Dealing with Uncertainty and Variability A Course on Physiologically Based Pharmacokinetic (PBPK) Modeling in Drug Development and Evaluation April 6-10, 2009 Center for Human Health Assessment Center for Drug Safety Sciences
Parameterizing the model: Parameter Estimation Issues Identifiability Colinearity (correlation) Sensitivity
Parameter optimization: Fitting the model to the data Issues: Multiple parameter fitting Colinearity Local Minima Weighting ObjectiveFunction = γ = heteroscedasticity parameter γ = 0 : absolute weighting constant variance linear plot follows mass γ = 2 : relative weighting constant coefficient of variance logarithmic plot follows rates ( Predicted - Observed ) Y Observed 2
Parameter colinearity (Correlation) Parameters in PBPK models are not always independent: VMaxC and KM VMaxC and KfC QCC and QPC Ignoring parameter interdependence can distort sensitivity analysis, uncertainty analysis and parameter optimization
Parameter sensitivity coefficient: (Log Normalized) Fractional Change in Model Output Fractional Change in Parameter If Normalized Coefficient >> 1, parameter error is amplified and model output may be highly uncertain
Monte Carlo uncertainty analysis Parameter A Parameter B PBPK Model Model Output Distribution Parameter X Parameter Distributions
Simple Monte Carlo analysis vs. Markov Chain Monte Carlo Simple Monte Carlo: parameters are repeatedly sampled from distributions provided by the investigator and distributions for outputs are generated with the model. Markov Chain Monte Carlo: information from prior distributions and fit of the model to selected data sets is combined in a hierarchical Bayesian framework to generate posterior distributions for parameters reflecting uncertainty and variability.
What is MCMC Analysis? Markov chain Monte Carlo (MCMC) analysis - an implementation of the hierarchical Bayesian statistical approach Monte Carlo analysis - drawing samples from distributions Markov chain - a series of random samples, each sample dependent only on the one before Heirarchy: population, study, individual
Hierarchical model Population Level M (mean) S 2 (variability) θ φ Subject Level PBPK Data σ 2 measurement/model error Figure adapted from Bois (2000) analysis of TCE model
Bayes Theorem P ( θ D ) = P ( θ ) P ( θ ) P ( D P ( D θ ) θ ) d θ Prior knowledge (parameter distributions) combined with new information (data) to obtain posterior distributions sample priors run model compute likelihoods MCMC algorithm (MCSim, developed by F. Bois) (http://toxi.ineris.fr/activites/toxicologie_quantitative/mcsim/mcsim.php#article3)
PBPK model calibration using hierarchical Bayesian analysis Prior Parameter Distributions Data MCMC Sampling PBPK Model Posterior Distributions Of Parameters MC Sampling Calibrated PBPK Model Posterior Distributions Of Dosemetrics
MCMC Samples drawn using a Markov chain 0.5 0-0.5-1 The chain forgets starting position and converges to a stationary, posterior distribution -1.5-2 -2.5 0 2000 4000 6000 8000 10000 12000 14000
Re-estimation of distribution for Muscle:Blood partition coefficient 1.8 1.6 1.4 lnpslw(1) Average Median 2 sd 1.8 1.6 1.4 Prior Posterior 1.2 1.2 lnpslw(1) 1 0.8 1 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 50 100 150 200 250 300 350 Average = 1.23268E+00 Stdev = 2.36992E-01 Median = 1.28613E+00 0-0.5 0 0.5 1 1.5 2
Distribution of Variance in Fractional Volume of the Fat Compartment 70 Prior 60 Posterior 50 40 30 20 10 0 0 0.02 0.04 0.06 0.08 0.1
Common MCMC problem: failure to converge 2.5 2.5 2 1.5 2 logvmaxc(1.3) 1 0.5 0 logvpr(1.14) 1.5 1-0.5-1 0.5-1.5 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 iteration 0 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 iteration
To MC or MCMC: That is the question Advantages of Markov Chain Monte Carlo: Combines parameter identification with variability analysis Allows estimation of variability distributions for selected parameters based on fit of model to multiple kinetic data sets Disadvantages of Markov Chain Monte Carlo: May give undue priority to kinetic data included in MCMC analysis (as opposed to other information regarding parameters) Parameter distributions may be inappropriately reestimated due to model error
Hattis s Laws of Uncertainty Analysis 1. Nearly all parameter distributions look lognormal, as long as you don t look too closely. 2. Any estimate of the uncertainty of a parameter value will always itself be more uncertain than the estimate of the parameter itself. 3. The application of standard statistical techniques to a single data set will nearly always reveal only a trivial proportion of the overall uncertainty in the parameter value.
Clewell s Law of Reciprocal Uncertainty The more important the parameter, the less certain will be its value.
Uncertainty/Variability analysis Maintaining mass balance During Monte Carlo sampling, the model code must assure that the selected blood flows to the tissues add up to the total blood flow, i.e.: ΣQ i = Q C
Uncertainty vs. Variability Definitions Uncertainty = limitation in estimating mean parameter value for population Variability = limitation in estimating extent to which individual parameter value may vary from population mean
Uncertainty vs. Variability Animal studies: Uncertainty dominates In-bred strains Multiple animals per dose group Human studies: Variability dominates Heterogeneous population Variability in exposure
PBPK model for Xenon exposure C I C X C V Alveolar Air Alveolar Blood Lung Tissue C A Venous Blood Arterial Blood Oral Dose Slowly Perfused Tissue C S ka Stomach Lumen ktssi Rapidly Perfused Tissue C R kasi Small Intestinal Lumen ktsli Fat Tissue C F kaui Upper Large Intestinal Lumen Gastrointestinal Tissue C G kali ktuli Lower Large Intestinal Lumen ktex Liver Tissue C L
Variability in G.I. uptake (Confidence intervals based on standard deviation) CT Hours
Uncertainty in G.I. uptake (Confidence intervals based on standard error of mean) CT Hours
Prediction of G.I. uptake across studies 3 log(atotp) 2 1 0 Accuracy of extrapolation reflects low uncertainty (SEM) in spite of high variability (SD) of individual data Non-fasting Predicted Fasting -1 0 2 4 6 8 10 Time (hours)
Variability in human P450 isozymes Isozyme Assay Variability (-fold) Inducible Evidence for Polymorphism Cancer Impact 1A1 -- -- yes no smoking 1A2 caffeine 50 yes no PAH 2C mephenytion -- -- yes (3% poor)? 2D6 debrisoquine -- -- yes? 2E1 chlorzoxazone 10 yes yes halogenated hydrocarbons 3A4 nifedipine 20 yes no aflatoxin-b1
Evaluating pharmacokinetic variability The pharmacokinetic variability across a population is a function of many chemical-specific, genetic, and physiological factors. Speculation regarding the overall variability in pharmacokinetic sensitivity based on the observed variability of individual pharmacokinetic factors can be highly misleading. Analysis using a PBPK model and Monte Carlo techniques provides a more reliable approach for estimating population pharmacokinetic variability.
Example: impact of CYP2C9 Polymorphism on Warfarin Internal Dose iv dose Plasma QC Kidney QKid Rapidly Perfused QRap Skin QSkn Slowly Perfused QSlw Liver QLiv oral dose VMax, KM, KMI PBPK Model for Warfarin (Gentry et al., 2002)
Metabolic Parameters for (S)-Warfarin for Three CYP2C9 Alleles Vmax (mg/hr/kg 3/4 ) Km (mg/l) Intrinsic Clearance Allele Reference Mean CV Mean CV (VmaxC/Km) CYP2C9*1 Haining et al., 1996 1 1.61 1.85 0.87 Takahashi et al., 1998b 2 2.13 0.0036 0.81 0.12 2.6 Sullivan-Klose et al., 1996 2 1.01 0.046 3.57 0.078 0.28 Rettie et al., 1994 3 3.2 1.26 2.5 Rettie et al., 1994 4 3.2 1.05 3 CYP2C9*2 Sullivan-Klose et al., 1996 2 1.26 0.031 3.85 0.056 0.33 Rettie et al., 1994 3 0.21 0.52 0.4 Rettie et al., 1994 4 0.36 0.65 0.55 Rettie et al., 1999 5 1.1 0.036 1.85 0.17 0.59 CYP2C9*3 Haining et al., 1996 1 0.31 9.24 0.034 Takahashi et al., 1998b 2 0.51 0.22 3.2 0.16 0.16 Sullivan-Klose et al., 1996 2 1.37 0.044 28.4 0.059 0.048 1 baculovirus/insect cell system, purified enzyme 2 yeast expression, microsomes 3 Hep G2 cells, cell lysate 4 Hep G2 cells, particulate preparation 5 expressed in insect cells, purified enzymes (Haber et al., 2002)
Average Prevalence of CYP2C9 Alleles in the U.S. Population Prevalence S1 homozygous 78% S1/S2 heterozygous 12% S1/S3 heterozygous 9% S2 homozygous 1% S2/S3 heterozygous 1% S3 homozygous 0.5% (Haber et al., 2002)
Descriptive Statistics of the AUC Distribution for (S)-Warfarin Case 1 (CYP2C9*1) Case 1 (CYP2C9*2) Case 1 (CYP2C9*3) Case 2 (Normal Population) Case 3 (Total Population) Mean 157 273 2670 202 311 Standard Error 5.28 4.73 52.7 11.1 21.9 Median 58.9 252 2680 83.6 104 Standard Deviation 167 149 1670 351 693 Sample Variance 27900 22300 2770000 123000 480000 Kurtosis 0.159 2.48 2.09 99.9 92.1 Skewness 1.34 0.822 1.03 7.44 8.21 Range 578 1180 11700 6280 9650 Minimum 22.8 20.1 333 10.5 10.1 95th Pctile 509 465 5610 731 1170 99th Pctile 555 692 7670 1380 2670 Maximum 601 1200 12100 6290 9660 Count 1000 1000 1000 1000 1000 * Case 1 Varying only the metabolism parameters defining the polymorphism, using the allele indicated Case 2 Varying all parameter except those defining the polymorphism Case 3 Varying all parameters, using U.S. population frequencies of each allele (Gentry et al., 2002)
Simulation of impact of genetic polymorphism on Warfarin internal dose Plasma Concentration (mg/l) 5 A. C Y P 2 C 9 * 1 A lle le 4 3 2 1 0 0 1000 2000 3000 4000 5000 6000 H o u rs Plasma Concentration (mg/l) 5 B. C Y P 2 C 9 * 2 A lle le 4 3 2 1 0 0 1000 2000 3000 4000 5000 6000 4000 5000 6000 H o u rs Plasma Concentration (mg/l) 5 C. C Y P 2 C 9 * 3 A lle le 4 3 2 1 0 0 1000 2000 3000 H o u rs
Simulation of impact of genetic polymorphism on Warfarin internal dose 250 Normal population Case 2 200 150 100 50 0 0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 Frequency of (S)-Warfarin AUC 1000 Total population Case 3 (S)-Warfarin AUC