Frédéric Y Bois
email: frederic.bois@utc.fr

Masoud Jamei
email: m.jamei@simcyp.com

Harvey J Clewell
email: hclewell@thehamner.org

Yves Frédéric

Bois

PBPK MODELLING OF INTER-INDIVIDUAL VARIABILITY IN THE PHARMACOKINETICS OF ENVIRONMENTAL CHEMICALS

Keywords: Advanced Dissolution, Absorption and Metabolism model ADME: Absorption, distribution, metabolism, excretion AUC Drug-drug interactions, Markov chain Monte Carlo, Monte Carlo simulations, PBPK models, Pharmacokinetics, Population models, Susceptible populations, Toxic interactions, Toxicokinetics, Variably assessment

Generic PBPK models, applicable to a large number of substances, coupled to parameter databases and QSAR modules, are now available for predictive modelling of inter-individual variability in the absorption, distribution, metabolism and excretion of environmental chemicals.

When needed, Markov chain Monte Carlo methods and multilevel population models can be jointly used for a Bayesian calibration of a PBPK model, to improve our understanding of the determinants of population heterogeneity and differential susceptibility. This article reviews those developments and illustrates them with recent applications to environmentally relevant questions.

INTRODUCTION

PBPK models are now commonly used in drug development and regulatory toxicology to predict the kinetics and metabolism of substances in the body, with a focus on the effective dose at the expected target site [START_REF] Barton | Characterizing uncertainty and variability in physiologically-based pharmacokinetic (PBPK) models: state of the science and needs for research and implementation[END_REF][START_REF] Bouvier D'yvoire | Physiologically-based kinetic modelling (PBK modelling): meeting the 3Rs agenda -The report and recommendations of ECVAM Workshop 63a[END_REF][START_REF] Edginton | Whole body physiologicallybased pharmacokinetic models: their use in clinical drug development[END_REF][START_REF] Loizou | Development of good modelling practice for physiologically based pharmacokinetic models for use in risk assessment: the first steps[END_REF].

The physiological basis of PBPK models makes them especially suited to explore, understand and predict the determinants of inter-or intra-individual variability in pharmacokinetics. Those translate into variability of target doses and can have direct consequences for therapeutic safety and the likelihood of toxicity, especially for compounds with narrow therapeutic windows or a steep dose-response for toxicity. Therefore, simulation of inter-individual variability has become an integral part of the assessment of pharmacokinetics in humans [START_REF] Bois | Applications of population approaches in toxicology[END_REF][START_REF] Rostami-Hodjegan | Simulation and prediction of in vivo drug metabolism in human populations from in vitro data[END_REF]. The mechanistic framework of PBPK models provides the capacity of predicting inter-individual variability in pharmacokinetics when the required information is adequately incorporated. This short review presents the state of the art on this question and illustrates the approach with two recent applications of the Simcyp software to environmentally relevant questions.

PBPK MODELING

When a chemical substance penetrates an animal body (following intentional administration or unintentional exposure), it is usually distributed to various tissues and organs by blood flow [START_REF] Gerlowski | Physiologically based pharmacokinetic modeling: principles and applications[END_REF][START_REF] Nestorov | Whole-body physiologically based pharmacokinetic models[END_REF]. Following its distribution to tissues, the substance can bind to various proteins and receptors, undergo metabolism, or can be eliminated unchanged.

The concentration versus time profiles of the xenobiotic in different tissues, or the amount of metabolites formed, are often used as surrogate markers of its internal dose or biological activity [START_REF] Andersen | What do we mean by[END_REF].

Mathematical models can be used to interpolate and extrapolate (predict) such concentrationtime profiles from data. Reported models range from simple compartmental [START_REF] Gibaldi | Pharmacokinetics[END_REF] to very sophisticated (Jamei et al. 2009a;[START_REF] Luecke | Windows® based general PBPK/PD modeling software[END_REF]. PBPK models are evolved compartmental models which tend to use realistic biological descriptions of the determinants that regulate the disposition of drugs in the body [START_REF] Andersen | Conclusions and future directions[END_REF]. Those models describe the body as a set of compartments corresponding to specific organs or tissues (e.g., adipose, bone, brain, gut, heart, kidney, liver, lung, muscle, skin, and spleen, etc.). Between compartments, the transport of substances is dictated by various physiological flows (blood, bile, pulmonary ventilation, etc.) or by diffusion [START_REF] Bois | An analysis of exposure rate effects for benzene using a physiologically based pharmacokinetic model[END_REF][START_REF] Gerlowski | Physiologically based pharmacokinetic modeling: principles and applications[END_REF]. Perfusionrate-limited kinetics applies when the tissue membrane presents no barrier to distribution.

Generally, this condition is likely to be met by small lipophilic substances. In contrast, permeability-rate kinetics applies when the distribution of the substance to a tissue is rate-limited by the drug's permeability across the tissue membrane. That condition is more common with polar compounds and large molecular structures. Consequently, the related PBPK models may exhibit different degrees of complexity. In the simplest and most commonly applied form (see Figure 1), each tissue is considered to be a well-stirred compartment in which the substance distribution is simply limited by blood flow. In such a model, any of the tissues can be a site of elimination. However, in Figure 1, it is assumed that the gut, liver and kidney are the only metabolising tissues and that excretion only happens in the kidney.

Building a PBPK model requires gathering a considerable amount of data which can be categorised in three groups: namely, the system's data (physiological, anatomical, biochemical data); drug-specific data; and the model structure, which refers to the arrangement of tissues and organs included in the model [START_REF] Rowland | Physiologically based pharmacokinetics in drug development and regulatory science: a workshop report[END_REF]). In a sense, PBPK modelling is an integrated systems approach to both understanding the pharmacokinetic behaviour of compounds and predicting concentration-time profiles in plasma and tissues. Additional details on PBPK modelling can be found elsewhere [START_REF] Gerlowski | Physiologically based pharmacokinetic modeling: principles and applications[END_REF][START_REF] Nestorov | Whole body pharmacokinetic models[END_REF][START_REF] Rowland | Physiologically based pharmacokinetics in drug development and regulatory science: a workshop report[END_REF]. Indeed, such a description of the body is approximate, if not rough, but a balance has to be found between precision (which implies complexity) and simplicity (for ease of use). Yet, the generic structure of a PBPK facilitates its application to any mammalian species as long as the related system data are used. Therefore, the same structural model can approximately be used for a human, a rat or a mouse.

The model structure can be described by a set of differential equations, with parameters representing blood flow rates, organ volumes etc., for which information is available in the published scientific literature or may be obtained in vitro [START_REF] Parrott | Application of full physiological models for pharmaceutical drug candidate selection and extrapolation of pharmacokinetics to man[END_REF]Woodruff and Bois 1993).

Numerical integration of that differential system computes the quantity and concentration of the drug considered in each compartment, as a function of time and exposure dose.

VARIABILITY VS. UNCERTAINTY

Inter-individual differences ("variability"), in the anatomical and physiological characteristics of humans or animal are glaring and a common experience. Such differences affect, for example, organ volumes and blood flow and translate quite naturally into variability in the pharmacokinetics of drugs and chemical substances from one individual to the next. Since pharmacokinetics determine, in part, effective dose and ensuing effects, pharmacokinetic variability has an impact on individual susceptibility. Two major, complementary, modelling approaches have been developed to understand, evaluate and predict that variability: A priori and a posteriori modelling.

A priori (bottom-up), purely predictive, modelling of variability can be undertaken either through deterministic descriptions of the determinants of variability or through stochastic simulations (Monte Carlo methods) [START_REF] Bois | Precision and sensitivity analysis of pharmacokinetic models for cancer risk assessment: tetrachloroethylene in mice, rats and humans[END_REF][START_REF] Clewell | Use of physiologically based pharmacokinetic modeling to investigate individual versus population risk[END_REF][START_REF] Clewell | Evaluation of the potential impact of age-and gender-specific pharmacokinetic differences on tissue dosimetry[END_REF][START_REF] Rostami-Hodjegan | Simulation and prediction of in vivo drug metabolism in human populations from in vitro data[END_REF][START_REF] Willmann | Whole-body physiologically-based pharmacokinetic population modelling of oral drug administration: inter-individual variability of cimetidine absorption[END_REF][START_REF] Willmann | Development of a physiology-based whole-body population model for assessing the influence of individual variability on the pharmacokinetics of drugs[END_REF]). Both approaches can also be combined. Deterministic modelling motivations stem, partly, from the fact that some of the differences between individuals are due to growth and other age-related changes, or sex, etc. Such changes can be explicitly modelled in life-time PBPK models [START_REF] Clewell | Evaluation of the potential impact of age-and gender-specific pharmacokinetic differences on tissue dosimetry[END_REF][START_REF] Edginton | Development and evaluation of a generic physiologically based pharmacokinetic model for children[END_REF]. Monte Carlo methods acknowledge (some may say: hide intellectual laziness behind) the fact that part of the differences between individuals seem to occur by chance or cannot be ascribed simply to easily modelled determinants such as age etc. In that case, those differences are considered as purely random and parameter values are described by statistical distributions, rather than point estimates or simple functions of time, sex or other covariates.

A posteriori (top-down) modelling of inter-individual variability is data-based, and was originally developed in pharmacokinetics, where human data were routinely generated through clinical trials. The aim here is to assess, for example, how across-subject variability of blood concentration for a substance can be explained by variability in renal excretion rate, bioavailability, metabolic rate, etc. That is best achieved through inverse simulation in the framework of multilevel statistical models ("population pharmacokinetic models") [START_REF] Bois | Applications of population approaches in toxicology[END_REF][START_REF] Sheiner | The population approach to pharmacokinetic data analysis: rationale and standard data analysis methods[END_REF]. That approach was first used with classical (minimal) compartmental models (Beal andSheiner 1980, 1982), but the added ingredients of MCMC simulations and Bayesian inference make it amenable to the treatment of PBPK models (Bois et al. 1996a;Bois et al. 1996b;Gelman et al. 1996). After unrolling the determinants of variability, forward, predictive, simulations can be performed to assess their impact in various exposure and risk or clinical treatment scenarios.

Uncertainty is different from variability, although their effects may be confounded and compounded. Uncertainty is essentially due to lack of knowledge and may have various sources.

For example, measurements are made only with finite precision; so when data are used to fit toxicokinetic or toxicodynamic models, there is always some uncertainty ("noise") about the estimated parameters values. Also, studying a limited population sample introduces an element of randomness when attempting to extrapolate the results to the whole population. Finally, our inability to describe or model precisely a system may result from our lack of understanding and simplifications or misspecifications of the fitted models translate into parametric uncertainty [START_REF] Bois | Applications of population approaches in toxicology[END_REF][START_REF] Bois | Uncertainty Analysis: The Bayesian Approach[END_REF]. While variability is by essence irreducible and unavoidable, since it is a state of Nature, uncertainty can be reduced by new experiments, or by a better understanding of the biology and better models. In a further twist, the fact that uncertainty is almost always present hampers a precise assessment of variability. Statistical "population" models are able to disentangle partly uncertainty from variability. These models will be reviewed in section 7, below.

A PRIORI PREDICTION OF VARIABILITY USING PBPK MODELING

Predictive variability assessment with PBPK models proceeds nowadays with a combination of mechanism and data based refinements of the model structure (e.g., using age-dependent compartment volumes) [START_REF] Clewell | Evaluation of the potential impact of age-and gender-specific pharmacokinetic differences on tissue dosimetry[END_REF][START_REF] Luecke | Windows® based general PBPK/PD modeling software[END_REF]) and stochastic modelling of the remaining random components of inter-individual differences. Numerical Monte Carlo methods [START_REF] Clewell | Evaluation of the uncertainty in an oral Reference Dose for methylmercury due to interindividual variability in pharmacokinetics[END_REF][START_REF] Price | Modeling interindividual variation in physiological factors used in PBPK models of humans[END_REF][START_REF] Spear | Modeling benzene pharmacokinetics across three sets of animal data: parametric sensitivity and risk implications[END_REF][START_REF] Woodruff | Structure and parametrization of toxicokinetic models: their impact on model predictions[END_REF], eventually hierarchical [START_REF] Bois | Analysis of PBPK models for risk characterization[END_REF]Bois et al. 1996a;Bois et al. 1996b), are the most widely used for the latter task. Those are best suited for complex models with many parameters. The basic assumption made by Monte Carlo simulations is that the randomness of the model state variables (e.g., blood concentration) is simply due to the randomness in the model parameter values. It is then enough to define statistical distributions for the parameters supposed to vary randomly in the population, and to sample the model parameter values from those distributions. The model is then run with the needed inputs and its outputs of interest are recorded. That sampling-running step constitutes a basic Monte Carlo iteration. As many iterations as needed can be performed to characterise precisely the statistical distribution of the recorded outputs. Additional details on Monte Carlo methods can be found in [START_REF] Ripley | Stochastic Simulation[END_REF].

Early attempts to use Monte Carlo methods and to simulate pharmacokinetic behaviour in "virtual populations" date back to the mid-1980s. Jackson et al. assessed the robustness of different experimental in vivo indices to detect and display genetic polymorphisms in human drugmetabolising activity [START_REF] Jackson | Pharmacokinetic-pharmacogenetic modelling in the detection of polymorphisms in xenobiotic metabolism[END_REF][START_REF] Jackson | Polymorphic drug oxidation: pharmacokinetic basis and comparison of experimental indices[END_REF]. Monte Carlo analyses of PBPK models were also described at that time [START_REF] Bois | Precision and sensitivity analysis of pharmacokinetic models for cancer risk assessment: tetrachloroethylene in mice, rats and humans[END_REF][START_REF] Portier | Variability of safe dose estimates when using complicated models of the carcinogenic process[END_REF]. These simulations were later expanded to show the effect of variability in absorption, distribution, metabolism and excretion (ADME) parameters on the power of single time point estimates for the assessment of metabolic activity [START_REF] Jackson | Backtracking booze with Bayes--the retrospective interpretation of blood alcohol data[END_REF], power analyses of bioequivalence measures (Bois et al. 1994a, b), the differentiation of parent drug and metabolite data in bioequivalence assessment [START_REF] Rostami-Hodjegan | Sensitivity of indirect metrics for assessing "rate" in bioequivalence studies--moving the "goalposts" or changing the "game[END_REF], the discriminatory power of different indices of in vivo enzyme activity and the optimisation of sampling to assess such activity [START_REF] Rostami-Hodjegan | Caffeine urinary metabolite ratios as markers of enzyme activity: a theoretical assessment[END_REF]. Coupled with Monte Carlo methods, PBPK modelling has been used to assess the quantitative impact of physiological and environmental factors on human variability in toxicokinetics and pharmacokinetics in other publications [START_REF] Bois | Comparison of three physiologically-based pharmacokinetic models of benzene disposition[END_REF][START_REF] Clewell | Use of physiologically based pharmacokinetic modeling to investigate individual versus population risk[END_REF]Jamei et al. 2009a;[START_REF] Nestorov | Modelling and simulation of variability and uncertainty in toxicokinetics and pharmacokinetics[END_REF][START_REF] Sato | The effect of environmental factors on the pharmacokinetic behaviour of organic solvent vapours[END_REF].

SOURCES OF VARIABILITY IN PHARMACOKINETICS

The overall inter-individual variability in pharmacokinetics can be simulated by considering the variability in key system parameters in PBPK modelling (Jamei et al. 2009a) (see Figure 2 for examples of clearance covariates). Details on the prediction of inter-individual variability in pharmacokinetics are discussed in detail below.

ABSORPTION

There are many routes for xenobiotics to enter into the body which can generally be divided into topical, enteral and parenteral categories. Some of these routes (oral, infusion, intravenous, intramuscular, transdermal and inhalation) are routinely used to administrate drugs. The oral, transdermal and inhalation routes, which are most relevant to environmental exposures, are briefly discussed in this section.

Oral administration is the most common and convenient route of drug administration. Yet, it is often associated with low bioavailability and high inter-individual variability. Oral bioavailability (F oral ) is defined as:

H G a oral F F f F (1)
where f a is the net fraction of dose absorbed from the intestinal tract, F G is the fraction of dose that escapes intestinal first-pass metabolism in the enterocytes, and F H is the fraction of dose that escapes hepatic first-pass metabolism. f a is discussed in this section and F G and F H will be discussed later.

Various factors can affect oral drug absorption. They can be divided into two categories: system (physiological and biological) factors and drug-related (physicochemical and pharmaceutical) factors. These factors can all contribute to the overall rate and extent of absorption, but mainly the system factors determine the inter-individual variability in absorption.

Oral Absorption

Key physiochemical properties, such as solubility and permeability are used in empirical methods to estimate the absorption potential of a drug [START_REF] Artursson | Correlation between oral drug absorption in humans and apparent drug permeability coefficients in human intestinal epithelial (Caco-2) cells[END_REF][START_REF] Dressman | Absorption potential: estimating the fraction absorbed for orally administered compounds[END_REF]. Physiological models, such as the CAT model, have been developed to simulate mechanistically drug absorption [START_REF] Yu | A compartmental absorption and transit model for estimating oral drug absorption[END_REF]. The CAT model has been further developed into the ACAT [START_REF] Agoram | Predicting the impact of physiological and biochemical processes on oral drug bioavailability[END_REF]) and the ADAM (Jamei et al. 2009b) models, which are respectively implemented within software Gastroplus (Simulation Plus Inc, California, USA, http://www.simulationsplus.com) and Simcyp Population-based ADME Simulator (Simcyp Ltd, Sheffield, UK, http://www.simcyp.com). In brief, these absorption models consist of physiologically based compartments corresponding to different segments of the gastrointestinal tract.

Gastric Emptying Time

The residence time of a drug in the stomach is an important factor determining the initiation of oral drug absorption. Variability in gastric emptying rates results in variable absorption rates and sometimes even variable absorption extents.

Intestinal Residence Time

Intestinal residence time can greatly affect oral drug absorption; particularly for drugs with low permeability. Yu and co-workers analyzed published data for low intestinal residence time in over 400 subjects and reported a mean value of 199 min with ranges of about 1 and 6 hours [START_REF] Yu | Compartmental transit and dispersion model analysis of small intestinal transit flow in humans[END_REF]. Intestinal residence time appears to be relatively less dependent upon the nature of the dosage form (liquid vs. solid) than gastric residence time [START_REF] Davis | Transit of pharmaceutical dosage forms through the small intestine[END_REF]). The presence of food appears not to influence intestinal transit [START_REF] Davis | Transit of pharmaceutical dosage forms through the small intestine[END_REF][START_REF] Fadda | Meal-Induced Acceleration of Tablet Transit Through the Human Small Intestine[END_REF].

Gastrointestinal pH

The regional pH in the gastrointestinal tract can influence drug solubility and hence the dissolution of solid dosage forms. Gastrointestinal pH may also affect drug permeability by influencing the balance between ionised and non-ionised moieties. Fallingborg and co-workers measured pH profiles along the gastrointestinal tract in 39 healthy volunteers and observed a range of values up to two pH units at the same site in different subjects [START_REF] Fallingborg | pH-profile and regional transit times of the normal gut measured by a radiotelemetry device[END_REF]).

In addition, the presence of food in the gastrointestinal tract can raise the pH in the stomach and the proximal part of the small intestine, due to the buffering capacity of proteins.

Transporters

Various transporters are expressed in the apical and basolateral membranes of intestinal epithelial cells [START_REF] Hilgendorf | Expression of thirty-six drug transporter genes in human intestine, liver, kidney, and organotypic cell lines[END_REF][START_REF] Koepsell | Organic cation transporters in intestine, kidney, liver, and brain[END_REF][START_REF] Koepsell | Polyspecific organic cation transporters: structure, function, physiological roles, and biopharmaceutical implications[END_REF][START_REF] Murakami | Intestinal efflux transporters and drug absorption[END_REF][START_REF] Tsuji | Carrier-mediated intestinal transport of drugs[END_REF]. Much attention has been given to the efflux transporters (e.g., P-gp, MDR1 (ABCB1), MRP2 (ABCC2) and BCRP (ABCG2)) at the apical (brushborder) membrane of the intestine, as they can limit the intestinal absorption of drugs administered orally. Von Richter et al. ( 2004) measured P-gp in the human small intestine and reported a marked inter-individual variation in the intestinal P-gp expression. Additional data were reported by several later publications [START_REF] Canaparo | Expression of CYP3A isoforms and P-glycoprotein in human stomach, jejunum and ileum[END_REF][START_REF] Mouly | P-glycoprotein increases from proximal to distal regions of human small intestine[END_REF].

Available data demonstrate that the expression levels of transporters vary along the gastrointestinal tract. [START_REF] Mouly | P-glycoprotein increases from proximal to distal regions of human small intestine[END_REF] found that relative P-gp levels increase progressively from the proximal to distal region of the small intestine. Other intestinal transporters may or may not follow the same pattern.

Inhalation

The large absorptive surface area, limited metabolic enzyme activity and active transporters in the pulmonary system make inhalation a favourable delivery strategy for systemic drugs with low bioavailability. In addition, many chemicals present in the air can get into body via the respiratory system. As for other routes of administration, the absorption kinetics in the lung tissues depend on both drug and system related parameters. Although inhalation is an established delivery strategy, the relationship between drug physicochemical properties and drug absorption kinetics in the lung has not been extensively investigated. In contrast to oral drug absorption little attention has been devoted to the question of how to optimize local drug absorption and retention in the lung [START_REF] Yu | Cell-Based multiscale computational modeling of small molecule absorption and retention in the lungs[END_REF]. The state of the art on this question is somewhat more advanced for occupational exposures to volatile substances [START_REF] Johanson | Modelling of respiratory exchange of polar solvents[END_REF][START_REF] Löf | Toxicokinetics of organic solvents: a review of modifying factors[END_REF]. Different aspects of lung physiology and formulation composition that influence the systemic delivery of inhaled therapeutics and recent advancement in inhaled drug delivery are reviewed in [START_REF] Patton | Inhaling medicines: delivering drugs to the body through the lungs[END_REF].

Dermal Absorption

The skin, the largest interface between the body and the environment, protects our body against chemical, physical, and microbial injury, loss of water, and other endogenous substances. It is also involved in the thermoregulation of the body and serves as an excretory organ [START_REF] Schaefer | Models for skin absorption and skin toxicity testing[END_REF]. Understanding skin absorption processes enables us to assess the safety aspects of chemicals, xenobiotics, and cosmetic formulations as well as optimally utilizing dermal drug delivery. Permeation of drug molecules across the skin occurs by passive diffusion according to the activity gradient [START_REF] Cleek | A new method for estimating dermal absorption from chemical exposure. 1. General approach[END_REF][START_REF] Krüse | Analysis, interpretation, and extrapolation of dermal permeation data using diffusion-based mathematical models[END_REF][START_REF] Mccarley | Pharmacokinetic models of dermal absorption[END_REF][START_REF] Potts | Predicting Skin Permeability[END_REF]. The outer skin layer, stratum corneum, forms a rate-controlling barrier for diffusion of most compounds. The predominant diffusional path for a molecule crossing the stratum corneum appears to be intercellular [START_REF] Hadgraft | Feasibility assessment in topical and transdermal delivery: mathematical models and in vitro studies[END_REF]. However this path is not exclusive and probably most molecules will pass through the stratum corneum by a combination of intercellular lipid domains, transcellular route and via the appendages (hair follicles, etc.) (Farahmand and Maibach 2009b).

Farahmand et al. investigated dermato-pharmacokinetic parameters of 12 transdermal patches and concluded that the serum concentration profile for transdermal therapeutic systems was affected by the physiological parameters, drug absorption and elimination. Therefore, in order to understand the variability in serum concentration it is necessary to take into account variability of each process involved (Farahmand and Maibach 2009a).

DISTRIBUTION

The distribution process refers to the reversible transfer of drug from one location to another within the body. Factors that determine the distribution pattern of a drug with time include delivery of drug to tissue by blood, ability to cross tissue membranes, binding within blood and tissues, partitioning into fat, and tissue uptake [START_REF] Rowland | Clinical pharmacokinetics: concepts and applications[END_REF]. The traditional description of the volume of distribution at steady state (V ss ) corresponds to the sum of the products of each tissue to plasma partition coefficient (K p:t ) and the respective tissue volume in addition to the plasma volume [START_REF] Sawada | Prediction of the volumes of distribution of basic drugs in humans based on data from animals[END_REF]:

t p t e p ss K V P E V V V : ) : ( (2)
where V p , V e and V t are the volumes of plasma, erythrocyte and tissue, respectively, E:P is erythrocyte-to-plasma coefficient. Physiological factors affecting drug distribution include tissue volumes, tissue composition, blood perfusion rates to the tissues, plasma protein concentrations, hematocrit, and the expression of transporter proteins. Drug-specific factors determining the distribution behaviour of a drug include its ionisation, ability to cross membranes, bind to plasma proteins, partition into red blood cells and fat, and its specific affinity to influx or efflux transporter proteins.

Direct determination of K p:t usually involves intravenous constant infusions to animals followed by an extraction and quantification of drugs from tissue homogenates (Lin et al. 1982a, b;[START_REF] Sawada | Prediction of the volumes of distribution of basic drugs in humans based on data from animals[END_REF], which is costly and time consuming. It is, therefore, of interest to predict K p:t values without conducting in vivo animal studies.

Several mechanistic equations have been proposed to predict the tissue affinities of volatile organic compounds [START_REF] Fiserova-Bergerova | Gases and their solubility: a review of fundamentals[END_REF][START_REF] Fiserova-Bergerova | Determination and prediction of tissue-gas partition coefficients[END_REF][START_REF] Fiserova-Bergerova | Predictable "individual differences" in uptake and excretion of gases and lipid soluble vapour -simulation study[END_REF]). Poulin and co-workers also developed mechanistic equations to predict the affinity of drugs for various tissues and organs [START_REF] Poulin | Prediction of adipose tissue: plasma partition coefficients for structurally unrelated drugs[END_REF][START_REF] Poulin | A priori prediction of tissue:plasma partition coefficients of drugs to facilitate the use of physiologically-based pharmacokinetic models in drug discovery[END_REF] and subsequently V ss predictions (Poulin and Theil 2002a) using Eq. 2. The species-specific tissue composition parameters can be found in the literature and have been summarised by Poulin and Theil (2002b). Corrections on these equations were later made by [START_REF] Berezhkovskiy | Volume of distribution at steady state for a linear pharmacokinetic system with peripheral elimination[END_REF]. More recently, Rodgers and co-workers extended and improved these equations by considering drug ionisation and incorporating more details on drug distribution inside tissues [START_REF] Rodgers | Physiologically based pharmacokinetic modeling 1: Predicting the tissue distribution of moderate-to-strong bases[END_REF][START_REF] Rodgers | Physiologically based pharmacokinetic modelling 2: predicting the tissue distribution of acids, very weak bases, neutrals and zwitterions[END_REF]. Briefly, those equations are based on the assumption that all drugs will dissolve in intra-and extracellular tissue water and partition into the neutral lipids and neutral phospholipids located within tissue cells.

Tissue Volumes and Tissue Blood Flows

Tissue volumes and blood flows are essential components of a PBPK model. Early publications reported representative physiological parameters values but did not indicate the biological variability associated with those data [START_REF] Davies | Physiological parameters in laboratory animals and humans[END_REF][START_REF] Williams | Reference values for resting blood flow to organs of man[END_REF].

Inter-individual variability on tissue volumes and tissue blood flows has been reported by later publications (de la [START_REF] De La Grandmaison | Organ weight in 684 adult autopsies: new tables for a Caucasoid population[END_REF][START_REF] Price | Modeling interindividual variation in physiological factors used in PBPK models of humans[END_REF].

Tissue Composition

Hematocrit refers to the percentage of total blood volume composed of red blood cells. It is influenced by factors including age, sex, seasonal influence, and habits of physical activity (Morse et al. 1947a;Morse et al. 1947b;[START_REF] Thirup | Haematocrit: within-subject and seasonal variation[END_REF]. Compared with men, women on average have lower hematocrit. Hematocrit ranges between 40%-54% in males and 38%-47% in females.

Drug protein binding is the reversible interaction of drugs with plasma proteins. The extent of protein binding is a function of drug and protein concentrations, the affinity constant for the drug-protein binding and the number of protein binding sites [START_REF] Grandison | Age-related changes in protein binding of drugs: implications for therapy[END_REF].

The major drug binding proteins in plasma are: albumin, AAG and lipoproteins. Albumin levels are generally decreased with age, whereas AAG levels are not significantly affected by age.

Transporters

Numerous drug transporters are found on the membranes of various tissues. These transporters can influence drug distribution into the tissues, particularly for drugs with low passive permeability. There is now increasing evidence to suggest that transporters may affect the volumes of distribution of certain drugs [START_REF] Grover | Effects of Drug Transporters on Volume of Distribution[END_REF]. For most drugs, however, transporters may not significantly influence the volume of distribution, but may still influence the local kinetics in certain tissues (e.g., brain, liver, etc) and cause pharmacological or toxicological consequences. Polymorphism has been identified in transporters, as reviewed by [START_REF] Ho | Transporters and drug therapy: implications for drug disposition and disease[END_REF].

METABOLISM

Drug metabolism reactions are generally grouped into 2 phases. Phase I metabolism includes oxidation, reduction, hydrolysis and hydration reactions. Phase II reactions use an endogenous compound, such as glucuronic acid, glutathione, or sulphate, for conjugation to the drug or its phase I-derived metabolite to produce a more polar end product that can be more readily excreted in bile or urine.

Although drug metabolism can take place in many organs, the liver has been long-recognised as the major site of metabolism for most drugs. More recently, the role of gut metabolism in firstpass metabolism has been increasingly recognised. The intestinal tissue is endowed with phase I and II enzymes, although at lower levels than those for the liver [START_REF] Pang | Modeling of intestinal drug absorption: roles of transporters and metabolic enzymes (for the Gillette Review Series)[END_REF]. Several CYP enzymes have been detected in the human small intestine, including CYP1A2, CYP2D6, CYP2E1, CYP2C8, CYP2C9, CYP2C19, CYP3A4, and CYP3A5 [START_REF] Paine | The human intestinal cytochrome P450 "Pie[END_REF]). Among them, CYP3A4 is the most prominent enzyme present in the human intestine [START_REF] Paine | The human intestinal cytochrome P450 "Pie[END_REF][START_REF] Paine | Characterization of interintestinal and intraintestinal variations in human CYP3A-dependent metabolism[END_REF]. [START_REF] Rane | Prediction of hepatic extraction ratio from in vitro measurement of intrinsic clearance[END_REF] successfully predicted in vivo hepatic metabolic clearance in rats based on in vitro data obtained from rat liver microsomes, taking into consideration the hepatic blood flow rate and the unbound fraction in blood. Since then, significant progresses have been achieved on predicting human hepatic metabolic clearance from a variety of in vitro systems, such as human liver microsomes, recombinant enzymes, and hepatocytes [START_REF] Galetin | Utility of recombinant enzyme kinetics in prediction of human clearance: impact of variability, CYP3A5, and CYP2C19 on CYP3A4 probe substrates[END_REF][START_REF] Houston | Utility of in vitro drug metabolism data in predicting in vivo metabolic clearance[END_REF][START_REF] Howgate | Prediction of in vivo drug clearance from in vitro data. I: impact of inter-individual variability[END_REF][START_REF] Iwatsubo | Prediction of species differences (rats, dogs, humans) in the in vivo metabolic clearance of YM796 by the liver from in vitro data[END_REF][START_REF] Obach | Prediction of human clearance of twenty-nine drugs from hepatic microsomal intrinsic clearance data: An examination of in vitro half-life approach and nonspecific binding to microsomes[END_REF][START_REF] Riley | A unified model for predicting human hepatic, metabolic clearance from in vitro intrinsic clearance data in hepatocytes and microsomes[END_REF].

Hepatic Metabolism

The unbound total hepatic intrinsic clearance (CLu int,H ) can be extrapolated from in vitro clearance determined in a variety of in vitro systems using scaling factors as described in Barter ( 2007) and according to the procedure described by Rostami-Hodjegan and Tucker (2007):

Recombinantly expressed enzymes:

weight Liver MPPGL rhEnZ Km abundance Enz rhEnz V ISEF n j n i i i i i i ji _ ) ( ) ( max CLu 1 1 H int, (3)
where there are i metabolic pathways for each of j enzymes; 'rh' indicates recombinantly expressed enzyme; Vmax is the maximum rate of metabolism by an individual enzyme; Km is the Michaelis constant; MPPGL is the amount of microsomal protein per gram of liver; and ISEF is a scaling factor that compensates for any difference in the activity per unit of enzyme between recombinant systems and hepatic enzymes [START_REF] Proctor | Predicting drug clearance from recombinantly expressed CYPs: intersystem extrapolation factors[END_REF] (5) where HPGL refers to hepatocellularity (millions of hepatocytes per gram of liver).

CLu int,H is then combined with other determinants to obtain total hepatic intrinsic clearance, using a liver model. Several hepatic clearance models have been developed to quantify the effects of hepatic blood flow, fraction unbound in blood, and hepatic intrinsic clearance on hepatic clearance [START_REF] Wilkinson | Clearance approaches in pharmacology[END_REF]). Among these models the well-stirred model (Eqs. 6 and 7) has been widely used for its mathematical simplicity and practicality, as shown below:

,H B H,B ,H B H,B H,B CLu fu Q CLu fu Q CL int int (6) ,H B H,B H,B H CLu fu Q Q F int (7)
where CL H,B is hepatic drug clearance based on whole blood drug concentration, Q H,B is hepatic blood flow, fu B is the free fraction of drug in blood.

As indicated by Eq. 6, inter-individual variability in CL H,B is influenced by the variability in three key parameters: Q H,B , fu B and CLu int,H .

Gut Metabolism

An operational model has been developed to predict first-pass metabolism in the gut. The "Q Gut " model (Eq. 8) [START_REF] Rostami-Hodjegan | The effects of portal shunts on intestinal cytochrome P450 3A activity[END_REF][START_REF] Yang | Prediction of Intestinal First-Pass Drug Metabolism[END_REF][START_REF] Yang | Prediction of ketoconazole interaction with midazolam, alprazolam and triazolam: incorporating population variability[END_REF]) retains the form of the "well-stirred" model but the flow term (Q Gut ) is a hybrid of both permeability through the enterocyte membrane and villous blood flow.

,G G Gut Gut G CLu fu Q Q F int (8)
where F G is intestinal availability (fraction of dose that escapes intestinal first-pass metabolism in the enterocytes), fu G is the fraction of drug unbound in the enterocyte and its value is close to is a hybrid of both permeability through the enterocyte membrane and villous blood flow:

perm villi perm villi Gut CL Q CL Q Q (9)
where CL perm is a clearance term defining permeability through the enterocyte and Q villi is villous blood flow [START_REF] Yang | Prediction of Intestinal First-Pass Drug Metabolism[END_REF].

EXCRETION

Excretion is the irreversible loss of the chemically unchanged drug. For most drugs, excretion occurs predominantly via the kidneys. However, some drugs and their metabolites are extensively excreted via the bile. Drug excretion can also happen via saliva, sweat, breast milk, and lungs, although their contributions to overall drug elimination are often small.

Renal Excretion

The kidney is the major site of drug excretion. Net renal drug excretion is a combination of three processesglomerular filtration, tubular secretion and tubular reabsorption, as described by Eq. 10. Glomerular filtration of a drug is a passive process that is dependent upon the unbound fraction of a drug in plasma (fu) and renal blood flow available for filtration, as described by Eq. 11. Tubular secretion occurs predominantly in the proximal tubule, and is mediated by several families of transporters. Tubular reabsorption of a drug can be a passive or an active transport process. Passive reabsorption may occur throughout the nephron. Active reabsorption occurs in the proximal tubule and, similar to tubular secretion, is energy-dependent, saturable, stereospecific and also likely to be associated with competitive drug interactions [START_REF] Tett | Principles and clinical application of assessing alterations in renal elimination pathways[END_REF].

Rate of Excretion = (Rate of Filtration + Rate of Active Secretion

)(1 -F R ) (10) Rate of Filtration = fu × GFR × C (11)
where F R is the fraction of drug reabsorbed from tubule lumen; fu is the fraction unbound in plasma; GFR is the glomerular filtration rate; and C is drug concentration in plasma.

The primary determinants of renal excretion include renal blood flow, plasma protein binding, urine flow, urine pH, and renal transporters [START_REF] Garrett | Pharmacokinetics and clearances related to renal processes[END_REF][START_REF] Tucker | Measurement of the renal clearance of drugs[END_REF].

Renal transporters play key roles in the secretion and reabsorption of many drugs and can significantly contribute to the variability in renal excretion of these compounds [START_REF] Lee | Transporters and renal drug elimination[END_REF]. Organic anion and organic cation transport systems are two major drug transport systems in the human kidney [START_REF] Dresser | Transporters involved in the elimination of drugs in the kidney: organic anion transporters and organic cation transporters[END_REF], and the effects of genetic variations in transporters on renal clearance have been investigated recently [START_REF] Wang | OCT2 polymorphisms and in-vivo renal functional consequence: studies with metformin and cimetidine[END_REF].

Biliary Excretion

Biliary excretion is one of the primary elimination routes for xenobiotics and the conjugate metabolites [START_REF] Arias | The biology of the bile canaliculus[END_REF]. Biliary excretion requires active secretory transport because drugs are transported across the biliary epithelium against a concentration gradient. Often drugs excreted into the bile undergo some degree of reabsorption along the intestine (enterohepatic circulation).

Ghibellini and co-workers used in vitro data obtained from sandwich-cultured human hepatocytes to predict the biliary clearance for three drugs, and the predicted values were significantly lower than in vivo data [START_REF] Ghibellini | In vitro-in vivo correlation of hepatobiliary drug clearance in humans[END_REF]). Biliary excretion is mediated by transporters in the canalicular membrane. Therefore, genetic variation in these transporters contributes to the inter-individual variability in biliary excretion. Several recent reviews have summarised the key transporters involved in hepatobiliary disposition of drugs [START_REF] Chandra | The complexities of hepatic drug transport: current knowledge and emerging concepts[END_REF][START_REF] Ghibellini | Methods to evaluate biliary excretion of drugs in humans: an updated review[END_REF].

CO-EXPOSURES

Part of the variability in metabolism observed in humans may also be due to uncontrolled coexposures to naturally occurring food-borne substances, environmental contaminants, therapeutic drugs, or chemical substances in the workplace. Drug-drug interactions are a well-known problem falling under that umbrella (Jamei et al. 2009a;[START_REF] Rostami-Hodjegan | Simulation and prediction of in vivo drug metabolism in human populations from in vitro data[END_REF], but its generalisation to many-substance exposure remains to be better explored. The necessary tools are becoming available under the auspices of systems biology (Bois 2010).

EXAMPLES OF APPLICATION OF A PRIORI MODELLING TO

CHEMICAL RISK ASSESSMENT

Risk assessments are performed to estimate the conditions under which individuals or populations may be harmed by exposure to environmental or occupational chemicals. In the absence of quantitative data in the human, this process is often dependent upon the use of animal and in vitro data to estimate human response. To reduce the uncertainty inherent in such extrapolations, there has been considerable interest in the development of PBPK models of toxic chemicals for application in quantitative risk assessments. PBPK models are effective tools for integrating internal dose assessment with diverse dose-response and mechanistic data in order to more accurately predict human risk [START_REF] Andersen | Physiologically based pharmacokinetics and the risk assessment process for methylene chloride[END_REF]. One of the more challenging issues that must be considered in performing a human health risk assessment is the heterogeneity among humans. This heterogeneity is produced by inter-individual variations in physiology, biochemistry, and molecular biology, reflecting both genetic and environmental factors, and results in differences among individuals in the biologically effective tissue dose associated with a given environmental exposure (pharmacokinetics) as well as in the response to a given tissue dose (pharmacodynamics).

There has sometimes been a tendency in risk assessments to use information on the variability of a specific parameter, such as inhalation rate or the in vitro activity of a particular enzyme, as the basis for expectations regarding the variability in dosimetry for in vivo exposures. However, whether or not the variation in a particular physiological or biochemical parameter will have a significant impact on in vivo dosimetry is a complex function of interacting factors. In particular, the structures of physiological and biochemical systems frequently involve parallel processes (e.g., blood flows, metabolic pathways, excretion processes), leading to compensation for the variation in a single factor. Moreover, physiological constraints may limit the in vivo impact of variability observed in vitro [START_REF] Johanson | Development of new technique for risk assessment using physiologically based toxicokinetic models[END_REF]. For instance, high affinity intrinsic clearance can result in essentially complete metabolism of all the chemical reaching the liver in the blood; under these conditions, variability in amount metabolized in vivo would be more a function of variability in liver blood flow than variability in metabolism in vitro. Thus it is often true that the whole (the in vivo variability in dosimetry) is less than the sum of its part (the variability in each of the pharmacokinetic factors). Because the parameters in a PBPK model have a direct biological correspondence, they provide a useful framework for determining the impact of observed variations in physiological and biochemical factors on the population variability in dosimetry within the context of a risk assessment for a particular chemical [START_REF] Clewell | Use of physiologically based pharmacokinetic modeling to investigate individual versus population risk[END_REF][START_REF] Price | Modeling interindividual variation in physiological factors used in PBPK models of humans[END_REF].

It is important at this point to remember the distinction made above between uncertainty and variability. Early attempts to distinguish the contributions of uncertainty and variability can be found in [START_REF] Bogen | Integrating uncertainty and interindividual variability in environmental risk assessment[END_REF] or [START_REF] Allen | Investigation of the impact of pharmacokinetic variability and uncertainty on risks predicted with a pharmacokinetic model for chloroform[END_REF]. Several studies have attempted to estimate the impact of parameter variability in PBPK models on risk assessment predictions using the Monte Carlo approach [START_REF] Allen | Investigation of the impact of pharmacokinetic variability and uncertainty on risks predicted with a pharmacokinetic model for chloroform[END_REF][START_REF] Clewell | Use of physiologically based pharmacokinetic modeling to investigate individual versus population risk[END_REF][START_REF] Clewell | Evaluation of the uncertainty in an oral Reference Dose for methylmercury due to interindividual variability in pharmacokinetics[END_REF][START_REF] Clewell | Incorporation of pharmacokinetics in non-carcinogenic risk assessment: Example with chloropentafluorobenzene[END_REF]). As will be discussed in the next section, the use of a hierarchical Bayesian approach and Markov chain Monte Carlo simulations makes it possible to refine prior estimates of parameter variability on the basis of experimental data. The hierarchical Bayesian approach is increasingly being used to characterize both the uncertainty and variability in PBPK model predictions [START_REF] Bois | Statistical analysis of Clewell et al. PBPK model of trichloroethylene kinetics[END_REF]Gelman et al. 1996;[START_REF] Hack | Bayesian population analysis of a harmonized physiologically based pharmacokinetic model of trichloroethylene and its metabolites[END_REF]Jonsson et al. 2001b;Jonsson and Johanson 2001a;[START_REF] Qiu | Bayesian analysis of a physiologically based pharmacokinetic model for perchloroethylene in humans[END_REF].

It is useful in that context to consider the total variability among humans in terms of three contributing sources: (1) the variation across a population of "normal" individuals at the same age, e.g., young adults; (2) the variation across the population resulting from their different ages, e.g., infants or the elderly; and (3) the variation resulting from the existence of subpopulations that differ in some way from the "normal" population, e.g., due to genetic polymorphisms. A fourth source of variability, health status, should also be considered, although it is frequently disregarded in environmental risk assessment. To the extent that the variation in physiological and biochemical parameters across these population dimensions can be elucidated, PBPK models can be used together with Monte Carlo methods to integrate their effects on the in vivo kinetics of a chemical exposure and predict the resulting impact on the distribution of risks (as represented by target tissue doses) across the population. The following examples illustrate the application of PBPK models to inform population variability of the three types described above.

Example 1: Population Variability

Acceptable exposures to environmental contaminants are typically defined using a single value, such as the USEPA's Reference Dose (RfD), which represents a daily ingestion rate considered to be without harm for most individuals. In the case of methylmercury, a PBPK model was used in a Monte Carlo analysis to provide information on the distribution of acceptable ingestion rates across the population [START_REF] Clewell | Site-specific reference dose for methylmercury for fish-eating populations[END_REF][START_REF] Clewell | Evaluation of the uncertainty in an oral Reference Dose for methylmercury due to interindividual variability in pharmacokinetics[END_REF]. That is in contrast to the regulatory approach based on a single point estimate obtained using conservative assumptions.

In that analysis, the maternal hair concentration associated with neurological effects in the offspring from an epidemiological study was converted to an expected distribution of daily ingestion rates across a population of U.S. women of childbearing age. The resulting distribution of acceptable daily ingestion rates (RfDs) ranged from approximately 0.3 to 1.1 μg/kg/day, with a population median (50th percentile) of 0.5 μg/kg/day. This population distribution was used to inform risk-benefit analysis for alternative risk management options for contaminated sediment.

In essence, the question here was: "How conservative are default conservative assumptions?".

The answer, given by Monte Carlo derived uncertainty and variability estimates, was in that case: "Very conservative".

Example 2: Age-Dependent Variability

The following example illustrates the use of PBPK modelling to investigate the impact of pharmacokinetic variability on risk for the case of age-dependent pharmacokinetics. Specifically, the question being evaluated in this example is how normal changes in pharmacokinetic parameters from birth, through childhood, and across adulthood affect the dosimetry for environmental exposures to chemicals. To this end, a previously developed PBPK model for isopropanol and its metabolite acetone [START_REF] Clewell | Development of a physiologically based pharmacokinetic model of isopropanol and its metabolite acetone[END_REF]) was adapted to simulate the physiological and biochemical changes in humans associated with growth and aging. In the agedependent model, all physiological and biochemical parameters change with time based on data from the literature [START_REF] Clewell | Evaluation of the potential impact of age-and gender-specific pharmacokinetic differences on tissue dosimetry[END_REF].

Figure 3 shows the results of using this age-dependent model to simulate continuous inhalation of isopropanol at 1 ppb, beginning at birth and continuing for 75 years [START_REF] Clewell | Evaluation of the potential impact of age-and gender-specific pharmacokinetic differences on tissue dosimetry[END_REF]). The model predicts that, for the same inhaled concentration, the blood concentrations achieved during early life are significantly higher than those achieved during adulthood. In the case of the metabolite acetone, however, it should be noted that production from isopropanol metabolism would be only a small fraction of endogenous production from ketogenesis. Obviously, these are only model predictions, but while waiting for a real-life epidemiological validation (which would be very costly and difficult to perform), they form a reasonable and transparent basis for immediate decision making.

Example 3: Genetic Polymorphism

The next example demonstrates the use of PBPK modelling, together with Monte Carlo techniques, to evaluate the impact of a genetic polymorphism for metabolism. In the example described here, the polymorphism of interest is for the enzyme paraoxonase. The PBPK model used in the analysis [START_REF] Gearhart | Variability of physiologically based pharmacokinetic (PBPK) model parameters and their effects on PBPK model predictions in a risk assessment for perchloroethylene (PCE)[END_REF]) describes exposure to parathion, its metabolism to paraoxon, and the inhibition of acetylcholinesterase by paraoxon. Paraoxonase is one of the enzymes responsible for the metabolic clearance of paraoxon. In vitro data on the two human alleles of paraoxonase (low and high activity) were used to develop distributions for the metabolism parameters in the PBPK model [START_REF] Gentry | An approach for the quantitative consideration of genetic polymorphism data in chemical risk assessment: examples with warfarin and parathion[END_REF]. Monte Carlo simulations were then performed to generate the resulting distribution of predicted blood concentrations of paraoxon across a population, considering the variability in other pharmacokinetic parameters.

Figure 4 displays the predicted distribution for the time-integrated (area under the curve) blood concentrations of paraoxon (mg-hr/L) across the sensitive population (dark bars), as compared to the "normal" population, following exposure to parathion at a dose of 0.033 mg/kg [START_REF] Gentry | An approach for the quantitative consideration of genetic polymorphism data in chemical risk assessment: examples with warfarin and parathion[END_REF]. The impact of genetic polymorphism is strongly dampened by rate-limiting pharmacokinetic effects, similarly to what was observed for methyl chloride [START_REF] Johanson | Development of new technique for risk assessment using physiologically based toxicokinetic models[END_REF]. The calculation performed here lead to internal dose levels very different from those which would be obtained using naive "all or nothing" guessed estimates based solely on qualitative genotype considerations.

A POSTERIORI ESTIMATION OF VARIABILITY USING BAYESIAN PBPK MODELING

Purely predictive modelling, as described above, can be cross-validated by confronting its predictions (e.g., for plasma concentration of a substance) to data obtained of a sample of individuals. In the ideal case, data and predictions agree on average and in terms of variability and no further model refinement is attempted. Yet, quite often, the predictions are less than "perfect" and may even be really poor (but publication bias reduces the visibility of that case).

That may be the occasion to improve the model through some calibration or data integration procedure and learn something about the true determinants of variability in a population [START_REF] Wakefield | The Bayesian modeling of covariates for population pharmacokinetic models[END_REF]. However, some care should be taken to properly disentangle uncertainty from variability in that calibration process, while retaining the prior physiological knowledge afforded by PBPK modelling (Gelman et al. 1996). The "naive" approach consists in fitting individual subjects' data separately, collecting the resulting individual parameter estimates and forming their average etc. That approach does not work and is actually incorrect if very precise parameter estimates cannot be obtained from the data [START_REF] Beal | Estimating population kinetics[END_REF][START_REF] Smith | The hierarchical Bayesian approach to population pharmacokinetic modelling[END_REF]. For ethical, feasibility or cost reasons, the data on individuals tend to be sparse in clinical pharmacokinetics or toxicokinetics. Such data usually lead to fairly uncertain parameter estimates and the so-called population approaches should then be used [START_REF] Beal | The NONMEM system[END_REF].

Population models, or multilevel models, were first introduced in the context of pharmacokinetic studies for drug development and evaluation [START_REF] Sheiner | The population approach to pharmacokinetic data analysis: rationale and standard data analysis methods[END_REF]. Their objective is to obtain, from data on individuals, a quantitative description of the variability of the kinetics of a compound within a large population. The same structural (e.g., PBPK) model is used describe the data for each subject, and that the model parameters differ randomly between subjects (see Figure 5) (Bois et al. 1996a;Bois et al. 1996b). Such randomness characterises variability can be described by a multivariate probability distribution. In population models, information of each subject is reinforced by "borrowing strength" from the other subjects' data and the overall estimation process is improved. Individuals' metabolic clearance, for example, can be assumed to be lognormally distributed around a "population mean", with a "population variance" which measures variability in the population. The population means and variances (one for each kinetic parameter supposed to vary between subjects) are aptly named "population" parameters. They are estimated during the model calibration (data fitting) together with the parameters of each subject [START_REF] Bois | Applications of population approaches in toxicology[END_REF].

A number of methods are available to calibrate population models [START_REF] Beal | Estimating population kinetics[END_REF][START_REF] Davidian | Smooth nonparametric maximum likelihood estimation for population pharmacokinetics, with application to quinidine[END_REF][START_REF] Mallet | Nonparametric maximum likelihood estimation for population pharmacokinetics, with application to cyclosporine[END_REF][START_REF] Racine-Poon | Population models[END_REF]. Bayesian approaches have emerged as the best suited for PBPK models, given the large amount of prior information they require [START_REF] Bernillon | Statistical issues in toxicokinetic modeling: a Bayesian perspective[END_REF]Gelman et al. 1996). But as we have seen above, a large amount of that information is already encoded as prior distributions, and updating those distributions with system's level data, is simply a matter of using Bayesian numerical methods such as Markov chain Monte Carlo simulations [START_REF] Gelman | Markov chain Monte Carlo methods in biostatistics[END_REF].

A number of applications of posterior Bayesian PBPK modelling have been published: on benzene (Bois et al. 1996b), butadiene [START_REF] Mezzetti | A Bayesian compartmental model for the evaluation of 1,3-butadiene metabolism[END_REF]), carbaryl (Nong et al. 2008), chloroform [START_REF] Lyons | Computational toxicology of chloroform: reverse dosimetry using Bayesian inference, Markov chain Monte Carlo simulation, and human biomonitoring data[END_REF], dichloromethane [START_REF] Bois | Analysis of PBPK models for risk characterization[END_REF][START_REF] David | Revised assessment of cancer risk to dichloromethane II. Application of probabilistic methods to cancer risk determinations[END_REF][START_REF] Johanson | Development of new technique for risk assessment using physiologically based toxicokinetic models[END_REF]Jonsson et al. 2001b;Jonsson andJohanson 2001a, 2003;[START_REF] David | Revised assessment of cancer risk to dichloromethane II. Application of probabilistic methods to cancer risk determinations[END_REF][START_REF] Marino | Probabilistic dose-response modeling: case study using dichloromethane PBPK model results[END_REF], methyl chloride (Jonsson et al. 2001a), methyl mercury [START_REF] Allen | Use of Markov chain Monte Carlo analysis with a physiologically-based pharmacokinetic model of methylmercury to estimate exposures in US women of childbearing age[END_REF], nanoparticles [START_REF] Péry | Development of a physiologically-based kinetic model for 99m-Technetium labelled carbon nanoparticles inhaled by humans[END_REF], tetrachloroethylene (Bois et al. 1996a;[START_REF] Chiu | Revisiting the population toxicokinetics of tetrachloroethylene[END_REF][START_REF] Covington | The use of Markov chain Monte Carlo uncertainty analysis to support a Public Health Goal for perchloroethylene[END_REF], toluene (Jonsson and Johanson 2001b;[START_REF] Vicini | Individual prior information in a physiological model of H-2(8)-toluene kinetics: an empirical Bayes estimation strategy[END_REF], and trichloroethylene [START_REF] Bois | Statistical analysis of Clewell et al. PBPK model of trichloroethylene kinetics[END_REF]. Extension to questions of optimal design [START_REF] Bois | Optimal design for a study of butadiene toxicokinetics in humans[END_REF], medical image analysis [START_REF] Brochot | Evaluation of antiangiogenic treatment effects on tumors microcirculation by Bayesian physiological pharmacokinetic modeling and magnetic resonance imaging[END_REF], or exposure reconstruction [START_REF] Allen | Use of Markov chain Monte Carlo analysis with a physiologically-based pharmacokinetic model of methylmercury to estimate exposures in US women of childbearing age[END_REF]) have also been proposed.

CONCLUSIONS

The state of the art on PBPK modelling of inter-individual variability has advanced to the point of being a mainstream commercial activity for drug development. We have shown how the concepts and tools now available (generic PBPK models, applicable to many substances, coupled to databases of parameter distributions and QSAR models; MCMC software routines for Bayesian data integration) can foster predictive toxicokinetics for environmental or occupational contaminants. Computation time is not an issue anymore, and the approach is being extended to toxicodynamics through the use of biology-motivated effect models, toward a true predictive toxicology applicable to very large number of chemicals.

FIGURE CAPTIONS

Figure 1 :

 1 Figure 1: Structure of a generic PBPK model of the mammalian body.

Figure 2 :

 2 Figure 2: The inter-correlation of covariates affecting drug clearance.

Figure 3 :

 3 Figure 3: Blood concentrations of isopropanol (IPA) and its metabolite acetone as a function of

Figure 4 :

 4 Figure 4: Paraoxon AUC distribution for the sensitive population compared to the AUC distribution for the "normal" population for a parathion dose of 0.033 mg/kg.

Figure 5 :

 5 Figure 5: Graphical representation of a toxicokinetic population model. Unknown quantities are

Figure 4 :

 4 Figure 4: Paraoxon AUC distribution for the sensitive population (black bars) compared to the AUC distribution for the "normal" population (white bars) in for a parathion dose of 0.033

  .

	CLu	int,	H		CLu	int	(	per	_	millions	_	hepatocyte	s	)	HPGL	Liver	_	weight
	Human liver microsomes:
	CLu	int,	H	CLu	int	(	per	_	mg	_	microsomes	)	MPPGL	Liver	_	weight	(4)
	Human hepatocytes:

in most cases(Yang et al.

2007), CLu int,G is the unbound total gut intrinsic clearance, and Q Gut

ACKNOWLEDGEMENTS

FYB was supported by the European Commission, 7 th FP project PREDICT-IV [grant agreement #202222] and the French Ministry for the Environment (PRG189_07_DRC03). MJ would like to thank the Simcyp team for its contributions to the development of the Simcyp® Populationbased Simulator and also appreciate the continued support of the Simcyp Consortium members and the support received from the PREDICT-IV project.

FIGURES


Frédéric Y Bois
email: frederic.bois@ineris.fr

Dany Habka

Céline Brochot

Yves Frédéric

Bois

An integrated modelling approach for in vitro to in vivo extrapolations

Keywords: Ciclosporin A, Diazepam, PBPK models, PK/PD models, Toxicity prediction

Predicting in vivo drug toxicity from in vitro testing requires modelling those processes that are not reproduced in vitro. The most obvious difference between the two settings is the lack of integrated absorption, distribution, metabolism, and excretion (ADME) that govern target tissue exposure in vivo. For equal input doses, the concentrations to which in vitro systems are exposed may not correspond to those found in vivo. The partners of the European project PREDICT-IV in charge of modelling develop prediction models for ADME processes and integrate them into generic physiologically based pharmacokinetic (PBPK) models. Examples of simulations of concentration-time profiles for diazepam and ciclosporin A, in various human or rat organs and tissues, are presented here. In vivo drugs' toxicity will be forecasted by coupling the predicted target tissue concentration profiles to dose-response relationships observed in vitro. Human variability will be evaluated for each component of the approach.

Introduction

It is now recognised that replacing animals by in vitro systems for assessment of medium to long-term toxicity of drugs or general chemicals requires an integration of pharmacokinetics [START_REF] Bouvier D'yvoire | Physiologically-based kinetic modelling (PBK modelling): meeting the 3Rs agenda -The report and recommendations of ECVAM Workshop 63a[END_REF]. Predicting in vivo drug toxicity from in vitro testing requires modelling the absorption, distribution, metabolism, and excretion (ADME) processes that are not reproduced in vitro.

For equal input doses, the concentrations to which in vitro systems are exposed may not correspond to the target tissue exposure experienced in vivo. INERIS, Simcyp Limited, the German Cancer Research Centre, Emergentec Biodevelopment GmbH, and ECVAM, which constitute the modelling team of the European project PREDICT-IV, will integrate prediction models for ADME processes into global generic physiologically based pharmacokinetic (PBPK) models able to simulate concentration-time profiles in human or rat blood and tissues. In vivo drugs' toxicity will be predicted by coupling the predicted target tissue concentration profiles to dose-response relationships observed in vitro.

Human variability will be evaluated for each component of the approach.

While the establishment of predictive dose-response relationships for neurotoxicity, liver and kidney toxicity is a very important part of the project, this article focuses on the progress made so far on PBPK modelling by PREDICT-IV.

PBPK models are increasingly used in drug development and regulatory toxicology to predict the kinetics and metabolism of substances in the body [START_REF] Barton | Characterizing uncertainty and variability in physiologically-based pharmacokinetic (PBPK) models: state of the science and needs for research and implementation[END_REF][START_REF] Bouvier D'yvoire | Physiologically-based kinetic modelling (PBK modelling): meeting the 3Rs agenda -The report and recommendations of ECVAM Workshop 63a[END_REF][START_REF] Loizou | Development of good modelling practice for physiologically based pharmacokinetic models for use in risk assessment: the first steps[END_REF]. In these models, the body is represented by a set compartments corresponding specific organs or tissues, and the transfers or transports of drugs are dictated by various physiological flows (blood, bile, pulmonary ventilation, etc) [START_REF] Bois | An analysis of exposure rate effects for benzene using a physiologically based pharmacokinetic model[END_REF]. A system of differential equations can be written, with parameters representing blood flow rates, organ volumes etc., for which information is available in the published scientific literature or may be obtainable in vitro [START_REF] Bois | Comparison of three physiologically-based pharmacokinetic models of benzene disposition[END_REF]Bois, 1993, Parrott, et al., 2005). Numerical integration of that differential system computes the quantity and concentration of the drug considered in each compartment, as a function of time and exposure dose. Indeed, such a description of the body is approximate, if not rough, but a balance has to be found between precision (and therefore complexity) and simplicity (ease of use).

We will briefly describe the global approach taken by PREDICT-IV to integrate experimental evidence for a predictive approach to toxicology. We will then describe Simcyp's generic PBPK model and the specific parameter setting adopted for ciclosporin A and diazepam in rats and humans. Modelling results for those two substances are presented and discussed.

Methods

General modelling framework

Fig. 1 presents graphically the integrated prediction and validation framework of PREDICT-IV. In vitro studies will provide the necessary data for the development of a set of ADME QSARs and multivariate dose-response models for cellular and organ-level effects. Those models will be coupled to a generic PBPK model and will provide an in silico pharmacokinetic-pharmacodynamic (PK/PD) tool predictive of medium-to long-term toxicity. In order to validate that tool, its predictions for plasma and tissue drug levels will be confronted to pharmacokinetic data obtained in humans or rats. We do not expect to get enough data relating plasma to pathological effects in humans. Therefore the pharmacodynamic component will be validated on the basis of already published rat data. We do not expect perfect predictions of pharmacokinetics or toxicity data and meta-analysis models will be developed to analyse and correct for the discrepancies.

Simcyp's generic PBPK models

The structure of the PBPK model developed by Simcyp for rats and humans is shown on Fig. 2. A set of ADME sub-models, able to predict substance-specific parameter values for a large range of molecules, is already coupled to this model [START_REF] Howgate | Prediction of in vivo drug clearance from in vitro data. I. Impact of inter-individual variability[END_REF], Jamei, et al., 2009a, Jamei, et al., 2009b, Jamei, et al., 2009c). Obviously, while the model structure remains the same, parameter values and corresponding absorption, distribution, metabolism and excretion (ADME) sub-models differ between species. Dosing can be by intra-venous injection (i.v.) or per os (p.o.) (the dermal and inhalation routes are also developed for humans).

The liver and gut are sites of metabolism. Renal excretion takes place in the kidney. Solving numerically the corresponding set of differential equations yields the time-course of the quantity or concentration of a given drug in any of the organs or tissues included in the model.

Specific parameters for diazepam in rats

Simcyp also provides default parameter values for diazepam with its rat model. A total in vivo clearance of 20.4 mL/min is reported [START_REF] Klotz | Pharmacokinetics and plasma protein binding of diazepam in man, dog, rabbit, guinea pig and rat[END_REF][START_REF] Igari | Prediction of diazepam disposition in the rat and man by a physiologically based pharmacokinetic model[END_REF]. To model the extra-hepatic metabolism of diazepam in rats, hepatic clearance was set equal to three-quarters of total clearance (CL iv = 0.75 CL tot = 15.3 mL/min; CL r = 0.25 CL tot = 5.1 mL/min).

The intravenous injection of 1.2 mg/kg used by [START_REF] Igari | Prediction of diazepam disposition in the rat and man by a physiologically based pharmacokinetic model[END_REF] in their experiments on Wistar was simulated.

Specific parameters for diazepam in humans

Diazepam is not part of the default substances provided by Simcyp's human model. We used the physico-chemical characteristics specified by Simcyp's rat model (molecular weight: 284.7 g/mol; log octanol to water partition coefficient: 2.9; monoprotic base with a pKa of 3.4). Blood binding parameters where obtained from the literature (fraction unbound in plasma, fu p : 0.022; blood to plasma ratio: 0.58) [START_REF] Klotz | The effects of age and liver disease on the disposition and elimination of diazepam in adult man[END_REF][START_REF] Klotz | Estimation of the blood-plasma concentration ratio of diazepam in the rat[END_REF][START_REF] Jones | Distribution of diazepam and nordiazepam between plasma and whole blood and the influence of hematocrit[END_REF]. Tissue-to-plasma concentration ratios (K p ) were automatically estimated by the corresponding ADME model. For in vivo clearance, CL iv , the published value of 1.596 L/h was used [START_REF] Klotz | The effects of age and liver disease on the disposition and elimination of diazepam in adult man[END_REF].

To simulate [START_REF] Klotz | The effects of age and liver disease on the disposition and elimination of diazepam in adult man[END_REF] data, a two-minute intravenous infusion of 0.1 mg/kg diazepam was simulated for a representative healthy volunteer, followed up for 72 hr.

Specific parameters for ciclosporin A in rats

Ciclosporin A was not one of the drugs predefined for in Simcyp's rat model. We used the physico-chemical properties provided by the Simcyp's human model (molecular weight: 1202 g/mol; log octanol to water partition coefficient: 4.3; neutral compound). Blood binding parameters where obtained from the literature (fraction unbound in plasma, fu p : 0.062; fraction unbound in blood, fu b : 0.05 for intra-venous doses up to 6 mg/kg and 0.094 for doses above) [START_REF] Kawai | Physiologically based pharmacokinetics of cyclosporine A: extension to tissue distribution kinetics in rats and scale-up to human[END_REF], Tanaka, et al., 2000). The tissue-to-plasma concentration ratios (or partition coefficients, K p ) (Tab. 1) were derived by multiplying published tissue-to-blood ratios (Tanaka, et al., 2000) by fu p / fu b .

In vivo plasma clearance values were derived from Tanaka et al., (2000) by multiplying the blood clearance by fu p / fu b : (CL iv : 0.82 mL/min for an intravenous dose of 1.2 mg/kg; 0.94 mL/min for 6 mg/kg; 0.35 mL/min for 30 mg/kg).

Three experiments on Sprague-Dawley rats with 2-minute intra-venous infusions of 1.2 mg/kg, 6 mg/kg and 30 mg/kg ciclosporin A, respectively, were simulated (Tanaka, et al., 2000).

Specific parameters for ciclosporin A in humans

Simcyp provides default parameter values for ciclosporin A pharmacokinetics in humans. We used the predefined "Healthy volunteers" population. To match the characteristics of the population studied by [START_REF] Gupta | Effect of food on the pharmacokinetics of cyclosporine in healthy subjects following oral and intravenous administration[END_REF], the reference body mass of the subjects was set to an average of 64 kg, with age ranging from 24 to 34 years, with 50% males. The duration of the simulated study was 24 hr after an oral administration of 10mg/kg of ciclosporin A. A first-order oral absorption model was used, with rate constant 0.91 h -1 [START_REF] Kawai | Physiologically based pharmacokinetics of cyclosporine A: extension to tissue distribution kinetics in rats and scale-up to human[END_REF].

For extrapolation to a larger population the age range was set from 20 to 70 years. To study variability, Monte Carlo simulations [START_REF] Bois | Precision and sensitivity analysis of pharmacokinetic models for cancer risk assessment: tetrachloroethylene in mice, rats and humans[END_REF][START_REF] Bois | Comparison of three physiologically-based pharmacokinetic models of benzene disposition[END_REF][START_REF] Bois | An analysis of exposure rate effects for benzene using a physiologically based pharmacokinetic model[END_REF][START_REF] Rostami-Hodjegan | Simulation and prediction of in vivo drug metabolism in human populations from in vitro data[END_REF] of 100 subjects were performed.

Results

Diazepam in rats and humans

A set of simulations was performed to compare the model predictions to actual data published on rat and human diazepam pharmacokinetics. Only minimal adjustments to the parameter values, has described in the Methods section, were made. Fig. 3 shows the results obtained for the rat. Only plasma, brain, liver and kidney data [START_REF] Igari | Prediction of diazepam disposition in the rat and man by a physiologically based pharmacokinetic model[END_REF] and model simulations are shown because these are the compartments of primary interest in PREDICT-IV. Similar results are obtained in the other tissues.

Fig. 4 presents similar results in human plasma. The experimental data of [START_REF] Klotz | The effects of age and liver disease on the disposition and elimination of diazepam in adult man[END_REF] for two healthy volunteers are slightly overestimated, but we have not attempted here to account for the heterogeneity of the human population. For humans, Fig. 6 shows the data of [START_REF] Gupta | Effect of food on the pharmacokinetics of cyclosporine in healthy subjects following oral and intravenous administration[END_REF], obtained after oral administration of 10 mg/kg of ciclosporin A to a healthy volunteer, in low fat or high fat diet conditions. Monte Carlo simulation of 100 similar subjects (diet not considered) were performed, and the average, 5 th and 95 th percentiles of the simulated values are plotted. Fig. 7 illustrates the very purpose of all this modelling exercise: predicting the concentration time-course of a substance (here ciclosporin A) in the brain, liver and kidney for a varied human population. We simulated 100 healthy subjects, aged from 20 to 70 years, taking a daily dose of 10 mg/kg of ciclosporin A for 60 days. Note that during the first week of treatment, liver exposure is the highest of the three, but brain exposure after 60 days is higher than liver or kidney exposure.

Ciclosporin A in rats and humans

Discussion

Predicting organ toxicity beyond acute effects is a significant scientific challenge.

The team of PREDICT-IV is taking an inclusive approach to the problem, coupling in vitro testing, in silico ADME, PBPK and dose-response models in an integrated predictive tool. An important aspect of the work will be the validation, or benchmarking, of the model predictions with a set of about 30 reference drugs, known for their neurotoxicity, hepatotoxicity or kidney toxicity. To that effect, Simcyp Ltd has already extended its human pharmacokinetic model and developed a rat model. The results presented here are indeed limited to only two compounds, diazepam and ciclosporin, but give an idea of the quality of the predictions obtained with the current model. We will test the approach with whole set of reference drugs, and we even expect improvements when additional, more precise, ADME properties prediction models will be included.

We have performed here Monte Carlo simulations to describe inter-individual variability only to a limited extent, and only for ciclosporin A. This type of simulations will be performed for more diverse populations, potentially more susceptible to organ toxicity. Accessing this type of results is difficult for in vitro testing alone, hence the synergy with mathematical modelling, which is one of the goals of PREDICT-IV.

The results presented here go only as far as the prediction of organ exposures after medium term repeated administration of a drug. We could have extended our simulations to longer times or different doses, but stable values were obtained for ciclosporin A after 60 days of dosing and we would have gained no more insight with additional follow-up. Obviously this may not be true for other compounds and will have to be assessed on a case by case basis. In any case, long term target organ exposure is the relevant measure of dose for the prediction of organ toxicity after sustained exposure. We have still to couple our PBPK model to doseresponse models for a complete PK/PD approach. The dose-response component will also add a time dimension of its own, but the work on correlated multidimensional dose-response models in PREDICT-IV has just begun.

We are focusing on local organ toxicity, for which the target tissue concentration can be considered as the relevant measure of dose. Systemic toxicity would be much more of a challenge, combining effects of the dysfunction of several organs. However, we are only a step away from being able to describe the toxic effects of co-administered substances, since our models have the capability to account for at least pharmacokinetic and metabolic interactions (Rostami-Hodjegan and Tucker, 2004, Bois, submitted). We are therefore confident in our ability to make significant contributions to the 3Rs agenda and to predictive clinical and environmental toxicology in the near future. Tanaka, C., Kawai, R., and Rowland, M. (2000). Dose-dependent pharmacokinetics of cyclosporin A in rats: events in tissues. Drug Metab. Dispos. 28, 582-589. Woodruff, T., and Bois, F.Y. (1993). Optimization issues in physiological toxicokinetic modeling -a case study with benzene. Toxicol. Lett. 69,[181][182][183][184][185][186][187][188][189][190][191][192][193][194][195][196].

Tables

Tab. 1: Tissue-to-blood partition coefficients used for ciclosporin A in rats. (a) The brain concentration was below the limit of detection in Tanaka et al., (2000). Left panel: Predictive component of project In vitro studies will provide the necessary data for the development of a set of ADME QSARs and multivariate dose-response models for cellular and organ-level effects. Those models will be coupled to a generic PBPK model for humans and rats and will provide an in silico pharmacokinetic-pharmacodynamic (PK/PD) tool predictive of medium-to long-term toxicity. Right panel: In order to validate that tool, its predictions for plasma and tissue drug levels will be confronted to pharmacokinetic data obtained in humans or rats. To relate plasma to pathological effects, the pharmacodynamic component will be validated on the basis of published animal (rat) data. The rat PBPK model was run to simulate the intra-venous administration of 1.25 mg/kg of diazepam to Wistar rats [START_REF] Igari | Prediction of diazepam disposition in the rat and man by a physiologically based pharmacokinetic model[END_REF]. The solid lines represent the predicted time-courses. The dots figure the corresponding data (average concentrations measured in 3 to 5 rats). The human PBPK model (solid line) was run to simulate the intra-venous administration of 0.1 mg/kg of diazepam to human healthy volunteers. The data correspond to those obtained in two subjects (circles and squares, respectively) by [START_REF] Klotz | The effects of age and liver disease on the disposition and elimination of diazepam in adult man[END_REF]. The rat PBPK model was run to simulate the intra-venous administration of 1.2, 6

Tissue

Figure legends

or 30 mg/kg of ciclosporin to Sprague-Dawley rats (Tanaka, et al., 2000). The solid lines represent the predicted time-courses. The dots (circles: 30 mg/kg dose;

squares: 6 mg/kg dose; diamonds: 1.2 mg/kg dose) figure the corresponding data (average concentrations measured in 3 rats). Left panel: Predictive component of project In vitro studies will provide the necessary data for the development of a set of ADME QSARs and multivariate dose-response models for cellular and organ-level effects. Those models will be coupled to a generic PBPK model for humans and rats and will provide an in silico pharmacokinetic-pharmacodynamic (PK/PD) tool predictive of medium-to long-term toxicity. Right panel: In order to validate that tool, its predictions for plasma and tissue drug levels will be confronted to pharmacokinetic data obtained in humans or rats. To relate plasma to pathological effects, the pharmacodynamic component will be validated on the basis of published animal (rat) data. The rat PBPK model was run to simulate the intra-venous administration of 1.25 mg/kg of diazepam to Wistar rats [START_REF] Igari | Prediction of diazepam disposition in the rat and man by a physiologically based pharmacokinetic model[END_REF]. The solid lines represent the predicted time-courses. The dots figure the corresponding data (average concentrations measured in 3 to 5 rats). The human PBPK model (solid line) was run to simulate the intra-venous administration of 0.1 mg/kg of diazepam to human healthy volunteers. The data correspond to those obtained in two subjects (circles and squares, respectively) by [START_REF] Klotz | The effects of age and liver disease on the disposition and elimination of diazepam in adult man[END_REF]. The rat PBPK model was run to simulate the intra-venous administration of 1.2, 6

or 30 mg/kg of ciclosporin to Sprague-Dawley rats (Tanaka, et al., 2000). The solid lines represent the predicted time-courses. The dots (circles: 30 mg/kg dose;

squares: 6 mg/kg dose; diamonds: 1.2 mg/kg dose) figure the corresponding data (average concentrations measured in 3 rats).

Fig. 5

 5 Fig.5shows the data obtained byTanaka et al. (2000) in Sprague-Dawley rats

Fig. 1 :

 1 Fig. 1: Graphical representation of the integrated prediction and validation framework of PREDICT-IV.

Fig. 2 :Fig. 3 :

 23 Fig. 2: Graphical representation of the physiologically based pharmacokinetic (PBPK) model of the human or rat body.

Fig. 4 :

 4 Fig. 4: Model predictions and published data on diazepam time-course in human plasma.

Fig. 5 :

 5 Fig. 5: Model predictions and published data on ciclosporin A pharmacokinetics in the rat.

Fig. 6 :

 6 Fig. 6: Modelled and observed ciclosporin A pharmacokinetics in human blood.

Fig. 7 :

 7 Fig. 7: Predictions of long-term ciclosporin A pharmacokinetics in humans.

  Fig. 3: Model predictions and published data on diazepam pharmacokinetics in the rat.

Fig. 4 :

 4 Fig. 4: Model predictions and published data on diazepam time-course in human plasma.

Fig. 5 :

 5 Fig. 5: Model predictions and published data on ciclosporin A pharmacokinetics in the rat.

Acknowledgements

This work was supported by the European Commission, 7 th FP project PREDICT-IV [grant agreement #202222] and the French Ministry for the Environment (PRG189_08_DRC03).

Venous Blood Arterial Blood