Statistical issues in toxicokinetic modeling : a … issues in toxicokinetic modeling : a Bayesian...

Statistical issues in toxicokinetic modeling : a Bayesian

perspective

Pascale Bernillon, Frederic Y. Bois

To cite this version:

Pascale Bernillon, Frederic Y. Bois. Statistical issues in toxicokinetic modeling : a Bayesianperspective. Environmental Health Perspectives, National Institute of Environmental HealthSciences, 2000, 108 (suppl. 5), pp.883-893. <ineris-00961852>

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Environmental Health Perspectives • VOLUME 108 | SUPPLEMENT 5 | October 2000 883

Exposure concentrations of chemicals in air,food, water, soil, dust, or other media withwhich populations are in contact are the dosescales most commonly used when assessinghealth effects for environmental or occupa-tional pollutants. Yet, external exposure isoften only a rough estimate for internal expo-sure delivered at the critical target in thebody. The latter is a more appropriate mea-sure of dose for mechanism-based risk assess-ments (1). For example, many chemicalsrequire metabolic conversion into chemicallyactive species before exhibiting toxicity, andthis conversion may be subject to saturationat high doses (2–5). In the presence of sat-urable activation or detoxification pathways,the relationship between administered (exter-nal) dose and delivered (internal) dose may benonlinear. This nonlinearity is of particularconcern when performing high-to-low-doseextrapolation of the overall exposure–response relationship. Indeed, a solution is tomeasure internal biomarkers of exposure, butsuch direct measurements may not be feasibleif the relevant biomarkers are not yet devel-oped. In this case, modeling approaches canbe advocated as an alternative. Toxicokinetic(TK) models, for example, are useful tools torelate external exposures to internal measuresof dose. TK models describe the behavior ofchemicals in the body, e.g. the processes ofabsorption, distribution, metabolism, andelimination. Two main classes of TK modelshave been developed: classical toxicokinetic

models and physiologically based toxico-kinetic (PBTK) models. The development ofPBTK models has been particularly activeduring the last 20 years, and their usefulnessfor chemical risk assessments is now firmlyestablished (5–10). Several major chemicalrisk assessments currently processed by theU.S. Environmental Protection Agencyinvolve the use of PBTK models.

The focus of PBTK modeling in riskassessment to date has been essentially onimproving the scientific basis for extrapolat-ing risk from animals to humans (11).Before the introduction of PBTK models,the key default assumption regarding inter-species extrapolation was that animal expo-sures could be converted to equivalenthuman exposures by a surface area correc-tion factor. However, several examples canbe found in the literature in which compara-tive modeling of distribution and metabo-lism in animals and humans has challengedthe use of simple surface area conversions forinterspecies extrapolations (12–15). Theability to tailor the parameter values ofPBTK models to a particular animal speciesoffers, in theory, a firm basis for interspeciesextrapolation. In practice, for most PBTKmodels some parameters remain difficult toextrapolate across species. For metabolicparameters, in particular, species-specific val-ues may be difficult to obtain experimen-tally. The “parallelogram approach” (16,17)may provide partial answers to this question,

but parameters typically still need to beadjusted through calibration of the modelwith pharmacokinetic data (usually mea-sured time courses of parent compound andmetabolites in biological media such asexhaled air, blood, urine, etc.).

If some uneasiness has been expressedabout the use of PBTK models (18), it wasessentially because of the lack of statisticalmethods for calibrating them, i.e., for adjust-ing their parameter values by taking toxicoki-netic data into account. Another issuegaining major attention in the scientific andregulatory communities is population vari-ability in toxicokinetic and metabolicprocesses (9,19–27). Considering this vari-ability and the uncertainty about many para-meters difficult to measure accurately, usingfixed parameter values or presenting resultsin the form of point estimates can be funda-mentally misleading (28). Obviously, thetask of a correct statistical treatment ofPBTK models is daunting, as we are facedwith large nonlinear models, small data sets,high uncertainty, and biological variability.Until recently, there was no method forrigourous statistical validation of PBTKmodels. For example, to our knowledge, thepredictions made by such models were neverpresented with meaningful confidence levels.Consequently, it was impossible to decidewhether the fits were acceptable, the modelsreasonable, and what confidence to place inthe results of extrapolation (29).

In response to this challenge, our groupbegan to examine the statistical issues associ-ated with the calibration and use of PBTKmodels and to develop methods to solve them.This article first briefly presents the structureof classical TK and PBTK models. We thendiscuss the issues of parameterization and cali-bration of these models and describe aBayesian approach to their statistical analysis.

Determining the relationship between an exposure and the resulting target tissue dose is a critical

issue encountered in quantitative risk assessment (QRA). Classical or physiologically based

toxicokinetic (PBTK) models can be useful in performing that task. Interest in using these models to

improve extrapolations between species, routes, and exposure levels in QRA has therefore grown

considerably in recent years. In parallel, PBTK models have become increasingly sophisticated.

However, development of a strong statistical foundation to support PBTK model calibration and use

has received little attention. There is a critical need for methods that address the uncertainties

inherent in toxicokinetic data and the variability in the human populations for which risk predictions

are made and to take advantage of a priori information on parameters during the calibration process.

Natural solutions to these problems can be found in a Bayesian statistical framework with the help

of computational techniques such as Markov chain Monte Carlo methods. Within such a

framework, we have developed an approach to toxicokinetic modeling that can be applied to

heterogeneous human or animal populations. This approach also expands the possibilities for

uncertainty analysis. We present a review of these efforts and other developments in these areas.

Appropriate statistical treatment of uncertainty and variability within the modeling process will

increase confidence in model results and ultimately contribute to an improved scientific basis for

the estimation of occupational and environmental health risks. Key words: Bayesian analysis,

hierarchical models, MCMC methods, toxicokinetic models, uncertainty, variability. — Environ

Health Perspect 108(suppl 5):883–893 (2000).

http://ehpnet1.niehs.nih.gov/docs/2000/suppl-5/883-893bernillon/abstract.html

This article is part of the monograph on MathematicalModeling in Environmental Health Studies.

Address correspondence to F.Y. Bois, INERIS, ParcALATA, BP 2, 60550 Verneuil en Hallate, France.Telephone: 33 03 44 55 65 96. Fax: 33 03 44 55 68 99.E-mail: [email protected]

Funding for this research was from Associationpour le Dévelopement de l’Informatique Médicale etl’Epidémiologie Clinique and Institut National del’Environnement Industriel et des Risques (ConventionIndustrielle de Formation par la Recherche onEnterprise # 96044).

Received 11 February 2000; accepted 8 August 2000.

Statistical Issues in Toxicokinetic Modeling: A Bayesian Perspective

Pascale Bernillon1 and Frédéric Y. Bois2

1B3E INSERM U444, Paris, France; 2Institut National de l’Environnement Industriel et des Risques (INERIS), Verneuil en Halatte, France

BERNILLON AND BOIS

884 VOLUME 108 | SUPPLEMENT 5 | October 2000 • Environmental Health Perspectives

Toxicokinetic Models

A variety of TK models have been developed;all are simplified representations of chemicaldisposition within the human/animal organ-ism. This section describes the structure andcharacteristics of classical compartmental andphysiologically based models, which are themain two classes of TK models found in theliterature. We then briefly present minimalphysiological models, which recently havebeen proposed by several investigators.

Classical Compartmental (TK) Models

In classical TK models, the body is repre-sented by several connected compartments(rarely more than three); each compartmentis a virtual space (i.e., compartments do notnecessarily reflect the anatomy of the speciesof interest) within which the chemical isassumed to be homogeneously distributed(30). Chemical transfers between compart-ments are described by a set of differentialequations. Typical parameters are compart-ment volumes, exchange rates between them,and clearance (elimination rate). The numberof compartments and the parameter valuesare inferred from fitting the model to TKdata. A procedure commonly adopted whenusing these models consists in first fitting asimple, usually one-compartment model. Ifthe fit is unsatisfactory, another compartmentis added to the model, and so on, until anacceptable fit is achieved. Parameter valuesderived from the analysis of classical compart-ments models are data dependent and are notintended to represent actual physiological vol-umes or flows in the organism (11). Forexample, the volume of distribution of a par-ticular compound can be much greater thanthe volume of the body (this is the case forchemicals with high affinities for tissue pro-teins) (31). These models thus are commonlyreferred as empirical or data-based models.

Classical compartmental models are reli-able tools to predict various surrogates of dosesuch as areas under the concentration–timecurve or the maximum concentration reachedin diverse biological media (e.g., exhaled air,venous blood, urine) when the objective is tointerpolate from the current data. They arewidely used in pharmacokinetic studies toinvestigate drug disposition in the body.Nevertheless, given their lack of physiologicalrelevance, these models are not indicated toextrapolate kinetic results between species,exposure routes, or exposure conditions. Forthese reasons, other modeling approaches,including physiologically-based models, havebeen investigated.

PBTK Models

In PBTK models, the body is subdivided intoa series of anatomical or physiological com-partments that correspond to specific organs

(liver, kidney, lung) or lumped tissue andorgan groups (fat, richly perfused, and slowlyperfused tissues) (32–34). Tissues or organsare lumped when they have, for example,similar blood flow and fat content and whenpartitioning of the substance of interestbetween them can be considered homoge-neous. Connections between compartmentsrepresent the blood or lymphatic circulationand chemical transfers between compart-ments are described by mass balance differen-tial equations. The time course of transportand transformations of a chemical throughthe various compartments can be simulatedvia the resolution of the model’s set of equa-tions for any given set of parameter values.PBTK models include three types of parame-ters: physiological parameters such as breath-ing rate, blood flows, and tissue volumes;physicochemical parameters, such as partitioncoefficients that represent the relative solubil-ity of a chemical in specific tissue; and bio-chemical parameters describing, for example,metabolic processes. The number of compart-ments to be included in the model dependson the objective of the study and on the pos-sible mode(s) of action and site(s) of toxicityof the chemical studied. In most cases, thekidney or the liver are usually individualizedwhen they are major organs of eliminationand/or metabolism. Other organs are alsoindividualized (e.g. bone marrow, brain,bone) when they are known sites of metabo-lism, storage, or toxicity. In many applica-tions, distribution among compartments issupposed to be limited by perfusion; once ina compartment, the chemical distributesinstantaneously and homogeneously through-out the entire volume of the compartment.More complex models describing the diffu-sion of the compound in subcompartmentsrepresenting vascular, interstitial, and intra-cellular spaces (diffusion-limited models)have also been developed (35–37).

Figure 1 illustrates a general four-compartment PBTK model for volatilechemicals, which we used to study tetra-chloroethylene (TETRA) kinetics in humans(38,39). In this model, TETRA distributionamong the four compartments (well-perfusedtissues, poorly perfused tissues, fat, liver) issupposed to be limited by blood perfusion, asis often the case for apolar solvents. TETRAabsorption is supposed to occur by inhalation;elimination takes place by exhalation andmetabolism. The only metabolically activecompartment included in the model is theliver. TETRA primary metabolism is describedby a saturable Michaelis-Menten mechanism.This model encloses 15 parameters.

Toxic effects induced by exposure tochemicals are commonly observed duringexperimental animal studies under conditionsof exposure different than typical human

occupational or environmental exposures.PBTK models are promising tools to solvesome of the critical extrapolation issuesencountered in chemical risk assessments(6,11,32,33,40–42):• for example, dose extrapolation can auto-

matically be achieved with a PBTK modelassuming the model correctly captures thelinear and nonlinear dynamics involved inthe transport and metabolism of thecompound studied.

• Interspecies extrapolation is resolved byassuming that the model structure is cor-rect for two or more species (the TETRAmodel described above, for example, isreasonable for any mammal). Simplychanging parameters to values specific tothe species of interest operates theextrapolation.

• Interroute (of exposure) extrapolationscan be performed. The model presentedabove (Figure 1) has only one route ofentry, the lung; it is possible to simulateintravenous injection by imposingincrease in venous blood concentration atany time or by an explicit additionalequation; ingestion has been modeledwith such models as a direct infusion intothe liver compartment or with additionalcompartments describing the gastro-intestinal tract; dermal absorption has alsobeen modeled by addition of a skincompartment (43,44).PBTK models can explain the toxico-

kinetic behavior of the compound studiedunder special conditions in humans such asphysical activity (45) or in specific targetedpopulations (children, aged, obese) by simplychanging the values of some physiologicalparameters. In addition, these models are

Figure 1. Schematic representation of a PBTK model

used for distribution and metabolism of TETRA (38). Q,

blood flows; V, compartment volumes; P, partition coeffi-

cients; VPR, ventilation over perfusion ratio; Vmax, maxi-

mum rate of metabolism; Km , Michaelis-Menten

parameter.

QalvExhaled air

VPR Pba

Qwp

Ve

no

us

blo

od

Well perfused

Poorly perfused

Fat

Liver

Metabolites

Vmax, Km

Vl, Pl

Vf, Pf

Vpp, Ppp

Vwp, Pwp

Qpp

Qf

Ql

STATISTICAL ISSUES IN TOXICOKINETIC MODELING

attractive when modeling complex situationssuch as developmental toxicity (36,46–52);unusual exposure scenarios such as delayedexposure of a nursing infant due to themother’s inhalation exposure can also beaddressed (53,54).

The above properties of PBTK modelscontribute to the development of more accu-rate estimates of risk. PBTK modeling tech-niques are now well developed. Nevertheless,such models involve a large number of para-meters (typically more than 20), each ofwhich is subject to some degree of uncer-tainty and variability, which are translatedinto PBTK model outputs. It is thus neces-sary to consider the statistical issues raised bysuch complexity.

Minimal Physiological Models

Minimal (or reduced) physiological modelsare PBTK models in which some of the com-partments (e.g., poorly and well-perfused tis-sues) have been lumped into a single centralcompartment. Reducing the number ofcompartments is a way to simplify the cali-bration process while partly preserving themodel’s physiological character. In the caseof benzene, it has been shown that a reducedthree-compartment PBPK model can be agood alternative to a five-compartmentPBPK model (23).

Figure 2 presents a minimal physiologi-cal TK model of TETRA kinetics in thehuman. TETRA kinetics in blood andexhaled air are described by a three-compart-ment TK model linked to a one-compart-ment submodel for the main TETRAmetabolite, trichloroacetic acid (TCA) (55).In the model, TETRA enters the body bypulmonary exchange, modeled by first-ordertransfers between air and the central com-partment. TETRA venous blood concentra-tions are assumed to be predicted by thecentral compartment concentrations. Thesecond and third compartments are a priorisupposed to represent fat and muscles,respectively. Metabolism of TETRA isassumed to take place in the central com-partment and is described by a linearprocess. A fraction of metabolized TETRAleads to TCA; the rest is made of the otherTETRA metabolites. TCA is formed fromTETRA metabolism but also from endoge-nous background reactions. Urinary elimi-nation of TCA is assumed to followfirst-order kinetics.

Reduced PBTK models may be promisingtools to analyze data when many subjects areinvolved. Recently, Fanning (56) used areduced PBTK model to analyze benzenekinetics data from a cohort of 60 Chineseworkers. In that model, the human body wasdivided into three compartments: a centralcompartment, fat, and liver. Thomaseth and

Salvan (57) consider a minimal physiologicalmodel to describe long-term kinetics of tetra-chlorodibenzo-p-dioxin in human tissuesusing data from a cohort of 359 subjects.

These examples simply illustrate that, aswith any modeling exercise, the developmentof a toxicokinetic model must be guided pri-marily by the question raised and the dataavailable to deal with it.

Statistical Considerations

PBTK models are considered an effectivemeans to predict target tissue dose, in particularbecause of the large amount of a priori infor-mation available in the literature on theirparameters. Nevertheless, calibration of TKmodels, even when they are physiologicallybased, is hardly feasible using only a prioriinformation. We mean by calibration theallocation of appropriate values to modelparameters to enable that model to behavelike the system it represents. In fact, eventhough reference values can be found in theliterature for physiological parameters such asorgan volumes or blood flows, experience hasshown that the calibration of virtually allPBTK models requires estimation of someparameters (in particular those describingmetabolism) using experimental TK data.Such data typically consist of measured par-ent and/or metabolite amounts or concentra-tions from various sites in the body (e.g.,blood, exhaled air, urine) at various times fol-lowing the beginning of a controlled exposureto the chemical studied.

We address, in the following subsections,the parameterization of TK models, their cali-bration, and finally, the predictive use ofthese models.

Parameterization

It is first necessary to pay careful attention tomodel parameterization (i.e., to definition ofthe model parameters) to facilitate their sub-sequent estimation. Many parameters of atoxicokinetic model (e.g., cardiac output,organ volumes, maximum rate of metabo-lism) tend to be correlated to some measureof body size. Furthermore, some parametersare correlated to each other (e.g., organ bloodflows with cardiac output). Neglecting thosecorrelations might lead to unrealistic physio-logical properties of the model, particularlywhen a distribution of behaviors is computed(e.g., through Monte Carlo simulations). It istraditional to implement an a priori deter-ministic modeling of known physiologicaldependencies between parameters by usingscaling functions and allometric (related-to-size) relationships (58–63). For example, vol-umes can be parameterized as fractions of thelean body weight, flows as fractions of cardiacoutput, and the maximum rate of metabolismin the liver as a power function of lean body

weight. Not all parameters need to be scaled.Unfortunately, there is no unique way toimplement scaling, and there does not appearto be a rigorous foundation for any particularchoice so far in the literature. We assert thatcovariate modeling should be guided by sta-tistical considerations, as it is essentially aregression problem, but this is a researchtopic still to be explored.

Another problem that can be solvedthrough proper parameterization is parameteridentifiability. Identifiability characterizesparameters for which reasonable estimates canbe obtained given experimental data (64–66).Parameter identifiability is a critical issue incompartmental kinetic modeling (67,68) andshould be examined before any attempt tocalibrate the model. Two types of identifiabil-ity can be distinguished: structural and statis-tical. The first is concerned with the formalmodel’s equations and parameters, i.e. thestructure of the model. As an example, con-sider the following model equation:

y = α × β × x, [1]

where α and β are unknown parameters toestimate from observations of x and y. If nei-ther α nor β appears separately in anothermodel-defining equation, α and β will not beseparately identifiable whatever the data usedto estimate them. Each one can take an infinite


Figure 2. Schematic representation of a linear three-

compartment model used for distribution and metabo-

lism of TETRA and distribution of TCA by Bernillon et al.

(55). The model considers 14 parameters; symbols are:

Falv, alveolar flow; V, volumes; K, rates of distribution

from central to peripheral compartments; P, partition

coefficients; CLTetra, TETRA metabolic clearance;

BgdTCA, endogenous TCA formation rate; KelTCA, TCA

elimination rate; fTCA, fraction of metabolized TETRA

leading to TCA; fTCA_u, fraction of total eliminated TCA

excreted into urine.

SecondperipheralTETRA

FirstperipheralTETRA

Falv /PcaFalv

Kcp2

Kcp2/Pp2c

Kcp1

Kcp1/Pp1c

Vperiph2 Vperiph1

Vc

CLTetra

1-fTCA fTCA

Other metabolite(s)

VTCA

Kel_TCA

fTCA_u1-fTCA_u

Other TCAeliminationpathways

TCA in urine

Central TCA

CentralTETRA

BgdTCA

BERNILLON AND BOIS


and unbounded number of values. Only theirproduct α × β can be reasonably estimated. αand β can become identifiable if an informa-tive prior has been specified for one of them ina Bayesian context. Structural identifiabilitycan also be ensured by an adequate parame-terization. Structural nonidentifiability wasencountered when calibrating the minimalPBTK model for TETRA kinetics depicted inFigure 2, with the two couples of parameters(Pp1c , Vperiph1) and (Pp2, Vperiph2). Their prod-ucts, Pp1c × Vperiph1 and Pp2c × Vperiph2, werethus the parameters actually estimated.Putting separate priors on Pp1c, Vperiph1, Pp2c,and Vperiph2 was another option we could havetaken. The second type of identifiability,called data-conditioned, or statistical, charac-terizes parameters that can be precisely esti-mated. For a parameter estimate, precision isthe inverse of uncertainty, and we treat thisquestion extensively in the next section. Inessence, statistical identifiability is ensured byeither adequate data to estimate all modelparameters or, in a Bayesian context, suffi-cient prior information to constrain thoseparameters (69). In a TK context, for exam-ple, statistical nonidentifiability may beencountered when experimental data do notinclude observations during or close to theabsorption phase. In that case, the parameterscharacterizing absorption will be poorly esti-mated given that data set. In a way, statisticalidentifiability is a milder form of structuralidentifiability that does not require reparame-terization to be treated. Reparameterizationcan still be useful in case of statistical identifi-ability, as it can hasten computations duringthe calibration process.

Calibration, Uncertainty, and Variability

Uncertainty and variability in the TKparameter estimates affect potentially all TKmodel predictions (e.g., estimates of tissuedoses); it is necessary to assess both to makepredictions useful for risk analysts and deci-sionmakers. This helps to determin the con-fidence to place in those predictions, toassess the range of internal dose likely tooccur within the population for which therisk is being evaluated, and finally, to weightthe decisions on the basis of TK modelpredictions (21,26).

It is important to distinguish betweenuncertainty and variability. Variability typi-cally refers to differences in the values ofmodel parameters among individuals (inter-individual variability) or across time within agiven individual (intraindividual variability).Variability may stem from genetic differences,lifestyles, physiological status, age, etc. (26).Uncertainty, on the other hand, essentially is aresult of lack of knowledge (70) and may havevarious sources. TK parameters are knownonly with finite precision. The use of

standard values, such as those in theInternational Commission on RadiologicalProtection report (71), tend to give a falseimpression of precision for physiologicalparameter values (and thus for model predic-tions). At best, such standard or default val-ues are approximate values of the average fora human population. There always will beuncertainty about their true value for a par-ticular group of animals or humans, and evenmore for a particular individual exposed(from which TK data may be available). Inaddition, most chemical-specific parameterstend to be imprecise; i.e., they may havebeen measured in vitro rather than in vivo orthey may be accessible only after fitting amodel to TK data. Measurement errors onthe experimental data, sparseness of those,and model simplifications or mispecificationsare translated into uncertainties in the para-meter values determined by statistical fittingto such data. Uncertainty may be reduced byfurther experiments, the design of which canbe formally optimized (72,73), or by a betterunderstanding of the actual processes understudy.

Variability is inherent in animal andhuman populations and cannot be reduced.Both variability and uncertainty are usuallypresent in TK data. For example, each TKdata set is specific to the subjects studied andextrapolating quantitative information fromsuch small groups to larger and different pop-ulations requires careful consideration. Infact, the primary interests of TK analysesrarely lie in the toxicokinetics of the com-pound studied in any particular individualbut resides instead in inferences both aboutaverage TK behavior of substances in largergroups (sensitive subpopulations, age groups,whole-country population, etc.) and quantifi-cation of the variability in such behavior. It isnecessary to take variability explicitely intoaccount within the calibration process. Datapooling (i.e., treating averaged data values asif they characterized a typical member of thepopulation) should be avoided, as it cangravely distort the data (74,75).

Given the usual sparseness of TK data andthe number of parameters involved in aPBTK model, conventional (e.g., uncon-strained least square) parameter estimation isgenerally not feasible. From a statistical pointof view, manually adjusting several parame-ters is not scientifically acceptable. Neither isthe commonly used alternative, which con-sists of statistically fitting only a subset of themodel parameters (assuming that the othersare exactly known and set to predefined refer-ence values). It is difficult to arrive at a cor-rect assessment of the uncertainty in modelpredictions through such an approach.Ignoring the uncertainty in a parameter valueimplicitly ignores covariances between this

particular parameter and all others. This leadsto distortions in the entire covariance struc-ture of parameter estimates (because covari-ance between two parameters vanishes whenone is fixed) and finally, to an underestima-tion of the uncertainty in model outputs(76). As an illustration, consider a simpletwo-dimensional case (Figure 3). Assumethat both parameter estimates, given thedata θ1, and θ2, are marginally (i.e., whenconsidered alone) normally distributed(Figure 3A, C) and that they are highly cor-related. The ellipse (Figure 3B) that outlinesthe 95% joint confidence region for θ1 andθ2 illustrates that covariance. A correct jointestimation of θ1 and θ2 should give bothmarginal and joint distributions closelyapproximating those. However, setting θ1,for convenience, to a fixed value, a, modifiesthe structure of covariance between θ1 andθ2, transforming the initial ellipse into a seg-ment of line, s. The 95% confidence intervalfor θ2, given θ1= a, now ranges from b to c(panel C). The resulting distribution of θ2

estimates is much narrower than the correctdistribution and its mean may not even bethe same, which could later lead to biasesand underdispersion in predictions. In higher(imagine 20) dimensions, the situationwould be considerably worse.

Three points can be made here: First, fix-ing the value of some parameters whileadjusting the other parameters is not satisfac-tory unless there is no correlation between thefixed and other parameters. Proving theabsence of such correlations is difficult andamounts to first estimating all parameterstogether (that is, considering all of themuncertain). It is simpler and often more judi-cious to stop at this first step and acknowl-edge the complete covariance between allparameters. Second, fixing θ1 for estimatingθ2 and then relaxing this constraint when

Figure 3. Illustration of covariance effects between two

parameter estimates. The correct joint and marginal

posteriors (i.e., data conditioned) are drawn as thick

lines. Setting θ1 to the value a before calibration leads

to biased and underdispersed estimates of θ2.

s a

cb

A

B C

θ1

θ2


performing predictions is incorrect, as thecovariance structure of θ1 and θ2 is disruptedin the process and the resulting marginal dis-tribution of θ2 is a conditional distributionon the assumed known value of θ1. Third, inmost cases, it is not sufficient to check theabsence of sensitivity of some of the modelpredictions to variations in θ1 values, becausewhat is important is the sensitivity of theentire model calibration process to θ1.Checking the sensitivity of some predictionsalso raises a number of questions. For exam-ple, which predictions should be checked?And what is the scientific value of a calibra-tion exercise which gives results only for aparticular application? Overall, ad hoc cali-brations are an impediment to the generalapplicability of PBTK models to various sce-narios of exposures and effects.

Confidence in Model Predictions

Models are usually developed to obtainpredictions of various quantities of interest,for example, the quantity of metabolitesformed for a given exposure to a specific com-pound. Ignoring both variability and uncer-tainty in model parameters renders TK modelpredictions almost useless for risk assessment.Several attempts have been made recently tostudy the propagation of variability anduncertainty in model parameters to modelpredictions using Monte Carlo simulationmethods (13,23 24,27,77,78). These meth-ods consist of: a) specifying a probability dis-tribution for each model parameter; b )sampling randomly each model parameterfrom its specified distribution; c) running themodel using the sampled parameter values,and computing various model predictions ofinterest. Repeating steps b) and c) a largenumber of times generates many differentvalues for the model predictions. Those val-ues can be used as samples to create his-tograms approximating the probabilitydistribution of any model prediction (79).

Monte Carlo methods can be useful inconducting global sensitivity analysis (25,80).It is also possible with such methods to dis-tinguish between variability and uncertaintyin model predictions when it is feasible toseparate the two for each model parameter(81). The validity of Monte Carlo methods isobviously highly dependent on the validity ofthe assumed parameter distributions. Thesedistributions should adequately characterizethe variability and/or uncertainty and covari-ance in the model parameters to predict therange of TK behavior that could be expectedin a population and to determine the confi-dence to place in model predictions (76).

Most Monte Carlo simulations of TKmodels performed to date assumed for conve-nience that all input parameters were inde-pendent—partial exceptions can be found in

Farrar et al. (20), Bois et al. (21), and Smithet al. (76). Most of the time this assumptionis not valid. Neglecting covariances typicallyleads to very large confidence intervals inmodel predictions, overestimating their actualspread (19).

A better and more comprehensiveapproach is to sample all parameter valuesfrom their joint probability distribution (39).Empirically specifying such a joint distribu-tion is a difficult task, particularly when someparameters have been estimated. In the nextsection we describe how a Bayesian statisticalanalysis can yield naturally such a joint distri-bution (called joint posterior distribution).Such an analysis leads to more relevant anduseful Monte Carlo simulations, taking intoaccount dependencies among all TK parame-ters when computing model predictions.

Multilevel Bayesian Analysis of TK ModelsThe above considerations led us to developand use multilevel models of uncertainty/vari-ability in a Bayesian framework for the statis-tical analysis of TK models. Those tools aredescribed in the following sections.

Multilevel Modeling

As stated above, inferring the TK behavior ofa compound in a population necessitatesextracting quantitative information fromindividual TK data. A hierarchical structuredistinguishing individual and population lev-els of variability is a convenient and efficientway to perform that task.

Population analyses were first introducedin the context of pharmacokinetic studies fordrug development and evaluation (74,82,83).Their advantages have long been discussed inthis context and a number of applications canbe found in the pharmacokinetic literature.Yuh et al. (84) published a comprehensivebibliography on methodological aspects andapplications of population models and adetailed review of these methods can befound in Davidian and Giltinian (85).However, until recently, very little attentionhas been paid to these approaches in a TKcontext. Before the work of Bois et al. (38,86)and Gelman et al. (39), the only attempt tolink population principles to TK modelingwas achieved by Droz et al. (87,88), but nofitting to experimental data was performed inthis exercice.

The objective of population models is toobtain from individual data a quantitativedescription of the variability of the kineticbehavior of a given compound within a pop-ulation. Such a model is illustrated inFigure 4. This hierarchical model has beenused for TETRA and benzene by Bois andcolleagues (38,39,86). The basic idea is thatthe same kinetic model f can describe the

concentration–time profiles of the compoundand its metabolites for each individual, andthat the model parameters may vary fromindividual to individual. Interindividual vari-ability is then described by assuming thateach individual's parameter set ψi arises inde-pendently from the other individual parame-ter sets, from a common multivariateprobability distribution.

The model presented in Figure 4 has twomajor components: the subject level and thepopulation level. At the subject level, chemicalconcentrations are measured on each of the Iindividuals studied (e.g., ni measurements forindividual i). Let: y1 = {yi1,yi 2, . . . , yin i

}denote the set of concentration measurementsmade on individual i, and ti = {ti1,ti2,...,tini

}the set of their associated sampling times; thekinetic model, denoted f, can predict concen-tration–time profiles for given exposure char-acteristics (E), individual TK parameters (ψi),and physiological covariables (ϕi) (e.g., bodyweight, age, sex, etc.). At this level, the mea-surement error model is defined. Various errormodels have been suggested to model the dif-ferences between observed and model-pre-dicted concentrations. One possiblelognormal error model frequently used forconcentration measurements is given by:

log yij = ln f (tij,Ei,ψi,ϕi) + εij, [2]

where the error terms εij are assumed to beindependent and normal random variables,


Figure 4. Graph of a hierarchical population model

describing dependencies between groups of variables.

Three types of nodes are featured: square nodes repre-

sent variables for which the values are known by obser-

vation, such as y i or ϕ i , or were fixed by the

experimenters, such as Ei and ti; circle nodes represent

unknown variables such as ψi , σ2, µ, Σ; the triangle

represents the deterministic TK model ƒ. A plain arrow

between two nodes indicates a direct statistical depen-

dency between the variables of those nodes, a dashed

arrow represents a deterministic link. Symbols are given

in the text.

subject i

Population µ

Eiti ψi ϕi

ƒ

yi

σ2

Σ

BERNILLON AND BOIS


with mean 0 and variance σ2. These errorterms may represent assay precision, modelmispecification, and/or random intraindivid-ual variability in model parameters andcovariables. σ2 can be set to a fixed knownvalue (e.g., when the analytical error isknown and modeling error assumed negligi-ble) or assumed to be a known function ofactual concentration in the biological sample;it can also be estimated along with the othermodel parameters.

At the population level, interindividualvariability is described by considering that thevectors of individual TK parameters ψ1,. . . , ψI are independent realizations from amultivariate distribution, F, with mean vectorµ and scale matrix Σ:

ψi ∼ F(µ,Σ), i = 1, . . . , I. [3]

The population parameters, µ and Σ, are apriori affected by uncertainty.

Within the population framework, infor-mation is gained on the average populationbehavior, but at the same time informationon each subject is reinforced by borrowingstrength from the other subjects. The lattereffect is particularly useful when few observa-tions are available per subject, which is typicalin pharmacokinetic and TK studies. Thisframework is flexible and, for example, can beextended to separate within-subject, between-subject, and measurement uncertainty.

A number of methods have been proposedto estimate kinetic parameters within a popu-lation framework. They can be separated intotwo broad classes, parametric and nonpara-metric. In the parametric case, the shape ofthe population distribution is assumed to beknown, and the population parameter valuesare quantities to estimate (e.g., multivariatenormal distribution with unknown vector ofmeans and variance/covariance matrix).Typically, assumed distributions for the popu-lation distribution F include multivariate nor-mal, lognormal, Student-t , or mixtures ofsuch distributions. In the nonparametricapproach, no assumption about the shape ofthe population distribution is made, and theentire distribution (both the shape and theparameters) is estimated from the populationdata (89–91). The parametric approach is themost commonly used because the data areoften too sparse for nonparametric estimation.

The above approaches rely typically onmaximum likelihood techniques (82,92,93)or on Bayesian principles (94,95). Bayesianapproaches prove to be very efficient whendealing with the complexities brought aboutby the large number of parameters of thehierarchical structure (a 10-parameter TKmodel applied to 50 subjects leads to morethan 500 parameters to estimate) and by thenonlinearities typically present in the subject-

specific kinetic models (94,95). They areparticularly appealing in the case of PBTKmodels, as they allow prior physiologicalinformation to be incorporated explicitelyinto the analysis (39).

The Bayesian Approach

A Bayesian statistical analysis allows combina-tion of two forms of information: priorknowledge about parameter values drawnfrom the scientific literature, and data fromTK experiments (96,97). In this section wefirst present the general principles of Bayesianstatistical analyses and then we explain howlikelihood and prior distributions can beelicited in the case of a population TK model;then, using the Metropolis Hastings (MH)sampler as an illustration, we explain howMarkov chain Monte Carlo (MCMC) algo-rithms proceed, and finally, we illustrate howinference and predictions can be made fromMCMC outputs.

General principles. In a Bayesian setting,all model unknowns, θ, are considered ran-dom variables. The probability distributionof an unknown model is interpreted in termsof degrees of belief about possible values ofthat quantity. Before conducting an experi-mental study, a prior probability distribu-tion, p(θ) is constructed to reflect currentknowledge of θ. This prior distribution isthen updated using the data values, y, toyield a posterior probability distribution ofmodel unknowns. Bayes’ theorem indicatesthat the posterior probability distribution ofθ given y, p(θ|y), is proportional to the prod-uct of the likelihood by the prior p(θ). In aBayesian framework, the likelihood of a par-ticular data set, given an assumed model andits parameter values, is the conditional prob-ability distribution of the observed data,p(y|θ). Bayes’ rule can be viewed as a formulathat shows how existing beliefs, formallyexpressed as probability distributions, aremodified by new information (98):

p(θ|y) ∝ p(y|θ) × p(θ). [4]

All inferences about θ or a particular set of itscomponents follow from the posterior distrib-ution p(θ|y): means, standard deviations(SDs), quantiles, correlations, and marginaldistributions are obtainable by integration ofp(θ|y). In the case of the above populationTK model, model unknowns, θ, comprise theindividual TK parameter vectors ψi, i = 1,. . . , I (denoted ψ in the following, for conve-nience), the population parameter vector ormatrix, µ and Σ, and a variance term for themeasurement errors, σ2. Hence, Equation 4takes the form:

p(ψ,µ,Σ,σ2|y) ∝ p(y|ψ,µ,Σ,σ2) × p(ψ,µ,Σ,σ2).[5]

It is possible to rewrite Equation 5 as aproduct of simpler conditional distributions,considering the following arguments, whichare legitimate in any similar hierarchicalmodel. First, arguments of conditional inde-pendence (99–101) allow us to simplify thefirst term of the right side of Equation 5:

p(y|ψ,µ,Σ,σ2) = p(y|ψ,σ2). [6]

Second, assuming that the experimentalerror variance is a priori independent of (ψ,µ, Σ) (which is reasonable in most applica-tions), and by the definition of conditionalprobabilities, the second term of the rightside of Equation 5 can be written as:

p(ψ,µ,Σ,σ2) = p(ψ,µ,Σ) × p(σ2)

= p(ψ|µ,Σ) × p(µ,Σ) × p(σ2), [7]

leading to the following expression for p(θ|y):

p(ψ,µ,Σ,σ2|y) ∝ p(y|ψ,σ2) ×p(ψ|µ,Σ) × p(µ,Σ) × p(σ2); [8]

finally, µ and Σ are usually assumed to be apriori independent, allowing the factorizationof p(µ, Σ):

p(ψ,µ,Σ,σ2|y) ∝ p(y|ψ,σ2) ×p(ψ|µ,Σ) × p(µ) × p(Σ) × p(σ2). [9]

Even though Equation 9 involves simplercomponents than Equation 5, it is impossibleto obtain an analytical expression for p(θ|y),in particular because of the nonlinear form ofthe TK model. Until recently, this intractabil-ity has been a strong impediment to practicalBayesian analyses. Fortunately, the recentadvent of MCMC methods has alleviated anumber of these difficulties. These methodsare powerful tools to provide samples of para-meter values from p(θ|y) even without knowl-edge of the analytical expression of p(θ|y)(102,103). They have largely contributed tothe recent proliferation of Bayesian analysesin applied statistics. These methods requirethe functional forms for p(y|ψ,σ2) (the likeli-hood), p(ψ|µ,Σ) (the population model perse), and p(µ), p(Σ), and p(σ2) (the priors).

Specification of likelihood and priordistributions. The choice of a likelihoodfunction embodies an error model and hasbeen discussed previously. For example, con-sider the 15-parameter (K = 15) PBTK modeldepicted in Figure 1 and the associated popu-lation model in Figure 4 (38). This modelwas calibrated using TETRA concentrationdata measured in the exhaled air and venousblood of I = 6 human volunteers exposed to70 and 144 ppm of TETRA for 4 hr (104).To specify a likelihood on the observed indi-vidual concentrations (ni observations forsubject i ), it was assumed that analytical


errors were independent and lognormallydistributed. The first-stage likelihoodp(y|ψ,σ2) was then:

[10]

The variance term σ2 was a vector with twocomponents: σ1

2 for the measurements inblood, and σ2

2 for the measurements inexhaled air; these measurements had differentexperimental protocols and were thus likelyto have different precisions.

Interindividual variability was describedby assuming that each component ψik of theindividual TK parameter sets ψi, was distrib-uted lognormally, with population para-meters µk and Σ 2

kk (in log-space). So thepopulation model p(ψ|µ,Σ) was given by:

[11]

and

lnψik ∼ N(lnµk,Σ2kk). [12]

The use of a probability distribution torepresent prior knowledge offers a unifiedway to account for precise as well as vagueinformation available before experiments.There is a large body of literature on the elic-itation of prior distributions (96,105,106).When little prior knowledge is available on aparticular parameter value, a noninformativedistribution (i.e., flat-shaped) can be used.Conversely, in the case of stronger priorknowledge, the distribution shape, location,and dispersion parameters should be chosento represent that information as appropri-ately as possible. As stated previously, there isusually very little prior knowledge of theparameter values for a particular individualbecause information obtainable from the lit-erature relates more often to average valuesor variances for a human population. Withina population model, such information can bedirectly used under the form of prior distrib-utions for the population parameters µ and Σ(Figure 4). For example, when a PBTKmodel is considered, reference values avail-able in the literature (71,107) for physiologi-cal parameters such as blood flows and organvolumes yield a sound basis for setting upinformative prior distributions. For chemi-cal-specific parameters, prior information isoften less precise but can still be obtainedfrom previously published experiments (forexample, in vitro determinations of partitioncoefficients or estimates of metabolic para-meters obtained from in vivo experiments).

In all cases, it is preferable when settinguncertainties on population parameters to beconservative and set the prior populationvariances higher rather than lower whenthere is ambiguity. To stay in physiologicalor biochemical plausible ranges, truncateddistributions can be used.

In the example cited above, all populationparameters were assumed a priori independent(after scaling functions had been applied). Theprior distribution assigned to each term, µk, ofthe population mean µ was a truncated nor-mal distribution (with parameters Mk and Sk

2

in log-space). So the prior on the mean popu-lation vector p(µ) was given by:

[13]

and

µk ∼ N(Mk ,Sk2) [14]

The prior distribution assigned to each popu-lation variance Σ2

kk was an inverse-gamma dis-tribution with shape parameter equal to one(to indicate large uncertainties) and scaleparameter ωk

2 (ωk2 was the prior estimate of

the true population variance). p(Σ) wastherefore given by:

[15]

where

Σ2kk ~ InvGamma(1,ωk

2) [16]

The quantities Mk, Sk2, and ωk

2 are calledhyperparameters. They directly embody priorknowledge. For each k, Mk relates to the loca-tion of the population mean µk, Sk

2 character-izes the (prior) uncertainty associated with thislocation; ωk

2 is used to represent prior beliefson the variability of the TK parameter ψk

within the population. Variability and uncer-tainty were thus formally distinguished. As anexample, consider the metabolic parameterVm, which represents the maximum rate ofTETRA metabolism. To reflect the large prioruncertainty associated with that parameter,exp(Sk

2) was set to 10, while exp(ωk2) was set

to 2. This indicates that this parameter wasbelieved to vary by a factor 2 in the popula-tion studied, but its true population mean wasuncertain by a factor of 10. It would be diffi-cult to express such uncertainties without anexplicit hierarchical model.

Finally, the prior distributions assignedto the experimental error variances σ1

2 andσ2

2 were standard noninformative priors forvariances (108):

p(σ2) = p(σ12,σ2

2) ∝ σ 1–2 × σ2

–2. [17]

Computational aspects—MCMCsampling. MCMC methods are iterativesampling schemes that provide directapproximations to a complex joint posteriordistribution (by random draws from it setsof parameter values). These methods consistin constructing a Markov chain that con-verges, as iterations progress, toward theposterior distribution, p(θ|y). VariousMCMC methods have been developed todate; they differ in ways the Markov chain isconstructed. The MH algorithm proved inour experience to be very efficient in dealingwith Bayesian population TK models.Briefly, it proceeds as follows: at the begin-ning, all model unknowns are assigned val-ues; for example, by sampling from theirrespective prior distribution. At each follow-ing iteration step, each component, θk, ofthe parameter vector θ is eventually updatedaccording to the following acception/rejec-tion rule: a proposed value, θk´, is sampledfrom a “proposal” distribution (e.g., normal,centered on the current value θk). The jointposterior density is then computed at θk andθk´ (up to a proportionality constant, usingEquation 9). Label these two density valuesπ and π´. If the ratio π´/π exceeds 1, thenew value θk´ is accepted and replaces θk ;otherwise, θk´ is accepted only with proba-bility π /π. In case of rejection of θk´, thevalue θk is kept. After (eventually) upatingall components of θ sequentially, their val-ues are recorded, which completes one itera-tion of the Markov chain. Many iterationsare typically needed. It has been shown thatunder some regularity conditions (i.e., theMarkov chain has to be irreducible, aperi-odic, and positive recurrent), the chain con-verges toward the distribution of interest,i.e., the joint posterior distribution. Thismeans that after a sufficient number of itera-tions, the samples generated by such aprocess can be considered approximate sam-ples from p(θ|y), and thus can be used tomake inferences. Implementation details(e.g., how to choose the proposal distribu-tion) and techniques to monitor convergenceare well described in the literature(102,103,109–111) and will not be discussedhere. Briefly, the main desired characteristicsof the proposal distribution are related toease of simulation and approximation of theposterior distribution.

MCMC methods provide joint distribu-tions of parameter estimates appropriate foruncertainty analyses because they account forthe full dependence structure between para-meter estimates. They are directly usable asinputs for uncertainty analysis of exposure–dose relationships.

MCMC methods can actually be viewedas extensions of the traditional or standardMonte Carlo methods for uncertainty analysis

p p kkk

K

Σ Σ( ) = ( )∏=

2

1

p p kk

K

µ µ( ) = ( )∏=1

p p ik k kkk

K

i

I

ψ µ ψ µ, , ,Σ Σ( ) = ( )∏∏==

2

11

p p y

N f t E

i ii=

I

ij ij

ni

i=

I

ij i i ij

ni

i=

I

y ψ σ ψ σ

ψ ϕ σ

, ,

ln , , , , .

2

1

2

11

2

11

( )∏ = ( )∏∏ =

( )[ ]∏∏

=

=


BERNILLON AND BOIS


(9). Figure 5 is an illustration of MCMCsampling, compared to simple Monte Carlosampling. In the former, the values drawn foreach parameter start from its prior distribu-tion and converge, as iterations progress, to adata-adjusted posterior distribution. In thecase of simple Monte Carlo sampling, the val-ues are always drawn from the prior distribu-tion. The use of uniform priors in a Bayesiananalysis leads to posteriors strictly propor-tional to the data likelihood and is equivalentto the frequentist maximum likelihood esti-mation. Conversely, if the data do not convey

information about some parameters, thecorresponding posteriors are equal to the priorsand this approach becomes equivalent to stan-dard Monte Carlo sampling from the priors.

Analyzing MCMC outputs. MCMC sam-pling methods provide joint multivariatesamples of parameter values from p(θ|y).Posterior means, SDs, and quantiles of anysingle (or set of) component(s) of θ can beestimated from these samples by computingtheir equivalent in the MCMC output. As anillustration, we present some results obtainedby calibrating, within a Bayesain hierarchicalframework, the TK model depicted in Figure2 with the data of Bernillon et al. (55). Thesedata consisted in concentration–time profiles

of TETRA in air and blood and TCA inblood and urine, followed for 1 week afterexposure for six volunteers exposed once ortwice to 1 ppm TETRA for 6 hr.

Histograms can be used to represent themarginal distribution of individual or popula-tion parameters. Figure 6A is a histogram ofthe estimates of TETRA clearance (mean,0.17 L/min1; SD, 0.05) for subject A. Figure6B is a histogram of the distribution of thepopulation mean of TETRA clearance (mean,0.16 L/min; SD, 0.03) for the six subjectsstudied. The distribution of the populationmean was slightly skewed.

In Bernillon et al. (55), four of the sixsubjects studied underwent repeated expo-sures to TETRA; intraindividual variabilitywas thus assessed in addition to interindivid-ual variability. This was achieved by addingone level to the hierarchy: the first level wasthe occasion level (where the observationswere available); at the individual level, occa-sion-specific parameters were supposed tovary among exposures around a subject-specific mean; at the population level, thesesubject-specific means were supposed to varyamong individuals. Figure 7 illustrates theposterior distributions of TETRA clearance ateach of these three levels: occasion, individualmean, and population mean. For display pur-poses, the distributions are approximated bynormal (occasion-level parameters) or lognor-mal (individual or population mean parame-ters) distributions, whose means and SDshave been computed on the posterior sam-ples. A subject's average clearance estimate isnot only influenced by his occasion-specificestimates but also by the population mean. Inother words, sharing information across indi-viduals and occasions occurs through thehierarchical structure.

In the six-subject sample studied,interindividual differences in clearancereached a factor of 1.2 (between subjects Aand C); intraindividual (i.e., interoccasion)variability reached a factor of 1.3 (subject C).Inference for a larger population is given bythe size of the posterior interindividual andintraindividual variances, whose correspond-ing coefficients of variation were estimated at33 and 18%, respectively. Using the popula-tion parameter estimates, it is possible to sim-ulate additional hypothetical individuals.

Correlations between parameters can alsobe studied through correlation matrices (esti-mated from the MCMC output) or simplescatterplots. Figure 8 illustrates the correla-tion between the occasion-specific estimatesof TETRA clearance and the TETRA cen-tral/air partition coefficient for subject A.These two parameters are negatively corre-lated (r = –0.33).

Distributions of model predictions of tox-icologic interest can also be easily obtained.

Figure 6. Posterior distributions of TETRA clearance for

subject A (A ) and mean population clearance (B )

obtained by calibrating the TK model depicted in Figure

2 to the data of Bernillon et al. (55). The histograms bin

3,000 parameter values sampled from their posterior

distribution.

Figure 5. Illustration of MCMC sampling compared to

simple Monte Carlo sampling. In MCMC sampling (dots),

the values drawn for any parameter θ start from the

prior distribution and converge, as iterations progress, to

a data-adjusted posterior distribution. The posterior den-

sity corresponds to the product of the prior density by

the data likelihood. In the case of simple Monte Carlo

sampling (crosses), the values are always drawn from

the prior distribution. Reproduced from Bois (119) with

permission from Environmental Health Perspectives.

101

100

10–1

10–2

10–3

10–4

0 200 400 600 800 1,000

Iteration

θ

Posterior

Prior

A

B

0 0.1 0.2 0.3 0.4 0.5

TETRA clearance (L/min)

Figure 7. Posterior distributions of TETRA clearance

parameters (occasion-specific, individual mean, and pop-

ulation mean) estimated by calibration of the TK model

depicted on Figure 2 with the data of Bernillon et al.

(55). The population mean distribution is denoted by µ

(thick line), individual means by unsubscripted capitals

A–F (thin lines), occasion-specific distribution values by

subscripted capitals A1–F2 (dashed lines). The mode of

the distribution of the population mean is indicated by

the vertical dashed line.

TETRA clearance (L/min)

0.05 0.15 0.25 0.35

F1

F2

Fµ

µE

E1E2

D

µD1

µ

C1C

B

B1

B2

A A1

C2

µ

µ


Figure 9 gives the posterior distributions ofthe fraction of TETRA metabolized by the sixindividuals studied, for each occasion ofexposure. These results were obtained byrunning the TK model once for each of 3,000vectors from the MCMC ouptuts for eachindividual and occasion-specific parameterssimulating a 1 ppm exposure to TETRA for 6hr. Large intraindividual variability isobserved following those of TETRA clearanceestimates (Figure 7). Estimates of the fractionof TETRA metabolized after 1 ppm 6-hrexposures range from 14% (subject C, occa-sion 2) to 25% (subject D, occasion 1). Thistends to confirm our results obtained byextrapolating the results obtained by calibra-tion of a PBTK model to high-exposure data(70 and 140 ppm) (38).

After calibration, residual (posterior)uncertainty is reflected by the range of para-meter estimates. For each parameter, an esti-mate of variability is given by the location ofthe corresponding (intraindividual, interindi-vidual) variance parameter. Note that thevariance parameters are themselves affected byuncertainty.

Sensitivity analyses. Bayesian analyses callfor the specification of prior distributions. Alegitimate question is the sensitivity of theresults (either posterior parameter estimatesor model predictions) with respect to thechoice of those priors. It is therefore recom-mended that diagnostics of sensitivity be per-formed. That is quite easy to do afterMCMC sampling, for example, by plottingand analyzing the correlations between resultsand sampled parameter values (39). The priorassumptions about the parameters thatstrongly influence the results can be furtherevaluated. It is possible, for example, to studythe effect of changing prior specifications.

The question of sensitivity to the priorscan also be answered generally. Note first thatthe choice of priors should always be trans-parent, i.e., fully stated, to allow useful dis-cussion of the results. Clearly, the choice ofpriors for noninfluential parameters shouldnot be a problem. For influential parameters,the priors used can either be vague or infor-mative (i.e., biologically motivated andknowledge based) and the posteriors distribu-tions can be either prior- or data-driven. Fourmajor cases can therefore be envisioned:• Both prior and posterior are vague (the

data brought no information). The resultswill be prior dependent and more infor-mation should be acquired either in formof prior knowledge or additional data.

• The prior is vague, but the posterior isinformative (the data are driving the esti-mates). In that case, the conclusionsshould be robust with respect to the prior.

• The prior is informative and reinforced bythe data (data and prior agree). Here also,

if the data are strongly informative, theconclusions should be robust. When therelative weight of prior and data is lessclearcut, further checks of sensitivity arewarranted.

• The prior is informative but the data con-flict with it. The conclusions will be priordependent, but in that case the model, theprior, and the data should be questionedto resolve the discrepancy.

Perspectives

The objective of this article was to present aBayesian statistical approach to the analysis ofpopulation TK models, which offers a num-ber of advantages over previously proposedapproaches. The proposed methodology is farfrom definitely established and several issuesstill must be resolved. For example, severalmodeling assumptions are involved, particu-larly within the population model, thatshould be further validated. Hopefully, expe-rience gained from population analyses usingsimpler models can be transposed to PBPKmodels. Another task is to reduce the associ-ated computational burden so as to quicklyexplore alternative models. Simple MCMCsamplers can sometimes converge slowly oreven fail to converge. Improved algorithmsrecently have been proposed (103,112–115)and should be tested. In addition, the defini-tion of prior distributions is not always aneasy task because of data accessibility.Although it is well known that TK parame-ters exhibit inter- or intraindividual variabil-ity in humans (and other species), the onlyvalues readily available and those commonlyused on physiological modeling are referencevalues for young Caucasian males. The use ofreference values artificially reduces estimatesof population variance. Information aboutpopulation variability may eventually befound by researching the original publica-tions, but even these often lack adequate sta-tistical treatment. Data about the shape of the

distributions are even harder to find. Adatabase giving population distributions ofimportant physiological parameter valuestogether with their correlations is needed.Such a database would be useful for all typesof physiological modeling and for both toxi-cants and drugs. Despite these limitations,experience has been sufficient to demonstratethat the Bayesian approach provides reason-able estimates of uncertainty and variability ofTK model parameters and predictions.

Is it worth making the computationaleffort implied by such an approach? Theanswer to that question is a matter of choiceand scientific judgment. A PBTK model withstandard parameter values might be sufficientfor simple exploratory analyses or for theoret-ical work on model structure. Alternatively,for the full analysis of a data set or for riskassessment purposes, a formal model calibra-tion along the lines presented here appearsnecessary. For example, in a scientific dataanalysis context, Smith (116) illustrated, inthe case of occupational exposure to gasoline,how PBTK and pharmacodynamic modelscould be combined with epidemiological datato evaluate concurrent reasonable hypothesesconcerning mechanisms of toxicity. The sta-tistical framework described in this article isconvenient in such a setting because distribu-tions of internal doses are more suitable thanaverage point estimates in comparing alterna-tive hypotheses. We have shown how toextend the methodology for optimal design-ing of occupational monitoring programs orTK studies (73).

This approach is also particularly suitedfor risk assessment and decision making. Forexample, population extrapolations can beaccomplished if the factors responsible forheterogeneity (e.g., body weight, breathingrate, metabolic constants) are among the listof model parameters. In this case, the popula-tion distributions obtained for these parame-ters, from fitting datasets with few volunteers,


Figure 8. Posterior correlations between TETRA clear-

ance and central/air TETRA partition coefficient esti-

mates (subject A) obtained by calibrating the TK model

depicted in Figure 2 to the data of Bernillon et al. (55).

The dots represent 3,000 pairs of parameter values from

the MCMC outputs. The correlation coefficient is esti-

mated at –0.33.

Figure 9. Prediction of the fraction of TETRA metabo-

lized after an exposure to 1 ppm TETRA for 6 hr by the

six subjects studied at each occasion. Box plots of 3,000

estimates are displayed. Each box encloses 50% of the

predictions; the lines extending from the top and bottom

of each box mark the 10th and 90th percentiles. The line

inside the box represents the median value.

0.40

0.35

0.30

0.25

0.20

0.15

0.10

0.055 6 7 8 9 10 11 12

Central/air TETRA partition coefficient

TE

TR

A c

lea

ram

ce

(L/

min

)

A1 B1 B2 C1 C2 D1 E1 E2 F1 F2

Fra

cti

on

of

TE

TR

A m

eta

bo

lize

d 0.35

0.30

0.25

0.20

0.15

0.10

BERNILLON AND BOIS


can be changed to better reflect the range ofvalues found in the general population or insensitive subpopulations (e.g., children).Alternatively, if these parameters are measuredfor each volunteer and introduced as fixedcovariates during the calibration procedure(117), they can be replaced by distributionswhen making predictions. In this way, thepredictions of interest (e.g., internal metabo-lite dose, fraction metabolized) can be extrap-olated beyond the group of individuals usedfor calibrating the model. Note that this statis-tical approach also can be used to calibrate theother (e.g., fate and transport, cancer effects)models used in risk assessment (118).

Finally, one of the challenges of toxico-logical modeling is the full exploitation ofthe numerous datasets collected during epi-demiological or occupational hygiene studies,generally in settings in which exposure con-centrations are unknown. Most of the time,exposure is represented in a considerably sim-plified and approximate manner, which can bemisleading. It is possible using the above statis-tical framework to consider exposure as one ofthe estimands. A major problem, however,resides in accounting fully for the uncertaintiesstemming from unknown time-varying expo-sures. The impact of particular functional formfor the time evolution of exposure has not yetbeen thoroughly studied and validated. We areconfident that as progress is made towardanswering such questions, TK modeling willbecome a more powerful and widespread toolfor toxicity and risk assessments.

REFERENCES AND NOTES

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