in Fig. 1.2 can be illustrated as shown in Fig. 1.3. The differential equations describing the compartment were given in Eqs. 1.15 and 1.16:

Fig. 1.3

A representative compartment with lumped interstitial and cellular regions

$$ \frac{{{V}_{b,j}}d{{C}_{b,j}}}{dt}={{Q}_{j}}(C-{{C}_{b,j}})-PA({{C}_{bf,j}}-{{C}_{tf,j}}) $$

(1.15)

$$ \frac{{{V}_{t,j}}d{{C}_{t,j}}}{dt}=PA({{C}_{bf,j}}-{{C}_{tf,j}})-{{r}_{j}}, $$

(1.16)

where Q _j is the blood flow rate for the $j$ compartment, PA is the product of the membrane permeability and the membrane area; subscript $b$ is for blood, $t$ for tissue, and $f$ for free drug concentration. The free and the total drug concentration can be correlated based on linear or nonlinear binding. The term ${{r}_{j}}$ represents the elimination rate (metabolism and/or excretion) of the drug from compartment j; it can occur at different regions of the compartment. The drug can be administered through oral, intravenous, or intramuscular routes. The route of administration can be incorporated to the PBPK model.

As shown in Eqs. 1.15 and 1.16, there are three types of information required to solve a PBPK model: (1) the anatomy and physiology of a specific species; (2) the physicochemical properties, such as binding and membrane permeability, that are drug specific; (3) metabolism and excretion that are both drug- and species-specific. The anatomical and physiological parameters are usually available. Extensive data such as weight and blood flow rate through each tissue for different species were provided by Brown et al. (1997). However, physicochemical data and metabolism information are limited and often rely on in vitro studies or in vivo tests that were carried out in different species.

1.3.2 Number of Compartments in a PBPK Model

The two questions that an analyst needs to ask himself/herself when developing a PBPK model are: (1) how many compartments are needed and (2) how much detail is required for that compartment? Extensive work from a typical four-compartment model with flow-limited assumption with or without extensive details for a particular targeted organ to more than ten compartments describing the whole body can be found in the literature (Andersen et al. 1984, 1987; Bischoff et al. 1968, 1970, 1975; Liu et al. 2005; Peters 2008; Peters and Hultin 2008; Ramsey and Andersen 1984; Wang et al. 1997). A general consideration of the number of compartments to choose from and the details of the model depend on these information: the target organ, the physicochemical and pharmacologic properties of the drug that determine the drug transfer in the body, and the PK time scale (Bischoff 1975).

1.3.3 Target Organ

The structure of a PBPK model starts with the anatomy of the body. As the drug concentration in a target organ or at the site of action is of interest, single compartment is often used to represent the target organ. A significant amount of work using the PBPK approach has been done for anticancer drugs, central nervous system , hepatic metabolism and xenobiotic inhalation (Andersen et al. 1984, 1987; Baxter et al. 1994; Chen and Gross 1979; Collins and Dedrick 1983; Pang and Durk 2010; Ramsey and Andersen 1984; Reddy et al. 2005).

1.3.4 Mass Transfer Phenomenon

Lumping is often used for PBPK model reduction. There are two levels of lumping: (1) at the organ level and (2) at the cellular level. Lumping at the cellular level was originally discussed in details by Bischoff and Brown in their work mentioned above (Bischoff 1966). Lumping at the organ level was extensively discussed from the late 1960s to the 1990s (Bischoff 1987; Bischoff and Dedrick 1968; Coxson and Bischoff 1987a, b; Gerlowski and Jain 1983; Nestorov et al. 1998). The basis for lumping is dependent on the mass transfer process and the physicochemical properties of a drug. The following section discusses the types of mass transfer function and the conditions for their applications.

1.3.4.1 Flow-Limited Assumption

The flow-limited assumption was made primarily due to the lack of information on membrane permeability. The criterion of flow limited was given as $\frac{P{{A}_{j}}}{{{Q}_{j}}}\gg 1$ (Bischoff 1975), that is, the membrane transfer is much faster than convection (from blood flow). Under this assumption, the free drug concentration in the tissue and in the blood is at equilibrium, ${{C}_{tf,j}}={{C}_{bf,j}}$. Therefore,

$$ {{C}_{t,j}}={{C}_{bf,j}}+\frac{{{R}_{\text{tot},j}}{{C}_{bf,j}}}{{{K}_{d,j}}+{{C}_{bf,j}}}. $$

(1.17)

For linear binding, Eq. 1.16 can be simplified as ${{C}_{t}}=R*{{C}_{b}}$, where$R$ is called the tissue to blood partition coefficient. Under flow-limited assumption, Eqs. 1.15 and 1.16 become

$$ \left({{V}_{t,j}}+\frac{{{V}_{b,j}}}{{{R}_{j}}}\right)\frac{\text{d}{{C}_{t,j}}}{\text{d}t}={{Q}_{j}}\left(C-\frac{{{C}_{t,j}}}{{{R}_{j}}}\right)-{{r}_{j}}$$

(1.18)

Or if it is expressed using ${{C}_{b}}$, Eq. 1.18 becomes

$$ ({{V}_{b,j}}+{{R}_{j}}{{V}_{t,j}})\frac{d{{C}_{b,j}}}{dt}={{Q}_{j}}(C-{{C}_{b,j}})-{{r}_{j}}$$

(1.19)

Equations 1.18 and 1.19 demonstrate that the concentration of a drug in a particular organ, ${{C}_{b,j}}$ or ${{C}_{t,j}}$, is determined by the value of $\frac{{{Q}_{j}}}{{{V}_{b,j}}+{{R}_{j}}{{V}_{t,j}}}$ and the elimination of that organ, $\frac{{{r}_{j}}}{{{V}_{b,j}}+{{R}_{j}}{{V}_{t,j}}}$. As such, lumping different organs or body regions depends on the blood flow rate through the organ, the partitioning of the drug between the blood and the tissue levels, and the elimination process of the organ (for an eliminating organ).

For noneliminating organs connected in parallel, the blood concentration entering those organs, $C$, is the same. Therefore, the blood concentration leaving the organ, ${{C}_{b,j}}$, and the concentration of the tissue, is determined by the ratio of $\frac{{{Q}_{j}}}{{{V}_{b,j}}+{{R}_{j}}{{V}_{t,j}}}$. The richly perfused organs with similar partition coefficient of the drug are usually lumped into a single compartment. The blood flow rate through the compartment is $\sum\nolimits_{j=1}^{n}{{{Q}_{j}}}$, and the volume of the compartment is $\sum\nolimits_{j=i}^{n}{{{V}_{b,j}}+{{R}_{j}}{{V}_{t,j}}}$. The same principle can be applied to poorly perfused organs with similar partition coefficient. For a lipophilic compound, higher partition coefficient in the adipose versus the lean tissue resulted in different profiles of the concentration of the drug in the tissue, and therefore, a separate compartment for the lean tissue or the adipose tissue is often required. In addition, an organ or a body region with significantly low blood flow rate and low partition coefficient of the drug in those regions can be omitted in the PBPK model. Whether an eliminating organ can be lumped with a noneliminating organ depends on the ratio of the blood flow rate to the clearance of that organ (Bischoff 1975; Nestorov et al. 1998).

For organs that are connected in series, such as the venous–lung–artery channel or the splanchnic organs, the blood concentration profile leaving the channel and returning to the vein is determined by the organ that has the longest transient time, or the organ that eliminates the xenobiotics. If the partition coefficient between the plasma and the lung tissue is small, the transient time of the lung is much smaller than those of the vein and the artery. The vein and the artery often can be lumped to one compartment without including the lung, $V={{V}_{artery}}+{{V}_{vein}}$, if the lung is not an eliminating organ. In the splanchnic channel, the splanchnic organs are often omitted, since the liver is the primary eliminating organ and the blood concentration leaving the channel is approximately represented by the liver. The gastrointestinal tract (GI) tract may be included to describe the absorption and/or reabsorption of the drug.

In general, a four-compartment lumped PBPK model, consisting of the blood compartment, the richly perfused compartment such as liver or kidney, the poorly perfused compartment such as the muscle, and a compartment representing the adipose tissue can adequately describe the drug disposition in the body. Other compartments may be added to describe the specific target organ as in the PBPK model to study tumor, wherein a separate compartment was incorporated to represent that organ where the tumor resides.

If the drug transfer across the membrane is fast enough compared to the mass transfer through convection (blood flow), the entire body can be modeled as a single-compartment model assuming that the blood concentration is at equilibrium with tissues at different regions. See the elimination-limited case below for the mathematical expression.

1.3.4.2 Membrane Limited

The opposite situation contrasting to the flow-limited mass transfer is the case wherein the membrane transfer is slow enough compared to the rate of the drug supply by blood flow, $\frac{P{{A}_{j}}}{{{Q}_{j}}}\ll 1$, so that the gradient of the drug concentration in the blood entering and leaving the compartment is negligible (Bischoff 1975). Therefore, ${{C}_{b,j}}=C$. Equations 1.15 and 1.16 for the compartment with membrane-limited transfer can be simplified as:

$$ {{V}_{t,j}}\frac{d{{C}_{t,j}}}{dt}=PA(C-{{C}_{tf,j}})-{{r}_{j}}. $$

(1.20)

If the drug transfer across the membrane of all regions of the body is slow enough that it can be considered negligible, the entire body can be modeled as a single-compartment model by only including the blood pool:

$$ {{V}_{b}}\frac{dC}{dt}=Input-{{r}_{j}}. $$

(1.21)

1.3.4.3 Elimination Limited

In their publication on the general solution of a two-compartment model, Bischoff and Dedrick introduced the concept of the elimination-limited assumption, where the mass transfer is much more rapid than the total elimination rate (Bischoff et al. 1970). The importance of introducing the elimination-limited concept is to simplify a PBPK model to a one-compartment model. The criteria for when a system follows the elimination-limited profile is given in their study through a two-compartment open model under a flow-limited assumption, that is, the drug distribution to the tissue through the blood flow rate (mass transfer through convection) is much faster than the rate of elimination. In the elimination-limited situation, the entire body can be lumped into a one-compartment model:

$$ \left({{V}_{b}}+\underset{j}{\overset{n}{\mathop \sum }}\,{{R}_{j}}{{V}_{tj}}\right)\frac{d{{C}_{b}}}{dt}=Input-{{r}_{j}}. $$

(1.22)

Equation 1.21 has the same mathematical expression as the one-compartment model in classical compartmental modeling. The difference is that Eq. 1.5 derived from PBPK model gives the meaning to ${{V}_{d}}$, which is equivalent to ${{V}_{b}}+\sum\nolimits_{j}^{n}{{{R}_{j}}{{V}_{t,j}}}$. In fact, the elimination-limited case does not necessary require flow-limited assumption. As long as the elimination rate is slow enough compared to both convection and membrane transfer, the elimination-limited case stands. This also explains why in covariate analysis in the population PK modeling, the volume of distribution often is related to body weight, as tissue volume is proportional to the body weight. The tissue concentration then can be easily calculated as $\frac{{{C}_{b}}}{{{R}_{t,j}}}$, where ${{C}_{b}}$ is the blood concentration and ${{R}_{t,j}}$ is the partition coefficient of the organ. A typical example can be found for 2,3,7,8-tetrachlorodibenzo-p-dioxin (TCDD), since TCDD is known to remain in the biological system for a very long time. The half-life in human is around 5–10 years. Table 1.1 lists the physiological data for a standard human with body weight of 70 kg and the estimated value of $\frac{1}{\lambda }$ with ${{\lambda }_{i}}\approx \frac{{{Q}_{i}}}{{{k}_{el}}}\frac{{{V}_{lip,b}}}{{{V}_{b}}{{V}_{lip}}}$ for a flow-limited case, or ${{\lambda }_{i}}\approx \frac{P{{A}_{i}}}{{{k}_{el}}}\frac{{{V}_{lip,b}}}{{{V}_{b}}{{V}_{lip}}}$ for a membrane-limited case, where the subscript lip refers to lipid content.

Table 1.1 Physiological parameters of TCDD for a standard 70-kg human

The lipid contents of each organ in Table 1.1 were calculated based upon the values from the literature (van der Molen et al. 1996). The elimination rate constant ${{k}_{el}}$ was estimated conservatively assuming a half-life of 5 years. The values shown in Table 1.1 suggested that the elimination-limited assumption was satisfied for TCDD. This one-compartment model through PBPK reduction was adopted in human risk assessment in the environmental toxicology (Thomaseth and Salvan 1998; van der Molen et al. 1996).

1.3.5 PK Time Scale

The PK time scale plays an important role in PBPK model development (Bischoff 1975; Dedrick and Bischoff 1980; Nestorov et al. 1998; Oliver et al. 2001). For a standard male or female, the time it takes to complete one blood circulation is about 1 min. For most of the drug acting in the scale of several minutes, hours, days or longer, it can be assumed that the blood in the circulation is a uniform pool. However, more details are required in the model for the rapidly eliminated drug, in a time scale of minutes. The sampling site could also be important. The following example illustrates the methodology in the selection of the number of compartments to use for building a PBPK model for a short-acting drug.

1.3.6 Example 1: A PBPK Model for a Contrast Agent for Ultrasound Imaging

The PBPK model developed for a contrast agent for ultrasound imaging (Wang et al. in preparation) is shown in Fig. 1.4. The model has detailed information on the cardiovascular circulation and pulmonary circulation, which included the vena cava, right heart, pulmonary vein, lungs, pulmonary artery, left heart, and aorta. The actual sampling site and the administration site had to be specified in the model to accurately describe the concentration of the agent in the blood circulation, the left and the right heart. Such detailed information became necessary due to the short time scale in its PK profile. For example, within 3 min following injection, blood concentration of the agent dropped tenfold. The left and the right sides of the heart are the target tissues for this contrasting agent. The lung is the eliminating organ. The adipose tissue and the lean tissue compartments were specified in the model, as the agent is a lipophilic compound. Coronary circulation was included in the model to evaluate whether coronary artery disease would have an impact on the PK of this agent. The viscera tissues consist of the kidney, the brain , the liver, etc. The blood flow rate per volume in these tissues is much faster than those of either adipose or lean tissues. Except for the lung, each compartment includes vascular and extravascular sub-compartments.

Fig. 1.4

Illustration of the PBPK model of the human body for a contrast agent used in ultrasound imaging

The lung is the primary eliminating organ for this compound. A heterogeneous compartment for the agent based on the anatomy of the lung and mass transfer is depicted in Fig. 1.5. As a static homogeneous lung compartment overpredicted the concentration in the alveolar gas phase during the absorption and under-predicted the concentrations during the elimination phase (Hutter et al. 1999), a heterogeneous lung model developed by Liguras and Bischoff (unpublished data), Frank (1982), and Bernards (1986) was adopted instead, as the one shown in Fig. 1.5.

Fig. 1.5

Structure of the physiological model of the lung

The lung was modeled using three compartments based on the physiology of the lungs, consisting of an upper dead space plus a lower dead space in series and a perfectly mixed alveolar region. The dead space is taken to be that of the large bronchial vessels, such that there is no mass transfer between the air in the lung and the capillary blood. The volume of the alveolar compartment changes with inhalation and exhalation. The mass transfer between the air in the lung and the capillary blood occurs across the alveolar-capillary membrane. Mathematical equations describing the PBPK model including the lung compartment are given in Eqs. 1.23–1.32.

1.3.6.1 Whole-Body PBPK Model

For the left and right sides of the heart, and other compartments except the lung, the mass balance equations have the following form:

$$ {{V}_{b,j}}\frac{d{{C}_{tb,j}}}{dt}={{Q}_{j}}({{C}_{a}}-{{C}_{tb,j}}), $$

(1.23)

where ${{V}_{b,j}}$ is the tissue blood volume, ${{C}_{bt,j}}$ refers to the concentration in the tissue blood, and ${{C}_{a}}$ is the drug concentration in the blood entering the tissue. For other tissues, the mass balance equations take the form of a flow-limited case.

1.3.6.2 Lung Compartment

The breathing pattern is described by the following equation:

$$ {{Q}_{\text{air}}}=0.5\cdot \omega \text{ }TV\sin (\omega t), $$

(1.24)

where ${{Q}_{\text{air}}}>0$ indicates an inhalation process,${{Q}_{\text{air}}}<0$ represents exhalation. In the following equations, all ${{Q}_{\text{air}}}$ are absolute values, and the inhalation and exhalation processes are identified by either a positive or a negative sign, respectively. The total dead space was modeled using two compartments.

For the upper dead space of the lung, the inhalation and exhalation processes were described by Eqs. 1.25 and 1.26, respectively, and the elimination rate for the lung was characterized by Eq. 1.27.

Inhalation:

$$ {{V}_{UPD}}\frac{d{{C}_{\text{UPD}}}}{dt}=-{{Q}_{air}}{{C}_{UPD}}$$

(1.25)

Exhalation:

$$ {{V}_{UPD}}\frac{d{{C}_{UPD}}}{dt}={{Q}_{air}}({{C}_{LPD}}-{{C}_{UPD}}) $$

(1.26)

$$ re{{x}_{LU}}={{Q}_{air}}{{C}_{UPD}}. $$

(1.27)

For the lower dead space of the lung, the inhalation and exhalation processes were also described separately using Eqs. 1.28 and 1.29, respectively:

Inhalation:

$$ {{V}_{LWD}}\frac{d{{C}_{LWD}}}{dt}={{Q}_{air}}({{C}_{UPD}}-{{C}_{LWD}}) $$

(1.28)

Exhalation:

$$ {{V}_{LWD}}\frac{d{{C}_{UPD}}}{dt}={{Q}_{air}}({{C}_{alv}}-{{C}_{LWD}}). $$

(1.29)

For the alveolar region, the volume of the alveoli is described by a sinusoidal function, given that the volume of the alveoli changes with the breathing pattern:

$$ {{V}_{alv}}={{V}_{alv,0}}+0.5\cdot TV(1-\cos (\omega t)). $$

(1.30)

And the corresponding inhalation process was characterized by coupled differential Eqs. 1.31 and 1.32:

$$ \frac{d{{V}_{alv}}{{C}_{alv}}}{dt}={{Q}_{air}}{{C}_{LWD}}+PA\left({{C}_{b,out}}-\frac{{{C}_{alv}}}{{{P}_{air}}}\right) $$

(1.31)

$$ {{V}_{bLu}}\frac{d{{C}_{b,out}}}{dt}=Q({{C}_{b,in}}-{{C}_{b,out}})-PA\left({{C}_{b,out}}-\frac{{{C}_{alv}}}{{{P}_{air}}}\right). $$

(1.32)

The exhalation process was also defined by coupled differential Eqs. 1.33 and 1.34:

$$ \frac{d{{V}_{alv}}{{C}_{alv}}}{dt}={{Q}_{air}}{{C}_{alv}}+PA\left({{C}_{b,\text{out}}}-\frac{{{C}_{alv}}}{{{P}_{air}}}\right) $$

(1.33)

$$ {{V}_{bLu}}\frac{\text{d}{{C}_{b,out}}}{\text{d}t}=Q({{C}_{b,in}}-{{C}_{b,out}})-PA\left({{C}_{b,out}}-\frac{{{C}_{alv}}}{{{P}_{\text{air}}}}\right). $$

(1.34)

The breathing frequency is $\omega =2\pi f$, where, f is the number of breaths/per minute. ${{V}_{alv,0}}$ is the functional residual capacity of alveoli; $Q$ is the cardiac output; $PA$ is the product of membrane permeability and area of membrane transfer. The subscripts refer to the following: UPD—upper dead space; LWD—lower dead space; alv—alveolar; b,in—blood entering the lung; b,out—blood leaving the lung; and VbLu is the blood volume in the lung tissue.

1.3.7 Sensitivity Analysis in PBPK Modeling

As shown in the example above, there are several different types of parameters in a PBPK model. Parameter values for tissue volume, blood volume, and blood flow rates were obtained from published results (Brown et al. 1997). These values usually represent a typical male or female individual. Other parameters such as partition coefficient of a compound between blood and the tissue are often estimated based on in vitro or scaled-up studies from animals to human. The remaining unknown parameters are then estimated by fitting the model to the observed data. Given the large number of parameters, it is critical to evaluate the impact of the uncertainty of those parameters on the disposition of the compound in the body. This is often done through a local (derivative) and global (Monte Carlo method) sensitivity analysis. We used the first example to illustrate the importance of this analysis.

In Example 1, the anatomical and physiological parameters related to the lung such as the volume of the alveoli, the dead space inside the lung, the functional residual capacity, the tidal volume, and the breathing frequency were obtained from Guyton’s textbook of physiology (Guyton and Hall 1996; Hall and Guyton 2011). The partition coefficient, P _ft = C _fat/C _blood, of 50, was estimated based on the oil/water partition ratio of the compound. According to the results reported for other lipophilic compounds such as dioxin or thiopental (Bischoff and Dedrick 1968; Wang et al. 1997), the partition coefficient for nonfatty tissue is approximately 10 % of that of the fat tissue. Therefore, the partition coefficients of other nonfatty tissue were assumed to be P _t = 5. Table 1.2 listed the parameter values for a typical 70-kg healthy subject.

Table 1.2 PBPK model parameters for a 70-kg healthy subject. (Parameter values from Guyton and Hall 1996)

There are three remaining unknown parameters, the partition coefficient between air and blood, P _air, and the two permeability values, PA _air and PA _t. Both individual fitting and mean value fitting were conducted. Figs. 1.6 and 1.7 present the fitting of the mean values at 0.3 mg/kg dose level. The fitted parameter values are given below:

Fig. 1.6

Mean observed and fitted blood concentrations following bolus injection (dose of 0.3 mL/kg; solid line is model-fitted values, symbols are observed concentrations, error bars represent SD)

Fig. 1.7

Mean cumulative lung elimination following bolus administration (dose of 0.3 mL/kg; solid line is model-fitted values, symbols are observed concentrations, error bars represent SD)

$$ \text{P}{{\text{A}}_{\text{air,}}}\text{=}\,\text{42}\text{.0}\pm\text{4}\text{.2}\,({{\text{m}}^{\text{3}}}\text{/min)} $$

$$ {{\text{P}}_{\text{air}}}\text{=}\,{{\text{C}}_{\text{gas}}}\text{/}{{\text{C}}_{\text{blood}}}\text{=}\,\text{106}\,{}\pm{}\,\text{50}. $$

The ratio of PA to the blood flow rate through that tissue were such that

PA_t/Q = 0.53 ± 0.15 (for the heart, the lung, and the viscera compartment)

PA_t/Q = 1.80 ± 0.22 (for the lean compartment and the fat tissue).

To assess the impact of parameters on the PK of the agent, Monte Carlo simulation was conducted. The low, median, and high values of a parameter based on physiological reality were selected. For example, the total cardiac output range of 0.975–1.625 indicates that the low value of the total cardiac output is 0.975 with a median value of 1.3 and an upper range of 1.625. The results of Monte Carlo simulation demonstrated that only cardiac output, tidal volume of the lung and permeability have a significant impact on the concentration in the left heart of the agent. Other parameters such as the coronary blood flow rate, the fat content (though it is a lipophilic compound), the total blood volume, the breathing frequency, and functional residual volume have negligible effects on the concentration in the left heart. This analysis suggested that fixing a large number of parameter values using published data would have little impact on the fitted values of those three parameters (Table 1.3). This sensitivity analysis also provided information on potential source of interindividual variability .

Table 1.3 Parameter effect of the contrasting agent disposition after sensitivity analysis

1.3.8 Application of PBPK Modeling

There are many similarities in the anatomy and physiology of mammalian species; for example, many physiological processes vary as the 0.7–0.8 power of body weight and the anatomic variables are proportional to the body weight (Hu and Hayton 2001; Peters 1986; Savage et al. 2004; West et al. 1997, 1999). As such, the physiological process per unit body weight or organ weight tends to decrease as body size increases. Physicochemical parameters, such as blood-to-tissue partition coefficient and plasma protein binding, are not expected to have a great variation across species. Hence, a major application of PBPK model was in the field of species scaling (Boxenbaum 1982; Boxenbaum and Ronfeld 1983; Dedrick 1973; Dedrick and Bischoff 1980; Mordenti and Chappell 1989). One of the main limitations in the extrapolation of a PBPK model, from one species to another, is when significant difference in metabolic pathways and enzyme activities exist. Since the first PBPK model with flow-limited assumption for thiopental (Bischoff and Dedrick 1968), extensive application has been seen in drug development and environmental toxicology (Dedrick 1973; Peters 2012; Reddy et al. 2005; Rowland et al. 1973).

Peters presented a generic 14-compartment PBPK model that includes one compartment for the stomach, seven compartments to describe the absorption of a substance from the small intestine, and another compartment for the colon (Peters 2008). The dissolution of an orally administered substance from the GI tract is dependent on the product of the dissolution parameter and the difference between the solubility of the drug and its concentration. The model, being used for the in vivo prediction based on in vitro measurements, requires parameter for solubility, the pH of the buffer that was used for solubility measurement, and the in vitro absorption rate constant from Caco-2 permeability. For the estimation of drug concentration leaving a specific organ, the model utilizes plasma protein-binding information and tissue: plasma partition coefficients based on the work by Poulin and Theil (Poulin et al. 2001; Poulin and Theil 2000, 2002a, b; Theil et al. 2003). Peters used the model to predict the PK parameters of several compounds (Peters 2008; Peters and Hultin 2008). The measure to assess the quality of fitting was based on the reduced ${{\chi }^{2}}$-statistics and mean fold error. The author concluded that a “generic and integrative PBPK approach of drug disposition as a tool for a priori simulations and mechanistic evaluations of pharmacokinetics has the potential to improve the selection and optimization of new drug candidates” (Peters 2008).

In recent years, PBPK modeling has been applied to drug development and regulation (Zhao et al. 2009, 2011, 2012). A comprehensive review on the application of PBPK modeling in drug development and regulatory review/submission was provided in a number of articles from the Food and Drug Administration (FDA; Huang 2012; Huang and Rowland 2012; Leong et al. 2012; Rowland et al. 2011; Zhao et al. 2011, 2012). The articles summarized the major advances in the predictability of key PK parameters in human from in vitro data, the availability of dedicated software platforms, and associated databases. Specific advances and contemporary challenges with respect to predicting the processes of drug absorption, distribution and clearance were reviewed, together with the ability to anticipate drug–drug interactions and the impact of age, genetics, diseases, and formulations on the PK of a drug. The value of this capability in selecting and designing appropriate clinical studies, its implications for cost-effective strategies, and a more holistic view of the application of PK across the preclinical and clinical drug development processes are considered. Finally, there is a greater focus on positioning PBPK within the drug development and approval paradigm, as well as its future application in personalized medicine .

1.4 Relationship Between Systemic and Tissue Clearance with PBPK Modeling Approach

1.4.1 Clearance Definition

Clearance is one of the most important concepts in PK. As Benet stated, “it allowed the field to develop a basic understanding and to make predictions as to how pathological and physiological changes would influence drug kinetics and drug dosing” (Benet 2010). By October 2009, there were more than 47,827 references found in PubMed under “drug clearance” (Benet 2010) . The clearance concept was originally developed to quantify the functional efficiency of the kidney in the removal of urea (Grehant 1904a, b) and was then extended to describe the elimination of xenobiotics through the liver (Lewis 1948). Early contributions on developing the clearance concept for the whole body were made by Benet, Rowland, and Wilkinson (Benet and Galeazzi 1979; Rowland 1972; Rowland et al. 1973; Wilkinson 1987; Wilkinson and Shand 1975).

Using the definition provided by Wilkinson (1987), “the most general definition of clearance is that it is a proportionality constant describing the relationship between a substance’s rate of transfer in amount per unit time, and its concentration, in an appropriate reference fluid.” This is illustrated in Fig. 1.8 for an elimination organ.

Fig. 1.8

Schematic representation of an eliminating organ using a mass balance approach

The mathematical expression for clearance (CL) is given as follows:

$$ CL=\frac{r(C)}{{{C}_{in}}}. $$

(1.35)

The extraction ratio reflecting the efficiency of an organ to remove a drug was defined as:

$$ E\cdot Q=\frac{CL}{Q}, $$

(1.36)

where $r(C)$ is the elimination rate, $C$ is the drug concentration inside the compartment, ${{C}_{in}}$ is the drug concentration entering the organ, and $Q$ is the blood flow rate through the organ.

The instantaneous clearance and extraction ratio from a blood or plasma concentration–time profile can be derived by applying mass balance to the eliminating organ, as in Fig. 1.8. Assuming that the organ is a homogeneous compartment such that ${{C}_{out}}=C$, where ${{C}_{out}}$ is the drug concentration leaving the organ, the mass balance equation is then

$$ V\frac{dC}{dt}=Q\cdot {{C}_{in}}-Q\cdot C-r(C) $$

(1.37)

given that $C=0$ at $t=0$.

By dividing Eq. 1.37 by $Q\cdot {{C}_{in}}$, the resulting expression is

$$ \frac{V}{Q\cdot {{C}_{in}}}\frac{dC}{dt}=1-\frac{C}{{{C}_{in}}}-\frac{r(C)}{Q\cdot {{C}_{in}}}. $$

(1.38)

The third term on the right side of Eq. 1.38 is the definition of extraction ratio. The expression for instantaneous extraction ratio and clearance can be derived as follows:

$$ E=1-\frac{C}{{{C}_{in}}}-\frac{V}{Q\cdot {{C}_{in}}}\frac{dC}{dt} $$

(1.39)

$$ CL=Q\left(1-\frac{C}{{{C}_{in}}}\right)-\frac{V}{{{C}_{in}}}\frac{dC}{dt} $$

(1.40)

Both Eqs. 1.39 and 1.40 show that, in general, instantaneous extraction ratio and organ clearance are time-dependent variables. Therefore, they have no definitive meaning unless the time when clearance is estimated is specified, or steady state is achieved.

At steady state, both $C$ and ${{C}_{in}}$ are constant, such that $\frac{dC}{dt}=0$. The steady-state extraction ratio and clearance can be derived from Eqs. 1.39 and 1.40:

$$ {{E}_{ss}}=1-\frac{C}{{{C}_{in}}}$$

(1.41)

$$ C{{L}_{ss}}=Q\left(1-\frac{C}{{{C}_{in}}}\right). $$

(1.42)

Since instantaneous clearance is of not much value for nonsteady-state situation, the mean clearance over time is more useful and can be derived by rearranging Eqs. 1.35 and 1.36 and integrating from 0 to infinity with respect to time:

$$ \int_{0}^{\infty }{E\cdot {{C}_{in}}dt=\int_{0}^{\infty }{\frac{r(C )}{Q}}dt} $$

$$ \int_{0}^{\infty }{CL\cdot {{C}_{\text{in}}}dt=}\int_{0}^{\infty }{r(C)dt}. $$

Applying the mean integration theorem to the equations above, there exists a value θ for clearance such that $\int_{0}^{\infty }{CL\cdot {{C}_{in}}dt=}\text{ }\theta \int_{0}^{\infty }{{{C}_{in}}dt}$. The value θ is the mean clearance over time from 0 to infinity. Equations for the computation of mean clearance and mean extraction ratio are shown in Eqs. 1.43 and 1.44:

$$ \overline{CL}=\frac{\mathop{\int }_{0}^{\infty }r(C)dt}{\mathop{\int }_{0}^{\infty }{{C}_{in}}dt} $$

(1.43)

$$ \overline{E}=\frac{\mathop{\int }_{0}^{\infty }r(C)dt}{Q\mathop{\int }_{0}^{\infty }{{C}_{in}}dt}. $$

(1.44)

Using the example of an eliminating organ in Fig. 1.8, integration of the mass balance Eq. 1.37 from t = 0 to infinity,

$$ V\int_{0}^{\infty }{dC=}\int_{0}^{\infty }{[Q\cdot {{C}_{in}}-Q\cdot C-r(C) ]}dt. $$

(1.45)

For limited dosage regimens, if $t\to \infty$, ${{C}_{in}}\to 0$ and therefore $C\to 0$. Together with the initial condition that $C=0$ at $t=0$, the left side of Eq. 1.45 is 0. Thus,

$$ 0=\int_{0}^{\infty }{[Q\cdot {{C}_{in}}-Q\cdot C-r(C) ]}dt. $$

(1.46)

By rearranging and dividing Eq. 1.46 by $\int_{0}^{\infty }{Q\cdot {{C}_{in}}dt}$, we obtain

$$ \frac{\mathop{\int }_{0}^{\infty }r(C )dt}{\mathop{\int }_{0}^{\infty }Q\cdot {{C}_{in}}dt}=1-\frac{\mathop{\int }_{0}^{\infty }C\text{ }dt}{\mathop{\int }_{0}^{\infty }{{C}_{in}}dt}. $$

(1.47)

The left side of Eq. 1.47 is the term for the mean extraction ratio (see Eq. 1.44). From Eq. 1.47, one can derive the relationship between mean extraction ratio and clearance:

$$ \overline{E}=\frac{\overline{CL}}{Q}=1-\frac{\mathop{\int }_{0}^{\infty }Cdt}{\mathop{\int }_{0}^{\infty }{{C}_{in}}dt} $$

(1.48)

$$ \overline{CL}=Q\left(1-\frac{\mathop{\int }_{0}^{\infty }Cdt}{\mathop{\int }_{0}^{\infty }{{C}_{in}}dt} \right). $$

(1.49)

For an eliminating organ, $\int_{0}^{\infty }{C\text{ }dt}<\int_{0}^{\infty }{{{C}_{in}}dt}$ holds true. One can deduce that the mean clearance is always smaller than the blood flow rate and the extraction ratio is smaller than one.

It has been observed that for some xenobiotics, the estimated extraction is greater than one, which is impossible as the amount of drug being removed cannot be greater than the amount supplied to the organ. Using the same eliminating organ shown in Fig. 1.8, if clearance is calculated using the blood samples leaving the specific organ, we would divide Eq. 1.46 by $\int_{0}^{\infty }{Q\cdot C\text{ }dt}$, where C is the drug concentration in the blood stream leaving the elimination organ. Following the same steps to derive Eq. 1.48, we obtain Eq. 1.50:

$$ \frac{{{\overline{CL}}_{out}}}{Q}=\frac{\mathop{\int }_{0}^{\infty }{{C}_{in}}dt}{\mathop{\int }_{0}^{\infty }Cdt}-1. $$

(1.50)

As one can see that for a high extraction drug, it is possible that $\frac{\mathop{\int }_{0}^{\infty }{{C}_{in}}dt}{\mathop{\int }_{0}^{\infty }Cdt}-1$ is greater than 1. And if $\frac{\mathop{\int }_{0}^{\infty }{{C}_{in}}dt}{\mathop{\int }_{0}^{\infty }Cdt}-1>1$, the extraction ratio defined in Eq. 1.48 is larger than 1, and the estimated clearance will exceed the blood flow rate. The conversion between ${{\overline{CL}}_{\text{out}}}$ and $\overline{CL}$ can be derived from Eqs. 1.48 and 1.50, such that

$$ \frac{\overline{CL}}{Q}=\frac{{{\overline{CL}}_{out}}}{Q+{{\overline{CL}}_{out}}}, $$

(1.51)

where ${{\overline{CL}}_{out}}$ refers to the clearance computed from the blood samples taken from the blood leaving the organ and $\overline{CL}$ is the clearance estimated from the blood entering the organ. The equation for extraction ratio, which is the left-hand side of Eq. 1.51, can be rearranged to obtain ${{\overline{CL}}_{out}}$ such that

$$ {{\overline{CL}}_{out}}=\frac{\overline{CL}}{1-\overline{E}}. $$

(1.52)

We can extrapolate the result to total body clearance (TBC) in relation to cardiac output. The TBC, in this case, can also be greater than cardiac output for a compound that is rapidly eliminated through the lungs.

1.4.2 Establishing a General Relationship between TBC and Organ Clearance

It is often misunderstood that TBC is the sum of the individual organ clearance . The contribution of the organ clearance to the TBC depends on the anatomy of the body. In this section, we derived a general mathematical expression of the TBC with the individual organ clearance through PBPK modeling (Wang, Lam and Bischoff, unpublished work).

Figure 1.9 illustrates a representative PBPK model. This model consists of seven tissues, namely vein, artery, lung, GI tract, liver, kidney, and the rest of the body. The first compartment C₁ represents the vein, second C₂ for the artery, third C₃ the lung, fourth C₄ the liver, fifth C₅ the kidney, sixth C₆ for the rest of the body, and seventh C₇ for the GI tract. In this example, an intravenous administration was utilized for the simplicity of the mathematical derivation. Assuming a flow-limited mass transfer, the equations representing the model based on mass balance are as shown in Eq. 1.53.

Fig. 1.9

A representative PBPK model used to illustrate that total body clearance is not necessarily the sum of all organ clearances

$$ \begin{aligned}& {{V}_{1}}\frac{d{{C}_{1\,(vein)}}}{dt}={{Q}_{1}}\cdot {{C}_{0}}-{{Q}_{1}}\cdot {{C}_{1}}+{{K}_{I}}(t) \\& {{V}_{2}}\frac{d{{C}_{2\,(artery)}}}{dt}={{Q}_{1}}\cdot {{C}_{3o}}-{{Q}_{1}}\cdot {{C}_{2}}\\& {{V}_{3}}\frac{d{{C}_{3\,(lung)}}}{dt}={{Q}_{1}}\cdot {{C}_{1}}-{{Q}_{1}}\cdot {{C}_{3o}}-{{r}_{3}}({{C}_{3}}) \\& {{V}_{4}}\frac{d{{C}_{4\,(liver)}}}{dt}={{Q}_{7}}\cdot {{C}_{7o}}+({{Q}_{4}}-{{Q}_{7}})\cdot {{C}_{2}}-{{Q}_{4}}\cdot {{C}_{4o}}-{{r}_{4}}({{C}_{4}}) \\& {{V}_{5}}\frac{d{{C}_{5\,(kidney)}}}{dt}={{Q}_{5}}\cdot {{C}_{2}}-{{Q}_{5}}\cdot {{C}_{5o}}-{{r}_{5}}({{C}_{5}}) \\& {{V}_{6}}\frac{d{{C}_{6\,(rest of body)}}}{dt}=6\cdot {{C}_{2}}-{{Q}_{6}}\cdot {{C}_{6o}}\\& {{V}_{7}}\frac{d{{C}_{7\,(GI tract)}}}{dt}={{Q}_{7}}\cdot {{C}_{2}}-{{Q}_{7}}\cdot {{C}_{7o}}-{{r}_{7}}({{C}_{7}}) \end{aligned} $$

(1.53)

where ${{K}_{I}}(t )$ is the drug input function, ${{r}_{i}}({{C}_{i}})$ represents the rate of elimination from an eliminating organ, and the subscript $o$ for out symbolizes the concentration leaving the organ. The initial conditions were assumed such that ${{C}_{i}}=0$ at $t=0$. The drug concentration at the joint point where the blood stream leaves the organ is represented by Eq. 1.54:

$$ {{Q}_{1}}\cdot {{C}_{0}}={{Q}_{4}}\cdot {{C}_{4o}}+{{Q}_{5}}\cdot {{C}_{5o}}+{{Q}_{6}}\cdot {{C}_{6o}}. $$

(1.54)

Given the relationship above, we solve for the venous compartment C₁ (see Appendix for details), assuming that the input function ${{K}_{I}}(t )=0$ and ${{C}_{i}}\to 0$ as $t\to \infty$:

$$ \int_{0}^{\infty }{{{Q}_{1}}({{C}_{1}}-{{C}_{3\text{o}}})dt+}\int_{0}^{\infty }{{{Q}_{4}}({{C}_{\text{p}}}-{{C}_{4\text{o}}})dt+}\int_{0}^{\infty }{{{Q}_{5}}({{C}_{2}}-{{C}_{5\text{o}}})dt+}\int_{0}^{\infty }{{{Q}_{7}}({{C}_{2}}-{{C}_{7\text{o}}})dt=}\int_{0}^{\infty }{{{K}_{\text{I}}}(t)dt}, $$

(1.55)

where ${{Q}_{4}}\cdot {{C}_{p}}={{Q}_{7}}\cdot {{C}_{7o}}+({{Q}_{4}}-{{Q}_{7}}){{C}_{2}}$.

Using Eq. 1.49 to compute the mean organ clearance, the generalized expression, except for the lung and liver, is shown in Eq. 1.56:

$$ {{\overline{CL}}_{i}}=\frac{{{Q}_{i}}\mathop{\int }_{0}^{\infty }({{C}_{2}}-{{C}_{io}})dt}{\mathop{\int }_{0}^{\infty }{{C}_{2}}dt}\in i\ne 3,4. $$

(1.56)

For the lung and liver compartments (C₃ and C₄), the resulting expressions for mean organ clearance are as follows:

$$ {{\overline{CL}}_{3}}=\frac{{{Q}_{1}}\mathop{\int }_{0}^{\infty }({{C}_{1}}-{{C}_{3o}})dt}{\mathop{\int }_{0}^{\infty }{{C}_{1}}dt} $$

(1.57)

$$ {{\overline{CL}}_{4}}=\frac{{{Q}_{4}}\mathop{\int }_{0}^{\infty }({{C}_{p}}-{{C}_{4o}})dt}{\mathop{\int }_{0}^{\infty }{{C}_{p}}dt}. $$

(1.58)

The total amount of drug elimination from the body, derived from solving the mass balance equations over the entire body, is $\int_{0}^{\infty }{\sum\nolimits_{i}{{{r}_{i}}(t)dt}={{Q}_{1}}}\int_{0}^{\infty }{({{C}_{1}}-{{C}_{0}})dt=}\int_{0}^{\infty }{{{K}_{I}}(t)dt}$. Following the expression in Eq. 1.43 for mean clearance, TBC or systemic clearance is derived as

$$ {{\overline{CL}}_{\text{total}}}=\frac{{{Q}_{1}}\mathop{\int }_{0}^{\infty }({{C}_{1}}-{{C}_{0}})dt}{\mathop{\int }_{0}^{\infty }{{C}_{1}}dt}=\frac{\mathop{\int }_{0}^{\infty }{{K}_{I}}(t)dt}{AUC}=\frac{Dose}{AUC}. $$

(1.59)

By substituting Eqs. 1.56–1.58 into Eq. 1.55, the relationship between TBC and organ clearance can be obtained as shown in Eq. 1.60 (see Appendix for derivation of Eq. 1.60):

$$ {{\overline{CL}}_{total}}={{\overline{CL}}_{lung}}+(1-{{\overline{E}}_{lung}})[{{\overline{CL}}_{liver}}+(1-{{\overline{E}}_{liver}}){{\overline{CL}}_{GI}}+{{\overline{CL}}_{kidney}}], $$

(1.60)

where ${{\overline{E}}_{i}}=\frac{{{\overline{CL}}_{i}}}{{{Q}_{i}}}$. A generalized relationship between TBC and organ clearance can then be established by extrapolating Eq. 1.60 to multiple eliminating organs:

$$ {{\overline{CL}}_{systemic}}={{\overline{CL}}_{lung}}+(1-{{\overline{E}}_{lung}})\left[{{\overline{CL}}_{liver\_channel}}+\underset{i}{\mathop \sum }\,{{\overline{CL}}_{i}}\right], $$

(1.61)

where ${{\overline{CL}}_{liver\_channel}}={{\overline{CL}}_{liver}}+(1-{{\overline{E}}_{liver}})\sum\nolimits_{j}{{{\overline{CL}}_{j}}}$, and$j$ indicates other organs except the liver in the liver channel. The subscript $i$ in Eq. 1.61 includes all other body organs except the lungs and organs in the liver channel.

The mathematical equation in Eq. 1.61 is broadly applicable not only to the linear systems but also to the nonlinear systems because the assumption made in the derivation of Eq. 1.61 was flow-limited mass transfer. From Eq. 1.61, one can deduce that the contribution of organ clearance to the systemic clearance can be derived by following the circulation scheme starting from the arteries. For those organs connected in parallel, the contribution of organ clearance to systemic clearance is additive. For the organs connected in series, the contribution of the ith organ clearance to systemic clearance needs to be corrected by a factor of $(1-{{\overline{E}}_{next\text{ }organ}})$. For example, following the blood circulation starting from the artery, the whole body can be viewed as consisting of two components connected in series, the rest of the body and the lung. Hence, in Eq. 1.61, the clearance of the portion that represents the rest of the body needs a correction factor of $(1-{{\overline{E}}_{lung}})$, as illustrated in Fig. 1.10a. The same is true for the liver channel such that the clearance of the GI tract or spleen also requires a correction factor $(1-{{\overline{E}}_{liver}})$, as represented in Fig. 1.10b.

Fig. 1.10

Schematic illustration of a the left heart–lung–right heart (H–L–H) circulatory system and b the liver channel

The PBPK model used in this discussion is a lumped total body model but represents a rather general description of the anatomical structure of the mammalian system. The expression describing TBC and the individual organ clearances can be easily derived based on the location of the eliminating organ by following the circulation starting from the arteries. For these organs that are connected in series, the contribution of the first organ clearance to TBC equals $(1-{{\overline{E}}_{i+1}}){{\overline{CL}}_{i}}$, whereas for organs connected in parallel, the contribution of organ clearance to TBC is additive. Even though the current discussion only considered the example of an intravenous administration, the conclusions derived from this scenario are broadly applicable to other routes of administration.

1.5 Population PK

PBPK models provide the quantitative information between the dose and the concentration of the xenobiotic agent at different regions of the body. Interindividual variability in PK can be quantified based on the physiological/biological difference among the target population. However, it is often impractical to apply PBPK models directly to humans. The nonlinear mixed effect model (also commonly referred to as population PK) incorporating both fixed effect (identified covariates representing the known source of variability) and random effect (unidentified source of variability) was introduced in the 1970s by Sheiner et al. (1977). This approach adopts the simplicity of the classical PK models but correlates the PK parameters to covariates such as body weight, age, gender, etc. to quantify the source of variability in PK. For example, the volume of distribution is often correlated to the body size, and clearance to body weight, creatinine clearance, and enzyme activities. Covariate analysis is primarily performed through statistical analysis, together with the information on physiology, pathology, metabolism, and clinical relevance. The population PK modeling has been widely applied to the analysis of clinical PK data, especially to the sparse PK samples from phase 2/3 trials. With this approach, the source of variability in PK can be identified using large pooled datasets. More importantly, population PK modeling has innovated the drug development through model-based approach (Bhattaram et al. 2005). From what used to be an unusable sparsely sampled blood concentration from phase 2/3 trials, population PK modeling made it possible to identify the PK characteristics in patients based on their demographic information, metabolic status, liver/kidney function, and disease status to support labeling. This ability to hone into a specific factor responsible for drug response is what makes population PK a valuable tool for personalized medicine . In a survey from the US FDA between 2000 and 2004, they reported that pharmacometric analyses were pivotal in regulatory decision making in more than half of the 42 new drug applications (NDA; Bhattaram et al. 2005). In the case of a failed trial of nesiritide, they concluded that “dose selection based on pharmacometric analysis could have saved 3 years of drug development time and 1 clinical trial.”

1.5.1 Population PK Model Development

A population PK model consists of the structural model and the covariate models. The structural model takes the form of the classical compartment model to describe the concentration profile for a typical subject, whereas the covariate model quantifies the sources of intersubject variability : the known (covariates) and unknown (first level random effect term) source of variability (Sheiner and Ludden 1992; Sheiner et al. 1977). The second level of variability is within-subject variability that is often described by additive or proportional error model. Sometimes, a third level variability, called inter-occasion variability, is introduced to describe the variability for the same individual when replicate samples were taken following different dosing occasions.

Population PK model development and key considerations were clearly discussed in Meibohm et al. (2005) and regulatory documents (EMA 2007; FDA 1999), which include base model and covariate model development, internal and/or external model validation. The base model development follows the same principles as classical compartmental PK model development. Covariate model development and model validation will be described briefly in the following sections.

1.5.2 Covariate Models

Covariate models quantify the source of intersubject variability contributing to a PK parameter. For continuous covariates, the covariate model usually takes the form of a linear or power function, which probably originated from the allometric scaling concept. Different ways of parameterization of the covariate model are often done using the reference value of the covariate such as the median value, as shown in Eqs. 1.62 and 1.63:

$$ TVP={{\theta }_{0}}+\sum{}_{i=1}^{m}{{{\theta }_{i}}(Co{{v}_{i}}-Re{{f}_{i}})} $$

(1.62)

$$ TVP={{\theta }_{0}}\prod{}_{i=1}^{m}{{{\left(\frac{Co{{v}_{i}}}{Re{{f}_{i}}}\right)}^{{{\theta }_{i}}}}}$$

(1.63)

where $TVP$ is the typical value of a model parameter, $Cov$ represents the covariates, $m$ is the number of covariates, and ${{\theta }_{i}}$ is the coefficient.

For categorical covariates such as binary, ordered, or nonordered, an indicator function ${{I}_{i}}(Co{{v}_{i}})$ is introduced such that if the covariate has a specific dummy variable value, a separate coefficient is designated:

$$ TVP={{\theta }_{0}}+\sum{}_{i=1}^{m}{{{\theta }_{i}}{{I}_{i}}(Co{{v}_{i}})} $$

(1.64)

$$ TVP={{\theta }_{0}}\prod{}_{i=1}^{m}{{{I}_{i}}{{(Co{{v}_{i}})}^{{{\theta }_{i}}}}}$$

(1.65)

and

$$ {{I}_{i}}(Co{{v}_{i}})=\left\{\begin{matrix} 1,\ ifi=i\\ 0,\ \text{otherwise}\end{matrix} \right. $$

(1.66)

Random effect is usually assumed to be log-normally or normally distributed, as given in Eq. 1.67, where $n$ follows a normal distribution with mean 0 and covariance matrix Ω:

$$ {{P}_{j}}=TVP\times \exp ({{\eta }_{j}})\quad \text{or}\quad {{P}_{j}}=TVP+{{\eta }_{j}}. $$

(1.67)

1.5.2.1 Allometric Scaling

One of the approaches for covariate model development is by applying allometric scaling principle. Thus, the body weight often contributes to clearance to the power of 0.75 and to the volume of distribution to the power of 1. The allometric scaling is also often used for scaling PK parameters obtained from the adult to the pediatric population.

1.5.2.2 Stepwise Regression

Stepwise regression is a common statistical method used in covariate model building. The algorithm includes forward addition, backward elimination, or combination of forward addition and backward deletion in stepwise fashion. This process is automated in Perl-Speaks-NONMEM. Each step of the model building process in the forward inclusion involves testing the effect of each covariate on the appropriate model parameter in a separate model run, such that the statistical significance of each covariate–parameter relationship is screened individually (univariate analysis). Covariates that reduce the objective function above a predefined significance level are added to the PK model. The backward elimination step starts with the final model from the forward inclusion step; the subsequent removal of each covariate is also based on a predefined difference in the objective function. The hypothesis testing to discriminate alternative hierarchical models is based on the likelihood ratio test, often at preset p-values for the forward inclusion and backward elimination of, e.g., 0.05 and 0.01, respectively. The differences in the objective function values of two alternative models is equivalent to − 2 log-likelihood, which follows a chi-squared distribution with n degrees of freedom, where n is the difference in the number of parameters in the hierarchical models. A difference of 3.84 and 6.64, for example, in the value of the objective function is considered significant under the likelihood-ratio test for n = 1 and p-values of 0.05 and 0.01, respectively.

1.5.2.3 Full Covariate Model

Because covariates often have collinearity and depending on the degree of correlation between covariates, the statistical inference approach such as stepwise methods could include a covariate that is not preferential. An algorithm of using full covariate model approach was proposed as an alternative for covariate model building (Agoram et al. 2006; Ravva et al. 2009). The decision on which covariate to include is based on exploratory graphics, scientific and clinical interest, mechanistic plausibility, or previous knowledge of these relationships. Covariates that are both statistically insignificant and clinically irrelevant can be dropped during covariate model development. The inferences on which covariate has clinical importance are then based on the magnitude of the estimated effect and the precision (Agoram et al. 2006). This approach is a simplification of the global model approach (Burnham and Anderson 2002) and is claimed to be the preferred choice when the goal is to estimate the magnitude of an effect (Harrell 2001). A hybrid approach can also be implemented, starting with a full covariate model from which covariates are tested using stepwise backward deletion.

1.5.2.4 Case Deletion to Determine Influential Individual

The statistical inferences based on maximum likelihood or likelihood ratio test are easily influenced by outliers or a few individuals (not necessarily outliers) in the data. Influential individual or a group of influential individuals can be evaluated using case-deletion diagnostics. The jackknife method evaluates how removing a specific individual affects the objective function values of the base model and the one with covariate, both models with all the data versus the one with the specific individual removed (Sadray et al. 1999):

$$ \Delta OF{{V}_{jackknife,i}}=(OF{{V}_{final,n}}-OF{{V}_{basic,n}})-(OF{{V}_{final,n-1}}-OF{{V}_{basic,n-1}}). $$

(1.68)

The algorithm fits both the covariate model and the base model to the dataset containing all the individuals and the dataset with the specific individual removed. The $\Delta OF{{V}_{jackknife,i}}$ value in Eq. 1.72 is obtained for each individual of the dataset. The “shark” plot with the number of subjects removed on the x-axis and the change in OFV on the y-axis and curves showing both positive and negative $\Delta OF{{V}_{i}}$ can be used as visual inspection for case-deletion diagnostics.

1.5.2.5 Covariate Identification Through PBPK Modeling

As mentioned in the section of PBPK modeling, lumping has been used to simplify the complicity of a PBPK model without losing the key physiological reality of the model. During the lumping process, covariates having impact on the PK can be identified. This approach requires a PBPK model that can be developed using preclinical data. Sensitivity analysis of the PBPK model can also assist in identifying the factors that could have significant impact on the disposition of a xenobiotic.

The example below demonstrating the advantage of using PBPK modeling to identify the covariate was from a collaboration between Bischoff and Stanski in the early 1990s (Bischoff 1992). It was observed that the amount of thiopental needed to be administered to elderly patients (about 70–80 years old) was much less than that used for a standard 30-year-old healthy male. Therefore, age could be considered as a covariate for dose adjustment for thiopental based on the classical PK and population PK models. To investigate the age effect on thiopental PK, a PBPK model (Bischoff and Dedrick 1968) was applied to the PK data obtained from clinical studies. An adipose tissue compartment was included in the PBPK model for thiopental, as this compound is highly lipophilic. The differences in cardiac outputs between young adult subjects at 30 years of age versus the geriatric patients at 70 years were taken into account in the PBPK model. By incorporating the physiological difference in cardiac output between the two age groups, the PBPK model captured well the difference in the exposure of thiopental for patients at age 30 versus those at 70 years, without having to incorporate age as a covariate of the model. This work was continued by Wada and colleagues to demonstrate the underlying mechanism behind the observed age effect in thiopental PK (Wada et al. 1997). Their results showed that the difference were due to the decline in cardiac output, followed by the increase in fat content with age. Since the cardiac output starts to decline at approximately 40 years of age, a nonlinear covariate equation to link clearance with age would be necessary. However, using the known data of age and gender differences in cardiac output and BMI (Brandfonbrener et al. 1955; Freedson et al. 1979; Guyton and Hall 1996), the change in clearance and volume of distribution of thiopental with age can be derived from these relationships.

1.5.2.6 Clinical Relevance in Covariate Model Development

Clinical relevance is a key consideration in covariate model assessment. In general, if the contribution of a covariate to the PK parameters resulted in less than 20 % difference in systemic exposure using the bioequivalence (BE) criteria, this covariate can be ignored or dropped even though it is shown to be statistically significant. Sometimes, if a drug has a large therapeutic window and the influential covariate determined from the PK analysis does not have any significant impact on clinical endpoints, this covariate can be removed. In contrasting situations, the lack of statistical significance does not necessarily indicate that the covariate tested is lacking impact on the clinical endpoints. For example, due to limited sample size (< 10 % of the subjects with a specific co-medication) or limited range of the covariate tested such as age, the impact of co-medication or age effect on the PK may not be statistically significant.

1.5.2.7 Power and Sample Size Calculation

Sample size (the total number of subjects and the sampling time per subject) is critical for population PK development. To be able to detect the interindividual and intraindividual variability , a minimum of two PK samples per subject is necessary. Ogungbenro and Aaron have demonstrated the minimal samples size requirement for a one- or two-compartmental PK model (Ogungbenro and Aarons 2008) based on the confidence interval of the PK parameters estimated.

Another statistical methodology called the Monte Carlo Mapped Power (MCMP) to determine the power and sample size calculation for covariate model development was introduced by Vong et al. (2012). Using the difference in individual’s objective function values between the reduced and full models ($\Delta iOFV=iOF{{V}_{reduced}}-iOF{{V}_{full}}$), the MCMP analysis tests for drug or covariate effect by the summation of the individual’s contribution to the overall objective function value in the likelihood ratio test (Vong et al. 2012). The MCMP uses the sum of the n $\Delta iOFV$ instead of the overall $\Delta OFV$ to base its statistical inference. The algorithm maps the statistical power over a specified sample size range. The MCMP method is simple to run without the need for correcting for type I error that is associated with stochastic simulations and estimations (Ette et al. 1998; Kowalski and Hutmacher 2001; Lee 2001).

1.5.2.8 Model Evaluation

Model evaluation is a key step in population PK model development. Diagnostic plots, bootstrap, shrinkage in the η (Savic and Karlsson 2009), prediction-corrected visual predictive check, internal validation through dataset split are often used to evaluate the model. In addition to those approaches, external validation with additional datasets is a preferable method when feasible.

1.5.3 Application of Population PK Model

Since Sheiner’s first and subsequent publications that established the population PK model methodology, population PK modeling and simulation together with information on disease progression , placebo response, dropout rates, as well as exposure–response (ER) of drug treatment, have been used in regulatory decision making, clinical trial waiver, as well as identification of design flaws and trial implementation problems prior to running a trial. These strategies have shown to decrease costs, improve the likelihood of achieving the trial goals, and generate conclusive findings (Brindley and Dunn 2009; Holford et al. 2010). Kimko and Peck recently edited a textbook on clinical trial simulation that encompasses diverse areas relevant to drug development such as metabolic disease, cardiovascular, infectious disease, oncology, and many other fields (Kimko and Peck 2011).

Yang et al. took an approach of incorporating a case–control comparison in the ER analysis to reduce the bias introduced by confounding risk factors when evaluating the recommended dosing regimen for trastuzumab in a registration trial (Yang et al. 2012). Their analysis suggested that patients with the lowest quartile of trastuzumab exposure did not benefit from addition of trastuzumab treatment to chemotherapy. However, contrary to the nonresponder hypothesis for this subgroup with the lowest quartile of tratuzumab, this subgroup appeared to be more sensitive to a higher trastuzumab exposure than the remaining 75 % population, suggesting that increasing trastuzumab exposure in the low-exposure subgroup may result in better overall survival (OS) benefit. This analysis justified the FDA recommendation of conducting postmarketing clinical trials to investigate a dosing regimen with higher exposure and acceptable safety in the identified subgroup and to prospectively evaluate whether this regime will result in acceptable OS benefit.

Other examples can be found from the approval of the 0.8 mg/kg once-weekly regimen of etanerceptin pediatric patients with juvenile rheumatoid arthritis . The clinical trial simulation confirmed that the 0.8 mg/kg once weekly yielded overlapping steady-state time–concentration profiles with that of 0.4 mg/kg SC twice weekly, leading to equivalent clinical outcomes (Yim et al. 2005).

The author (Wang) utilized model-based approach to apply for waiver of clinical pharmacology trial for a novel tyrosine kinases inhibitor that was under development in oncology. The absolute bioavailability of the tablet was required in the NDA submission. However, to conduct a clinical trial to obtain the absolute bioavailability was difficult, since the intravenous dosage formulation needed to be developed, and the study can only be conducted in cancer patients due to genotoxicity. A population PK model of the PK datasets obtained from several phase I dose escalation trials with either oral solution or tablet was developed with the formulation as one of the covariates. Covariate model test demonstrated that the formulation was not a statistically significant covariate. The health authority accepted this approach and a standalone absolute bioavailability trial was no longer required. The example listed above demonstrates how the model-based approach can cut development costs, as well as improve trial designs to come to conclusive findings.

1.6 PD Models for Continuous Response Variables

PD often refers to as the body’s response to drug. Derendorf et al. defined PD as “a broad term that is intended to include all pharmacological actions, pathophysiological effects and therapeutic responses, both beneficial and adverse, of an active drug ingredient, therapeutic moiety, and/or its metabolite(s) on the various systems of the body from subcellular effects to clinical outcomes. Pharmacodynamic studies can provide information about a drug’s mechanism of action or about its dose-response relationship where response can be expressed as a direct or indirect measure of efficacy and/or safety of the drug” (Derendorf et al. 2000). As collecting biomarker information is becoming common in clinical trials, modeling the exposure and biomarker responses has become critical in model-based drug development .

PK/PD studies intend to link the dose–exposure profile relationship with the PD response, in particular, the time course of the pharmacological/pathophysiological effects (Derendorf et al. 2000). Integrated PKPD models are categorized according to the manner in which the PK and PD data are related. The two types of basic PD models that are often used to establish PKPD relationships are the direct and indirect response models . Based on receptor theory, the response of a drug is triggered by the free drug concentration at the site of action. Since the systemic blood or plasma drug concentration samples were collected during a trial, while the effect or response is dependent on the concentration at the effect site, a delay in response might be observed when linking the blood/plasma concentration to the drug response. If the effect is further downstream in the process, a longer delay in response could be observed.

1.6.1 Direct response PD Model

In the direct response model, the linear, E _max, or sigmoidal E _max models are often used. Equation 1.69 is the expression for the sigmoidal E _max model. When γ equals 1, it is the called the E _max model:

$$ E(C)={{E}_{0}}\pm \frac{{{E}_{\max{}}}C{{(t)}^{\gamma }}}{EC_{50}^{\gamma }+C{{(t)}^{\gamma }}}, $$

(1.69)

where${{E}_{0}}$ is the baseline of the response and ${{E}_{\max }}$ is the maximum response. $C$ is the concentration of drug and $C(t )=\frac{\text{Dose}}{V}{{\text{e}}^{-\frac{CL}{V}t}}$ is the drug concentration at which 50 % of the maximum response is achieved and $\gamma$ is the sigmoidicity factor that determines the steepness in the linear portion of the curve. The direct response relationship assumes that the processes involved in the drug transfer to the site of action and eliciting the response is rapid enough compared to the disposition of a drug. Thus, for the same drug concentration, the response elicited by the drug is the same, regardless of the time to reach that drug concentration. As shown in the left panel of Fig. 1.11, the concentration C1 in the ascending phase of the concentration–time profile with the equivalent concentration level at C2 in the descending phase has a corresponding response R1 in the ascending phase of the response profile and the equivalent response level R2 in the descending phase. In other words, there is only one value for the response corresponding to one value for the drug concentration, as shown in the right panel of Fig. 1.11.

Fig. 1.11

Relationship between drug concentration and response in a direct response relationship

1.6.2 Indirect Response Model

It is often the case in pharmacology that the effect of the pharmacological agent is lagging behind the drug concentration–time course, where, the response versus concentration does not exhibit a one-to-one relationship, often called a hysteresis loop (counterclockwise hysteresis) , as shown in the right panel of Fig. 1.12 . The temporal dissociation between the time courses of drug concentration and effect results in the hysteresis pattern and is likely caused by a distributional delay between the drug concentrations in the plasma and the effect site (Derendorf et al. 2000).

Fig. 1.12

Relationship between drug concentration and response in an indirect response relationship

The two different approaches often used to describe the observed delay in the plasma concentration and drug response are the effect compartment model originally proposed by Sheiner et al. (1979) and the indirect response model with differential equation by Jusko (Dayneka et al. 1993; Jusko and Ko 1994).

1.6.2.1 Effect-Compartment PD Model

The effect-compartment model links the drug effect to the drug concentration of a hypothetical effect compartment (Sheiner et al. 1979), instead of the drug concentration in the systemic circulation. In the effect-compartment model, it was assumed that the drug amount entering the effect compartment is negligible, so that the plasma concentration of the central compartment can be described without the mass transfer between the central compartment and the effect compartment. The drug concentration in the effect compartment is at equilibrium with that of the central compartment. The equilibration process between the plasma drug concentration and the effect site is determined by the first-order rate constant ${{k}_{e0}}$ that also describes the loss of drug from the effect compartment (Derendorf et al. 2000). The illustration in Fig. 1.13 shows a schematic representation of the link model, where EC refers to the effect compartment. When the drug response links to the drug concentration of the effect compartment, the hysteresis observed will be collapsed.

Fig. 1.13

Schematic representation of the effect compartment linking to a pharmacodynamic model

Equations for calculating the effect compartment concentration can be found in Gabrielsson and Weiner (2000). For example, for a one-compartment model with bolus injection, the plasma concentration can be calculated using Eq. 1.4, and the effect compartment concentration can be expresses as:

$$ {{C}_{e}}=\frac{Dose\cdot {{k}_{e0}}}{{{V}_{e}}\left({{k}_{e0}}-\frac{CL}{V} \right)}\left(\exp \left(-\frac{CL}{V}\cdot t \right)-\exp (-{{k}_{e0}}\cdot t \right) ), $$

(1.70)

where ${{C}_{e}}$ is the drug concentration in the effect compartment, ${{V}_{e}}$ is the volume of the effect compartment.

1.6.2.2 Indirect Response Models

The indirect response models was developed based on the receptor theory and signal transduction, where a series of delay could occur during those processes caused by indirect-response mechanism such as a synthesis or dissipation of an endogenous substance or response mediator. Dayneka et al. proposed four basic models for indirect PD response (Dayneka et al. 1993; Jusko and Ko 1994). The generalized form of the indirect response models in the absence of drug is described as follows:

$$ \frac{dR}{dt}={{k}_{in}}-{{k}_{out}}R, $$

(1.71)

where ${{k}_{in}}$ is the zero-order constant for the production of the response, and ${{k}_{out}}$ refers to the first-order rate constant for the dissipation of response. A biological system should stay at steady state under normal condition when no drug intervention is applied. Therefore, at baseline ${{R}_{o}}=\frac{{{k}_{in}}}{{{k}_{out}}}$.

The four indirect response models under drug intervention are shown below:

$$ \frac{dR}{dt}={{k}_{in}}I(t)-{{k}_{out}}R $$

(1.72)

$$ \frac{dR}{dt}={{k}_{in}}-{{k}_{out}}I(t)R $$

(1.73)

$$ \frac{dR}{dt}={{k}_{in}}S(t)-{{k}_{out}}R $$

(1.74)

$$ \frac{dR}{dt}={{k}_{in}}-{{k}_{out}}S(t)R, $$

(1.75)

where $I(t )=1-\frac{{{C}_{p}}}{{{C}_{p}}+I{{C}_{50}}}$ represents the classical inhibitory function and $S(t)=1+\frac{{{E}_{\max{}}}\cdot {{C}_{p}}}{E{{C}_{50}}+{{C}_{p}}}$ is the stimulation process using an E _max model. The schematic representations of the indirect response models showing effect on the synthesis and dissipation processes are presented in Fig. 1.14 .

Fig. 1.14

Schematic representation of an indirect response—stimulation or inhibition of synthesis (top) and dissipation rates (bottom)

The indirect response model was used to fit to the data of the inhibition of prothrombin complex activity by warfarin , an oral anticoagulant used in thrombophlebities and pulmonary embolism (Jusko and Ko 1994). It blocks the vitamin K epoxide reductase, an enzyme that reduces vitamin K epoxide to vitamin K, which is a cofactor for carboxylation of the clotting factor such as factor II, VII, IX, and X. The blockade of the reductase activity by warfarin leads to the inhibition of coagulation, measured by the prothrombin time. It is assumed that the clotting factors are synthesized with a zero-order rate constant, ${{k}_{in}}$, and degraded with a first-order rate constant, ${{k}_{out}}$.

1.7 PD Models for Noncontinuous Response

Data collected from clinical trials, which are not continuous but categorical variables, can be dichotomous, ordinal scaled (e.g., none/mild/moderate/severe), or censored data (e.g., time to recurrence of a disease). Logistic regression and survival models are usually applied to describe the probability of events. Recently, Markov chain models to estimate event probability were also applied in pharmacometrics (Bizzotto et al. 2011; Lacroix et al. 2009; Sy et al. 2013a).

1.7.1 Time to Event

In the time to event analysis, the time of origin in pharmacometric analysis is usually the start of treatment. If the endpoint is some events, such as the occurrence of an adverse event, relapse of a disease, death, etc., the observations, which are the difference between the time of the specific event and the time from the origin, are referred to as time to event data or survival times. The distribution of time to event data is usually not normal and the data are often “censored.” Right censoring refers to data that the specific event of interest has not yet occurred when the subjects leave the study. Left censoring can occur in clinical studies when we know that the event of interest has already occurred at the observation time, but it is not known exactly when. For example, a patient may be tested positive for a specific disease but the exact time of the disease onset is unknown.

The survivor function, S(t), is defined as the probability that the event of interest has not occurred by duration t, such that

$$ F(t)=\Pr \left\{T\le t \right\}=F(t ). $$

(1.76)

where T denotes the time of an event; Pr stands for probability. The probability that the event has occurred by duration t, F(t), is defined as the complement of the survivor function, which is 1-S(t). The survivor function is related to the hazard function $h(t)$ and the cumulative hazard $H(t)$, as defined in Eqs. 1.77–1.79 (Collett 1994). One can obtain the hazard function by dividing the event density function $f(t)$ by the survivor function.

$$ S(t )=\Pr \left\{T>t \right\}=1-F(t )=\exp [-H(t ) ] $$

(1.77)

$$ h(t)=\underset{\Delta \to 0}{\mathop{\lim }}\,\frac{\Pr \left\{t<T\le t+\Delta |T>t \right\}}{\Delta }=\frac{f(t)}{S(t)}=-\frac{\partial \log S(t)}{\partial t} $$

(1.78)

$$ H(t)=-\log S(t). $$

(1.79)

Figure 1.15 shows representative examples of cumulative distribution, probability density, survival, and hazard functions.

Fig. 1.15

Example of baseline hazard and the corresponding probability density functions for exponential and Weibull distribution

With nonparametric and semi-parametric methods, namely the Kaplan–Meier estimate of the survivor function and the Cox proportional hazard model, which is an extension of the Kaplan–Meier method, the form of the baseline hazard is not specified. The form of the covariate relationship, however, is specified in the Cox proportional hazard model.

With parametric models, both the hazard function and the effect of covariates are explicitly defined. Holford argued for using a parametric model for the hazard function because “hazard is the way to introduce biological mechanism to the survival model and understanding the variability of time to event distributions” (Holford 2013). Table 1.4 lists the density, hazard, and survivor functions for the commonly used parametric models. Figure 1.15 shows the example of the probability density function and the corresponding hazard function for the exponential and Weibull distributions. In the exponential example, the hazard is a constant over time. This may not be the case in most clinical situations.

Table 1.4 Relationship between density, survival, hazard, and likelihood functions for exponential, Weibull, and log-logistic parametric models

The Weibull model is more flexible as well as more generalized than the exponential model; the hazard rates are monotonic in the sense that the hazard is either increasing, decreasing, or constant over time. The hazard for a Weibull cannot be a combination of increasing and decreasing trends. The shape parameter, $p$, determines the trend. When $p<1$, the hazard is decreasing monotonically. For $p>1$, the hazard is increasing with time. With $p=1$, the Weibull becomes an exponential function and the hazard is constant. The log-logistic model allows for nonmonotonic hazards. The shape parameter, $p$, determines the trend such that if $p>1$, then the hazard increases and then declines whereas if $p<1$, the hazard has a decreasing trend.

The Kaplan–Meier type estimate is useful for determining the appropriate hazard function to use. The ratio of the number of events (${{\delta }_{j}}$) and the number of individuals at risk at the time ($({{V}_{b,j}}+{{R}_{j}}{{V}_{t,j}})\frac{\text{d}{{C}_{b,j}}}{\text{d}t}={{Q}_{j}}(C-{{C}_{b,j}})-{{r}_{j}}$njτj) is the hazard in the time interval from ${{t}_{j}}$ to ${{t}_{j+1}}$:

$$ h(t)=\frac{{{\delta }_{j}}}{{{n}_{j}}{{\tau }_{j}}}, $$

(1.80)

where ${{n}_{j}}$ is the number of individual who has not had the event and therefore at risk of the event and ${{\tau }_{j}}$ is the time interval computed as ${{t}_{j+1}}-{{t}_{j}}$. The plot of time versus $h(t)$ provides a visual inspection of the trend of the hazard function.

The proportional hazard is one of the methodologies to introduce and evaluate a nontime-varying covariate effect, the other being accelerated failure time, which will not be discussed. Assuming a set of nontime-varying covariate vector $X=[{{X}_{1}},{{X}_{2}},\cdots,{{X}_{n}}]$, the proportional hazard function can be setup as

$$ h(t,X )={{h}_{0}}(t)\exp ({{\beta }^{T}}X), $$

(1.81)

where ${{h}_{0}}(t)$ is the baseline hazard that depends on $t$ but not $X$ and ${{\beta }^{T}}X$ is ${{\beta }_{1}}{{X}_{1}}+{{\beta }_{2}}{{X}_{2}}+\cdots +{{\beta }_{n}}{{X}_{n}}$. A covariate, for example the presence of a specific disease, has an effect on the hazard and the coefficient, ${{\beta }_{disease}}=1.39$. The relative risk for individuals with the disease is approximately fourfold (since $\exp{}(1.39)=4$) that of the healthy individual. We recently applied the proportional hazard model in pharmacogenomics to evaluate the influence of CYP3A5 and ABCB1 polymorphisms on the renal transplant patient’s relative risks for adverse events associated with tacrolimus (Sy et al. 2013b). The study used a marginal proportional hazard model with common baseline hazard to adjust for possible correlations between multiple incidents of adverse events, given that each patient can have multiple adverse events which were considered competing risks (Sy et al. 2013b; Wei et al. 1989; Wei and Glidden 1997). The marginal semi-parametric model is not without its criticism. The most frequent concern being raised is its assumption that each individual is considered to be at risk of all recurrent events from the start (Metcalfe and Thompson 2007). This assumption apparently would result in estimates that exceed those provided by alternative approaches. However, the marginal approach is considered to be the lesser of the two evils, with the alternative being one that does not consider a marginal model for repeated events from the same individual. For the parametric approach, the frailty model where the random effect has a multiplicative effect on the hazard can be used to handle recurrent events coming from the same individual. As pointed out by Hougaard, the limitation of the frailty model is the standard assumption of using a gamma distribution for frailty which puts more importance on late events (Hougaard 1995).

One can treat frailty as multiplicative of the hazard term such that

$$ h({{t}_{ij}}|{{\textbf{X}}_{ij}},{{\nu }_{i}})={{h}_{0}}({{t}_{ij}}){{\nu }_{i}}\exp (\beta {{\textbf{X}}_{ij}}), $$

(1.82)

where $j$ refers the individual and i is the subgroup, and the frailty term is ${{\nu }_{i}}=\exp ({{W}_{j}}\psi)$. ${{W}_{j}}$ is the “frailty” sampled from a distribution with mean 0 and a variance 1. If $\psi$ is 0, we have a standard proportional hazard. The hazard rate above is conditional on both the covariates and the frailty term and so is the survivor function,

$$S\left( {{t_{ij}}{\rm{|}}{{\bf{X}}_{ij}},{\nu _i}} \right) = \exp \left( { - \int_0^t {h(u|\nu )du} } \right) = \exp \left( { - \nu \int_0^t {h\left( u \right)du} } \right).$$

(1.83)

Before we obtain the marginal survivor function, we shall introduce the gamma distribution. The density for the gamma distribution is given by

$$ g(\nu,\alpha,\beta)=\frac{1}{{{\beta }^{\alpha }}{}\Gamma{}(\propto)}{{\nu }^{\propto -1}}{{e}^{-\nu /\beta }}, $$

(1.84)

where $\propto =\frac{1}{\theta }$, $\beta =\theta$, and the gamma integral is given by ${}\Gamma{}(\propto)=\int_{0}^{\infty }{{{\nu }^{\propto -1}}{{e}^{-\nu }}}$. By adopting the gamma distribution, $g(\nu)$, the expected survivor function can be derived:

$$S(t) = E\left[ {S\left( {{t_{ij}}{{\left| {\bf{X}} \right.}_{ij}},{\nu _i}} \right)} \right] = E\left[ {\exp \left( { - \int_0^t {h(u|\nu )du} } \right)} \right] = {\rm{L}}\left[ {\exp \left( {\int_0^t {h\left( u \right)du} } \right)} \right],$$

(1.85)

where ${{V}_{LWD}}\frac{d{{C}_{LWD}}}{dt}={{Q}_{air}}({{C}_{UPD}}-{{C}_{LWD}})$L is the Laplace transformation to integrate out the distribution of the frailty term.

Using the Weibull model as an example, the marginal Weibull survivor function with gamma frailty is such that

$$ S(t )={{[1+\theta {{(\lambda t )}^{p}}]}^{-1/\theta }}$$

(1.86)

and the Weibull hazard with gamma frailty is equal to

$$ h(t )=\lambda p{{(\lambda t)}^{p-1}}{{[S(t) ]}^{\theta }}. $$

(1.87)

The frailty model is applicable in the clinical setting. For example, when a population is heterogeneous, it is likely that the population composition over time will consist of the more robust individuals as the frail ones failed. In such case, the overall population hazard is declining while individual hazards increase. The frailty term allows for the overall population hazard to decrease regardless of the individual hazard shape. The frailty model is more suited for the population approach in this respect.

For time-varying covariates, which are very applicable in the pharmacometric setting wherein the effect of drug concentration on the risk or hazard is a dynamic variable, the hazard should vary over time. Holford provided a tutorial explaining how the treatment effect can be incorporated to the hazard function to evaluate the dynamic drug time course on the hazard over time (Holford 2013).

1.7.2 Logistic Regression

A logistic regression is suitable for establishing relationship between the outcome of binary response data and explanatory variables (predictors). The probability of having an event is defined as:

$$ {}\pi(x)=\frac{\exp (L(x) )}{1+\exp (L(x) )} $$

(1.88)

where ${}\pi{}(x )$ is called logistic function with values between 0 and 1. $L(x)$ is a linear function of predictors, $L(x)={{\beta }_{0}}+{{\beta }_{1}}x+\cdots +{{\beta }_{i}}{{x}_{i}}$, where ${{\beta }_{0}}$ is the intercept and ${{\beta }_{1,}}\ldots {{\beta }_{i}}$ are the coefficients, and $x$ represents the predictors, such as drug concentration (Heiberger and Holland 2004; Venables et al. 1994).

We take the probability of no event, which is one subtract the previous probability, ${}\pi(x)$:

$$ 1-\pi (x)=\frac{1+\exp (L(x) )-\exp (L(x) )}{1+\exp (L(x) )}=\frac{1}{1+\exp (L(x) )}. $$

(1.89)

The odds describe the relative risk such that:

$$ \frac{\pi (x)}{1-\pi (x)}=\frac{\exp (L(x) )/[1+\exp (L(x) ) ]}{1/[1+\exp (L(x) ) ]}=\exp (L(x) ). $$

(1.90)

Taking the natural logarithm of the odds above gives the $logit(L)$:

$$ logit(L )=\log \left(\frac{\pi (x)}{1-\pi (x)} \right)=L(x). $$

(1.91)

The logit is no longer bounded and its value can take from – ¥ to + ¥. It is important to note that the error around the logit follows a binomial distribution rather than a normal distribution.

The generalized logistic regression model extends the analysis to multiple categorical response data or multinomial responses (Agresti 1999). The approach is to model cumulative logits by comparing each response category with baseline such that

$$\log \frac{{{\pi _i}\left( {\bf{x}} \right)}}{{{\pi _{baseline}}({\bf{x}})}} = {\beta _{i0}} + {\bf{\beta }}{{\bf{'}}_{\bf{i}}}{\bf{x}},$$

(1.92)

where the subscript $i$ represents the $i-1$ levels of response categories plus the baseline, ${{\pi }_{i}}(\textbf{x})=P(Y=i\left| \textbf{x} \right.)$ and $\sum\nolimits_{i}{{{\pi }_{i}}(\textbf{x})=1}$. This approach is often called the proportional odds assumption (McCullagh 1980; Peterson and Harrell 1990):

$${\rm{logit}}\left[ {P(Y \le i\left| {\bf{x}} \right.} \right] = {\beta _{i0}} + {\bf{\beta 'x}}$$

(1.93)

The comparison between two responses is then

$$ \log \frac{{{\pi }_{a}}(\textbf{x})}{{{\pi }_{b}}(\textbf{x})}=\log \frac{{{\pi }_{a}}(\textbf{x})}{{{\pi }_{baseline}}(\textbf{x})}-\log \frac{{{\pi }_{b}}(\textbf{x})}{{{\pi }_{baseline}}(\textbf{x})}. $$

(1.94)

Sheiner in 1994 used the proportional odds model with individual specified random effects for the analysis of a 4-degree pain scale. The nonlinear mixed-effects model of ordered categorical PD data is mostly based on the proportional odds model and has been widely used for the evaluation of both efficacy and adverse events (Cullberg et al. 2005; Gomeni et al. 2001; Gupta et al. 1999; Johnston et al. 2003; Knibbe et al. 2002; Kowalski et al. 2003; Lunn et al. 2001; Mandema and Stanski 1996; Mould et al. 2001, 2002; Olofsen et al. 2005; Xie et al. 2002; Zingmark et al. 2003).

Kjellsson et al. presented a differential odds model to circumvent the assumption with the proportional odds model that the size of the predictor effect is the same for the log odds for all categories (Kjellsson et al. 2008). They argued that the assumption is valid for categories on a continuous scale but would not hold for categories based on ranking scale. Even though historically, the partial proportional odds model has been used to allow for variable sizes of predictor effect, some of the categories within the model have odds that are proportional to each other. While the predictor function is identical for the proportional odds model, the differential odds model allowed this function to vary. The model was implemented using cumulative probabilities so that the correct probability value can be allocated to a specific score or category (for example, mild score is more likely than severe score for a response such as sedation). Figure 1.16 illustrates the predictor versus log cumulative odds in proportional and differential odds model. In the upper left-hand corner, the slopes of the log cumulative odds versus the predictor graph are the same for the proportional odds model. The remaining graphs illustrate the log-cumulative odds versus predictor trend with differential odds model. The E _max-shaped predictor effects in the upper right all had positive slopes whereas mixtures of positive and negative slopes are possible with the alternative implementation.

Fig. 1.16

Logit functions based on proportional odds versus differential odds models and their effects on the cumulative log odds. Examples include proportional odds model with linear predictor effects of the same slope (upper left), differential odds model using E _max-shaped predictor with variable E _max values (upper right), linear predictor with variable but positive slopes (lower left), and linear function with both positive and negative slopes (lower right). α ₂ for baseline logit with score ≥ 2, α ₃ and α ₂ for baseline-shifted logit with score ≥ 3 and ≥ 4, respectively. (Image from Kjellsson et al. 2008, used with permission)

1.7.3 Markov Chain

Another methodology in pharmacometrics that is gaining popularity for the analysis of categorical response variable is the Markov chain model , which has been applied in clinical studies. A Markov chain process is a probability model in which the distribution of future outcomes depends only on the current state and not on the whole history; this is often referred to as the memoryless property of a Markov chain (Bass 2011). In other words, the probability of a certain state to occur in the following time interval is only dependent on the state in the current time frame. With this definition, we suppose that a process in state $i$ has a fixed probability ${{P}_{ij}}$ that it will be in the next state $j$, such that

$$ P\left\{{{X}_{n+1}}=j\left| {{X}_{n}}\right.=i \right\}={{P}_{ij}}, $$

(1.95)

where the set $\left\{{{X}_{n}},n=0,1,2,\cdots\right\}$ is a stochastic process of finite possible outcomes and ${{P}_{ij}}$ is often referred to as transition probability.

For this discussion, we shall use the example of early and late tacrolimus-related adverse event occurrence in stable pediatric renal allograft recipients after transplantation (Sy et al. 2013a). The transition probabilities were defined based on two states: without (state 0) and with (state 1) adverse event. A Markov chain model was chosen in that study because the observations may not be independent since it was assumed that the occurrence of an adverse event is related to the drug concentration. The current state of the patient was conditioned on his previous visit. The transition probabilities were: ${{P}_{00}}$ for those who did not report an adverse event at a particular visit given no adverse event in the previous visit; ${{P}_{01}}$ if the patient without adverse event in the previous visit reported an adverse event in the current visit; ${{P}_{10}}$ for patients with an adverse event in the previous visit but no adverse event in the current one; and ${{P}_{11}}$ if an adverse event occurred on both visits. The transition probabilities adhered to Markov properties such that the sum of the transition probabilities from the specific state is 1:

$$ \sum\nolimits_{j=0}^{1}{{{P}_{ij}}=1}. $$

(1.96)

The function that is utilized for the estimation of the transition probabilities varies depending on the study needs. Kemp and Kamphuisen simulated human hypnograms using a Markov chain model (Kemp and Kamphuisen 1986). Karlsson et al. parameterized the transition probabilities through binary logistic function to describe sleep data (Karlsson et al. 2000). Bizzotto et al. utilized a multinomial logistic function to characterize the time course of transition probabilities between sleep stages in insomniac patients (Bizzotto et al. 2010, 2011). Ouellet et al. used a logistic regression wherein the logit function is linear to estimate the transition probability of having an adverse event due to dizziness in subjects who were administered a selective glycine transporter 1 inhibitor (Ouellet et al. 2011). An example in Ross probability textbook adopted Poisson probability density function (pdf) for counting process in the estimation of transition probability (Ross 2006). The Poisson pdf assumes that the magnitude of the variance is identical to its mean. However, many counting processes show greater variability than that predicted by the Poisson model. Troconiz et al. explored mixed Markov elements and Poisson distribution to evaluate overdispersion in the variance of a Poisson distribution (Troconiz et al. 2009). There are numerous other implementations of the transition probability which we cannot possibly list all of them in this introductory chapter. When selecting whether or not to use a Markov model, Karlsson suggested that the Markov model is more suitable for consecutive same-state observations, which are typical for sleep patterns, as an example (Karlsson et al. 2000).

1.8 Disease Progression Model

The natural time course of a disease is often not one that is static but becomes progressively worse if left untreated. The disease trajectory is not constant, unlike the common assumption that is taken when using the E _max model wherein the baseline E₀ is static. Even as early as the 1970s, investigators reported longitudinal studies of the natural history of non-Hodgkin’s lymphoma stages and coronary artery stenosis (Fuller et al. 1975; Rosch et al. 1976). A disease progression model describes how an indicator for the disease or a clinically relevant endpoint changes in time. For the purpose of modeling disease progression, the approach has been applied in degenerative diseases such as Alzheimer’s disease (Holford and Peace 1992a, b), schizophrenia (Kimko et al. 2000), and diabetic neuropathy (Bakris et al. 1996; Bjorck et al. 1992; Crepaldi et al. 1998; Gall et al. 1993; Lewis et al. 1993; Parving et al. 1995).

Most of the disease progression models are empirical that describe the disease trajectory rather than its physiological background. The linear model has the following general form of equation that characterizes the disease as changing linearly with time:

$$ S(t)={{S}_{0}}+\alpha t, $$

(1.97)

where $S(t)$ represents the disease status at a specific time $t$, ${{S}_{0}}$ is the baseline that can be constant or a time-dependent function (e.g., sinusoidal function to characterize circadian rhythm), and $\propto$ is the slope of the linear process. Therapeutic interventions, including placebo, can change the trajectory of a disease process. Interventions are generally classified as either symptomatic or disease modifying. Let $f(T)$ be the function to characterize the effect of treatment or intervention. In the case of symptomatic treatment, the effect of intervention would shift the disease baseline but not change the slope whereas disease-modifying interventions would change the rate of disease progression , as shown in Eqs. 1.98 and 1.99, respectively (Mould 2007; Mould et al. 2007; Schmidt et al. 2011):

$$ S(t )={{S}_{0}}+f(T)+\alpha t $$

(1.98)

$$ S(t )={{S}_{0}}+(f(T)+\alpha)t $$

(1.99)

From the two equations above, the symptomatic interventions have different effect on the disease status from disease-modifying interventions. The disease trajectory will revert to its natural progression rate $\alpha$ once the treatment is discontinued, regardless of the type of treatment. A third type of intervention is one that exerts completely cure and reverses the disease status back to pre-disease state. This type of intervention may best be characterized by a model that incorporates both symptomatic and disease-modifying effect:

$$ S(t )={{S}_{0}}+{{f}_{1}}(T )+({{f}_{2}}(T)+\alpha)t $$

(1.100)

In the examples listed previously of applications of disease progression modeling approach in specific therapeutic areas, the linear disease progression was used by Kimko et al. to study the effect of quetiapine fumarate, an antischizophrenic agent, on schizophrenia status based on the Brief Psychiatric Rating Scale (Kimko et al. 2000).

Nonlinear functions have also been used as disease progression model. Pors-Nielsen and Friberg used an exponential model to describe the effect of estrogen/progestin treatment on osteoporosis (Pors Nielsen et al. 1994). Grantham et al. also used a similar model to describe the increase in renal volume in autosomal dominant polycystic kidney disease (Grantham et al. 2008). Pillai et al. utilized an indirect response type model to investigate biomarker response to ibandronate (Pillai et al. 2004).

1.9 Systems Pharmacology

Molecular biology evaluates single genes and proteins while systems biology combines the complex interactions at all levels of a biological system. By viewing all levels of biological information in the process, scientists are able to determine important properties of the system. Mathematical models of biological processes help describe time-dependent kinetic behavior and causality. The mechanistic approach to modeling systems’ biological processes is based on sound biological principles with prior knowledge about the biochemical network involved. The variables and parameters are related to a physiological or cellular process where the information is obtained from an in vitro or physiological experiment. This approach gives the scientist a holistic view of the biological system. The study of mechanism of drug action on the system itself also becomes more precise.

One of the systems level models applied in drug development is that of the regulation of glucose. Landersdorfer and Jusko provided an excellent review of application of modeling in diabetes, with a specific focus on modeling drug effects (Landersdorfer and Jusko 2008). In 1979, Bergman et al. developed a minimal model that included three coupled differential equations to describe the intravenous glucose tolerance test (Bergman et al. 1979). Using several feedback control mechanisms, the model couples insulin $I$ and glucose $G$ regulation and also introduces an additional unobserved insulin component $X$ to describe the delay between insulin release and the response characterized by the reduction in blood glucose (Fig. 1.17):

Fig. 1.17

Structure and equations of the minimal model of insulin-glucose feedback and control (top) and example of glucose and insulin concentrations after an IV glucose tolerance test fitted by the minimal model. (Image from Landersdorfer and Jusko 2008, used with permission)

$$ \begin{aligned}& \frac{\text{d}G}{\text{d}t}=-({{S}_{G}}+X(t ) )\cdot G(t )+{{S}_{G}}\cdot {{G}_{B}}\\& \frac{\text{d}X}{\text{d}t}={{p}_{2}}\cdot (I(t )-{{I}_{B}})-{{p}_{3}}\cdot X(t) \\& \frac{\text{d}I}{\text{d}t}=\gamma \cdot (G(t )-h )t-n\cdot I(t ) \end{aligned} $$

(1.101)

The baseline values were $h$, ${{G}_{B}}$ and ${{I}_{B}}$ and parameters were ${{S}_{G}}$, ${{p}_{2}}$, ${{p}_{3}}$, $\gamma$, and $n$. The initial conditions were such that

$$ \begin{aligned}& G(0)=\frac{Dose}{{{V}_{d}}}+{{G}_{B}}\\& X(0)=0 \\& I(0)={{I}_{B}}\end{aligned} $$

(1.102)

Though this model is widely used, it is not without its problems. The model does not allow both insulin and glucose to be fitted simultaneously (Pacini and Bergman 1986). The additional unobserved insulin effect compartment $X$ is unbounded and can increase indefinitely when both glucose and insulin parts of the model were estimated simultaneously (De Gaetano and Arino 2000). Another issue with the model is that it does not take into account the first and second insulin phases (Agerso and Vicini 2003). This model is applicable for diagnostic test but limited for drug evaluation.

In the study of glucagon-like peptide-1 analog NN221, Agerso and Vicini introduced a Gaussian term to describe the first-phase insulin secretion (Agerso and Vicini 2003). The resulting equation for $I$ was such that

$$ \frac{dI}{dt}=\beta (t)+\gamma \cdot (G(t)-h)t-n\cdot I(t), $$

(1.103)

where $\beta (t)$ is an empirical Gaussian function that accounts for the amplitude of the first-phase insulin as well as duration of this process. Mager et al. (2004) further modified the $\gamma$ parameter to an E _max model to include a drug effect for another GLP-1 analog, exenatide (Mager et al. 2004). This model was used in the analysis of data from hyperglycemic clamp study in healthy subjects and diabetic patients.

Indirect response type models have been applied to study the effects of various antidiabetic agents on glucose and insulin. Benincosa and Jusko evaluated rosiglitazone effects wherein both fasting plasma glucose (FPG) and hemoglobin A1c (HbA1c) were measured (Benincosa and Jusko 1999). The glycosylation of hemoglobin was described by a second-order process that is proportional to the FPG concentrations and is dependent on the ratio of the steady-state HbA1c and FPG. The elimination of HbA1c is a first-order process. Hamren et al. modified the model for tesaglitazar such that the glycosylation process also takes into account the erythrocyte lifespan and utilizes several transit compartments to describe the aging process of erythrocytes (Hamren et al. 2008).

Given that diabetes is a chronic disease that becomes progressively worse, the models should also examine disease progression for long-term studies of antidiabetic drugs. Frey et al. investigated the effect of sustained-release gliclazide on FPG over 10 weeks to 1 year period (Frey et al. 2003). In patients who responded to the treatment, FPG levels initially declined and then slowly increased whereas the nonresponders’ FPG levels continued to increase in the natural disease progression process. The authors utilized an empirical linear model with an intercept and a positive slope to describe the disease progression, measured by FPG over time:

$$ FPG(t)=baseline+\alpha \cdot t-{{E}_{t}}, $$

(1.104)

where α is the slope of the disease progression process, ${{E}_{t}}$ is the predicted treatment effect at the time when the treatment was administered, and baseline is the predicted baseline FPG. Since gliclazide was assumed to only alleviate the symptoms of the disease without modifying the disease itself, the effect of the drug was to shift the curve without affecting the rate of disease progression that is characterized by the α term.

De Winter et al. modeled the worsening of β cell activity ($B$) and insulin sensitivity ($S$) using a different disease progression model approach (de Winter et al. 2006):

$$ \begin{aligned}& B=\frac{1}{1+\exp ({{b}_{0}}+{{r}_{b}}\cdot t)} \\& S=\frac{1}{1+\exp ({{s}_{0}}+{{r}_{s}}\cdot t)} \end{aligned} $$

(1.105)

where baselines were ${{b}_{0}}$ and ${{s}_{0}}$ and rates of disease progression were ${{r}_{b}}$ and ${{r}_{s}}$. Their investigation also included a model for HbA1c, which was fitted simultaneously along with FPG and insulin levels.

More sophisticated models of the whole body have been developed and focused on tissues and organs that are relevant in diabetes. These models were used to simulate virtual patients and predict clinical trial outcomes. As these complex models are relatively difficult to develop, many assumptions are made and model parameters are often taken from literature. Examples of these models include the Archimedes, the Entelos Metabolism, and T1Dm PhysioLab®. As these models are proprietary, there is a lack of transparency in the model equations and parameter values (Herman 2003).

Another therapeutic area where systems biology and pharmacology models are extensively used is in oncology. Biochemical reactions and signaling pathways are often described by differential equations that characterize a chemical reaction. Most of these processes involve a complex network of chemical and biochemical reactions. The law of mass action is the common convention that is used to describe the rate at which chemical entities interact to form a different combination. The computational model for heregulin-induced p-ErBB3 signaling and the effect of antibody inhibitors pertuzumab and lapatinib utilizes such convention (McDonagh et al. 2012; Schoeberl et al. 2009). The following chemical reactions were listed for ErbB2/3-bispecific antibody binding and receptor cross-linking:

$$ E2+BsAb\leftrightarrow E2:BsAb $$

(1.106)

$$ BsAb+E3\leftrightarrow BsAb:E3 $$

(1.107)

$$ E2+BsAb:E3\leftrightarrow E2:BsAb:E3 $$

(1.108)

$$ E2:BsAb+E3\leftrightarrow E2:BsAb:E3, $$

(1.109)

where E2 and E3 represent ErbB2/3, respectively, and BsAb refers to a bi-specific antibody. The chemical reactions listed in Eqs. 1.106–1.109 are reversible processes and the reaction schema can be generalized:

$$ A+B\xrightarrow{k}C\xrightarrow{{{k}'}}A+B, $$

(1.110)

where k is the forward reaction rate and k’ is the reversible rate. The rate of change for each of the above species can be written as follows:

$$ \begin{aligned}& \frac{d[A]}{dt}={{k}^{'}}[C ]-k[A][B] \\& \frac{d[B]}{dt}={{k}^{'}}[C ]-k[A][B] \\& \frac{d[C]}{dt}=k[A ][B ]-{{k}^{'}}[C] \end{aligned} $$

(1.111)

As one can see, writing out the differential equations for the chemical reaction in Eq. 1.111 becomes a tedious effort, especially when the biological system has many players. Some systems biology tools allow user to create reaction processes and will automatically create the corresponding differential equations in the background (Maiwald and Timmer 2008).

In identifying targets, McDonagh et al. utilized computation modeling and cell signaling insights to develop specific targeted antibodies that are capable to destabilize the over expression of ErbB2 by inhibiting ErbB3 signaling (McDonagh et al. 2012). They identified that ErbB3/heregulin activation plays a critical role in ErbB2-positive refractory disease and that the synergy can be achieved in combination therapies involving ErbB3 inhibitor and ErbB2 therapies (McDonagh et al. 2012). System pharmacology models are slowly being adopted in drug development settings and the examples show promising prospect for systems pharmacology to become part of mainstream pharmacometric analyses.

1.10 Software

As software facilitates PKPD modeling and simulation , we will discuss the available software packages that have been used for different types of analyses. For noncompartmental analysis (or statistical moment approach), WinNonlin is most commonly used in the pharmaceutical industry. Alternately, the PK package in R is a basic PK package that performs noncompartmental analysis of PK data as well. A brief description of this tool can be found on the CRAN project website (http://cran.r-project.org/web/views/Pharmacokinetics.html). For nonlinear mixed effect modeling, NONMEM is the most commonly used software in the drug development setting. The Non-Parametric Adaptive Grid algorithm using NPAG (USC*PACK) is common for modeling population PK data in the therapeutic drug monitoring setting. The other software for mixed effect modeling are ADAPT that is based on important sampling algorithm, MONOLIX with the Markov Chain Monte Carlo (MCMC) stochastic approximation expectation maximization (SAEM). BUGS is a software package for performing Bayesian inference using Gibbs sampling. The user specifies a statistical model by stating the relationships between related variables. The software determines an appropriate MCMC scheme (based on the Gibbs sampler) for analyzing the specified model. The user then controls the execution of the scheme. There are two main versions of BUGS, namely WinBUGS and OpenBUGS. The recent release of NONMEM has incorporated the algorithms in the other software packages mentioned above. SimCyp includes a population-based simulation system using PBPK model.

For systems biology modeling, the Matlab platform and specialized toolboxes that were built on top of the Matlab platform can handle large and complex models that may contain hundreds of coupled differential equations. The large models can be slow in Matlab. Specialized Matlab toolbox such as Potterswheel (URL: potterswheel.de) utilizes chemical reaction scheme and builds the corresponding differential equations in C language to speed up the analysis and fitting processes (Maiwald and Timmer 2008). Curated systems biology models are available from library consortiums that are publicly accessible (e.g., http://www.ebi.ac.uk/biomodels-main) and can be ported to specialized software packages through Systems Biology Markup Language (SBML).

1.11 Conclusion

Over the past decade, pharmacometrics has become a discipline that is frequently utilized in academia, worldwide regulatory agencies, and the biopharmaceutical industry. Cost cutting and improvement in drug development will come from creative application of pharmacometric modeling approach.

The US FDA has emphasized the importance of model-based drug development wherein PK/PD models to characterize drug efficacy and safety are being developed for both preclinical and clinical data. The agency strongly supports this program, “Pharmacometric analyses, we believe, are valuable to gain insights into the data across drugs and to plan future development. The model and simulation approaches should not be viewed as substitutes to conducting clinical trials in all instances. Also, such quantitative analyses should not be primarily used to ‘rescue’ failed trials for seeking approval. Where appropriate, the FDA accepted simulation results” (Bhattaram et al. 2005).

Appendices

Appendix

Derivation of Eq. 1.55

Implicit in the assumption that the concentration of drug in the system that is being cleared is constant or at steady state, we assumed that the rate of change in the drug concentration is 0 and thus set all the left-hand sides in the list of equations in Eq. 1.53 to 0.

$$ 0={{V}_{1}}\frac{d{{C}_{1(vein)}}}{dt}={{Q}_{1}}\cdot {{C}_{0}}-{{Q}_{1}}\cdot {{C}_{1}}+{{K}_{I}}(t) $$

(1.112)

$$ 0={{V}_{2}}\frac{d{{C}_{2(artery)}}}{dt}={{Q}_{1}}\cdot {{C}_{3o}}-{{Q}_{1}}\cdot {{C}_{2}}$$

(1.113)

$$ 0={{V}_{3}}\frac{d{{C}_{3(lung)}}}{dt}={{Q}_{1}}\cdot {{C}_{1}}-{{Q}_{1}}\cdot {{C}_{3o}}-{{r}_{3}}({{C}_{3}}) $$

(1.114)

$$ 0={{V}_{4}}\frac{d{{C}_{4(liver)}}}{dt}={{Q}_{7}}\cdot {{C}_{7o}}+({{Q}_{4}}-{{Q}_{7}})\cdot {{C}_{2}}-{{Q}_{4}}\cdot {{C}_{4o}}-{{r}_{4}}({{C}_{4}}) $$

(1.115)

$$ 0={{V}_{5}}\frac{d{{C}_{5(kidney)}}}{dt}={{Q}_{5}}\cdot {{C}_{2}}-{{Q}_{5}}\cdot {{C}_{5o}}-{{r}_{5}}({{C}_{5}}) $$

(1.116)

$$ 0={{V}_{6}}\frac{d{{C}_{6(rest\text{ }of\text{ }body)}}}{dt}={{Q}_{6}}\cdot {{C}_{2}}-{{Q}_{6}}\cdot {{C}_{6o}}$$

(1.117)

$$ 0={{V}_{7}}\frac{d{{C}_{7(GI\text{ }tract)}}}{dt}={{Q}_{7}}\cdot {{C}_{2}}-{{Q}_{7}}\cdot {{C}_{7o}}-{{r}_{7}}({{C}_{7}}). $$

(1.118)

From Eq. 1.112, we integrate

$$ \int_{0}^{\infty }{{{Q}_{1}}\cdot {{C}_{1}}dt=}\int_{0}^{\infty }{({{Q}_{1}}\cdot {{C}_{0}}+{{K}_{I}}(t) )dt}. $$

(1.119)

By applying Eq. 1.54 and substituting ${{Q}_{1}}\cdot {{C}_{0}}$ with ${{Q}_{4}}\cdot {{C}_{4o}}+{{Q}_{5}}\cdot {{C}_{5o}}+{{Q}_{6}}\cdot {{C}_{6o}}$ to Eq. 1.119, the resulting expression is

$$ {{Q}_{1}}\int_{0}^{\infty }{{{C}_{1}}dt=}\int_{0}^{\infty }{({{Q}_{4}}\cdot {{C}_{4o}}+{{Q}_{5}}\cdot {{C}_{5o}}+{{Q}_{6}}\cdot {{C}_{6o}})dt+}\int_{0}^{\infty }{{{K}_{I}}\text{(t)}dt}. $$

(1.120)

Rearrangement of Eq. 1.120 by adding and subtracting $({{Q}_{7}}\cdot {{C}_{7o}}+({{Q}_{4}}-{{Q}_{7}}){{C}_{2}}+{{Q}_{5}}\cdot {{C}_{5o}}+{{Q}_{6}}\cdot {{C}_{2}})$ to Eq. 1.120 such that the net result is 0 will yield the following expression:

$$ {{Q}_{1}}\int_{0}^{\infty }{{{C}_{1}}dt=}\int_{0}^{\infty }{[({{Q}_{7}}\cdot {{C}_{7o}}+({{Q}_{4}}-{{Q}_{7}}){{C}_{2}}+{{Q}_{5}}\cdot {{C}_{5o}}+{{Q}_{6}}\cdot {{C}_{2}})-({{Q}_{7}}\cdot {{C}_{7o}}+({{Q}_{4}}-{{Q}_{7}}){{C}_{2}}+{{Q}_{5}}\cdot {{C}_{5o}}+{{Q}_{6}}\cdot {{C}_{2}}) ]dt+}\int_{0}^{\infty }{({{Q}_{4}}\cdot {{C}_{4o}}+{{Q}_{5}}\cdot {{C}_{5o}}+{{Q}_{6}}\cdot {{C}_{6o}})dt+}\int_{0}^{\infty }{{{K}_{I}}(t)dt}. $$

(1.121)

From Eq. 1.117, $\int_{0}^{\infty }{({{Q}_{6}}\cdot {{C}_{2}}-{{Q}_{6}}\cdot {{C}_{6o}})dt=0}$. Given that ${{Q}_{7}}\cdot {{C}_{7o}}+({{Q}_{4}}-{{Q}_{7}}){{C}_{2}}$ represents the total amount of drug entering the liver from the artery and the gut which is also C_p, wherein${{Q}_{4}}\cdot {{C}_{p}}={{Q}_{7}}\cdot {{C}_{7o}}+({{Q}_{4}}-{{Q}_{7}}){{C}_{2}}$, we can then simplify Eq. 1.121 as follows:

$$ {{Q}_{1}}\int_{0}^{\infty }{{{C}_{1}}dt=}\int_{0}^{\infty }{({{Q}_{4}}+{{Q}_{5}}+{{Q}_{6}}){{C}_{2}}dt-}\left\{\int_{0}^{\infty }{{{Q}_{4}}({{C}_{p}}-{{C}_{4o}})dt+}\int_{0}^{\infty }{{{Q}_{5}}({{C}_{2}}-{{C}_{5o}})dt+}\int_{0}^{\infty }{{{Q}_{7}}({{C}_{2}}-{{C}_{7o}})dt} \right\}+\int_{0}^{\infty }{{{K}_{I}}\text{(t)}dt}. $$

(1.122)

Furthermore, $\int_{0}^{\infty }{({{Q}_{4}}+{{Q}_{5}}+{{Q}_{6}}){{C}_{2}}dt={{Q}_{1}}}\int_{0}^{\infty }{{{C}_{2}}dt}$, since ${{Q}_{1}}={{Q}_{4}}+{{Q}_{5}}+{{Q}_{6}}$ and from Eq. 1.113, ${{Q}_{1}}\underset{0}{\overset{\infty }{\mathop \int }}\,{{C}_{3o}}dt$, then Eq. 1.122 can be rewritten as Eq. 1.55:

$$ \int_{0}^{\infty }{{{Q}_{1}}({{C}_{1}}-{{C}_{3o}})dt+}\int_{0}^{\infty }{{{Q}_{4}}({{C}_{p}}-{{C}_{4o}})dt+}\int_{0}^{\infty }{{{Q}_{5}}({{C}_{2}}-{{C}_{5o}})dt+}\int_{0}^{\infty }{{{Q}_{7}}({{C}_{2}}-{{C}_{7o}})dt=}\int_{0}^{\infty }{{{K}_{I}}(t)dt}. $$

Derivation of Eq. 1.60

Using the expression for mean clearance, the rearrangement of Eqs. 1.55 through 1.59 will result in the following:

$$ {{\overline{CL}}_{systemic}}={{\overline{CL}}_{3}}+\left(\frac{\mathop{\int }_{0}^{\infty }{{C}_{p}}dt}{\mathop{\int }_{0}^{\infty }{{C}_{1}}dt} \right){{\overline{CL}}_{4}}+\left(\frac{\mathop{\int }_{0}^{\infty }{{C}_{2}}dt}{\mathop{\int }_{0}^{\infty }{{C}_{1}}dt} \right)({{\overline{CL}}_{5}}+{{\overline{CL}}_{7}}). $$

(1.123)

We set ${{Q}_{4}}\cdot {{C}_{p}}={{Q}_{7}}{{C}_{7o}}+({{Q}_{4}}-{{Q}_{7}}){{C}_{2}}$ to ${{Q}_{4}}({{C}_{2}}-{{C}_{p}})={{Q}_{7}}({{C}_{2}}-{{C}_{7o}})$. Then integrate the latter expression from 0 to infinity, ${{Q}_{4}}\int_{0}^{\infty }{({{C}_{2}}-{{C}_{p}})dt={{Q}_{7}}}\int_{0}^{\infty }{({{C}_{2}}-{{C}_{7o}})dt}$ and divide both sides of the equation by $\int_{0}^{\infty }{{{C}_{2}}dt}$ to obtain $\frac{{{Q}_{4}}\mathop{\int }_{0}^{\infty }({{C}_{2}}-{{C}_{p}})dt}{\mathop{\int }_{0}^{\infty }{{C}_{2}}dt}=\frac{{{Q}_{7}}\mathop{\int }_{0}^{\infty }({{C}_{2}}-{{C}_{7o}})dt}{\mathop{\int }_{0}^{\infty }{{C}_{2}}dt}$. The previous equation can be rewritten as $1-\frac{\mathop{\int }_{0}^{\infty }{{C}_{p}}dt}{\mathop{\int }_{0}^{\infty }{{C}_{2}}dt}=\frac{{{Q}_{7}}\mathop{\int }_{0}^{\infty }({{C}_{2}}-{{C}_{7o}})dt}{{{Q}_{4}}\mathop{\int }_{0}^{\infty }{{C}_{2}}dt}$. The ratio consisting the integrating components on the left side of the equation is the expression for the extraction ratio for tissue compartment 7 for the GI tract, ${{\overline{E}}_{7}}=\frac{\mathop{\int }_{0}^{\infty }({{C}_{2}}-{{C}_{7o}})\text{d}t}{\mathop{\int }_{0}^{\infty }{{C}_{2}}\text{d}t}$. Therefore,

$$ \frac{\mathop{\int }_{0}^{\infty }{{C}_{p}}dt}{\mathop{\int }_{0}^{\infty }{{C}_{2}}dt}=1-\frac{{{Q}_{7}}}{{{Q}_{4}}}{{\overline{E}}_{7}}$$

(1.124)

From Eq. 1.113 above, ${{Q}_{1}}\int_{0}^{\infty }{{{C}_{2}}dt={{Q}_{1}}}\int_{0}^{\infty }{{{C}_{3}}dt}$, and from Eq. 1.57, ${{\overline{CL}}_{3}}=\frac{{{Q}_{1}}\mathop{\int }_{0}^{\infty }({{C}_{1}}-{{C}_{3o}})dt}{\mathop{\int }_{0}^{\infty }{{C}_{1}}dt}=\frac{{{Q}_{1}}\mathop{\int }_{0}^{\infty }({{C}_{1}}-{{C}_{2}})dt}{\mathop{\int }_{0}^{\infty }{{C}_{1}}dt}$. Thus,

$$ \frac{\mathop{\int }_{0}^{\infty }{{C}_{2}}dt}{\mathop{\int }_{0}^{\infty }{{C}_{1}}dt}=1-\frac{{{\overline{CL}}_{3}}}{{{Q}_{1}}}=1-{{\overline{E}}_{lung}}$$

(1.125)

In the following step, multiply Eq. 1.124 by A125:

$$ \frac{\mathop{\int }_{0}^{\infty }{{C}_{p}}dt}{\mathop{\int }_{0}^{\infty }{{C}_{1}}dt}=\left(1-\frac{{{\overline{CL}}_{3}}}{{{Q}_{1}}}\right)\left(1-\frac{{{\overline{CL}}_{7}}}{{{Q}_{4}}}\right). $$

(1.126)

Since ${{\overline{CL}}_{7}}={{Q}_{7}}\cdot {{\overline{E}}_{7}}$. In the following step, we substitute Eqs. 1.126 and 1.125 to Eq. 1.123, ${{\overline{CL}}_{systemic}}={{\overline{CL}}_{3}}+\left(1-\frac{{{\overline{CL}}_{3}}}{{{Q}_{1}}}\right)\left(1-\frac{{{\overline{CL}}_{7}}}{{{Q}_{4}}}\right){{\overline{CL}}_{4}}+\left(1-\frac{{{\overline{CL}}_{3}}}{{{Q}_{1}}}\right)({{\overline{CL}}_{5}}+{{\overline{CL}}_{7}})={{\overline{CL}}_{3}}+(1-{{\overline{E}}_{3}})[{{\overline{CL}}_{4}}+(1-{{\overline{E}}_{4}}){{\overline{CL}}_{7}}+{{\overline{CL}}_{5}}].$