PHARMACEUTICAL STATISTICS

Pharmaceut. Statist. 2008; 7: 36–41

Published online 5 March 2007 in Wiley InterScience

(www.interscience.wiley.com) DOI: 10.1002/pst.263

Modeling attainment of steady state of

drug concentration in plasma by means

of a Bayesian approach using MCMC

methods

Paul Jordan1,*,y, Hadassa Brunschwig1,2 and Eric Luedin1

1Pharma Development, Clinical Science, F. Hoffmann-La Roche Ltd., Basel, Switzerland2Biometrics, Serono S.A. International, Geneva, Switzerland

The approach of Bayesian mixed effects modeling is an appropriate method for estimating both

population-specific as well as subject-specific times to steady state. In addition to pure estimation, the

approach allows to determine the time until a certain fraction of individuals of a population has

reached steady state with a pre-specified certainty. In this paper a mixed effects model for the

parameters of a nonlinear pharmacokinetic model is used within a Bayesian framework. Model fitting

by means of Markov Chain Monte Carlo methods as implemented in the Gibbs sampler as well as the

extraction of estimates and probability statements of interest are described. Finally, the proposed

approach is illustrated by application to trough data from a multiple dose clinical trial. Copyright #

2007 John Wiley & Sons, Ltd.

Keywords: pharmacokinetics; steady state; Bayes; mixed effects; Markov Chain Monte Carlo;

Gibbs sampling

1. INTRODUCTION

In phase I of clinical development the time toattainment of steady state of a new regimen isroutinely assessed. The time to attainment ofsteady state is the time – usually expressed as thenumber of days – that is needed until the plasma

concentration of a new drug is stabilized, i.e. doesnot show an increasing trend by drug accumula-tion. If a drug is applied at regular dosingintervals, drug residues from preceding doses areaccumulated. Stabilization of the concentrationoccurs when the amount of drug eliminated duringthe dosing interval equals the amount that wasentered. In order to assess the time to steady stateblood is sampled at a certain time point withineach dosing interval e.g. at the time before the nextdosing [1]. Following the drug concentrations atyE-mail: paul.jordan@roche.com

*Correspondence to: Paul Jordan, Pharma Development,Clinical Science, F. Hoffmann-La Roche Ltd., PDMB, B570/413, CH-4070 Basel, Switzerland.

Copyright # 2007 John Wiley & Sons, Ltd.

these time points, steady state is attained whenthese concentrations do not show increasing trendanymore due to drug accumulation but onlyrandom variability around a mean value thatremains constant over time.

In order to simplify the presentation in this paperwe will always use time to next dosing as the timepoint within the dosing interval where blood samplingtakes place and we denote the according drugconcentrations by trough concentrations. The con-cept is naturally also applicable to other time points.

Hoffman et al. [2] propose a nonlinear mixedeffects modeling approach to estimate the time tosteady state within a frequentist framework. Thisapproach is an appropriate method as it allows forsubject-specific as well as population-specific esti-mates of the time to steady state both in data-richand data-sparse situations. However, it does notallow for probability statements in the sense of howcertain one can be that e.g. 80% of the individualsof the population have reached steady state.

Based on the model used in [2] this paperpresents a Bayesian inference approach usingMarkov Chain Monte Carlo (MCMC) methodsto estimate population-specific as well as subject-specific parameters. This approach yields compar-able estimates to the proposed method in [2] butenables to draw additional conclusions such as‘with 95% certainty at least 80% of the individualsin the population have reached steady state’.

In the following section, the Bayesian hierarch-ical model is developed taking into account themodel in [2] and adding the probabilistic compo-nent of the certainty about an estimate. Theprocess of model fitting is described in Section 3.In Section 4, the Bayesian hierarchical model isillustrated by applying it to trough data from amultiple dose clinical trial. Section 5 containsconclusions and further comments.

2. METHODOLOGY

2.1. Pharmacokinetic model

Let us consider the simple case where the sameamount of a drug is given m times in constant

dosing intervals of duration o: In analogyto [2] the time course of the trough concentra-tions CðtÞ; t ¼ j � o; j ¼ 1; 2; . . . ;m in a one-compartment model is described by the followingformula:

CðtÞ ¼ Cssð1� e�KtÞ ð1Þ

where K is the elimination rate defined as theratio of clearance and volume of the drug andis assumed to be constant over time and Css is thedrug concentration at asymptotic steady state.The asymptotic steady-state concentration Css isnever fully achieved in finite time. For practicalpurposes it may be adequate to assume attain-ment of steady state as soon as 90% of Css arereached. We thus define time to steady-state as thetime when 90% of the asymptotic steady-state concentration Css has been reached;this definition is also in accordance with [2].The time to steady state for an individual i,denoted by t

ðiÞ0:9; can then be derived from equation

(1) as follows:

tðiÞ0:9 ¼

lnð0:1Þ�K ðiÞ

ð2Þ

2.2. Statistical model

As pharmacokinetic drug behavior varies betweenindividuals, the following hierarchical model issuggested. The model allows for the subject-specific as well as population-specific parametersbased on the pharmacokinetic model of Section2.1. The measured individual concentrations C

ðiÞt ;

t ¼ j � o; j ¼ 1; 2; . . . ;m are written as

CðiÞt ¼ CðiÞss 1� exp

lnð0:1Þt

tðiÞ0:9

! !eEðiÞt ð3Þ

The within-subject error is denoted by EðiÞt � Nð0; tðiÞE Þwhere tðiÞE is the individual error precision definedas t ¼ 1=s2: Volume and the elimination constantK of the drug are assumed to be log-normallydistributed and therefore so is t

ðiÞ0:9: In agreement

with common knowledge and practice in pharma-cokinetics we assumed a log-normal distributionfor CðiÞss as well.

Copyright # 2007 John Wiley & Sons, Ltd. Pharmaceut. Statist. 2008; 7: 36–41DOI: 10.1002/pst

Modeling attainment of steady state of drug concentration 37

In the Bayesian framework [3], parameters CðiÞssand t

ðiÞ0:9 are treated as random variables with:

1CðiÞss � NðCpopss ; tCss

Þ; lnðtðiÞ0:9Þ � Nðtpop0:9 ; tt0:9 Þ ð4Þ

where Cpopss and tpop0:9 are the population parameters

and tCssand tt0:9 the respective precisions.

2.3. Probabilistic considerations

By computing the posterior distribution of theindividual times to steady state, t

ðiÞ0:9; we can extract

an estimate for any quantile qb½tðiÞ0:9� of the

distribution. Taking, for example qb½tðiÞ0:9�; with b ¼

0:8 an estimate will result in a time at which 80%of the individuals of the population have attainedsteady state. Treating qb½t

ðiÞ0:9� as a parameter of

interest, we can include it as node in the MCMCalgorithm (described in Section 3) and thus deriveits posterior distribution as well. From this poster-ior distribution of qb½t

ðiÞ0:9�; a quantile qg½qb½t

ðiÞ0:9��

can be calculated. This quantile qg½qb½tðiÞ0:9�� can be

interpreted as the time, where, with g� 100%certainty, at least b� 100% of the individuals in thepopulation have attained steady state.

3. MODEL FITTING WITH MCMCAND GIBBS SAMPLER

For statistical inference purposes one is usuallyinterested in the distributions of individual modelparameters, i.e. in the respective marginal distribu-tions. This implies high-dimensional integrations ofthe joint posterior distribution. MCMC methodsreplace such high-dimensional integrals by creatingrandom samples from the joint distribution. Hav-ing a random sample from the multidimensionaljoint distribution, one automatically obtains ran-dom samples for each parameter in the model, i.e.random samples for each marginal distribution.With a sufficiently large sample, all features of theunderlying distribution can be estimated in astraightforward manner with any precision needed.

The details of the sampling process based onMCMC are described e.g. in Casella and George[4], Brooks [5], Smith and Roberts [6] and Gilks

et al. [7]. MCMC methods are implemented in thesoftware package WinBUGS [8].

4. EXAMPLE

The approach presented in Sections 2 and 3 isillustrated by application to data of a phase Imultiple dose clinical trial. Trial subjects consistedof 34 healthy male volunteers aged between 18 and65 who were randomly assigned a treatment groupof drug doses A ¼ 400mg and B ¼ 800mg; suchthat 17 subjects were in each treatment group.Each subject was administered the correspondingdrug dose on a daily basis over a dosing interval of13 days. The measurements of trough drugconcentrations in plasma for all patients and alldays were reported and used as the example data.Due to confidentiality reasons the data andconditions of the trial have been made anonymous.

The population estimator was defined to be the80th percentile ðb ¼ 0:8Þ of the posterior distribu-tion for the time to steady state. The desired degreeof certainty for the population estimator was set tog ¼ 0:9:

The model in (3) was used as a first approach tothe data and to demonstrate the developed methodin Sections 1–3. This model serves as a firstapproach to the data and suffices for our aim toobtain a first idea about the estimates and forillustrating our method. Model (3) then becomes

CðiÞt ¼ CðiÞss 1� exp

lnð0:1Þt

tðiÞ0:9

! !eEðiÞt ;

i ¼ 1; . . . ; 17; t ¼ 1; . . . ; 13 ð5Þ

where the parameters and the within-subject errorare as described in Section 2.2.

The joint density of CðiÞss and tðiÞ0:9 is then

lnðCðiÞss Þ

lnðtðiÞ0:9Þ

!� N

Cpopss

tpop0:9

!;T

!

T ¼ O�1; O ¼s2Css

sCss ;t0:9

sCss;t0:9 s2t0:9

!ð6Þ

Copyright # 2007 John Wiley & Sons, Ltd. Pharmaceut. Statist. 2008; 7: 36–41DOI: 10.1002/pst

38 P. Jordan et al.

where T is the precision matrix, O the covariancematrix of ðlnðCðiÞss Þ; lnðt

ðiÞ0:9ÞÞ

t and Cpopss ; T, O and tpop0:9

are the population parameters. We modeled thedistribution of lnðCðiÞss Þ and lnðtðiÞ0:9Þ as a jointmultivariate distribution in order to keep thedistributional assumptions as general as possible.The priors for the population parameters and theindividual error precision were set as

lnðCpopss Þ � Nð3:5; 0:001Þ; lnðtpop0:9 Þ � Nð2; 0:01Þ

tðiÞE � Gð0:01; 0:01Þ; T �WðV ; 2Þ ð7Þ

where W is the Wishart distribution and V reflectsthe order of magnitude of O:

Note that the priors were selected to be littleinformative in order to emphasize the influence ofthe likelihood on the posterior [3] and since nospecific prior knowledge was assumed aboutthe parameters: low precisions were specified forthe normal distributions and the gamma distribu-tion for the precision was chosen such thatnearly all mass is close to 0. The degrees offreedom for the Wishart distribution were set to beas small as possible (i.e. 2, the rank of O) torepresent vague prior knowledge about theprecision matrix of the interesting parameters.We used a crude estimate for V by first assumingindependence between the two parameters lnðCðiÞss Þand lnðtðiÞ0:9Þ: We then obtained first guesses abouttheir variances by fitting model (5) with lnðCðiÞss Þand lnðtðiÞ0:9Þ independently normally distributedand choosing non-informative gamma priors fortheir precisions [8].

The 80th percentile of the posterior distributionof t

ðiÞ0:9; specified as q80; was included as parameter

in the computation using the assumption ofnormally distributed lnðtðiÞ0:9Þ in (6). Thus, q80 can

simply be defined as follows:

q80 ¼ expðlnðtpop0:9 Þ þ z80 � st0:9Þ ð8Þ

where z80 is the 80th percentile of the standardnormal distribution and st0:9 is the populationstandard deviation.

The data of each treatment group were fittedseparately using WinBUGS version 1.4.1 [8] andthe package R2WinBUGS as implemented in R2.3.1 [9]. A separate fit was chosen for each dose inorder to obtain a first idea about the populationtime to steady state under each dose. The samenon-informative priors were used for each dosegroup since no dose-specific assumptions weremade. However, the crude estimates for V wereobtained separately for each dose group and wereintroduced in the prior for the precision matrix.

For each parameter, a burn-in of 2000 sampleswas performed and then further 10 000 samplesdrawn. The sampling history was checked forstability and the posterior distributions for con-vergence in distribution using WinBUGS. Further-more, we checked quantile–quantile plots andresidual plots for the adequacy of this model as afirst approach to the data. Figure 1 shows thefitted curves and estimated individual steady statefor all subjects using (5).

The resulting posterior distributions of thepopulation parameters of interest tpop0:9 and q80 forboth treatment groups are summarized in Table I.For a comparison, we calculated the estimatesfrom a mixed effects model using the R functionnlme [10]. The resulting estimates from this fit,added to Table I in brackets, are similar to theestimates using the Bayesian approach.

Table I. Summary of the posterior distributions for the interesting parameters.

Treatment Parameter Mean Standard deviation Median 90th Percentile

A tpop0:9 3.37 (3.60) 0.26 (0.36) 3.35 4.02q80 4.21 0.98 4.04 5.18

B tpop0:9 3.72 (3.94) 0.53 (0.54) 3.66 4.62q80 5.91 2.02 5.49 7.95

Values in brackets are from a fit with a mixed model for comparison. Numbers in bold are population times to steady state for group Aand B, respectively, with 90% certainty.

Modeling attainment of steady state of drug concentration 39

Copyright # 2007 John Wiley & Sons, Ltd. Pharmaceut. Statist. 2008; 7: 36–41DOI: 10.1002/pst

The hierarchical modeling approach usingGibbs sampling yielded the following result(numbers in bold in Table I): for treatment groupA, at least 80% of the population has attainedsteady state on day 6 with 90% certainty; fortreatment group B at least 80% of the population

has attained steady state on day 8 with 90%certainty. Figure 2 shows the number of subjectshaving attained steady state for each treatmentgroup: treatment group A attains overall steadystate earlier than treatment group B. We have thusobtained estimates of the day to steady state for

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Figure 1. Individually fitted curves and estimates of time to steady state. Dots indicate concentration values,

solid lines the fitted curve. The estimated 90% of theoretical steady state and its corresponding time to steady

state are represented by dashed lines.

Copyright # 2007 John Wiley & Sons, Ltd. Pharmaceut. Statist. 2008; 7: 36–41DOI: 10.1002/pst

40 P. Jordan et al.

each dose group, based on a first model for thedata, and can express the certainty about theseestimates.

5. CONCLUSIONS

In analogy to nonlinear mixed effects modeling, aBayesian hierarchical model is proposed usingMCMC and Gibbs sampling for assessing the timeto steady state. The drug concentration is modeleddependent on the parameter for the time to steadystate. The population estimator for the time tosteady state, which is any pre-defined quantile ofthe posterior distribution for the time to steadystate, is included as parameter in the model. Inaddition to mixed effects modeling, the Bayesianframework allows for the indication of estimateswith a predefined degree of certainty by extractingthe respective quantile from the posterior distribu-tion of the population estimator. This approach isanalogous to the frequentist method of toleranceintervals [11]. Similar to nonlinear mixed effectsmodeling, the model can be extended to allow forsubpopulations or covariates. Finally, the methodis not model dependent and is applicable to

possibly more complex pharmacokinetic relation-ships than proposed by (1).

The implementation of the presented method issimplified by using software packages such as R(with the libraries R2WinBUGs and BRugs) andWinBUGS. The programs used in the abovesections are available upon request.

As in nonlinear mixed effects modeling, theBayesian hierarchical model using MCMC andGibbs sampling provides adequate fits and reason-able estimates for most applications.

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3. Gelman A, Carlin JB, Stern HS, Rubin DB.Bayesian data analysis (2nd edn). Chapman &Hall/CRC Press: London, Boca Raton, FL, 2004.

4. Casella G, George E. Explaining the Gibbs sampler.The American Statistician 1992; 46:167–174.

5. Brooks SP. Markov Chain Monte Carlo methodand its application. The Statistician 1998; 47:69–100.

6. Smith AFM, Roberts GO. Bayesian computationvia the Gibbs sampler and related Markov ChainMonte Carlo methods. Journal of the RoyalStatistical Society. Series B (Methodological) 1993;55:3–23.

7. Gilks WR, Richardson S, Spiegelhalter DJ. MarkovChain Monte Carlo methods in practice. Chapman &Hall: London, 1996.

8. Spiegelhalter DJ, Thomas A, Best NG. WinBUGSversion 1.4: user manual. MRC Biostatistics Unit:Cambridge, 2003.

9. R Development Core Team. R: A language andenvironment for statistical computing. R Foundationfor Statistical Computing: Vienna, 2005; ISBN3-900051-07-0, URL http://www.R-project.org.

10. Pinheiro JC, Bates DM. Mixed-effects models in S-plus. Springer: New York, 2000.

11. Kendall M, Stuart A. The advanced theory ofstatistics: inference and relationship. Charles Griffinand Co. Ltd: London, 1979.

DAY

Num

ber

of S

ubje

cts

Hav

ing

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

0

2

4

6

8

10

12

14

16

18

Dose

800 mg400 mgR

each

ed S

tead

y S

tate

Figure 2. Number of subjects having reached steady

state versus time. The dashed line represents treatment

group A, the solid line treatment group B.

Modeling attainment of steady state of drug concentration 41

Copyright # 2007 John Wiley & Sons, Ltd. Pharmaceut. Statist. 2008; 7: 36–41DOI: 10.1002/pst