Pros and Cons of Bayesian Pharmacometric Modeling Using BUGS

Pros and Cons of BayesianPharmacometric Modeling Using BUGSWilliam R GillespieMetrum InstituteAAPS Annual MeetingNovember 8–12, 2009c○2009 Metrum Institute Bayesian Modeling Using BUGS AAPS 2009 1 / 26

Bayesian modelingBayesian modelingTwo distinguishing core notionsThe two core notions that distinguish Bayesian analysis are:1 Unknown quantities are represented as probability distributions,and2 A formal mechanism exists for combining prior knowledge andnew data.c○2009 Metrum Institute Bayesian Modeling Using BUGS AAPS 2009 2 / 26

Bayesian modelingBayesian modelingTwo distinguishing core notionsThe two core notions that distinguish Bayesian analysis are:1 Unknown quantities are represented as probability distributions,and2 A formal mechanism exists for combining prior knowledge andnew data.(1) provides for rigorous quantitative description of uncertainty inmodel parameters and predictions.c○2009 Metrum Institute Bayesian Modeling Using BUGS AAPS 2009 2 / 26

Bayesian modelingBayesian modelingTwo distinguishing core notionsThe two core notions that distinguish Bayesian analysis are:1 Unknown quantities are represented as probability distributions,and2 A formal mechanism exists for combining prior knowledge andnew data.(1) provides for rigorous quantitative description of uncertainty inmodel parameters and predictions.(2) permits inferences that consider both prior knowledge and newdata.c○2009 Metrum Institute Bayesian Modeling Using BUGS AAPS 2009 2 / 26

Bayesian modelingTwo distinguishing core notionsBasic elements of Baysian inferenceJust as with maximum likelihood methods, we begin with a modeldescribing the relationship between the data y and theunknown-valued parameters θ — a likelihood function p (y|θ).c○2009 Metrum Institute Bayesian Modeling Using BUGS AAPS 2009 3 / 26

Bayesian modelingTwo distinguishing core notionsBasic elements of Baysian inferenceJust as with maximum likelihood methods, we begin with a modeldescribing the relationship between the data y and theunknown-valued parameters θ — a likelihood function p (y|θ).Prior knowledge (or belief) about model parameters θ isquantitatively described in terms of a probability distribution — aprior distribution p (θ).c○2009 Metrum Institute Bayesian Modeling Using BUGS AAPS 2009 3 / 26

Bayesian modelingTwo distinguishing core notionsBasic elements of Baysian inferenceJust as with maximum likelihood methods, we begin with a modeldescribing the relationship between the data y and theunknown-valued parameters θ — a likelihood function p (y|θ).Prior knowledge (or belief) about model parameters θ isquantitatively described in terms of a probability distribution — aprior distribution p (θ).Bayes rule provides a rigorous basis for quantitative statisticalinference that considers both prior knowledge and new data viathe posterior distribution: p (θ|y) ∝ p (y|θ) p (θ)c○2009 Metrum Institute Bayesian Modeling Using BUGS AAPS 2009 3 / 26

Bayesian modelingTwo distinguishing core notionsBasic elements of Baysian inferenceJust as with maximum likelihood methods, we begin with a modeldescribing the relationship between the data y and theunknown-valued parameters θ — a likelihood function p (y|θ).Prior knowledge (or belief) about model parameters θ isquantitatively described in terms of a probability distribution — aprior distribution p (θ).Bayes rule provides a rigorous basis for quantitative statisticalinference that considers both prior knowledge and new data viathe posterior distribution: p (θ|y) ∝ p (y|θ) p (θ)Use the posterior distribution for inferences regarding parametervalues.Use the posterior predictive distribution for inferences regardingfuture observations:∫p (y new |y) = p (y new |θ) p (θ|y) dθc○2009 Metrum Institute Bayesian Modeling Using BUGS AAPS 2009 3 / 26

Bayesian modelingBarriers to Bayesian pharmacometric applicationsBarriers to more widespread use of Bayesianpharmacometric modelingCan be much more computationally intensiveLack of adequate support for PK and PD modelsPKBugs provides built-in models limited to linear compartmentalmodels with 1–3 compartments, but itDoes not appropriately handle time-dependent covariatesOnly allows input into 1 compartment: central or absorptionSupport for ODEs available (WBDIFF, MCSim and OpenBUGS) butsubstantial programming is needed to apply it to complicated eventhistories, e.g., typical multiple dose dataIn short, there is was no real equivalent to PREDPPc○2009 Metrum Institute Bayesian Modeling Using BUGS AAPS 2009 4 / 26

Computational tools/methods for Bayesian modelingMCMC simulationComputational methods for Bayesian modelingMarkov chain Monte Carlo (MCMC) simulationSimulation methods for performing high dimensional integrationrequired for Bayesian modeling.Simulate random variables from the posterior distributions ofinterest, e.g., means and variances of the population distributionsof PK/PD parameters.Inferences follow directly from the distributions of simulatedparameter valuesPoint estimates from mean, median or mode.Posterior intervals from percentiles, e.g., 95% interval from the 2.5and 97.5 percentiles.c○2009 Metrum Institute Bayesian Modeling Using BUGS AAPS 2009 5 / 26

Computational tools/methods for Bayesian modelingComputer programs implementing Bayesian MCMCComputer programs implementing Bayesian MCMCMCMC methods for Bayesian modeling have been addedrecently to some existing pharmacometric tools, e.g.,NONMEM VIIS-AdaptAdvantagesPK/PD model libraries and model-specification languagesMCMC and less computationally-demanding algorithms within asingle platformDisadvantagesLimited stochastic model structure2 levels of random variation + prior distributionsVery limited choice of distributions: normal for IIV, normal-Wishart forpriorc○2009 Metrum Institute Bayesian Modeling Using BUGS AAPS 2009 6 / 26

Computational tools/methods for Bayesian modelingComputer programs implementing Bayesian MCMCComputer programs implementing Bayesian MCMCMore general-purpose software tools implementing MCMC forBayesian modeling include:WinBUGSSAS MCMC procedureOpenBUGSMicrosoft Infer.NETJAGSAdvantagesFlexible stochastic model structureNo limit on levels of variabilityChoice of many distributionsCan easily combine models with very different stochastic structuresDisadvantagesNo specialized support for pharmacometrics applicationsLess flexible specification of structural model (esp. for BUGSvariants)Less computationally-demanding algorithms not available within thesame platformc○2009 Metrum Institute Bayesian Modeling Using BUGS AAPS 2009 7 / 26

Computational tools/methods for Bayesian modelingBUGS language MCMC programsBUGS language MCMC programsWinBUGS, OpenBUGS and JAGS use the BUGS model specificationlanguage.BUGS = Bayesian analysis Using Gibbs SamplingSame basic syntaxSome differences in available functions and distributionsc○2009 Metrum Institute Bayesian Modeling Using BUGS AAPS 2009 8 / 26

Computational tools/methods for Bayesian modelingBUGS language MCMC programsBUGS language MCMC programsWinBUGS 1.4.3Windows application only available as executable codeWell-tested stable code. Probably the best version for productionuse at this time.No active development. Bug fixes only.Both graphical and text model specificationData format is R-like.May be run from R via the R2WinBUGS package (and others).OpenBUGS 3.0.5JAGSc○2009 Metrum Institute Bayesian Modeling Using BUGS AAPS 2009 9 / 26

Computational tools/methods for Bayesian modelingBUGS language MCMC programsBUGS language MCMC programsWinBUGS 1.4.3Uses Gibbs sampling with sampling from the conditionaldistributions according to:Continuous distributionsConjugateDirect sampling using standard algorithmsLog-concave Derivative-free adaptive rejection samplingRestricted range Slice samplingUnrestricted range MetropolisDiscrete distributionsFinite upper bound InversionShifted Poisson Direct sampling using standard algorithmsFrom D Spiegelhalter, A Thomas, N Best, D Lunn. WinBUGS User Manual.Version 1.4 (2003).OpenBUGS 3.0.5JAGSc○2009 Metrum Institute Bayesian Modeling Using BUGS AAPS 2009 9 / 26

Computational tools/methods for Bayesian modelingBUGS language MCMC programsBUGS language MCMC programsWinBUGS 1.4.3OpenBUGS 3.0.5Probable successor to WinBUGS 1.4.*Source code availablePrimarily a Windows application, but a non-GUI version is availablefor Linux systems with Intel/AMD processorsActive development. Currently a beta version.Release of a stable version imminent.More functions and distributions.Improved existing samplers and added moreDifferent approach to selecting samplersJAGSc○2009 Metrum Institute Bayesian Modeling Using BUGS AAPS 2009 9 / 26

Computational tools/methods for Bayesian modelingBUGS language MCMC programsBUGS language MCMC programsWinBUGS 1.4.3OpenBUGS 3.0.5JAGSJAGS = Just Another Gibbs SamplerWritten in C++, so is more platform-independentUsually run from R via the rjags packagec○2009 Metrum Institute Bayesian Modeling Using BUGS AAPS 2009 9 / 26

Computational tools/methods for Bayesian modelingMore on WinBUGSBUGS model specification languageNot a procedural languageModel specification similar to statistical literature conventionsDoes not explicitly describe a sequence of calculationsStatement order is largely unimportantStochastic nodes a.k.a. probability distributionsLogical nodes a.k.a. deterministic functionsVery flexible w.r.t. stochastic structure of a modelLack of control structures (true loops & if-then-else) and controlover calculation order can make specification of complexdeterministic functions difficult (but there is a solution)c○2009 Metrum Institute Bayesian Modeling Using BUGS AAPS 2009 10 / 26

Computational tools/methods for Bayesian modelingA simple BUGS modelMore on WinBUGSy i ∼ N `ŷ i , σ 2´ model{“loop” overŷ i = a + bx iPrior distributions:✘✾✘ ✘✘ ✘ ✘ the datafor ( i in 1:10){a ∼ N (0, 10000)# model for observed data (likelihood function)b ∼ N (0, 10000)y[ i ] ∼ dnorm(ymean[i],tau)σ ∼ Uniform (0, 10000)# simulation of new ”observations”, i.e., samples from# the posterior predictive distributionypred[i ] ∼ dnorm(ymean[i],tau)# expected value of y[i] given x[i]stochastic ✟✚ ✟✟✟✟✟✟✯✚✚✚✚✚✚✚❃ ymean[i]

Computational tools/methods for Bayesian modelingMore on WinBUGSAvailable distributions (WinBUGS 1.4.*)ContinuousDiscreteUnivariate Multivariate Univariate Multivariatebeta Dirichlet Bernoulli multinomialchi-squared normal binomialdouble exponential Student-t categoricalgamma Wishart negative binomialgeneralized gammapoissonlog-normallogisticnormalparetoStudent-tuniformWeibullc○2009 Metrum Institute Bayesian Modeling Using BUGS AAPS 2009 12 / 26

Computational tools/methods for Bayesian modelingMore on WinBUGSAvailable distributions (WinBUGS 1.4.*)ContinuousDiscreteUnivariate Multivariate Univariate Multivariatebeta Dirichlet Bernoulli multinomialchi-squared normal binomialdouble exponential Student-t categoricalgamma Wishart negative binomialgeneralized gammapoissonlog-normallogistic More flexibilitynormalWinBUGS also has a large set of built-inparetofunctions.Student-tUsers may program custom functionsuniformand distribution via the WBDevWeibullpackage.c○2009 Metrum Institute Bayesian Modeling Using BUGS AAPS 2009 12 / 26

BUGSModelLibraryDescriptionBUGSModelLibraryhttp://bugsmodellibrary.googlecode.comBUGSModelLibrary is a prototype PKPD model library for use withWinBUGS 1.4.3. The current version includes:Specific linear compartmental models:One compartment model with first order absorptionTwo compartment model with elimination from and first orderabsorption into central compartmentGeneral linear compartmental model described by a matrixexponentialGeneral compartmental model described by a system of first orderODEsOpen-source software developed and distributed free of charge byMetrum Institute.c○2009 Metrum Institute Bayesian Modeling Using BUGS AAPS 2009 13 / 26

BUGSModelLibraryDescriptionBUGSModelLibraryThe models and data format are based onNONMEM/NMTRAN/PREDPP conventions including:Recursive calculation of model predictionsThis permits piecewise constant covariate valuesBolus or constant rate inputs into any compartmentHandles single dose, multiple dose and steady-state dosinghistoriesImplemented NMTRAN data items include:TIME, EVID, CMT, AMT, RATE, ADDL, II, SSc○2009 Metrum Institute Bayesian Modeling Using BUGS AAPS 2009 14 / 26

BUGSModelLibraryBuilt-in modelsBUGSModelLibraryBuilt-in modelsPre-compiled models are currently limited to one and twocompartment linear PK models with or without first orderabsorption.OneCptModel(time, amt, rate, ii, evid, cmt, addl, ss, theta)TwoCptModel(time, amt, rate, ii, evid, cmt, addl, ss, theta)WinBUGSmodelfunction argument parametersmodel name names in thetaone compartment OneCptModel time, amt, rate, ii, CL, V 2 , k a − CL , F V 1 , F 2 ,2model with first orderevid, cmt, addl, ss t lag1 , t lag2absorptiontwo compartmentmodel with first orderabsorptionTwoCptModel time, amt, rate, ii,evid, cmt, addl, ssCL, Q, V 2 , V 3 , k a − λ 1 ,F 1 , F 2 , F 3 , t lag1 , t lag2 ,t lag2c○2009 Metrum Institute Bayesian Modeling Using BUGS AAPS 2009 15 / 26

BUGSModelLibraryBuilt-in modelsBUGS model using a built-in BUGSModelLibrary functionmodel{for ( i in 1:nSubjects){# Inter-patient variation in PK parameters# theta[1:5] = (CL, Q, V1, V2, ka - lambda1)logtheta[ i , 1:5] ∼ dmnorm(logthetaMean[i, 1:5], omega.inv[1:5, 1:5])for ( j in 1:5){log(theta[ i , j ])

BUGSModelLibraryBuilt-in modelsBUGS model using a built-in BUGSModelLibrary functionvalue10.610.410.210.09.89.69.42.12.01.91.81.716.015.515.014.514.0373635343332110105100950.1060.1040.1020.1000.0980.0960.094200 400 600 800 1000MCMC sampleCLHatDkaHatQHatV1HatV2Hatsigmachain123density2.01.51.00.50.01.21.00.80.60.40.20.00.140.120.100.080.060.040.020.00CLHat9.4 9.6 9.8 10.010.210.410.6QHat14.0 14.5 15.0 15.5 16.0V2HatMean SD Median 2.5%ile 97.5%ile Effective NcCL 10.1 0.201 10.1 9.69 10.5 1740bQ 14.8 0.331 14.8 14.2 15.5 647cV 1 34.8 0.983 34.8 32.9 36.7 130cV 2 103 2.86 103 97.7 109 345k â− λ 1 1.91 0.0704 1.91 1.77 2.05 109σ 0.1 0.00226 0.0999 0.0957 0.104 16405432100.40.30.20.10.015010050DkaHat1.7 1.8 1.9 2.0 2.1V1Hat32 33 34 35 36 37sigma095 100 105 1100.0940.0960.0980.100.1020.1040.106valueMarginal posteriordistributions of modelparameters estimatedby MCMC simulation.c○2009 Metrum Institute Bayesian Modeling Using BUGS AAPS 2009 17 / 26

BUGSModelLibraryBuilt-in modelsBUGS model using a built-in BUGSModelLibrary functionplasma concentration1201008060402005004003002001000●●●● ● ● ●●●●●●●●●●●●●●●● ● ● ●● ●●●●● ●●5 mg●●●●●●●●●●●●● ● ● ●●●●●20 mg●●●●●●●●● ●●●●●● ●●●●●●● ●●●●●●●● ●●●●● ●●●● ● ● ● ● ●●● ● ●●●0 5 10 15 200 5 10 15 20time (h)Posterior median and 95% posterior prediction25020015010050010008006004002000●●●10 mg●●●●● ● ●●● ●●●●●●●●●●●● ●●●●●●●●●●●●●● ● ● ● ● ●● ● ●●●●●●●●●●●●●● ●●●●●● ●● ● ●●●●●●●●● ●●●●●●interval overlayed on observed data40 mg●●●●●●●●●● ●● ● ● ●●●● ●●●●Simulation-baseddiagnostics, e.g.,posterior-predictivechecking, may be done aspart of the fitting.Separate simulation orboot-strapping steps arenot necessary.c○2009 Metrum Institute Bayesian Modeling Using BUGS AAPS 2009 18 / 26

BUGSModelLibraryUser-programmed modelsBUGSModelLibraryUser-programmed modelsLinear compartmental modelsUser specifies the non-zero elements of the rate constant matrix.Linear ODE’s are solved using matrix exponential methods.General compartmental modelsUser specifies the ODE’s.The ODE’s are solved using either a Runge-Kutta 4th/5th ordermethod or LSODA, the Livermore Solver for Ordinary Differentialequations with Automatic method switching for stiff and nonstiffproblems.Both cases require user specification of a rate constant matrix orODE’s in a template Component Pascal procedure that must becompiled using the BlackBox Component Builder 1.5.c○2009 Metrum Institute Bayesian Modeling Using BUGS AAPS 2009 19 / 26

BUGSModelLibraryUser-programmed modelsComponent Pascal code for matrix exponential modelPROCEDURE UserKMatrix(IN theta: ARRAY OF REAL; nCmt: INTEGER):POINTER TO ARRAY OF ARRAY OF REAL;VARkMatrix: POINTER TO ARRAY OF ARRAY OF REAL;i , j : INTEGER;CL, Q, V2, V3, ka, ke0, k10, k12, k21: REAL;BEGINNEW(kMatrix,nCmt,nCmt);FOR i := 0 TO nCmt−1 DO;FOR j := 0 TO nCmt−1 DO;kMatrix[ i , j ] := 0;END; END;CL := theta[0]; Q := theta[1]; V2 := theta[2];V3 := theta[3]; ka := theta[4]; ke0 := theta[5];k10 := CL/V2; k12 := Q/V2; k21 := Q/V3;(∗ Assign nonzero rate constants ∗)kMatrix[0,0] := -ka;kMatrix[1,0] := ka;kMatrix[1,1] := -(k10+k12);kMatrix[1,2] := k21;kMatrix[2,1] := k12;kMatrix[2,2] := -k21;kMatrix[3,1] := ke0;kMatrix[3,3] := -ke0;RETURN kMatrix;END UserKMatrix;x ′ =⎡⎢⎣⎤−k a 0 0 0k a − (k 10 + k 12 ) k 21 0⎥0 k 12 −k 21 0 ⎦ x0 k e0 0 −k e0c○2009 Metrum Institute Bayesian Modeling Using BUGS AAPS 2009 20 / 26

BUGSModelLibraryUser-programmed modelsComponent Pascal code for general ODE modelPROCEDURE UserDerivatives(IN theta, x: ARRAY OF REAL;numEq: INTEGER; t: REAL; OUT dxdt: ARRAY OF REAL) ;VARCL, Q, V2, V3, ka, kout, yeff0, EC50, k10, k12, k21, conc: REAL;BEGINCL := theta[0]; Q := theta[1]; V2 := theta[2];V3 := theta[3]; ka := theta[4];kout := theta[5]; yeff0 := theta[6]; EC50 := theta[7];k10 := CL/V2; k12 := Q/V2; k21 := Q/V3;(∗ ODE’s excluding piecewise constant ∗)(∗ input rates in the data set ∗)dxdt[0] := -ka * x[0];dxdt[1] := ka * x[0] - (k10 + k12) * x[1] + k21 * x[2];dxdt[2] := k12 * x[1] - k21 * x[2];conc := x[1]/V2;dxdt[3] := kout * (yeff0 - (1 - conc/(EC50+conc))*(x[3]+yeff0))END UserDerivatives;c○2009 Metrum Institute Bayesian Modeling Using BUGS AAPS 2009 21 / 26

BUGSModelLibraryDevelopment plansBUGSModelLibrary development plansMore built-in modelsPort to OpenBUGSTool to simplify and automate specification and compiling ofuser-programmed modelsRefining program architecture and efficiencyMore extensive testingc○2009 Metrum Institute Bayesian Modeling Using BUGS AAPS 2009 22 / 26

Pros and consWhy Bayesian modeling?of Bayesian modelingAbility to combine prior knowledge with new datain a manner that appropriately accounts foruncertainty in the prior informationFully Bayesian treatment of uncertainty in parameters andpredictionsInferences about parameters and predictions easily expressed interms of the posterior distributionsNo series or linear approximations of the likelihood functionImproved estimation performanceParticularly for generalized hierarchical models, e.g, models forcategorical datac○2009 Metrum Institute Bayesian Modeling Using BUGS AAPS 2009 23 / 26

Pros and consof Bayesian modelingWhy not Bayesian modeling?Computation timeAdditional work required to specify and assess sensitivity to priordistributions.Learning time.c○2009 Metrum Institute Bayesian Modeling Using BUGS AAPS 2009 24 / 26

Pros and consPros and cons of BUGSof BUGSProsMuch more flexible stochastic structureNo restriction on number of levels of random effectsMany built-in distribution functions plus ability to add moreEasier to combine models for multiple types of data, e.g.:Individual patient data + summary dataPreclinical + clinical dataAvailability of pharmacometrics modeling tools, e.g.,BUGSModelLibrary.c○2009 Metrum Institute Bayesian Modeling Using BUGS AAPS 2009 25 / 26

Pros and consof BUGSPros and cons of BUGSProsMuch more flexible stochastic structureNo restriction on number of levels of random effectsMany built-in distribution functions plus ability to add moreEasier to combine models for multiple types of data, e.g.:Individual patient data + summary dataPreclinical + clinical dataConsAvailability of pharmacometrics modeling tools, e.g.,BUGSModelLibrary.Lack of less computationally-demanding methods within the sameplatform to support exploratory modeling, e.g., estimation ofposterior modes.Component Pascal programming required for development ofmore complex structural models.c○2009 Metrum Institute Bayesian Modeling Using BUGS AAPS 2009 25 / 26

And now for a bit of self-deprecating humorFan mail from some frequentistsBayesian (bey’ -zhuhn) n. 1. Result ofbreeding a statistician with a clergyman toproduce the much sought after “honeststatistician” a . 2. One who asks you whatyou think before a clinical trial in order to tellyou what you think afterwards b . 3. Onewho, vaguely expecting a horse, andcatching a glimpse of a donkey, stronglybelieves he has seen a mule c .a anonymousb Stephen Senn. Statistical Issues in Drugdevelopment, 2nd Edition. Wiley, 2008. p. 51.c ibid. p. 46c○2009 Metrum Institute Bayesian Modeling Using BUGS AAPS 2009 26 / 26