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Modeling. Jerry L. Campbell Jr., Rebecca A. Clewell, P. Robinan Gentry, ..... mcsim.php .... PBPK model, as well as for a model developer seeking to have his.
Chapter 18 Physiologically Based Pharmacokinetic/Toxicokinetic Modeling Jerry L. Campbell Jr., Rebecca A. Clewell, P. Robinan Gentry, Melvin E. Andersen, and Harvey J. Clewell III Abstract Physiologically based pharmacokinetic (PBPK) models differ from conventional compartmental pharmacokinetic models in that they are based to a large extent on the actual physiology of the organism. The application of pharmacokinetics to toxicology or risk assessment requires that the toxic effects in a particular tissue are related in some way to the concentration time course of an active form of the substance in that tissue. The motivation for applying pharmacokinetics is the expectation that the observed effects of a chemical will be more simply and directly related to a measure of target tissue exposure than to a measure of administered dose. The goal of this work is to provide the reader with an understanding of PBPK modeling and its utility as well as the procedures used in the development and implementation of a model to chemical safety assessment using the styrene PBPK model as an example. Key words: PBPK, Styrene, Pharmacokinetics

1. Introduction Pharmacokinetics is the study of the time course for the absorption, distribution, metabolism, and excretion of a chemical substance in a biological system. Implicit in any application of pharmacokinetics to toxicology or risk assessment is the assumption that the toxic effects in a particular tissue can be related in some way to the concentration time course of an active form of the substance in that tissue. Moreover, except for pharmacodynamic differences between animal species, it is expected that similar responses will be produced at equivalent tissue exposures regardless of animal species, exposure route, or experimental regimen (1–3). Of course the actual nature of the relationship between tissue exposure and response, particularly across species, may be quite complex, and exceptions to the rule of tissue dose equivalence are numerous. Brad Reisfeld and Arthur N. Mayeno (eds.), Computational Toxicology: Volume I, Methods in Molecular Biology, vol. 929, DOI 10.1007/978-1-62703-050-2_18, # Springer Science+Business Media, LLC 2012

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Nevertheless, the motivation for applying pharmacokinetics is the expectation that the observed effects of a chemical will be more simply and directly related to a measure of target tissue exposure than to a measure of administered dose. 1.1. Compartmental Modeling

One of the first general descriptions of pharmacokinetic modeling was presented by Teorell (4, 5). The model consisted of a number of compartments representing specific tissues. The concentration of chemical in each compartment was described by a mass balance equation in which the rate of change of the amount of chemical in a compartment was determined from the rates at which the chemical entered and left the compartment in the blood as well as, when appropriate, the rate of clearance of the chemical in that compartment. Unfortunately, in order to obtain an analytical solution of the resulting system of differential equations, Teorell found it necessary to make a number of simplifying assumptions. These assumptions led to a solution in the form of a sum of exponential terms, and thus the “classical” compartmental modeling approach still used today was born. Over the years, Teorell’s association of the model compartments with specific tissues has to a large extent been lost, and compartmental modeling as currently practiced is largely an empirical exercise. In this empirical approach, data on the time course of the chemical of interest in blood (and perhaps other tissues, urine, etc.) are collected. Based on the behavior of the data, a mathematical model is selected which possesses a sufficient number of compartments (and therefore parameters) to describe the data. The compartments do not in general correspond to identifiable physiological entities but rather are described in abstract terms. An example of a simple twocompartment mathematical model of this type is shown in Fig. 1. This particular model consists of a “central” compartment, characterized by concentrations measured in the blood (but not considered to actually represent only the blood), and a “deep” compartment representing unspecified tissues communicating with the central compartment as described by kinetic parameters, k12 and k21, which themselves have no obvious physiological or biochemical interpretation. Similarly, the volume of the central compartment and the uptake and clearance parameters (ka and ke) are empirically determined by the analysis or fitting of experimental data sets. ka

CENTRAL

uptake

ke clearance

k12

k21

DEEP

Fig. 1. Simple compartmental pharmacokinetic model.

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The advantage of this modeling approach is that there is no limitation to fitting the model to the experimental data. If a particular model is unable to describe the behavior of a particular data set, additional compartments can be added until a successful fit is obtained. Since the model parameters do not possess any intrinsic meaning, they can be freely varied to obtain the best possible fit, and different parameter values can be used for each data set in a related series of experiments. Statistical tests can then be employed to compare the values of the parameters used, for example at two administered dose levels, in order to determine whether any apparent differences are statistically significant (6). Once developed, these models are useful for interpolation and limited extrapolation of the concentration profiles which can be expected as experimental conditions are varied. If the model parameters vary with dose, this information can provide evidence for the presence of nonlinearities in the animal system, such as saturable metabolism or binding. At this point, however, one of the serious disadvantages of the empirical approach becomes evident. Since the compartmental model does not possess a physiological structure, it is often not possible to incorporate a description of these nonlinear biochemical processes in a biologically appropriate context. For example, in the case of inhalation of chemicals subject to high-affinity, low-capacity metabolism in the liver, an important determinant of metabolic clearance at low inhaled concentrations is the fact that only the fraction of the chemical in the blood reaching the liver is available for metabolism (1). Without a physiological structure it is not possible to correctly describe the interaction between bloodtransport of the chemical to the metabolizing organ and the intrinsic clearance of the chemical by the organ. 1.2. Physiologically Based Modeling

Physiologically based pharmacokinetic (PBPK) models differ from conventional compartmental pharmacokinetic models in that they are based to a large extent on the actual physiology of the organism. Instead of compartments defined solely by the experimental data, actual organ and tissue groups are described using weights and blood flows obtained from the literature. Moreover, instead of composite rate constants determined solely by fitting data, measured physical–chemical and biochemical constants of the compound can often be used. To the extent that the structure of the model reflects the important determinants of the kinetics of the chemical, PBPK models can predict the qualitative behavior of an experimental time course data set without having been based directly on it. Refinement of the model to incorporate additional insights gained from comparison with experimental data yields a model which can be used for quantitative extrapolation well beyond the range of experimental conditions on which it was based. In particular, a properly validated PBPK model can be used to perform

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Biochemical Constants

Biochemical Constants

Model Formulation

Simulation

Refine Model

Compare to Kinetic Data

Validate Model

Design / Conduct Critical Experiments

Extrapolation to Humans

Fig. 2. Flowchart of the biologically motivated PBPK modeling approach to chemical risk assessment.

the high-to-low dose, dose-route, and interspecies extrapolations necessary for estimating human risk on the basis of animal toxicology studies (7–15). A number of excellent reviews have been written on the subject of PBPK modeling (16–20). The basic approach is illustrated in Fig. 2. The process of model development begins with the definition of the chemical exposure and toxic effect of concern, as well as the species and target tissue in which it is observed. Literature evaluation involves the integration of available information about the mechanism of toxicity, the pathways of chemical metabolism, the nature of the toxic chemical species (i.e., whether the parent chemical, a stable metabolite, or a reactive intermediate produced during metabolism is responsible for the toxicity), the processes involved in absorption, transport and excretion, the tissue partitioning and binding characteristics of the chemical and its metabolites, and the physiological parameters (e.g., tissue weights and blood flow rates) for the species of concern (i.e., the experimental species and the human). Using this information, the investigator develops a PBPK model which expresses mathematically a conception of the animal–chemical system. In the model, the various time-dependent biological processes are described as a system of simultaneous differential equations. A mathematical model of this form can easily be written and exercised using

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commonly available computer software (21). The specific structure of the model is driven by the need to estimate the appropriate measure of tissue dose under the various exposure conditions of concern in both the experimental animal and the human. Before the model can be used in human risk assessment it has to be validated against kinetic, metabolic, and toxicity information and, in many cases, refined based on comparison with the experimental results. The model itself can frequently be used to help design critical experiments to collect data needed for its own validation. The chief advantage of a PBPK model over an empirical compartmental description is its greater predictive power. Since fundamental biochemical processes are described, dose extrapolation over ranges where saturation of metabolism occurs is possible (22). Since known physiological parameters are used, a different species can be modeled by simply replacing the appropriate constants with those for the species of interest, or by allometric scaling (23–25). Similarly, the behavior for a different route of administration can be determined by adding equations which describe the nature of the new input function (21, 26). The extrapolation from one exposure scenario, say a single 6 h exposure, to another, e.g., a repetitive 6 h exposure, 5 days a week for the life of the animal, is relatively easy and only requires a little ingenuity in writing the equations for the dosing regimen in the kinetic model (27, 28). Since measured physical–chemical and biochemical parameters are used, the behavior for a different chemical can quickly be estimated by determining the appropriate constants. An important result is the ability to reduce the need for extensive experiments with new chemicals (12). The process of selecting the most informative experimental data is also facilitated by the availability of a predictive pharmacokinetic model (29). Perhaps the most desirable feature of a physiologically based model is that it provides a conceptual framework for employing the scientific method in which hypotheses can be described in terms of biological processes, predictions can be made on the basis of the description, and the hypothesis can be revised on the basis of comparison with experimental data. The trade-off against the greater predictive capability of physiologically based models is the requirement for an increased number of parameters and equations. However, values for many of the parameters, particularly the physiological ones, are already available in the literature (30–35), and in vitro techniques have been developed for rapidly determining the compound-specific parameters (36–39). An important advantage of PBPK models is that they provide a biologically meaningful quantitative framework within which in vitro data can be more effectively utilized (40). There is even a prospect that predictive PBPK models can someday be developed based almost entirely on data obtained from in vitro studies.

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Some of the best examples of successful PBPK modeling efforts were performed to support the clinical use of chemotherapeutic drugs, e.g., methotrexate (41) and cisplatin (42) (see (43) for a review). There are also a large number of good examples of PBPK models which describe the kinetics of important environmental contaminants, including methylene chloride (8, 44, 45), trichloroethylene (46–48), chloroform (49, 50), 2-butoxyethanol (51), kepone (52), polybrominated biphenyls (53), polychlorinated biphenyls (54) and dibenzofurans (55), dioxins (56, 57), lead (58–62), arsenic (63, 64), methylmercury (65), atrazine (66, 67), acrylonitrile (68–70), perchlorate (71–76), and BTEX components (77–81). The U.S. EPA is currently compiling a compendium of PBPK models including source code. This should be available online within the next year.

2. Materials Currently there exists a very diverse group of modeling software packages that vary in both complexity and range of application. Because of this diversity, there is a software package suitable for every level of user from the expert to the first-time modeler. However, not all modeling packages are created equal, and some of the more user-friendly software can lack the capabilities of the more complex programs. Consequently, no single software package available can meet all needs of all users, and the diversity and complexity of the programs can often make converting a model from one package to another rather difficult. Table 1 provides a list of some of the available software packages that may be useful for PBPK modeling (82). An additional list of pharmacokinetic software is located at http://boomer.org/pkin/soft.html. However, not all of the software listed on this website is suitable for PBPK modeling. The most commonly used software packages for PBPK modeling have included Advanced Continuous Modeling Language (ACSL) (now acslX), Berkeley Madonna, MATLAB, MATLAB/ Simulink, ModelMaker, and SCoP. Table 2 provides a summary of the features of each of these followed by further information. acslX is an updated version of the widely used ACSL software. It has graphical as well as text interface with automatic linkage to the integration algorithm. In particular, a Pharmacokinetic Toolkit in the graphic code interface makes it possible to build PBPK models by connecting predefined tissue code blocks. The software allows for the use of discrete blocks and script files and automatically sorts equations in the derivative block. The model may be compiled into either C/C++ or Fortran, although C++ is now the preferred compiler, and may be debugged interactively.

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Table 1 Representative list of available software packages Package

Source

Website

General-purpose high-level scientific computing software. These high-level programming language packages are very general modeling tools that are not specifically designed for PBPK modeling, but offer more complexity acslX AEgis Technologies http://www.acslX.com Group, Inc. Berkeley Madonna University of California http://www.berkeleymadonna.com at Berkeley GNU octave University of Wisconsin http://www.octave.org MATLAB/Simulink The MathWorks, Inc. http://www.mathworks.com MLAB Civilized Software, Inc. http://www.civilized.com Biomathematical modeling software. Packages that were specifically designed for modeling biological systems and some are user-friendly. Their usefulness in PBPK modeling is determined by their graphical interfaces, computational speed, and language flexibility and may provide mixed-effects (population) capabilities allowing for the analysis of sparse data sets ADAPT II Biomedical Simulations http://bmsr.usc.edu Resource, USC MCSim INERIS http://toxi.ineris.fr/activites/ toxicologie_quantitative/mcsim/ mcsim.php ModelMaker ModelKinetix http://www.modelkinetix.com http://www.globomaxservice.com NONMEM University of California at San Francisco and Globomax Service Group SAAM II SAAM Institute, Inc. http://www.saam.com SCoP Simulation Resources, http://www.simresinc.com Inc. Stella High Performance http://www.hps-inc.com Systems, Inc. WinNonlin Pharsight Corp. http://www.pharsight.com WinNonMix Pharsight Corp. http://www.pharsight.com Toxicokinetic software. These packages were designed specifically for PBPK and PBTK modeling and are extremely flexible. They are based on modeling languages developed in the aerospace industry for modeling complex systems SimuSolv Dow chemical Not maintained or subject to further development Physiologically based custom-designed software. Custom-designed proprietary software programs specifically for biomedical systems or applications that provide a high level of biological detail but are not easily customized GastroPlus Simulations Plus, Inc. http://www.simulations-plus.com Pathway prism Physiome Sciences, Inc. http://www.physiome.com Physiolab Entelos, Inc. http://www.entelos.com SimCYP Simcyp, Ltd. http://www.simcyp.com

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Table 2 Comparison of modeling software features Feature

Berkeley acslXc, d Madonnac, d

MATLAB/ MATLABc, d Simulinkc, d

Model makerc, d

SCoPd

Graphical interface

Y

Y

N

Y

Y

N

Text interface

Y

Y

Y

N

Y

Y

Automatic linkage to integration algorithm

Y

Y

N

Y

Y

Y

Discrete blocks

Y

N

N

Y

Y

N

Scripting

Y

N

Y

Y

N

N

Code sorting

Y

Y

N

Y

Y

N

Choice of target language

Y

N

N

N

N

N

Interactive model debugging

Y

Y

N

Y

N

N

Optimization

Y

Y

Ya

Y

Y

Y

b

Y

b

Y

Y

b

Y

Y

b

Y

Y

N

Y

b

Monte Carlo

Y

b

Units checking

N

N

N

N

N

Y

Database of physiological values

N

N

N

N

N

N

Compiled (faster)

Y

Y

Ya, e

N

N

Y

Interpreted (more convenient)

N

N

Y

Y

Y

N

Sensitivity analysis

Y

b

Y

a

Extra cost Can perform through the use of user-developed model code or script files c Must contact vendor for price. Price may depend upon the type of license d Student and/or academic licenses are available e With separate compiler b

An optimization program is also included. Sensitivity analysis and Monte Carlo analysis can currently be conducted with script files and there is some capability built into the package that allows these analyses to be conducted with the aid of a using a gui. Berkeley Madonna has many of the same features as acslX; however, it does not allow for the use of discrete blocks or script files. It does currently have both an optimization and sensitivity analysis feature, but does not have a built-in Monte Carlo capability. MATLAB has a text interface, but not a graphical interface. It does allow for the use of script files but not discrete blocks. It does not sort the code in the model so the user must be careful regarding

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the order of statements in the code. This statement order problem can be problematic in the case of PBPK models, which require simultaneous solution of multiple differential equations, and can complicate conversion from a software package that automatically sorts the model equations. An optimization package is available through an add-on toolbox, but sensitivity analysis and Monte Carlo analysis must be performed through the use of script files. A large variety of user-built model code blocks are available at the MATLAB web site. The model code may be compiled through the purchase and use of a MATLAB Compiler, but the user has no choice of the target language used. Simulink is an add-on to MATLAB that offers a graphic interface but no text interface. The use of discrete blocks is also added with the use of Simulink. Since Simulink uses only a graphical interface, there is no code to be viewed. Simulink adds a graphical debugger to MATLAB and an optimizer. ModelMaker has many of the same features as acslXtreme and Berkeley Madonna. It has both a graphical and a text interface with automatic linkage to the integration algorithm and allows for the use of discrete blocks but not the use of script files. ModelMaker also provides the capabilities for optimization, sensitivity analysis, and Monte Carlo analysis. SCoP has only a text interface with automatic linkage to the integration algorithm. It does not allow for the use of discrete blocks or script files and it does not automatically sort the code. Optimization and sensitivity analysis capabilities are included, but not Monte Carlo analysis capabilities. An additional modeling software package, not included in Table 2, is designed specifically to support Markov Chain Monte Carlo (MCMC) simulation. This program, MCSim, has been used to reestimate parameter distributions for PBPK models on the basis of the agreement between model predictions and measured data from kinetic studies. MCSim is available for free download from the website listed in Table 1. Its proper use requires expertise in programming and statistics. Each of the software packages described above provides different features that would recommend them for a particular user. From the viewpoint of a risk assessor wanting to apply an existing PBPK model, as well as for a model developer seeking to have his model used in a risk assessment, the key requirement is the ability to readily evaluate (verify) the model, reproduce (validate) the capability of the model to simulate key data sets, and document its application for the risk assessment. The necessary documentation includes a model definition (preferably code-based) that can be reviewed to verify the mathematical correctness of the model, a description of the parameters that should be used to run the model for comparison with validation data, and a description of the

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parameters used to calculate the dose metrics for the risk assessment. The most important characteristics of a language for model evaluation are verifiable code, self-documentation, and ease of use. The feature that is most important with regard to selfdocumentation is scripting, which allow the model developer to create procedures consisting of sequences of commands that, for example, set model parameters, run the model, and plot the model predictions against the appropriate data set. Other features which contribute to ease of model evaluation include viewable model definition code, code sorting, and automatic linkage to integration algorithms. The modeling language that has seen the most widespread use in PBPK modeling is the ACSL, which is currently implemented as acslX. The ACSL language has also served as the basis for a variety of older packages, including SimuSolv (from Dow Chemical, no longer supported), ACSL/Tox (from Pharsight, no longer supported), and ERDM (currently used by the USEPA). The automatic code sorting provided by ACSL allows code to be grouped functionally (liver, lung, fat, etc.) rather than in program order, greatly simplifying model development and improving readability of the code. The graphic code block capability in acslX is particularly attractive for PBPK modeling because it provides the ability to create a model by connecting functional units (e.g., tissues) in a graphic environment, while at the same time creating a model definition in ACSL code that can be reviewed for model verification. The scripting capability, which permits both MATLAB-like m-files, greatly expedites the comparison of the model with multiple data sets that is generally required for PBPK modeling of data from multiple species and routes of exposure. The scripting capability also makes it possible to document the use of a model in a risk assessment, since m-scripts or command file procedures can be written that set the model parameters and run the model for each of the dose metrics required for the risk assessment, as well as for each of the data sets used for model evaluation/validation. Berkeley Madonna provides a particularly intuitive, flexible platform for model development, and has been very popular in academic settings. The software provides automatic code sorting, the ability to automatically convert between code and graphic model descriptions, and automatic compilation, greatly simplifying and expediting model development, debugging, and verification. Conversion of models between ACSL and Berkeley Madonna is relatively straightforward. However, the lack of a scripting capability makes comparison of the model with data fairly cumbersome, particularly in situations where a large number of data sets are being modeled. The lack of scripting also makes it more difficult to document the actual use of a model in a risk assessment and greatly complicates model evaluation.

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MATLAB is a very powerful and flexible software package that is particularly attractive in research activities. Its main drawback is that its use requires significant expertise in programming. MATLAB/Simulink avoids this drawback by providing a graphical interface, and is very popular with engineers in the automotive and electronic industries. However, verification of a Simulink model by a nonengineer is hampered by the lack of any model definition code. That is, the model is specified only by a “wiring diagram” that shows the connections between blocks built up from basic mathematical functions (adders, multipliers, integrators, etc.). Conversion of a model between MATLAB and Simulink, or between one of these programs and another software package, can be very difficult. ModelMaker, which is popular in Europe, provides surprisingly broad functionality at a relatively low price. Its only serious drawback is the lack of a scripting capability. SCoP, which along with ACSL was one of the first languages to be used for PBPK modeling, continues to be used due to its familiarity and low cost. However, its lack of scripting or code sorting, and its DOS-based, menu-driven run-time interface can make model evaluation more difficult.

3. Methods 3.1. General Concepts

The methods will begin with a description of the seminal PBPK model published by Ramsey and Andersen in 1984 and lead into the elements necessary for successful model development, refinement and validation. The experience of Ramsey and Andersen serves as a useful example of the advantages of the PBPK modeling approach. In this case, blood and tissue time-course curves of styrene had been obtained for rats exposed to four different concentrations of 80, 200, 600, and 1,200 ppm (83). Data were obtained during a 6 h exposure period and for 18 h after cessation of the exposure. The initial analysis of these data had been based on a simple compartmental model, similar to the model shown in Fig. 1, which had a zero-order input related to the amount of styrene inhaled, a two-compartment description of the rat, and linear metabolism in the central compartment. The compartmental model was successful with lower concentrations but was unable to account for the more complex behavior at higher concentrations (note the different behavior of the data at the two concentrations shown in Fig. 3). In an attempt to provide a more successful description, a PBPK model was developed with a realistic equilibration process for pulmonary uptake and Michaelis–Menten saturable metabolism in the

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Styrene Concentration (mg/L)

450

Rat 600 ppm

80 ppm

Hours

Fig. 3. Model predictions (solid lines) and experimental blood styrene concentrations in rats during and after 6 h exposures to 80 and 600 ppm styrene. The thick bars represent the chamber air concentrations of styrene and are shown to highlight the nonlinearity of the relationship between administered and internal concentrations. The model (Fig. 1.3) contains sufficient biological realism to predict the very different behaviors observed at the two concentrations.

liver. A diagram of the PBPK model that was used by Ramsey and Andersen (1984) (22) to describe styrene inhalation in both rats and humans is shown in Fig. 4. In this diagram, the boxes represent tissue compartments and the lines connecting them represent blood flows. The model contained several “lumped” tissue compartments: fat tissues, poorly perfused tissues (muscle, skin, etc.), richly perfused tissues (viscera), and metabolizing tissues (liver). The fat tissues were described separately from the other poorly perfused tissues due to their much higher partition coefficient for styrene, which leads to different kinetic properties, while the liver was described separately from the other richly perfused tissues due to its key role in the metabolism of styrene. Each of these tissue groups was defined with respect to their blood flow, tissue volume, and their ability to store (partition) the chemical of interest. Although the model diagram in Fig. 4 shows a lung compartment, a steady-state approximation for the equilibration of lung blood with alveolar air was used in the mathematical formulation of the model to eliminate the need for an actual lung tissue compartment. This simple model structure, with realistic constants for the physiological, partitioning, and metabolic parameters, very accurately predicted the behavior of styrene in both fat and blood of the rat at all concentrations. Fig. 3 compares the model-predicted time course in the blood with the experimental data for the highest and lowest exposure concentrations in the rat studies. The structure of the PBPK model for styrene reflects the generic mammalian architecture. Organs are arranged in a parallel system of blood flows with total blood flow through the lungs.

18 Physiologically Based Pharmacokinetic/Toxicokinetic Modeling QAlv

Alveolar Space

CInh QT CVen

CVF

451

QAlv CAlv

Lung Blood

QT CArt

Fat Tissue Group

QF CArt QM

Muscle Tissue Group CVM

CVR

CVL

CArt

Richly Perfused Tissue Group

Liver [Metabolizing Tissue Group] VMax

QR CArt

QL CArt Metabolites

KM

Fig. 4. Diagram of a physiologically based pharmacokinetic model for styrene. In this description, groups of tissues are defined with respect to their volumes, blood flows (Q ), and partition coefficients for the chemical. The uptake of vapor is determined by the alveolar ventilation (QALV), cardiac output (QT), blood–air partition coefficient, and the concentration gradient between arterial and venous pulmonary blood (CART and CVEN). Metabolism is described in the liver with a saturable pathway defined by a maximum velocity (Vmax) and affinity (Km). The mathematical description assumes equilibration between arterial blood and alveolar air as well as between each of the tissues and the venous blood exiting from that tissue.

This model can easily be scaled-up to examine styrene kinetics for other mammalian species. In the case of styrene, exposure experiments had also been conducted with human volunteers (84). In order to model this data, the PBPK model parameters were changed to human physiological values, the human blood–air partitioning was determined from human blood samples, and the metabolism was scaled allometrically so that capacity (Vmax) was related to basal metabolic rate (body weight raised to the 0.7 power) and affinity (Km) was the same in the human as in the rat, 0.36 mg/L. Ramsey (84) measured both venous blood and exhaled air concentrations in these human volunteers. Although the rat PBPK model was developed for blood and fat, not for exhaled air, the physiologically based description automatically provides information on expected exhaled air concentrations. It was straightforward then to predict expected

Styrene Concentration (mg / L)

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80 ppm Blood

Exhaled Air

Hours

Fig. 5. Model predictions and experimental blood and exhaled air concentrations in human volunteers during and after 6 h exposures to 80 ppm styrene. The model is identical to that used for rats (Fig. 1.4). The model parameters have been changed to values appropriate for humans on the basis of physiological and biochemical information, and have not been adjusted to improve the fit to the experimental data.

Styrene Concentration (mg / L)

452

376 216 51

Hours

Fig. 6. Model predictions and experimental exhaled air concentrations in human volunteers following 1 h exposures to 51, 216, and 376 ppm styrene. The model is the same as Fig. 1.5.

exhaled air concentrations in humans and compare the predictions with the concentrations measured during the experiments (Fig. 5). A similar comparison of the model’s predictions with another human data set from (85) also demonstrated the ability of the PBPK structure to support extrapolation of styrene kinetics from the rat to the human (Fig. 6).

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This basic PBPK model for styrene has several tissue groups which were lumped according to their perfusion and partitioning characteristics. In the mathematical formulation, each of these several compartments is described by a single mass-balance differential equation. It would be possible to describe individual tissues in each of the lumped compartments, if necessary. This detail is usually unnecessary unless some particular tissue in a lumped compartment is the target tissue. One might, for example, want to separate brain from other richly perfused tissues if the model were for a chemical that had a toxic effect on the central nervous system (86–88). Other examples of additional compartments include the addition of placental and mammary compartments to model pregnancy and lactation (89–91). The interactions of chemical mixtures can even be described by including compartments for more than one chemical in the model (92–94). Increasing the number of compartments does increase the number of differential equations required to define the model. However, the number of equations does not pose any problem due to the power of modern desktop computers. On the other hand, as the number of compartments in the PBPK model increases, the number of input parameters increases correspondingly. Each of these parameters must be estimated from experimental data of some kind. Fortunately, the values of many of these can be set within narrow limits from nonkinetic experiments. The PBPK model can also help to define those experiments which are needed to improve parameter estimates by identifying conditions where the sensitivity of the model to the parameter is the greatest (95). The demand that the PBPK fit a variety of data also restricts the parameter values that will give a satisfactory fit to experimental data. For example, the styrene model (described above) was required to reproduce both the high and low concentration behaviors, which appeared qualitatively different, using the same parameter values. If one were independently fitting single curves with a model, the different parameter values obtained under different conditions would be relatively uninformative for extrapolation. As the renowned statistician George Box has said, “All models are wrong, and some are useful.” Even a relatively complex description such as a PBPK model will sometimes fail to fit reliable experimental data. When this occurs, the investigator needs to think how the model might be changed, i.e., what extra biological aspects must be added to the physiological description to bring the predictions in line with experimental observation? In the case of the work with styrene cited above, continuous 24 h styrene exposures could not be modeled with a time-independent maximum rate of metabolism, and induction of enzyme activity had to be included to yield a satisfactory representation of the observed kinetic behavior (96). When a PBPK model is unable to adequately describe kinetic data, the nature of the discrepancy can provide the investigator with

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additional insight into time dependencies in the system. This insight can then be utilized to reformulate the biological basis of the model and improve its fidelity to the data. The resulting model may be more complicated, but it will still be useful if the pertinent kinetic constants can be estimated for human tissues. Indeed, as long as the model maintains its biological basis the additional parameters can often be determined directly from separate experiment, rather than estimated by fitting the model to kinetic data. As the models become more complex, they necessarily contain larger numbers of physiological, biochemical, and biological constants. The crucial task during model development is to keep the description as simple as possible and to ensure the identifiability of new parameters that are added to the model; every attempt should be made to obtain or verify model parameters from experimental studies separate from the modeling exercises themselves (97). The following section explores some of the key issues associated with the development of PBPK models. It is meant to provide a general understanding of the basic design concepts and mathematical forms underlying the PBPK modeling process, and is not meant to be a complete exposition of the PBPK modeling approach for all possible cases. It must be understood that the specifics of the approach can vary greatly for different types of chemicals, e.g., volatiles, nonvolatiles, and metals, and for different applications. Model building is an art, and is best understood as an iterative process in the spirit of the scientific method (97). The literature articles cited in the introductory section include examples of successful PBPK models for a wide variety of chemicals and provide a wealth of insight into various aspects of the PBPK modeling process. They should be consulted for further detail on the approach for applying the PBPK methodology in specific cases. 3.3. Tissue Grouping

The first aspect of PBPK model development that will be discussed is determining the extent to which the various tissues in the body may be grouped together. Although tissue grouping is really just one aspect of model design, which is discussed in the next section, it provides a simple context for introducing the two alternative approaches to PBPK model development: “lumping” and “splitting” (Fig. 7). In the context of tissue grouping, the guiding philosophy in the lumping approach can be stated as follows: “Tissues which are pharmacokinetically and toxicologically indistinguishable may be grouped together.” In this approach, model development begins with information at the greatest level of detail that is practical, and decisions are made to combine physiological elements (tissues and blood flows) to the extent justified by their similarity. The common grouping of tissues into richly (or rapidly) perfused and poorly (or slowly) perfused on the basis of their perfusion rate (ratio of blood flow to tissue volume) is an example of the lumping approach. The contrasting philosophy of splitting is

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Splitting Body Body / Liver Rapid / Slow / Liver Rapid / Slow / Liver / Fat Only a Few Tissues Grouped All Tissues and Organs Separate

Fig. 7. The role of lumping and splitting processes in PBPK model development.

as follows: “Tissues which are pharmacokinetically or toxicologically distinct must be separated.” This approach starts with the simplest reasonable model structure and increases the model’s complexity only to the extent required to reproduce data on the chemical of concern for the application of interest. Splitting requires the greater initial investment in data collection and, if taken to the extreme, could paralyze model development. Lumping, on the other hand, is more efficient but runs a greater risk of overlooking chemicalspecific determinants of chemical disposition. The description of fat tissue in the PBPK model of styrene described in the previous section can be used to provide an example of the different approach associated with the two philosophies. In the splitting approach, which is the approach used by Ramsey and Andersen (22), a single fat compartment was used initially with volume, blood flow and partitioning parameters selected to represent all adipose tissues in the body. Clearly, there are actually a number of distinguishable adipose tissues, including inguinal, perirenal, and brown fat, among others, which may have different partitioning and kinetic characteristics for styrene. However, since this single-compartment treatment provided an adequate description of the available data on the kinetics of styrene in the fat and blood, no attempt was made to split the fat tissue group into multiple compartments. For a more lipophilic chemical, polychlorotrifluoroethylene oligomer, on the other hand, it was not possible to adequately reproduce fat and blood kinetic data using a single fat compartment (98); therefore, the fat compartment was split into two parts: “perirenal fat” and “other fat tissues,” resulting in an acceptable simulation of the observed kinetic behavior. The splitting process just described can be contrasted with a lumping approach, in which the PBPK model would initially be designed to include separate compartments for all physiologically distinguishable fat tissues. Partition coefficients for each of the fat

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tissues would be determined experimentally, and the volume and blood flow for each fat tissue would be estimated. If it were then determined that the kinetic characteristics of the various fat tissues were not sufficiently different to justify retaining separate compartments, they would be lumped together by appropriately combining the individual parameter values (adding the volumes and blood flows and averaging the partition coefficients). 3.3.1. Criteria for Grouping Tissues

There are two alternative approaches for determining whether tissues are kinetically distinct or should be lumped together. In the first approach, the tissue rate-constants are compared. The rate-constant (kT) for a tissue is similar to the perfusion rate except that the partitioning characteristics of the tissue are also considered: kT ¼ Q T =ðP T  V T Þ; where QT ¼ the blood flow to the tissue (L/h), PT ¼ the tissue–blood partition coefficient for the chemical, VT ¼ the volume of the tissue (L). Thus the units of the tissue rate-constant are the same as for the perfusion rate, h1, but the rate-constant more accurately reflects the kinetic characteristics of a tissue for a particular chemical. It was the much smaller rate-constant for fat in the case of a lipophilic chemical such as styrene that required the separation of the fat compartment from the other poorly perfused tissues (muscle, skin, etc.) in the PBPK model for styrene (22). The second, less rigorous, approach for determining whether tissues should be lumped together is simply to compare the performance of the model with the tissues combined and separated. This approach is essentially the reverse of the example given above for splitting of the fat compartment. The reliability of this approach depends on the availability of data under conditions where the tissues being evaluated would be expected to have an observable impact on the kinetics of the chemical. Sensitivity analysis can sometimes be used to determine the appropriate conditions for such a comparison (95).

3.4. Model Design Principles

There is no easy rule for determining the structure and level of complexity needed in a particular modeling application. The wide variability of PBPK model design for different chemicals can be seen by comparing the diagram of the PBPK model for methotrexate (41), shown in Fig. 8, with the diagram for the styrene PBPK model shown in Fig. 3. Model elements which are important for a volatile, lipophilic chemical such as styrene (lung, fat) do not need to be considered in the case of a nonvolatile, water soluble compound such as methotrexate. Similarly, while kidney excretion and enterohepatic recirculation are important determinants of the kinetics of methotrexate, only metabolism and exhalation are

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Plasma QL - QG

QG Liver r

G.I. Tract

Biliary Secretion τ

r1

τ

r2

τ

Gut Absorption

C1

r3

C2

C3

C4

Feces

Gut Lumen QK Kidney

Urine QM Muscle

Fig. 8. PBPK model for methotrexate (Bischoff et al. 1971).

significant for styrene. The decision of which elements to include in the model structure for a specific chemical and application draws on all of the modeler’s experience and knowledge of the animal– chemical system. The alternative approaches to tissue grouping discussed above are actually just reflections of the two competing criteria which must be balanced during model design: parsimony and plausibility. The principle of parsimony simply states that a model should be as simple as possible for the intended application (but no simpler). This “splitting” philosophy is related to that of Occam’s Razor: “Entities should not be multiplied unnecessarily.” That is, structures and parameters should not be included in the model unless they are needed to support the application for which the model is being designed. For example, if a model is developed to describe inhalation exposure to a chemical over periods from hours to years, as in the case of the styrene model discussed earlier, it is not necessary to describe transient, breath-by-breath behavior of chemical uptake and exhalation in the lung. On the other hand, if the model is being developed to predict initial inhalation uptake of the chemical at times on the order of minutes, this level of detail clearly might be justified (98, 99). The desire for parsimony in model development is driven not only by the desire to minimize the number of parameters whose values must be identified, but also by the recognition that as the number of parameters increases, the potential for unintended interactions between parameters increases disproportionately. A generally accepted rule of software engineering warns that it is relatively easy to design a computer program which is too complicated to be

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completely comprehended by the human mind. As a model becomes more complex, it becomes increasingly difficult to validate, even as the level of concern for the trustworthiness of the model should increase. Countering the desire for model parsimony is the need for plausibility of the model structure. As discussed in the introduction, it is the physiological and biochemical realism of PBPK models that gives them an advantage for extrapolation. The credibility of a PBPK model’s predictions of kinetic behavior under conditions different from those under which the model was validated rests to a large extent on the correspondence of the model design to known physiological and biochemical structures. In general, the ability of a model to adequately simulate the behavior of a physical system depends on the extent to which the model structure is homomorphic (having a one-to-one correspondence) with the essential features determining the behavior of that system. For example, if the model of styrene had not included a description of saturable metabolism, it would not have been able to adequately simulate the kinetics of styrene at both low and high doses using a single parameterization. 3.4.1. Model Identification

The process of model identification begins with the selection of those model elements which the modeler considers to be minimum essential determinants of the behavior of the particular animal–chemical system under study, from the viewpoint of the intended application of the model. Comparison with appropriate data, relevant to the intended purpose of the model, then can provide insights into defects in the model which must be corrected either by reparameterization or by changes to the model structure. Unfortunately, it is not always possible to separate these two elements. In models of biological systems, estimates of the values of model parameters will always be uncertain, due both to biological variation and experimental error. At the same time, the need for biological realism unavoidably results in models that are “overparameterized”; that is, they contain more parameters than can be identified from the kinetic data the model is used to describe. As an example of the interaction between model structure and parameter identification, the two metabolic parameters, Vmax and Km, in the model for styrene discussed earlier could both be identified relatively unambiguously in the case of the rat. Indeed, as pointed out previously, the inclusion of capacity-limited metabolism in the model was necessary in order to reproduce the available data at both low and high exposure concentrations. In the case of the human, however, data was not available at sufficiently high concentrations to saturate metabolism. Therefore, only the ratio, Vmax/ Km, would actually be identifiable. The use of the same model structure, including a two-parameter description of metabolism, in the human as in the rat was justified by the knowledge that similar

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enzymatic systems are responsible for the metabolism of chemicals such as styrene in both species. However, if the model were to be used to extrapolate to higher concentrations in the human, the potential impact of the uncertainty in the values of the individual metabolic parameters would have to be carefully considered. Model identification is the selection of a specific model structure from several alternatives, based on conformity of the models’ predictions to experimental observations. The practical reality of model identification in the case of biological systems is that regardless of the complexity of the model there will always be some level of “model error” (lack of homomorphism) which will result in systematic discrepancies between the model and experimental data. This model structural deficiency interacts with deficiencies in the identifiability of the model parameters, potentially leading to misidentification of the parameters or misspecification of structures. This most dangerous aspect of model identification is exacerbated by the fact that, in general, adding equations and parameters to a model increases the model’s degrees of freedom, improving its ability to reproduce data, regardless of the validity of the underlying structure. Therefore, when a particular model structure improves the agreement of the model with kinetic data, it can only be said that the model structure is “consistent” with the kinetic data; it cannot be said that the model structure has been “proved” by its consistency with the data. In such circumstances, it is imperative that the physiological or biochemical hypothesis underlying the model structure is tested using nonkinetic data. 3.5. Elements of Model Structure

The process of selecting a model structure can be broken down into a number of elements associated with the different aspects of uptake, distribution, metabolism, and elimination. In addition, there are several general model structure issues that must be addressed, including mass balance and allometric scaling. The following section treats each of these elements in turn.

3.5.1. Storage Compartments

Naturally, any tissues which are expected to accumulate significant quantities of the chemical or its metabolites need to be included in the model structure. As discussed earlier, these storage tissues can be grouped together to the extent that they have similar time constants. Three storage compartments were included in the styrene model described above: fat tissues, richly perfused tissues, and poorly perfused tissues. The generic mass balance equation for storage compartments such as these is (Fig. 9):

QT CA

Tissue

Fig. 9. Blood flow through a storage compartment.

QT CVT

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dA T =dt ¼ Q T  C A  Q T  C VT ; where AT ¼ the mass of chemical in the tissue (mg), QT ¼ the blood flow to (and from) the tissue (L/h), CA ¼ the concentration of chemical in the arterial blood reaching the tissue (mg/L), CVT ¼ the concentration of the chemical in the venous blood leaving the tissue (mg/L). Thus this mass balance equation simply states that the change in the amount of chemical in the tissue with respect to time (dAT/dt) is equal to the difference between the amount of chemical entering the tissue and the amount leaving the tissue. We can then calculate the concentration of the chemical in the storage tissue (CT) from the amount in the tissue and the tissue volume (VT): C T ¼ A T =V T : In PBPK models, it is common to assume “venous equilibration”; that is, that in the time that it takes for the blood to perfuse the tissue, the chemical is able to achieve its equilibrium distribution between the tissue and blood. Therefore, the concentration of the chemical in the venous blood can be related to the concentration in the tissue by the equilibrium tissue–blood partition coefficient (PT): C VT ¼ C T =P T : Therefore we obtain a differential equation in AT: dA T =dt ¼ Q T  C A  Q T  A T =ðP T  V T Þ: If desired, we can reformulate this mass balance equation in terms of concentration: dA T =dt ¼ dðC T  V T Þ=dt ¼ C T  dV T =dt þ V T  dC T =dt: If (and only if) VT is constant (i.e., the tissue does not grow during the simulation), dVT/dt ¼ 0, and: dA T =dt ¼ V T  dC T =dt, so we have the alternative differential equation: dC T =dt ¼ Q T  ðC A  C T =P T Þ=V T : This alternative mass balance formulation, in terms of concentration rather than amount, is popular in the pharmacokinetic literature. However, in the case of models with compartments that change volume over time it is preferable to use the formulation in terms of amounts in order to avoid the need for the additional term reflecting the change in volume (CT  dVT/dt). Depending on the chemical, many different tissues can potentially serve as important storage compartments. The use of a fat storage compartment in the styrene model is typical of a lipophilic chemical. The gut lumen can also serve as a storage site for chemicals subject to enterohepatic recirculation, as in the case of

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methotrexate. Important storage sites for metals, on the other hand, can include the kidney, red blood cells, intestinal epithelial cells, skin, bone, and hair. Transport to and from a storage compartment does not always occur via the blood, as described above; for example, in some cases the storage is an intermediate step in an excretion process (e.g., hair, intestinal epithelial cells). As with methotrexate, it may also be necessary to use multiple compartments in series, or other mathematical devices, to model plug flow (i.e., a time delay between entry and exit from storage). 3.5.2. Blood Compartment

The description of the blood compartment can vary considerably from one PBPK model to another depending on the role the blood plays in the kinetics of the chemical being modeled. In some cases the blood may be treated as a simple storage compartment, with a mass balance equation describing the summation (S) of the venous blood flows from the various tissues and the return of the total arterial blood flow (QC) to the tissues, as well as any urinary clearance (Fig. 10): X  Q T  C T =P T  Q C  C B  K U  C B ; dA B =dt ¼ where AB ¼ the amount of chemical in the blood (mg), QC ¼ the total cardiac output (L/h), CB ¼ the concentration of chemical in the blood (mg/L), KU ¼ the urinary clearance (L/h). For some chemicals, such as methotrexate, all of the chemical is present in the plasma rather than the red blood cells, so plasma flows and volumes are used instead of blood. For other chemicals it may be necessary to model the red blood cells as a storage compartment in communication with the plasma via diffusion-limited transport. Note that if the blood is an important storage compartment for a chemical, it may be necessary to carefully evaluate data on tissue concentrations, particularly the richly perfused tissues, to determine whether chemical in the blood perfusing the tissue could be contributing to the measured tissue concentration. For still other chemicals, such as styrene, the amount of chemical actually in the blood may be relatively unimportant. In this case, instead of having a true blood compartment, a steady-state approximation can be used to estimate the concentration in the blood at any time. Assuming the blood is at steady-state with respect to the tissues: dA B =dt ¼ 0:

Blood

QC CB

KU

Fig. 10. Blood flow to tissue and urinary clearance.

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Therefore, solving the blood equation for the concentration:   X Q T  C T =P T CB ¼ : QC 3.5.3. Metabolism/ Elimination

The liver is frequently the primary site of metabolism for a chemical. The following equation is an example of the mass balance equation for the liver in the case of a chemical which is metabolized by two pathways (Fig. 11): d A L =dt ¼ Q L  ðC A  C L =P L Þ  kF  C L  V L =P L  V max  C L =P L =ðK m þ C L =P L Þ: In this case, the first term on the right-hand side of the equation represents the mass flux associated with transport in the blood and is identical to the case of the storage compartment described previously. The second term describes metabolism by a linear (firstorder) pathway with rate constant kF (h1) and the third term represents metabolism by a saturable (Michaelis–Menten) pathway with capacity Vmax (mg/h) and affinity Km (mg/L). If it were desired to model a water soluble metabolite produced by the saturable pathway, an equation for its formation and elimination could be added to the model (Fig. 12): dA M =dt ¼ Rstoch  V max  C L =P L =ðK m þ C L =P L Þ  ke  A M C M ¼ AM =V D ; where AM ¼ the amount of metabolite in the body (mg), Rstoch ¼ the stoichiometric yield of the metabolite times the ratio of its molecular, Weight to that of the parent chemical, ke ¼ the rate

QL CA

QL

Liver

CVL

k

VMax, KM

F

Fig. 11. Liver compartment metabolizing a chemical by two pathways.

VMax, KM Metabolite

ke

Fig. 12. Metabolite formation and elimination compartment.

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constant for the clearance of the metabolite from the body (h1), CM ¼ the concentration of the metabolite in the plasma (mg/L), VD ¼ the apparent volume of distribution for the metabolite (L). 3.5.4. Metabolite Compartments

In principle, the same considerations which drive decisions regarding the level of complexity of the PBPK model for the parent chemical must also be applied for each of its metabolites, and their metabolites, and so on. As in the case of the parent chemical, the first and most important consideration is the purpose of the model. If the concern is direct parent chemical toxicity and the chemical is detoxified by metabolism, then there is no need for a description of metabolism beyond its role in the clearance of the parent chemical. The models for styrene and methotrexate discussed above are examples of parent chemical models. Similarly, if reactive intermediates produced during the metabolism of a chemical are responsible for its toxicity, as in the case of methylene chloride, a very simple description of the metabolic pathways might be adequate (8). The cancer risk assessment model for methylene chloride described the rate of metabolism for two pathways: the glutathione conjugation pathway, which was considered responsible for the carcinogenic effects, and the competing P450 oxidation pathway, which was considered protective. On the other hand, if one or more of the metabolites are considered to be responsible for the toxicity of a chemical, it may be necessary to provide a more complete description of the kinetics of the metabolites themselves. For example, in the case of teratogenicity from all-trans-retinoic acid, both the parent chemical and several of its metabolites are considered to be toxicologically active; therefore, in developing the PBPK model for this chemical it was necessary to include a fairly complete description of the metabolic pathways (100). Fortunately, the metabolism of xenobiotic compounds often produces metabolites which are relatively water soluble, simplifying the description needed. In many cases, such as the production of trichloroacetic acid from trichloroethylene (46–48), a classical one-compartment description may be adequate for describing the metabolite kinetics. An example of such a description was provided earlier. In other cases, however, the description of the metabolite (or metabolites) may have to be as complex as that of the parent chemical. An example of such a case is the PBPK model for parathion (88), in which the model for the active metabolite, paraoxon, is actually more complex than that of the parent chemical.

3.5.5. Target Tissues

Typically, a PBPK model used in toxicology or risk assessment applications will include compartments for any target tissues for the toxic action of the chemical. The target tissue description may in some cases need to be fairly complicated, including such features as in situ metabolism, binding, and pharmacodynamic processes in order to provide a realistic measure of biologically effective tissue exposure (57). For example, whereas the lung compartment in the

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styrene model was represented only by a steady-state description of alveolar vapor exchange, the PBPK model for methylene chloride that was applied to perform a cancer risk assessment (8) included a two-part lung description in which alveolar vapor exchange was followed by a lung tissue compartment with in situ metabolism. This more complex lung compartment was required to describe the dose–response for methylene chloride induced lung cancer, which was assumed to result from the metabolism of methylene chloride in lung clara cells. In other cases, describing a separate compartment for the target tissue may be unnecessary. For example, the styrene model described above could be used to relate acute exposures associated with neurological effects without the necessity of separating out a brain compartment. Instead, the concentration or AUC of styrene in the blood could be used as a metric, on the assumption that the relationship between brain concentration and blood concentration would be the same under all exposure conditions, routes, and species, namely, that the concentrations would be related by the brain–blood partition coefficient. In fact, this is probably a reasonable assumption across different exposure conditions in a given species. However, while tissue–air partition coefficients for volatile lipophilic chemicals appear to be similar in dog, monkey, and man (101), human blood–air partition coefficients appear to be roughly half of those in rodents (102). Therefore, the human brain–blood partition would probably be about twice that in the rodent. Nevertheless, if the model were to be used for extrapolation from rodents to humans, this difference could easily be factored into the analysis as an adjustment to the blood metric, without the need to actually add a brain compartment to the model. A fundamental issue in determining the nature of the target tissue description required is the need to identify the toxicologically active form of the chemical. In some cases, a chemical may produce a toxic effect directly, either through its reaction with tissue constituents (e.g., ethylene oxide) or through its binding to cellular control elements (e.g., dioxin). Often, however, it is the metabolism of the chemical that leads to its toxicity. In this case, toxicity may result primarily from reactive intermediates produced during the process of metabolism (e.g., chlorovinyl epoxide produced from the metabolism of vinyl chloride) or from the toxic effects of stable metabolites (e.g., trichloroacetic acid produced from the metabolism of trichloroethylene). The specific nature of the relationship between tissue exposure and response depends on the mechanism of toxicity, or mode of action, involved. Some toxic effects, such as acute irritation or acute neurological effects, may result primarily from the current concentration of the chemical in the tissue. Other toxic effects, such as tissue necrosis and cancer, may depend on both the concentration and duration of the exposure. For developmental effects, the chemical time course may also have to be convoluted with the window of

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susceptibility for a particular gestational event. The selection of the dose metric, that is, the active chemical form for which tissue exposure should be determined and the nature of the measure to be used—e.g., peak concentration (Cmax) or area under the concentration (AUC)–time profile—is the most important step in a pharmacokinetic analysis and a principal determinant of the structure and level of detail that will be required in the PBPK model. 3.5.6. Uptake Routes

Each of the relevant uptake routes for the chemical must be described in the model. Often there are a number of possible ways to describe a particular uptake process, ranging from simple to complex. As with all other aspects of model design, the competing goals of parsimony and realism must be balanced in the selection of the level of complexity to be used. The following examples are meant to provide an idea of the variety of model code which can be required to describe the various possible uptake processes. A B0 ¼ Dose  BW,

Intravenous Administration

where AB0 ¼ the amount of chemical in the blood at the beginning of the simulation (t ¼ 0), Dose ¼ administered dose (mg/kg), BW ¼ animal body weight (kg). or, in the case where a steady-state approximation has been used to eliminate the blood compartment:   C B ¼ Q L  C VL þ    þ Q F  C VF þ kIV =QC, where kIV ¼ Dose  BW=t IV ; ¼ 0 ðt>t IV Þ

ðt1.0 in absolute value represent amplification of input error and would be a cause for concern. An alternative approach is to conduct a Monte Carlo analysis, as described below, and then to perform a simple correlation analysis of the model outputs and input parameters. Both methods have specific advantages. The analytical sensitivity coefficient most accurately represents the functional relationship of the output to the specific input under the conditions being modeled. The advantage of the correlation coefficients is that they also reflect the impact of interactions between the parameters during the Monte Carlo analysis.

3.9.5. Uncertainty Analysis

There are a number of examples in the literature of evaluations of the uncertainty associated with the predictions of a PBPK model using the Monte Carlo simulation approach (10, 117, 118). In a Monte Carlo simulation, a probability distribution for each of the PBPK model parameters is randomly sampled, and the model is run using the chosen set of parameter values. This process is repeated a large number of times until the probability distribution for the desired model output has been created. Generally speaking, 1,000 iterations or more may be required to ensure the reproducibility of the mean and standard deviation of the output distributions as well as the 1st through 99th percentiles. To the extent that the input parameter distributions adequately characterize the uncertainty in the inputs, and assuming that the parameters are reasonably independent, the resulting output distribution will provide a useful estimate of the uncertainty associated with the model outputs.

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In performing a Monte Carlo analysis it is important to distinguish uncertainty from variability. As it relates to the impact of pharmacokinetics in risk assessment, uncertainty can be defined as the possible error in estimating the “true” value of a parameter for a representative (“average”) person. Variability, on the other hand, should only be considered to represent true interindividual differences. Understood in these terms, uncertainty is a defect (lack of certainty) which can typically be reduced by experimentation, and variability is a fact of life which must be considered regardless of the risk assessment methodology used. An elegant approach for separately documenting the impact of uncertainty and variability is “two-dimensional” Monte Carlo, in which distributions for both uncertainty and variability are developed and multiple Monte Carlo runs are used to convolute the two aspects of overall uncertainty. Unfortunately, in practice it is often difficult to differentiate the contribution of variability and uncertainty to the observed variation in the reported measurements of a particular parameter (118). Due to its physiological structure, many of the parameters in a PBPK model are interdependent. For example, the blood flows must add up to the total cardiac output and the tissue volumes (including those not included in the model) must add up to the body weight. Failure to account for the impact of Monte Carlo sampling on these mass balances can produce erroneous results (95, 117). In addition, some physiological parameters are naturally correlated, such as cardiac output and respiratory ventilation rate, and these correlations should be taken into account during the Monte Carlo analysis (118). 3.9.6. Collection of Critical Data

As with model development, the best approach to model evaluation is within the context of the scientific method. The most effective way to evaluate a PBPK model is to exercise the model to generate a quantitative hypothesis; that is, to predict the behavior of the system of interest under conditions “outside the envelope” of the data used to develop the model (at shorter/longer durations, higher/lower concentrations, different routes, different species, etc.). In particular, if there is an element of the model which remains in question, the model can be exercised to determine the experimental design under which the specific model element can best be tested. For example, if there is uncertainty regarding whether uptake into a particular tissue is flow or diffusion limited, alternative forms of the model can be used to compare predicted tissue concentration time courses under each of the limiting assumptions under various experimental conditions. The experimental design and sampling time which maximizes the difference between the predicted tissue concentrations under the two assumptions can then serve as the basis for the actual experimental data collection. Once the critical data has been collected, the same model can also be used to support a more quantitative experimental

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inference. In the case of the tissue uptake question just described, not only can the a priori model predictions be compared with the observed data to test the alternative hypotheses, but the model can also be used a posteriori to estimate the quantitative extent of any observed diffusion limitation (i.e., to estimate the relevant model parameter by fitting the data). If, on the other hand, the model is unable to reproduce the experimental data under either assumption, it may be necessary to reevaluate other aspects of the model structure. The key difference between research and analysis is the iterative nature of the former. It has wisely been said, “If we knew when we started what we had to do to finish, they’d call it search, not research.”

4. Example The previous sections have focused on the process of designing the PBPK model structure needed for a particular application. At this point the model consists of a number of mathematical equations: differential equations describing the mass balance for each of the compartments and algebraic equations describing other relationships between model variables. The next step in model development is the coding of the mathematical form of the model into a form which can be executed on a computer. The discussion in this section will be couched in terms of a particular software package, acslX. 4.1. Mathematical Formulation

Mathematically, a PBPK model is represented by a system of simultaneous linear differential equations. The model compartments are represented by the differential equations that describe the mass balance for each one of the “state variables” in the model. There may also be additional differential equations to calculate other necessary model outputs, such as the area under the AUC in a particular compartment, which is simply the integral of the concentration over time. The resulting system of equations is referred to as simultaneous because the time courses of the chemical in the various compartments are so interdependent that solving the equations for any one of the compartments requires information on the current status of all the other compartments; that is, the equations for all of the compartments must be solved at the same time. The equations are considered to be linear in the sense that they include only first-order derivatives, not due to any pharmacokinetic considerations. This kind of mathematical problem, in which a system is defined by the conditions at time zero together with differential equations describing how it evolves over time, is known as an initial value problem, and matrix decomposition methods are used to obtain the simultaneous solution.

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A number of numerical algorithms are available for solving such problems. They all have in common that they are step-wise approximations; that is, they begin with the conditions at time zero and use the differential equations to predict how the system will change over a small time step, resulting in an estimate of the conditions at a slightly later time, which serves as the starting point for the next time step. This iterative process is repeated as long as necessary to simulate the experimental scenario. The more sophisticated methods, such as the Gear algorithm (named after the mathematician, David Gear, who developed it) use a predictor–corrector approach, in which the corrector step essentially amounts to “predicting backwards” after each step forward, in order to check how closely the algorithm is able to reproduce the conditions at the previous time step. This allows the time step to be increased automatically when the algorithm is performing well, and to be shortened when it is having difficulty, such as when conditions are changing rapidly. However, due to the wide variation of the time constants (response times) for the various physiological compartments (e.g., fat vs. richly perfused), PBPK models often represent “stiff” systems. Stiff systems are characterized by state variables (compartments) with widely different time constants, which cause difficulty for predictor–corrector algorithms. The Gear algorithm was specifically designed to overcome this difficulty. It is therefore generally recommended that the Gear algorithm be used for executing PBPK models. An implementation of the Gear algorithm is available in most of the advanced software packages. Regardless of the specific algorithm selected, the essential nature of the solution, as stated above, will be a step-wise approximation. However, all of the algorithms made available in computer software are convergent; that is, they can stay arbitrarily close to the true solution, given a small enough time step. On modern personal computers, even large PBPK models can be run to more than adequate accuracy in a reasonable timeframe. 4.2. Model Coding in ACSL

The following sections contain typical elements of the ACSL code for a PBPK model, interspersed with comments, which will be written in italics to differentiate them from the actual model code. The first section describes the model definition file, which by convention in ACSL is given a filename with the extension CSL. The model used as an example in the following sections is a simple, multiroute model for volatile chemicals, similar to the styrene model discussed earlier, except that is also has the ability to simulate closed-chamber gas uptake experiments.

4.3. Typical Elements in a Model File

An acslX source file follows the structure defined in the Standard for Continuous Simulation Languages (just like there is a standard for C++). Thus, for example, there will generally be an INITIAL block

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defining the initial conditions followed by a DYNAMIC block which contains DISCRETE and/or DERIVATIVE sub-blocks that define the model. In addition, conventions which have been generally adopted by the PBPK modeling community (most of which started with John Ramsey at Dow Chemical during the development of the styrene model) help to improve the readability of the code. The following file shows typical elements of “Ramseyan code.” The first line in the code must start with the word PROGRAM (a remnant of ACSL’s derivation from FORTRAN). PROGRAM Lines starting with an exclamation point (and portions of lines to the right of one) are ignored by the ACSL translator and can be used for comments: ! Developed for ACSL Level 10 ! by Harvey Clewell (KS Crump Group, ICF Kaiser Int’l., Ruston, LA) ! and Mel Andersen (Health Effects Research Laboratory, USEPA, RTP, NC) The first section of an ACSL source file is the INITIAL block, which is used to define parameters and perform calculations that do not need to be repeated during the course of the simulation: INITIAL ! Beginning of preexecution section Only parameters defined in a CONSTANT statement can be changed during a session using the SET command: LOGICAL CC ! Flag set to .TRUE. for closed chamber runs ! Physiological parameters (rat) CONSTANT QPC ¼ 14. ! Alveolar ventilation rate (L/hr) CONSTANT QCC ¼ 14. ! Cardiac output (L/hr) CONSTANT QLC ¼ 0.25 ! Fractional blood flow to liver CONSTANT QFC ¼ 0.09 ! Fractional blood flow to fat CONSTANT BW ¼ 0.22 ! Body weight (kg) CONSTANT VLC ¼ 0.04 ! Fraction liver tissue CONSTANT VFC ¼ 0.07 ! Fraction fat tissue !—————Chemical specific parameters (styrene) CONSTANT PL ¼ 3.46 ! Liver/blood partition coefficient CONSTANT PF ¼ 86.5 ! Fat/blood partition coefficient CONSTANT PS ¼ 1.16 ! Slowly perfused tissue/blood partition CONSTANT PR ¼ 3.46 ! Richly perfused tissue/blood partition CONSTANT PB ¼ 40.2 ! Blood/air partition coefficient CONSTANT MW ¼ 104. ! Molecular weight (g/mol) CONSTANT VMAXC ¼ 8.4 ! Maximum velocity of metabolism (mg/hr-1kg) CONSTANT KM ¼ 0.36 ! Michaelis–Menten constant (mg/L) CONSTANT KFC ¼ 0. ! First order metabolism (/hr-1kg) CONSTANT KA ¼ 0. ! Oral uptake rate (/hr)

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!—————Experimental parameters CONSTANT PDOSE ¼ 0. ! Oral dose (mg/kg) CONSTANT IVDOSE ¼ 0. ! IV dose (mg/kg) CONSTANT CONC ¼ 1000. ! Inhaled concentration (ppm) CONSTANT CC ¼ .FALSE.! Default to open chamber CONSTANT NRATS ¼ 3. ! Number of rats (for closed chamber) CONSTANT KLC ¼ 0. ! First order loss from closed chamber (/hr) CONSTANT VCHC ¼ 9.1 ! Volume of closed chamber (L) CONSTANT TINF ¼ .01 ! Length of IV infusion (hr) It is an understandable requirement in ACSL to define when to stop and how often to report. The parameter for the reporting frequency (“communication interval”) is assumed by the ACSL translator to be called CINT unless you tell it otherwise using the CINTERVAL statement. The parameter for when to stop can be called anything you want, as long as you use the same name in the TERMT statement (see below), but the Ramseyan convention is TSTOP: CONSTANT TSTOP ¼ 24. ! Length of experiment (hr) The following parameter name is generally used to define the length of inhalation exposures (the name LENGTH is also used by some): CONSTANT TCHNG ¼ 6. ! Length of inhalation exposure (hr) The INITIAL block is a useful place to perform logical switching for different model applications, in this case between the simulation of closed-chamber gas uptake experiments and normal inhalation studies. It is also sometimes necessary to calculate initial conditions for one of the integrals (“state variables”) in the model (the initial amount in the closed chamber in this case): IF (CC) RATS ¼ NRATS ! Closed chamber simulation IF (CC) KL ¼ KLC IF (.NOT.CC) RATS ¼ 0. ! Open chamber simulation IF (.NOT.CC) KL ¼ 0. ! (Turn off chamber losses so concentration in chamber remains constant) IF (PDOSE.EQ.0.0) KA ¼ 0. ! If not oral dosing, turn off oral uptake VCH ¼ VCHC-RATS*BW ! Net chamber air volume (L) AI0 ¼ CONC*VCH*MW/24450. ! Initial amount in chamber (mg) After all the constants have been defined, calculations using them can be performed. In contrast to the DERIVATIVE block (of which more later), the calculations in the INITIAL block are performed in

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the order written, just like in FORTRAN, so a variable must be defined before it can be used. Note how allometric scaling is used for flows (QC, QP) and metabolism (VMAX, KFC). Also note how the mass balance for the blood flows and tissue volumes is maintained by the model code. Runtime changes in the parameters for fat and liver are automatically balanced by changes in the slowly and richly perfused compartments, respectively. The fractional blood flows add to one, but the fractional tissue volumes add up to only 0.91, allowing 9% of the body weight to reflect nonperfused tissues: !————Scaled parameters QC ¼ QCC*BW**0.74 ! Cardiac output QP ¼ QPC*BW**0.74 ! Alveolar ventilation QL ¼ QLC*QC ! Liver blood flow QF ¼ QFC*QC ! Fat blood flow QS ¼ 0.24*QC-QF ! Slowly perfused tissue blood flow QR ¼ 0.76*QC-QL ! Richly perfused tissue blood flow VL ¼ VLC*BW ! Liver volume VF ¼ VFC*BW ! Fat tissue volume VS ¼ 0.82*BW-VF ! Slowly perfused tissue volume VR ¼ 0.09*BW-VL ! Richly perfused tissue volume VMAX ¼ VMAXC*BW**0.7 ! Maximum rate of metabolism KF ¼ KFC/BW**0.3 ! First-order metabolic rate constant DOSE ¼ PDOSE*BW ! Oral dose IVR ¼ IVDOSE*BW/TINF ! Intravenous infusion rate An END statement is required to delineate the end of the initial block: END ! End of initial se-ction The next (and often last) section of an ACSL source file is the DYNAMIC block, which contains all of the code defining what is to happen during the course of the simulation: DYNAMIC ! Beginning of execution section ACSL possesses a number of different algorithms for performing the simulation, which mathematically speaking consists of solving an initial value problem for a system of simultaneous linear differential equations. (Although it is easier to just refer to it as integrating.) Available methods include the Euler, Runga-Kutta, and AdamsMoulton, but the tried and true choice of most PBPK modelers is the Gear predictor–corrector, variable step-size algorithm for stiff systems, which PBPK models often are. (Stiff, that is): ALGORITHM IALG ¼ 2

! Use Gear integration algorithm

NSTEPS NSTP ¼ 10

! Number of integration steps in communication interval

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MAXTERVAL MAXT ¼ 1.0e9

! Maximum integration step size

MINTERVAL MINT ¼ 1.0e-9

! Minimum integration step size

CINTERVAL CINT ¼ 0.01

! Communication interval

One of the structures which can be used in the DYNAMIC block is called a DISCRETE block. The purpose of a DISCRETE block is to define an event which is desired to occur at a specific time or under specific conditions. The integration algorithm then keeps a lookout for the conditions and executes the code in the DISCRETE block at the proper moment during the execution of the model. An example of a pair of discrete blocks which are used to control repeated dosing in another PBPK model are shown here as an example: DISCRETE DOSE1 ! Schedule events to turn exposure on and off daily INTERVAL DOSINT ¼ 24. ! Dosing interval !(Set interval larger than TSTOP to prevent multiple exposure) IF (T.GT.TMAX) GOTO NODOSE IF (DAY.GT.DAYS) GOTO NODOSE CONC ¼ CI ! Start inhalation exposure TOTAL ¼ TOTAL + DOSE ! Administer oral dose TDOSE ¼ T ! Record time of dosing SCHEDULE DOSE2 .AT. T + TCHNG ! Schedule end of exposure NODOSE..CONTINUE DAY ¼ DAY + 1. IF (DAY.GT.7.) DAY ¼ 0.5 END ! of DOSE1 DISCRETE DOSE2 CONC ¼ 0. ! End inhalation exposure END ! of DOSE2 Within the DYNAMIC block, a group of statements defining a system of simultaneous differential equations is put in a DERIVATIVE block. If there is only one it does not have to be given a name: DERIVATIVE ! Beginning of derivative definition block The main function of the derivative block is to define the “state variables” which are to be integrated. They are identified by the INTEG function. For example, in the code below, AI is defined to be a state variable which is calculated by integrating the equation defining the variable RAI, using an initial value of AI0. For most of the compartments, the initial value is zero. !———————CI ¼ Concentration in inhaled air (mg/L) RAI ¼ RATS*QP*(CA/PB-CI) - (KL*AI) ! Rate equation AI ¼ INTEG(RAI,AI0) ! Integral of RAI CI ¼ AI/VCH*CIZONE ! Concentration in air CIZONE ¼ RSW((T.LT.TCHNG).OR.CC,1.,0.) CP ¼ CI*24450./MW ! Chamber air concentration in ppm

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Any experienced programmer would shudder at the code shown above, because several variables appear to be used before they have been calculated (for example, CIZONE is used to calculate CI and CI is used to calculate RAI. However, within the derivative block, writing code is almost too easy because the translator will automatically sort the statements into the proper order for execution. That is, there is no need to be sure that a variable is calculated before it is used. The down side of the sorting is that you cannot be sure that two statements will be calculated in the order you want just because you place them one after the other. Also, because of the sorting (as well as the way the predictor–corrector integration algorithm hops forward and backward in time), IF statements will not work right. The RSW function above works like an IF statement, setting CIZONE to 1. whenever T (the default name for the time variable in ACSL) is Less Than TCHNG, and setting CIZONE to 0. (and thus turning off the exposure) whenever T is greater than or equal to TCHNG. The following blocks of statements each define one of the compartments in the model. These statements can be compared with the mathematical equations described in the previous sections of the manual. One of the advantages of models written in ACSL following the Ramseyan convention is that they are easier to comprehend and reasonably self-documenting. !——MR ¼ Amount remaining in stomach (mg) RMR ¼ -KA*MR MR ¼ DOSE*EXP(-KA*T) Note that the stomach could have been defined as one of the state variables: MR ¼ INTEG(RMR,DOSE) But instead the exact solution for the simple integral has been used directly. Similarly, instead of defining the blood as a state variable, the steady-state approximation is used: !——CA ¼ Concentration in arterial blood (mg/L) CA ¼ (QC*CV + QP*CI)/(QC + (QP/PB)) AUCB ¼ INTEG(CA,0.) !——AX ¼ Amount exhaled (mg) CX ¼ CA/PB ! End-alveolar air concentration (mg/L) CXPPM ¼ (0.7*CX+0.3*CI)*24450./MW ! Average exhaled air concentration (ppm) RAX ¼ QP*CX AX ¼ INTEG(RAX,0.) !——AS ¼ Amount in slowly perfused tissues (mg) RAS ¼ QS*(CA-CVS) AS ¼ INTEG(RAS,0.) CVS ¼ AS/(VS*PS)

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CS ¼ AS/VS !——AR ¼ Amount in rapidly perfused tissues (mg) RAR ¼ QR*(CA-CVR) AR ¼ INTEG(RAR,0.) CVR ¼ AR/(VR*PR) CR ¼ AR/VR !——AF ¼ Amount in fat tissue (mg) RAF ¼ QF*(CA-CVF) AF ¼ INTEG(RAF,0.) CVF ¼ AF/(VF*PF) CF ¼ AF/VF !——AL ¼ Amount in liver tissue (mg) RAL ¼ QL*(CA-CVL)-RAM + RAO AL ¼ INTEG(RAL,0.) CVL ¼ AL/(VL*PL) CL ¼ AL/VL AUCL ¼ INTEG(CL,0.) !——AM ¼ Amount metabolized (mg) RAM ¼ (VMAX*CVL)/(KM + CVL) + KF*CVL*VL AM ¼ INTEG(RAM,0.) !——AO ¼ Total mass input from stomach (mg) RAO ¼ KA*MR AO ¼ DOSE-MR !——IV ¼ Intravenous infusion rate (mg/h) IVZONE ¼ RSW(T.GE.TINF,0.,1.) IV ¼ IVR*IVZONE !——CV ¼ Mixed venous blood concentration (mg/L) CV ¼ (QF*CVF + QL*CVL + QS*CVS + QR*CVR + IV)/ QC !——TMASS ¼ mass balance (mg) TMASS ¼ AF + AL + AS + AR + AM + AX + MR !——DOSEX ¼ Net amount absorbed (mg) DOSEX ¼ AI + AO + IVR*TINF-AX Last, but definitely not least, you have to tell ACSL when to stop: TERMT(T.GE.TSTOP) ! Condition for terminating simulation END ! End of derivative block END ! End of dynamic section Another kind of code section, the TERMINAL block, can also be used here to execute statements that should only be calculated at the end of the run. END ! End of program

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4.4. Model Evaluation

The following section discusses various issues associated with the evaluation of a PBPK model. Once an initial model has been developed, it must be evaluated on the basis of its conformance with experimental data. In some cases, the model may be exercised to predict conditions under which experimental data should be collected in order to verify or improve model performance. Comparison of the resulting data with the model predictions may suggest that revision of the model will be required. Similarly, a PBPK model designed for one chemical or application may be adapted to another chemical or application, requiring modification of the model structure and parameters. It is imperative that revision or modification of a model is conducted with the same level of rigor applied during initial model development, and that structures are not added to the model with no other justification than that they improve the agreement of the model with a particular data set. In addition to comparing model predictions to experimental data, model evaluation includes assessing the plausibility of the model input parameters, and the confidence which can be placed in extrapolations performed by the model. This aspect of model evaluation is particularly important in the case of applications in risk assessment, where it is necessary to assess the uncertainty associated with risk estimates calculated with the model.

4.5. Model Revision

An attempt to model the metabolism of allyl chloride (119) serves as an excellent example of the process of model refinement and validation. As mentioned earlier, in a gas uptake experiment several animals are maintained in a small, enclosed chamber while the air in the chamber is recirculated, with replenishment of oxygen and scrubbing of carbon dioxide. A small amount of a volatile chemical is then allowed to vaporize into the chamber, and the concentration of the chemical in the chamber air is monitored over time. In this design, any loss of the chemical from the chamber air reflects uptake into the animals. After a short period of time during which the chemical achieves equilibration with the animals’ tissues, any further uptake represents the replacement of chemical removed from the animals by metabolism. Analysis of gas uptake data with a PBPK model has been used successfully to determine the metabolic parameters for a number of chemicals (103). In an example of a successful gas uptake analysis, (108) described the closed chamber kinetics of methylene chloride using a PBPK model which included two metabolic pathways: one saturable, representing oxidation by Cytochrome P450 enzymes, and one linear, representing conjugation with glutathione (Fig. 22). As can be seen in this figure, there is a marked concentration dependence of the observed rate of loss of this chemical from the chamber. The initial decrease in chamber concentration in all of the experiments results from the uptake of chemical into the animal

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Fig. 22. Gas uptake experiment. Concentration (ppm) of methylene chloride in a closed, recirculated chamber containing three Fischer 344 rats. Initial chamber concentrations were (top to bottom) 3,000, 1,000, 500, and 100 ppm. Solid lines show the predictions of the model for a Vmax of 4.0 mg/h/kg, a Km of 0.3 mg/L, and a first-order rate constant of 2.0/h/kg, while symbols represent the measured chamber atmosphere concentrations.

tissues. Subsequent uptake is a function of the metabolic clearance in the animals, and the complex behavior reflects the transition from partially saturated metabolism at higher concentrations to linearity in the low concentration regime. The PBPK model is able to reproduce this complex behavior with a single set of parameters because the model structure appropriately captures the concentration dependence of the rate of metabolism. A similar analysis of gas uptake experiments with allyl chloride using the same model structure was less successful. The smooth curves shown in Fig. 23 are the best fit that could be obtained to the observed allyl chloride chamber concentration data assuming a saturable pathway and a first-order pathway with parameters that were independent of concentration. Using this model structure there were large systematic errors associated with the predicted curves. The model predictions for the highest initial concentration were uniformly lower than the data, while the predictions for the intermediate initial concentrations were uniformly higher than the data. A much better fit could be obtained by setting the first-order rate constant to a lower value at the higher concentration; this approach would provide a better correspondence between the data and the model predictions, but would not provide a basis for extrapolating to different exposure conditions. The nature of the discrepancy between the PBPK model and the data for allyl chloride suggested the presence of a dosedependent limitation on metabolism not included in the model structure. This indication was consistent with other experimental evidence indicating that the conjugative metabolism of allyl chloride depletes glutathione, a necessary cofactor for the linear

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Fig. 23. Model failure. Concentration (ppm) of allyl chloride in a closed, recirculated chamber containing three Fischer 344 rats. Initial chamber concentrations were (top to bottom) 5,000, 2,000, 1,000, and 500 ppm. Symbols represent the measured chamber atmosphere concentrations. The curves represent the best result that could be obtained from an attempt to fit all of the data with a single set of metabolic constants using the same closed chamber model structure as in Fig. 4.1.

conjugation pathway. The conjugation pathway for reaction of methylene chloride and glutathione regenerates glutathione, but in the case of allyl chloride glutathione is consumed by the conjugation reaction. Therefore, to adequately reflect the biological basis of the kinetic behavior, it was necessary to model the time dependence of hepatic glutathione. To accomplish this, the mathematical model of the closed chamber experiment was expanded to include a more complete description of the glutathione-dependent pathway. The expanded model structure used for this description (120) included a zero-order production of glutathione and a first-order consumption rate that was increased by reaction of the glutathione with allyl chloride; glutathione resynthesis was inversely related to the instantaneous glutathione concentration. This description provided a much improved correspondence between the data and predicted behavior (Fig. 24). Of course, the improvement in fit was obtained at the expense of adding several new glutathione-related parameters to the model. To ensure that the improved fit is not just a consequence of the additional parameters providing more freedom to the model for fitting the uptake data, a separate test of the hypothesis underlying the added model structure (depletion of glutathione) was necessary. Therefore the expanded model was also used to predict both allyl chloride and hepatic glutathione concentrations following constant concentration inhalation exposures. Model predictions for end-exposure hepatic glutathione concentrations compared very favorably with actual data obtained in separate experiments (Table 5).

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Table 5 Predicted glutathione depletion caused by inhalation exposure to allyl chloride Depletion (mM) Concentration (ppm)

Observed

0

7,080  120

7,088

10

7,290  130

6,998a

0

7,230  80

7,238a

100

5,660  90

5,939

0

7,340  180

7,341a

1,000 0 2,000

970  10 6,890  710 464  60

Predicted

839 6,890a 399

Note: Glutathione depletion data were graciously supplied by John Waechter, Dow Chemical Co., Midland, Michigan a For the purpose of this comparison, the basal glutathione consumption rate in the model was adjusted to obtain rough agreement with the controls in each experiment. This basal consumption rate was then used to simulate the associated exposure

Fig. 24. Cofactor depletion. Symbols represent the same experimental data as in Fig. 23. The curves show the predictions of the expanded model, which not only included depletion of glutathione by reaction with allyl chloride, but also provided for regulation of glutathione biosynthesis on the basis of the instantaneous glutathione concentration, as described in the text.

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To reiterate the key points of this example: 1. A PBPK model which had successfully described experimental results for a number of chemicals was unable to reproduce similar kinetic data on another chemical. 2. A hypothesis was developed that depletion of a necessary cofactor was affecting metabolism. This hypothesis was based on: (a) The nature of the discrepancy between the model predictions and the kinetic data. (b) Other available information about the nature of the chemical’s biochemical interactions. 3. The code for the PBPK model was altered to include additional mass balance equations describing the depletion of this cofactor, and its resynthesis, as well as the resulting impact on metabolism. 4. The modification to the model was then tested in two ways: (a) By testing the ability of the new model structure to simulate the kinetic data that the original model was unable to reproduce. (b) By testing the underlying hypothesis regarding cofactor depletion against experimental data on glutathione depletion from a separate experiment. Both elements of testing the model, kinetic validation and mechanistic validation, are necessary to provide confidence in the model. Unfortunately, there is a temptation to accept kinetic validation alone, particularly when data for mechanistic validation are unavailable. It should be remembered, however, that the simple act of adding equations and parameters to a model will, in itself, increase the flexibility of the model to fit data. Therefore, every attempt should be made to obtain additional experimental data to provide support for the mechanistic hypothesis underlying the model structure. References 1. Andersen ME (1981) Saturable metabolism and its relation to toxicity. Crit Rev Toxicol 9:105–150 2. Monro A (1992) What is an appropriate measure of exposure when testing drugs for carcinogenicity in rodents? Toxicol Appl Pharmacol 112:171–181 3. Andersen ME, Clewell HJ, Krishnan K (1995) Tissue dosimetry, pharmacokinetic modeling, and interspecies scaling factors. Risk Anal 15:533–537 4. Teorell T (1937) Kinetics of distribution of substances administered to the body. I. The

extravascular mode of administration. Arch Int Pharmacodyn 57:205–225 5. Teorell T (1937) Kinetics of distribution of substances administered to the body. I. The intravascular mode of administration. Arch Int Pharmacodyn 57:226–240 6. O’Flaherty EJ (1987) Modeling: an introduction. National Research Council. In: Pharmacokinetics in risk assessment. Drinking water and health, vol 8. National Academy Press, Washington DC, pp. 27–35 7. Clewell HJ, Andersen ME (1985) Risk assessment extrapolations and physiological modeling. Toxicol Ind Health 1(4):111–131

18 Physiologically Based Pharmacokinetic/Toxicokinetic Modeling 8. Andersen ME, Clewell HJ, Gargas ML, Smith FA, Reitz RH (1987) Physiologically based pharmacokinetics and the risk assessment for methylene chloride. Toxicol Appl Pharmacol 87:185–205 9. Gerrity TR, Henry CJ (1990) Principles of route-to-route extrapolation for risk assessment. Elsevier, New York 10. Clewell HJ, Jarnot BM (1994) Incorporation of pharmacokinetics in non-carcinogenic risk assessment: Example with chloropentafluorobenzene. Risk Anal 14:265–276 11. Clewell HJ (1995) Incorporating biological information in quantitative risk assessment: an example with methylene chloride. Toxicology 102:83–94 12. Clewell HJ (1995) The application of physiologically based pharmacokinetic modeling in human health risk assessment of hazardous substances. Toxicol Lett 79:207–217 13. Clewell HJ, Gentry PR, Gearhart JM, Allen BC, Andersen ME (1995) Considering pharmacokinetic and mechanistic information in cancer risk assessments for environmental contaminants: examples with vinyl chloride and trichloroethylene. Chemosphere 31:2561–2578 14. Clewell HJ, Andersen ME (1996) Use of physiologically-based pharmacokinetic modeling to investigate individual versus population risk. Toxicology 111:315–329 15. Clewell HJ III, Gentry PR, Gearhart JM (1997) Investigation of the potential impact of benchmark dose and pharmacokinetic modeling in noncancer risk assessment. J Toxicol Environ Health 52:475–515 16. Himmelstein KJ, Lutz RJ (1979) A review of the application of physiologically based pharmacokinetic modeling. J Pharmacokinet Biopharm 7:127–145 17. Gerlowski LE, Jain RK (1983) Physiologically based pharmacokinetic modeling: principles and applications. J Pharm Sci 72:1103–1126 18. Fiserova-Bergerova V (1983) Modeling of inhalation exposure to vapors: uptake distribution and elimination, vol 1 and 2. CRC, Boca Raton 19. Bischoff KB (1987) Physiologically based pharmacokinetic modeling. National Research Council. In: Pharmacokinetics in Risk Assessment. Drinking water and health, vol 8. National Academy Press, Washington, DC, pp. 36–61 20. Leung HW (1991) Development and utilization of physiologically based pharmacokinetic models for toxicological applications. J Toxicol Environ Health 32:247–267

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21. Clewell HJ, Andersen ME (1986) A multiple dose-route physiological pharmacokinetic model for volatile chemicals using ACSL/PC. In: Cellier FD (ed) Languages for continuous system simulation. Society for Computer Simulation, San Diego, pp 95–101 22. Ramsey JC, Andersen ME (1984) A physiological model for the inhalation pharmacokinetics of inhaled styrene monomer in rats and humans. Toxicol Appl Pharmacol 73:159–175 23. Adolph EF (1949) Quantitative relations in the physiological constitutions of mammals. Science 109:579–585 24. Dedrick RL (1973) Animal scale-up. J Pharmacokinet Biopharm 1:435–461 25. Dedrick RL, Bischoff KB (1980) Species similarities in pharmacokinetics. Fed Proc 39:54–59 26. McDougal JN, Jepson GW, Clewell HJ, MacNaughton MG, Andersen ME (1986) A physiological pharmacokinetic model for dermal absorption of vapors in the rat. Toxicol Appl Pharmacol 85:286–294 27. Paustenbach DJ, Clewell HJ, Gargas ML, Andersen ME (1988) A physiologically based pharmacokinetic model for inhaled carbon tetrachloride. Toxicol Appl Pharmacol 96:191–211 28. Vinegar A, Seckel CS, Pollard DL, Kinkead ER, Conolly RB, Andersen ME (1992) Polychlorotrifluoroethylene (PCTFE) oligomer pharmacokinetics in Fischer 344 rats: development of a physiologically based model. Fundam Appl Toxicol 18:504–514 29. Clewell HJ, Andersen ME (1989) Improving toxicology testing protocols using computer simulations. Toxicol Lett 49:139–158 30. Bischoff KB, Brown RG (1966) Drug distribution in mammals. Chem Eng Prog Symp 62 (66):33–45 31. Astrand P, Rodahl K (1970) Textbook of work physiology. McGraw-Hill, New York 32. International Commission on Radiological Protection (ICRP) (1975) Report of the task group on reference man. ICRP Publication 23 33. Environmental Protection Agency (EPA) (1988) Reference physiological parameters in pharmacokinetic modeling. EPA/600/6-88/ 004. Office of Health and Environmental Assessment, Washington, DC 34. Davies B, Morris T (1993) Physiological parameters in laboratory animals and humans. Pharm Res 10:1093–1095 35. Brown RP, Delp MD, Lindstedt SL, Rhomberg LR, Beliles RP (1997) Physiological parameter values for physiologically based

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