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Quantifying Uncertainty in Predictions of HepaticClearanceJames A. Rogers, Jayson Wilbur, Susan Cole, Paul W. Bernhardt, Jaye Lynn Bupp, MorganJ. Lennon, Nathan Langholz & Christopher Paul SteinerPublished online: 24 Jan 2012.

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Quantifying Uncertainty in Predictions ofHepatic Clearance

James A. ROGERS, Jayson WILBUR, Susan COLE,

Paul W. BERNHARDT, Jaye Lynn BUPP, Morgan J. LENNON,

Nathan LANGHOLZ, and Christopher Paul STEINER

Preclinical predictions of human pharmacokinetic pa-rameters are routinely used in pharmaceutical researchand development. In particular, pharmacokinetic predic-tions are critical in the decision to advance a potentialdrug to the clinic, to determine appropriate dosing reg-imens for first-in-human studies, and as a component oftranslational pharmacology models. Although the associ-ated biological and mathematical models have been ex-tensively discussed in the pharmacokinetic literature, rel-atively little work has been done to explicitly relate theestimation error of these methods to the underlying ex-perimental variability. This article proposes and evaluatesBayesian models for this purpose.

We apply our methodology to a dataset describing bothpreclinical and clinical pharmacokinetic experimentationfor 12 different anonymized drugs. For each drug and foreach preclinical mode of prediction, a credible intervalis computed and compared against estimates obtained bydirect experimentation with human subjects in the clinic.We conclude that many apparent translational differencesmay be readily explained as a function of experimentalerror.

We view this problem as representative of a largerclass of statistical problems in translational medicine,where the mathematics of translation from one species toanother requires multiple experimentally estimated scal-ing factors.

Key Words: Bayesian; Pharmacokinetics; Preclinical; Transla-

tional.

1. Introduction

1.1 General Background

Pharmacokinetics is the study of the absorption, distri-bution, metabolism, and excretion of drugs in the body.One important pharmacokinetic parameter is clearance(CL), the rate at which an organ clears a unit of bloodcontaining drug. Units for clearance units may be, for ex-ample, liters per minute. In many cases, clearance is themost important determinant of the therapeutic dose andpossible toxic dose of a drug and is fundamental to therisk–benefit characterization of a drug.

As a matter of established drug development practiceand by regulatory requirement, the clearance of any in-vestigational new drug is extensively studied in the clinic.Moreover, prior to the clinical evaluation of pharma-cokinetics, preclinical methodology is routinely appliedto predict human pharmacokinetic parameters, includingclearance. Preclinical predictions of drug pharmacokinet-ics will generally be considered in the decision to ad-vance a drug to the clinic, and once the decision is madeto advance a drug to the clinic those predictions will beused to determine appropriate dosing regimens for thefirst study of the drug in humans.

It is important at the outset to distinguish between un-certainty in the predicted value of clearance for an “aver-age human,” and the population variability that is antici-pated around that average value. The importance of pop-ulation pharmacokinetic variability is well recognized, as

c© American Statistical AssociationStatistics in Biopharmaceutical Research

2011, Vol. 3, No. 4DOI: 10.1198/sbr.2011.09019

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reflected in FDA guidance on the topic (Guidance for In-dustry 1999). With no intent to minimize the importanceof population variability it is our opinion (and the opinionof others; see, e.g., Nestorov et al. 2002) that estimationerror associated with a preclinical prediction for an “aver-age human” may also be substantial and worthy of atten-tion. This latter uncertainty arises from various sourcesexperimental error, and is the focus of this article.

1.2 Systemic Clearance and Hepatic Clearance

Systemic clearance (CL) describes how quickly thedrug would leave the blood if administered as an intra-venous (IV) bolus. If an individual is in fact administeredan IV bolus of a drug, if linear pharmacokinetics canbe assumed, and if appropriately timed blood samplesare taken following administration and assayed for drugconcentration, then systemic clearance for that individ-ual may be directly estimated as dose/AUC, where AUCis the area under the concentration-versus-time curve(Rowland and Tozer 1995). Estimation of systemic clear-ance based on data from orally dosed individuals re-quires an adjustment based on an estimate of the drug’sbioavailability. All in vivo data presented here are in factbased on IV bolus dosing and so require no such adjust-ment.

This article focuses specifically on clearance at-tributable to metabolism in the liver, or hepatic clearance.All of the anonymized drugs discussed in this article arebelieved to be cleared primarily via metabolism by liverenzymes known as “P450s,” or “CYPs.” This restrictedfocus allows us to use the notation CLH , CLR, and CLD

to denote estimates of systemic clearance in human, rat,and dog, respectively, with the understanding that sys-temic clearance can also be interpreted as hepatic clear-ance in this context.

For our purposes, it turns out to be notationally andarithmetically convenient to express CLH per unit ofbody weight, for example, liters per minute per kilogramof body weight, while letting CLR and CLD express ab-solute rates for an average animal, in units such as litersper minute. A fourth estimate of clearance, based on hu-man liver microsomes and denoted CLM , is directly re-lated to hepatic clearance per se. We define our estimandof primary interest to be θ = E[CLH ]. The estimandsµR = E[CLR], µD = E[CLD], and µM = E[CLM] are func-tionally related to θ as described in subsequent sections.

Note that throughout this article, only numerical su-perscripts are to be interpreted as exponents. Symbolicsuperscripts are used to indicate variables and species.

1.3 Fractions Unbound

Most drugs are at least partially protein bound in bothblood cells and plasma. Drug molecules that are pro-tein bound are effectively sheltered from enzymatic ac-

tivity, so that only the unbound fraction of a drug can becleared via hepatic metabolism. Letting fu denote an esti-mate of the fraction unbound, the derived quantity CL/fuestimates a conceptual rate at which the drug would becleared from the blood if none of the drug moleculeswere protein bound. We use fuH , fuR, and fuD to denoteestimates of the fractions of drug that are not proteinbound in plasma, in human, rat, and dog, respectively.Their estimands we denote ψ f u(H), ψ f u(R), and ψ f u(D),respectively.

Values for ψ may be estimated based on in vitro ex-perimentation. The drug is incubated with a plasma sam-ple under appropriate conditions, ultrafiltration or equi-librium dialysis is used to separate the bound and un-bound fractions, and the amount of drug in each fractionis assayed. In the most basic scenario, a single biologicalsample is processed and assayed only one time, resultingin a single n = 1 estimate of fraction unbound. Replica-tion is discussed at the end of this section.

An estimate of the fraction unbound in blood cells (asopposed to the fraction in plasma described above), isrequired by the “well-stirred model” methodology de-scribed below, and there is no readily available technol-ogy for directly assaying this fraction. One may, how-ever, estimate the ratio of the total (bound and unbound)concentration of drug in blood cells over the total con-centration of drug in plasma using conceptually straight-forward techniques: one incubates a drug under appro-priate conditions in a whole blood sample, centrifugesto separate the plasma from the blood cells, and assayseach fraction. The resulting ratio we denote BP, and itsexpected value we denote ψBP. Again, in the most ba-sic scenario, a single sample may be processed and as-sayed only one time, resulting in a single n = 1 estimateof ψBP, with possibilities for replication discussed be-low. The unbound concentration of drug is then assumedto be the same in both blood cells and plasma (as alreadynoted, it is the unbound fractions that are not assumed tobe the same, owing to differential binding in blood cellsversus plasma). As a result, the ratio ψ f u(H)/ψBP maybe assumed to be equal to the fraction unbound in bloodcells.

There are many levels of replication that could be em-ployed in the estimation of ψ f u(H) and ψBP. For example,a very limited form of replication of fu would replicateonly at the assay level, using separate aliquots of the in-cubated and separated fractions, whereas a much morecomplete form of replication would replicate the entireprocess using distinct plasma samples from distinct indi-viduals. We do not attempt here to characterize the na-ture of the replication that is actually used in practice,although this is clearly of critical importance. For ourpresent purposes we proceed as if replication, where itexists, is at a level adequate to characterize all of the rele-vant sources of experimental variability. One of our goals

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in developing this statistical methodology is to make anexplicit quantitative connection between the degree ofreplication and the precision of the estimate of clearancethat ultimately results. Our hope is that this coherent sta-tistical framework will help illuminate the value of statis-tical design principles in all of the experiments associatedwith preclinical predictions of clearance.

1.4 Allometric Scaling

As described above, systemic clearance may be di-rectly estimated for each individual that is administeredan IV bolus of a drug. In this manner, systemic clearancemay be estimated for each of several animals of a givenpreclinical species. Typical choices of preclinical speciesare rat and dog. Taking rat as an example, the samplemean of systemic clearance values of three rats, CLR

1 ,CLR

2 , CLR3 might be taken as an estimate of µR, the pop-

ulation mean of systemic clearance in rat. This value canthen used in conjunction with an estimate of fuR (basedon the in vitro experiment described above) to obtain anestimate of unbound clearance, µR/ψ f u(R), a conceptualrate inferred to be the systemic clearance that would beobserved in rat if there were no plasma protein binding.

Unbound clearance is thought to be translatable fromthe preclinical species to human via allometric scaling(Adolph 1949). That is, an approximate power relation-ship between body weight and unbound clearance (inter-species) is used to scale unbound clearance from the pre-clinical species to human. Noting that θγ(BW) is the totalclearance (in liters per minute, not liters per minute perkilogram of body weight) of an average human, a typicaluse of “single-species” allometry would use the follow-ing relationship:

(θγBW (H))/ψ f u(H)

µR/ψ f u(R)=

[γBW (H)

γBW (R)

]3/4

, (1)

where γBW (H) and γBW (R) denote an average body weightfor humans and rats, respectively. The term “single-species scaling” refers to the fact that the exponent in theabove equation is fixed at a particular value, in this case3/4, whereas in “multi-species scaling” the value of theexponent is estimated from data (Hosea et al. 2009).

1.5 The Well-Stirred Model

Under the assumptions described above, clearance ofdrug from the blood is mediated entirely by enzymes inthe liver known as “P450” or “CYP” enzymes. Micro-somes are small vesicles that may be isolated from liverhomogenate via centrifugation, and which contain theP450 enzymes. Microsomes are therefore used as an invitro model for liver metabolic function. A fixed concen-tration of drug is incubated with microsomes in multiple

test tubes, allowing the metabolism of the drug to occuruntil the reaction is quenched. The reaction is quenchedat different times, so that one may ultimately producea concentration versus time curve from the experiment.This curve is used to obtain a single estimate of CLM . Asin preceding sections, this value is divided by an estimateof f uM to obtain an estimate of µM/ψ f u(M), conceptu-ally the clearance rate that could be achieved by the mi-crosomal system in the absence of binding. This quantityis sometimes referred to as “intrinsic clearance.” Sincea fixed and somewhat arbitrary quantity of microsomalprotein is used in the in vitro system, scaling is neces-sary to obtain the values expected in a whole organism(Ito and Houston 2005; Shiran et al. 2006). To this end,the initial estimate of intrinsic clearance per milligram ofmicrosomal protein is multiplied by the average numberof milligrams of protein per gram of liver, and then by theaverage number of grams of liver per kilogram of bodyweight. The resulting quantity is meant to reflect the rateof clearance that could be achieved by an “average liver”in the absence of any binding and assuming that the liverenzymes could act on all of the drug at once. The lat-ter assumption is unrealistic, since only a fraction of thedrug is presented to the liver, via blood cells, at any givenmoment. Letting γQ denote the average rate of hepaticblood flow in humans, the following equation, resultingfrom the “well-stirred model,” reflects the flow of bloodinto and out of the liver, and also introduces the effect ofprotein binding in the blood cells:

θ =

(γQ/γBW (H)

)×(

µM/ψ f u(M))×(

ψ f u(H)/ψBP)

(γQ/γBW (H)

)+(µM/ψ f u(M)

)×(ψ f u(H)/ψBP

) .

(2)

1.6 Quantifying Uncertainty: Current Practice andProposed Methodology

Given estimates for all quantities in Equation (1) ex-cept θ , or given all quantities in Equation (2) except θ ,one can solve to obtain an estimate of θ itself, and thisis indeed the conventional approach. While this conven-tional computation is reasonable and easily understood, aserious shortcoming is the lack of standard errors or con-fidence intervals associated with the predictions. Statis-tical properties of this conventional methodology are noteasily assessed using popular statistical software, sinceeach prediction is a function of multiple datasets, eachfrom a totally different type of experiment. For exam-ple, as described above, a rat-based prediction requiresone dataset to produce the estimate CLR, one dataset togenerate the estimate fuR, and one dataset to generate theestimate fuH . A dog-based prediction must similarly besupported by three separate datasets, and a microsomalprediction is a function of four different datasets, each

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Table 1. Number of observations for each variable for each of the 12 compounds. Numbers in bold were reported as means of three observations

Compound CLH CLR CLD CLM f uH f uR f uD f uM BP

1 3 3 8 4 7 4 4 2 92 22 3 4 2 2 4 3 2 23 17 3 4 4 22 10 5 9 34 14 1 1 2 2 1 1 3 25 26 3 3 4 5 3 3 1 36 16 3 1 1 13 2 16 1 27 8 3 3 3 4 2 2 1 18 9 6 5 4 2 2 1 1 19 20 1 4 2 2 5 3 1 410 20 4 1 1 2 11 40 6 211 12 3 3 4 11 14 12 2 112 16 2 4 3 5 7 5 1 1

from a totally different kind of experiment (one for CLM ,one for fuM , one for fuH , and one for BP). We thereforedevelop associated Bayesian models that may be used tocompute Bayesian credible intervals, reflecting the prop-agation and cancellation of error from the various exper-imental inputs.

2. Data

We analyze data collected internally by Pfizer GlobalResearch and Development. The data represent 12 dif-ferent drugs, presented here anonymously, all of whichwere tested in human, rat, and dog using IV bolus dosing,all of which were presumed to be cleared primarily byP450 metabolism, and all of which had been tested in thein vitro microsomal systems described above. The datawere generated in the course of individual research anddevelopment programs for each of the drugs, and wereaggregated retrospectively. As no prospective effort wasmade to standardize the design and execution of these ex-periments, some heterogeneity of approaches is to be ex-pected. This is evident from the varying degree of repli-cation for different compounds (see Table 1). One mayreasonably speculate that the level of replication may alsohave differed across compounds (e.g., in some cases re-assaying multiple aliquots of the same sample, in somecases assaying different samples). While this would beextremely important, it is beyond the scope of our cur-rent effort to investigate this (refer to the discussion ofreplication in Section 1.3).

Table 2 displays the means of the data. Note that CLM

does not have a value for compound 12; rather, the valuehas been determined to be <7.182 mL/min/kg, the lowerlimit of quantitation for the assay. The implementation ofour proposed models accommodates this censoring of thedata in a probabilistically correct manner.

3. Final Models

For the sake of brevity and clarity, we first describe thefull final models that were ultimately selected, reservingour discussion of alternative models and model selectionfor subsequent sections.

In general, all candidate models (including the finalmodel) attempted to use priors that reflected known phys-iological and measurement boundaries but were other-wise “noninformative.” For example, priors for true meanclearance had support only on the positive real line, butwere assigned large variances in an attempt to obtain es-timates that would be approximately equal to likelihood-based estimates.

Exceptions to the use of “noninformative” priors weremade for parameters corresponding to “known con-stants,” that is, true average body weights γBW (H), γBW (R),and γBW (D) and true average rate of hepatic blood flowγQ. We chose to model these “known constants” as ran-dom variables since none of them is known with abso-lute precision. Nonetheless, our priors reflect the reason-able degree of certainty with which these parameters areknown. As noted in the model description below, nega-tive values for “γ ” parameters were disallowed by lefttruncation of the Normal distributions at zero. Techni-cally, this truncation results in distribution means greaterthan the nominal mean. However, in all cases it is only avery extreme portion of the left tail that is truncated, sothat the resulting distributions have means that are notpractically different from the nominal values. We reit-erate that modeling of population variability, while ul-timately essential, is not a goal in our current effort; thevariances of these priors reflect uncertainty about pop-ulation means, not variability of measurements within apopulation.

In the following model description, drugs are indexedby the subscript i = 1, . . . ,12 and replicates of a given

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Table 2. Arithmetic means of observations for each variable for each of the 12 compounds

Compound CLH CLR CLD CLM f uH f uR f uD f uM BP

1 5.867 53.833 11.600 11.600 0.031 0.053 0.142 0.820 0.6422 12.600 26.500 13.125 36.000 0.705 0.120 0.363 0.920 0.8053 9.800 55.400 131.650 60.716 0.070 0.061 0.113 0.789 3.3334 20.076 34.000 15.000 39.200 0.686 0.640 0.750 0.943 0.7055 9.800 66.733 26.000 14.151 0.044 0.147 0.087 0.240 1.1286 5.376 397.267 12.000 54.100 0.010 0.021 0.059 0.450 0.8007 17.902 16.133 28.833 130.095 0.114 0.239 0.277 0.900 0.5908 34.762 139.500 76.200 25.610 0.560 0.630 0.610 1.000 2.7809 6.400 110.000 29.000 25.468 0.435 0.403 0.472 1.000 0.87510 2.600 15.863 5.530 16.727 0.079 0.062 0.075 0.754 0.95011 9.800 64.967 32.925 41.178 0.064 0.133 0.266 0.810 0.54012 2.200 20.150 65.781 <7.182 0.584 0.831 0.853 1.000 0.700

variable for a given drug are indexed by the subscript j(with the range of j differing by variable and by drug).Except where noted, it is implied that all likelihood defi-nitions pertain to all i and j, and independence is impliedfor different i and j, conditional on the parameters ap-pearing in the right hand side of the distributional state-ment. For example, we use:

CLHi j ∼ N(θi,σ

2(H)i )

as shorthand for:

{CLH

i j

∣∣θi,σ

2(H)i

iid∼ N(θi,σ

2(H)i ) ∀ j

}independently ∀ i.

3.1 Final Model for Observed Human Clearance

3.1.1 Likelihood

CLHi j ∼ N(θi,σ

2(H)i ).

3.1.2 Priors

θi ∼ Gamma(1.0001,0.1)

1/σ2(H)i ∼ Gamma(1.0001,0.1).

Here and throughout, we use the “rate” parameteriza-tion of the Gamma distribution, for which the mean of thedistribution is the ratio of the first parameter (“shape”) tothe second (“rate”) parameter (as opposed to the “scale”parameterization, for which the mean of the distributionis the product of the two parameters). In this case the re-sulting prior mean for θ is approximately 10, which isa reasonable “typical” clearance rate in liters per minuteper kilogram of body weight. The shape parameter for theGamma distribution is set to 1.0001 rather than 1 in orderto remove the support of the distribution at θ = 0. Thisnumerical convenience is employed for subsequently de-fined Gamma distributions as well.

3.2 Final Model for Rat Data

3.2.1 Likelihoods

CLRi j ∼ N(µR

i ,σ2(R)i ),

where µR =(θγBW (H))ψ f u(R)

(γBW (H)

γBW (R)

)3/4ψ f u(H)

fuRi j ∼ Beta

(ψ f u(R)

i τ f u(R),(

1−ψ f u(R)i

)τ f u(R)

)

fuHi j ∼ Beta

(ψ f u(H)

i τ f u(H)i ,

(1−ψ f u(H)

i

)τ f u(H)

i

).

The expression for µR results from solving equation 1for this parameter. Note also that the Beta distributionsare parameterized so that E[fu] = ψ . Note also that thelack of subscript on τ f u(R): a common variance for fuM

was assumed for all drugs. This issue is discussed furtherin the section on alternative models.

3.2.2 Priors

θi ∼ G(1.0001,0.1)

1/σ2(R)i ∼ Gamma(1.0001,0.1)

ψ f u(R)i ∼ U(0,1)

π(

τ f u(R))

=1

(1 + τ f u(R))2

ψ f u(H)i ∼ U(0,1)

π(

τ f u(H)i

)=

1

(1 + τ f u(H))2

γBW (H)i ∼ N(70,42), left truncated at 0

γBW (R)i ∼ N(0.25,0.0252), left truncated at 0

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3.3 Final Model for Dog Data

The model for dog-based predictions is essentially thesame as that for rat-based predictions, the only differ-ence being the value of the prior mean for average bodyweight. For dog we use:

γBW (D) ∼ N(12.5,12), left truncated at 0.

3.4 Final Model for Microsomal Data

3.4.1 Likelihoods

CLMi j ∼ N(µM

i ,σ2(M)i ),

where µM =θψ f u(M) γQ

γBW (H)(

γQ

γBW (H) −θ)(

ψ f u(H)

ψBP

)

fuMi j ∼ Beta

(ψ f u(M)

i τ f u(M),(

1−ψ f u(M)i

)τ f u(M)

)

fuHi j ∼ Beta

(ψ f u(H)

i τ f u(H)i ,

(1−ψ f u(H)

i

)τ f u(H)

i

)

BPi j ∼ Gamma

((ψBP

i )2

σ 2(BP)i

,ψBP

i

σ2(BP)i

)

.

The expression for µM results from solving Equation(2) for this parameter. Similar to the Beta likelihoods forfractions unbound, the Gamma likelihood for BP is pa-rameterized so that E[BP] = ψBP. As with the rat anddog models, note there is no subscript on τ f u(M), that is,a common variance was assumed for all drugs for fuM .

As discussed in Section 2, microsomal clearance forcompound 12 was assayed only one time and the resultwas below the limit of quantitation. Since there is no spe-cific value for CLM

12,1 on which to condition, we insteadconditioned on the event {0< CLM

12,1 < 7.182}.

3.4.2 Priors

θi ∼ Gamma(1.0001,0.1)

1/σ 2(M)i ∼ Gamma(1.0001,0.1)

ψ f u(M)i ∼ U(0,1)

π(

τ f u(M))

=1

(1 + τ f u(M))2

ψ f u(H)i ∼ U(0,1)

π(

τ f u(H)i

)=

1

(1 + τ f u(H))2

ψBPi ∼ Gamma(0.01,0.01)

1/σ 2(BP)i ∼ Gamma(1.0001,0.1)

γQ ∼ N(1400,502), left truncated at 0

γBW (H) ∼ N(70,42), left truncated at 0

An additional component of our prior belief isP(γQγBW (H) > θ) = 1, reflecting a physiological con-straint under the well-stirred model. Note that µM takesa positive value if and only if γQγBW (H) > θ . Our tech-nique for implementing this constraint is described inSection 5.

4. Alternative Models

The model described in Section 3 was chosen fromamong several candidate models. We now describethe other candidate models considered, as well as ourrationale for selecting our final model.

4.1 Likelihoods

Use of a Normal distribution for CLH is reasonablywell supported by exploratory data analysis (see Fig-ure 1); however, our dataset does not include sufficient

Figure 1. Distribution of CLH for representative low, medium, and high clearance drugs.

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Table 3. Likelihoods considered for clearance. These distributionalalternatives were considered for CLH , CLR, CLD, and CLM . All threedistributions are parameterized such that E[CL] = µ and var[CL] = σ2.

Likelihoods

CL∼ N(µ ,σ2)CL∼ Gamma

(µ2/σ2,µ/σ2

)

CL∼ Log-Normal(log(µ)− log((1 + µ2/σ2)/2),

log((1 + µ2/σ2)/2))

replication to make this determination with confidence.Still fewer replicates are available to assess the distribu-tional assumptions for CLR, CLD, and CLM (see Table 1).We therefore compared our final model to similar modelsusing Gamma and log-normal likelihoods for these vari-ables. Parameterizations are provided in Table 3.

4.2 Priors

Our rationale for choosing a Gamma prior for θ is thatit provided no support for negative values and permit-ted easy manipulation of the informativeness (precision)of the prior while maintaining the desired mean of ap-proximately 10 liters per minute per kilogram of bodyweight. Hyperparameters α and β were varied to achievea range of variances. A variance of approximately 100(using α = 1.0001 and β = 0.1) was sufficiently uninfor-mative that the resulting estimates were generally closeto the conventionally computed estimates (and so wouldpresumably also be close to likelihood based estimates).Gamma distributions with larger variances (e.g., 1000,using α = 0.1 and β = 0.01) were evaluated; however,these unnecessarily assigned increased density to com-pletely implausible values for clearance and resulted inwider credible intervals, yet with very little further re-duction of the apparent bias.

In addition to the Gamma priors described above, trun-cated Normal priors (left truncated at zero) for θ werealso evaluated; however, these resulted in final estimatesthat appeared to have a consistently positive bias relativeto conventional estimates.

4.3 Variance Pooling

As described in Section 3, variances for fuR, fuD, andfuM were pooled in the final model by assuming commonvalues for τ parameters. Models specifying distinct τ val-ues for each drug were also considered. For example, thenonpooled variant of the model for rat fraction unboundis:

fuRi j ∼ Beta

(ψ f u(R)

i τ f u(R)i ,

(1−ψ f u(R)

i

)τ f u(R)

i

)

π(

τ f u(R)i

)= 1/(1 + τ f u(R)

i )2,

which differs from the final model in the use of i as asubscript on τ f u(R).

5. Computation

Models were fit using the WinBUGS software, version1.4 (Lunn et al. 2000). Data summary and visualizationwere carried out in the R language (R Team 2008).

As an approximate implementation of the constraint{γQγBW (H) > θ}, the likelihood for CLM

i was modifiedas follows:

CLMi ∼ N(µ∗(M)

i ,σ 2(M)i ),

where

µ∗(M)i =

{µM

i γQi γBW (H)

i > θi

107 γQi γBW (H)

i ≤ θi

Since all of the observed values of CLMi are extremely

unlikely under the scenario µ∗(M) = 107, the effect of thismodel implementation is to strongly favor values of θ ,γQ, and γBW (H) such that µM = µ∗(M), that is, parametervalues that are physiologically possible under the well-stirred model.

6. Model Checking and Model Selection

6.1 Posterior Predictive Checks

Informal posterior predictive checks were performedfor all parameters for all models by examining his-tograms of posterior predictive samples with referencelines for observed values superimposed. In general, ob-served values for CLH , CLR, CLD, and CLM were consis-tent with posterior predictive distributions from all vari-ants of the associated models. Consequently, no mod-els were discarded on the basis of posterior predictivechecks. Some inconsistencies did arise between the pos-terior predictive distribution and observed values for fuH ,fuR, fuD, and BP, and in particular fuM . Most problemswere associated with observed fu values close to or equalto one.

6.2 Deviance Information Criterion

Deviance information criterion (DIC) values for themodels we considered are provided in Table 4. As lowerDIC values are considered better (e.g., see Spiegelhalter

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Table 4. DIC values for model comparison. The data above show theDeviance Information Criterion for the clearance model when τ f u(R),τ f u(D), and τ f u(M) were pooled for the 12 compounds.

Likelihoods Normal Gamma Log-normal

CLH 812.94 799.70 806.56CLR −261.17 −255.41 −192.90CLD −549.91 −533.64 −498.78CLM −270.39 −252.20 −195.87

et al. 2002), the values suggest some preference againstuse of a log-normal likelihood for CLR, CLD, and CLM .Beyond this, there appears to be only negligible differen-tiation based on DIC.

6.3 “Operational” Accuracy and Precision

To rigorously assess the statistical accuracy and pre-cision of our methodology, Monte Carlo simulation un-der known states of “truth” would be required. In the ab-sence of such simulation, we may nonetheless investigatea sort of operational accuracy and precision. Specifically,we may treat the estimates based on human data alone as“truth,” as indeed these estimates are generally the bench-marks against which preclinical predictions are assessed.One must acknowledge in this assessment that both thetranslational model (i.e., the biological assumptions re-lating the results from the different biological systems)and the statistical model are being assessed simultane-ously. While this is a limitation (since a failure in thecombined methodology cannot be easily attributed to ei-ther the statistical or biological methodology alone), thiscombined assessment is quite relevant from an appliedperspective.

Noting that the posterior intervals for θ based on thehuman data alone are generally extremely tight aroundthe associated point estimates, we take these human-based point estimates (i.e., the posterior medians) as ouroperational definition of the true value of θ , for the pur-pose of evaluating the bias associated with the preclini-cal predictions. That is, letting θ̂H , θ̂R, θ̂D, θ̂M denote theestimates of θ based respectively on human data only,rat data only, dog data only, and microsomal data only,our operational definitions of bias for the three preclinicalmethodologies are: BiasR = θ̂R− θ̂H , BiasD = θ̂D− θ̂H ,BiasM = θ̂M− θ̂H , where these quantities may be evalu-ated on a per-compound basis. To obtain relative opera-tional bias, we divide the operational bias by θ̂H .

Similarly, the posterior variances of the preclinical es-timates may be considered as an operational measure ofimprecision. These variances are clearly a function of thenature and extent of experimental replication, so that alarge posterior variance does not necessarily reflect inher-

ent imprecision associated with the translational models,nor does it necessarily reflect statistical inefficiency. In-deed, as mentioned previously, a goal of this research ef-fort is to establish a clear quantitative link between repli-cation and experimental design on the one hand, and re-sultant statistical imprecision on the other. For currentpurposes, posterior variances are only indicative of theperformance of the combined methodology as practiced.Posterior variances were expressed as relative standarderrors, that is, relative to θ̂H .

7. Results

Ninety-five percent credible intervals for θ for each ofthe 12 compounds are displayed in Figure 2. One may saygenerally that statistical imprecision, as reflected by thewidth of these confidence intervals, accounts for muchof the between-species/between-systems differences inpoint estimates. There are some obvious exceptions tothis, for example the microsomal estimate and the dog-based estimate for compound three are clearly discordanteven after accounting for statistical imprecision.

Operational relative bias and relative standard errorsare displayed in Figures 3 and 4 on a per-compoundbasis. One notes that both the relative bias and relativestandard errors are smaller in magnitude for the in vitromethodology than for either of the in vivo methodolo-gies. Though not definitive (being based on only 12 com-pounds), one also notes an apparent slight negative biasof the in vitro methodology. Such a bias would generallybe expected, since no compounds are likely to be 100%cleared by P450 metabolism.

As described in Section 1, the conventional methodfor computing clearance predictions is to solve Equations(1) and (2) for θ and substitute all unknowns with corre-sponding sample means. While there is no reason, froma statistical perspective, to treat the conventional estima-tion method as a “gold standard,” nonetheless it wouldgenerally be undesirable for our credible intervals to failto contain the conventionally computed estimates. Withthe exception of the rat-based prediction for compoundsix, all conventional estimates fell within the 95% poste-rior credible intervals. For the case of compound six, thenon-rat data generally appear to corroborate the Bayesianestimate rather than the conventional estimate.

8. Discussion

We have proposed a Bayesian statistical model relat-ing the variability and replication of multiple experimen-tal inputs to the aggregate statistical imprecision in pre-clinical predictions of hepatic clearance. We believe thisis a novel contribution to translational pharmacokinetics,

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Figure 2. A graphical display of the 95% credible intervals based on the four models.

as we are not aware of any Bayesian models that havebeen previously published for this purpose. Our approachis somewhat similar in spirit to both Monte Carlo simu-lation methodologies and fuzzy set methodologies thathave been proposed (Nestorov et al. 2002); however, webelieve that ours is the only methodology to date that con-ditions on the underlying data using explicit likelihoods.

The cross-species comparisons presented in Figure 2indicate that deficiencies remain in either the physiolog-ical hypotheses or in our statistical characterization ofvariability. We are not aware of definitive physiologicalexplanations for any of the discrepancies, and given thelack of replication in the estimation of some scaling pa-rameters it seems entirely plausible that certain aspects ofour statistical model are misspecified. As indicated at theoutset of our manuscript, we hope that the introduction ofa coherent statistical framework will motivate increaseduse of replication in these experiments, thereby enablingfuture refinement of some of the more speculative aspectsof the data likelihood.

Potential opportunities for using our model include:

• Bayesian PK and PK/PD modeling. A number of

software applications now enable Bayesian PK andPK/PD modeling, wherein the user may specify aprior distribution for certain parameters. The poste-rior distribution for clearance obtained with preclin-ical data using our model could then be used as aprior in the analysis of early phase clinical data.

• Preclinical resource allocation. Our model providesa framework wherein the ultimate practical signif-icance of changes to experimental practice can beevaluated. For example, one can investigate the rel-ative merits of more precisely estimating a fractionunbound (whether by increased replication or by useof alternative technology) versus more precisely es-timating a blood-plasma partitioning coefficient.

• Preclinical hypothesis generation and assessment.The concordance or discordance of various preclin-ical predictions is used as a guide in evaluating hy-potheses about the pharmacokinetics of a compound(e.g., if the microsomal prediction appears to belower than an in vivo prediction, this may generatehypotheses of non-P450 mediated clearance mech-anisms). It is therefore helpful to know whether ap-

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Figure 3. Operational relative bias for combined biological/statistical methodology.

parent differences in preclinical predictions is easilyexplained simply in terms of statistical imprecision.

Although we have not evaluated it here, we note thatthere is a natural meta-analytic extension of our method-ology that could be used synthesize information frommultiple preclinical predictions. Since allometric scalingis often applied to two preclinical species, a total of threepredictions will typically be generated: rat, dog, and invitro. The separate estimates must eventually be synthe-sized or in some way reduced to a single overall preclini-cal prediction, but a precise means of doing so is not wellestablished. In current practice, confidence in the finalprediction is generally expressed qualitatively, in consid-eration of the concordance of the three different estimatesas well as historical experience with similar compounds.As a potential improvement over this, one could mergeall three statistical models into a single preclinical model

that simultaneously related all three sources of informa-tion to the single parameter θ . Concordance/discordanceof the different preclinical estimates could be defined, forexample, according to whether the posterior distributionis unimodal or multimodal. Where synthesis of the dif-ferent sources of information is deemed appropriate, thecombined meta-analytic model could be used to obtaina single aggregate point estimate and associated credibleinterval.

Finally, we suggest this work is illustrative of thegeneral utility of Bayesian methodology in problemsof translational pharmacology. Quantitative translationalmodels generally employ one or more “scaling factors,”that is, “constants” required to convert a parameter fromone species/system to another. In our application, frac-tions unbound, blood-plasma coefficients, average liversize, and other “constants” could all be considered scal-ing factors. Importantly, many scaling factors are deter-

Figure 4. Relative standard errors.

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mined experimentally, and all scaling factors have un-certainty associated with them. Bayesian modeling pro-vides a natural framework for integrating these multiplesources of uncertainty.

Acknowledgments

This research was conducted as part of the Worcester Polytechnic In-stitute (WPI) Research Experience for Undergraduates (REU) programin Industrial Mathematics and Statistics. Paul Bernhardt, Jaye Bupp,Morgan (Gieseke) Lennon, Nathan Langholz, and Christopher Steinercontributed to this research as undergraduate participants in the WPIREU program, funded in part by the National Science Foundation, De-partment of Defense, and Pfizer Global Research and Development.

[Received July 2009. Revised May 2010.]

References

Adolph, E.F. (1949), “Quantitative Relations in the Physiological Con-stitutions of Mammals,” Science, 109, 579–585. 517

Guidance for Industry (1999), “Population Pharmacokinetics,” Techni-cal Report, Food and Drug Administration, Center for Drug Evalu-ation and Research, Center for Biologics Evaluation and Research.516

Hosea, N.A., Collard, W.T., Cole, S., Maurer, T.S., Fang, R.X., Jones,H., Kakar, S.M., Nakai, Y., Smith, B.J., Webster, R., and Beaumont,K. (2009), “Prediction of Human Pharmacokinetics from Preclin-ical Information: Comparative Accuracy of Quantitative PredictionApproaches,” Journal of Clinical Pharmacology, 49, 513–533. 517

Ito, K., and Houston, J.B. (2005), “Prediction of Human Drug Clear-ance from In vitro and Preclinical Data using Physiologically Basedand Empirical Approaches,” Pharmaceutical Research, 22, 103–112. 517

Lunn, D., Thomas, A., Best, N., and Spiegelhalter, D. (2000),“Winbugs—A Bayesian Modelling Framework: Concepts, Struc-ture, and Extensibility,” Statistics and Computing, 10, 325–337. 521

Nestorov, I., Gueorguieva, I., Jones, H.M., Houston, B., and Rowland,M. (2002), “Incorporating Measures of Variability and Uncertaintyinto the Prediction of In vivo Hepatic Clearance from In vitro Data,”Drug Metabolism and Disposition, 30, 276–282. 516, 523

R Team (2008), R: A Language and Environment for Statistical Com-puting, Vienna, Austria: R Foundation for Statistical Computing.See http://www.R-project.org. 521

Rowland, M., and Tozer, T.N. (1995), Clinical Pharmacokinetics:Concepts and Applications (3rd ed.), Baltimore, MD: Williamsand Wilkins. See http://www.loc.gov/catdir/enhancements/ fy0809/94026305-d.html. 516

Shiran, M.R., Proctor, N.J., Howgate, E.M., Rowland-Yeo, K., Tucker,G.T., and Rostami-Hodjegan, A. (2006), “Prediction of MetabolicDrug Clearance in Humans: In vitro-In vivo Extrapolation vs Allo-metric Scaling,” Xenobiotica, 36, 567–580. 517

Spiegelhalter, D.J., Best, N.G., Carlin, B.P., and van der Linde, A.(2002), “Bayesian Measures of Model Complexity and Fit,” Journalof the Royal Statistical Society, 64, 583–639. 521

About the Authors

James A. Rogers is Principal Scientist, Metrum Re-search Group, Tariffville, CT (E-mail for correspondence:jimr@metrumrg.com). Jayson Wilbur is Principal Statistician, In-strumentation Laboratory, Lexington, MA. Susan Cole is Asso-ciate Research Fellow, Pfizer Global Research and Develop-ment, Sandwich, UK. Paul W. Bernhardt is a Student, Depart-ment of Statistics, North Carolina State University, Raleigh, NC.Jaye Lynn Bupp is Survey Statistician, Department of Defense,Monterey, CA. Morgan J. Lennon is a Student, Department ofStatistics, North Carolina State University, Raleigh, NC. NathanLangholz is a Student, Statistics Department, University of Cali-fornia, Los Angeles, Los Angeles, CA. Christopher Paul Steineris Student, Department of Economics, University of California,San Diego, La Jolla, CA.

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