Hi Hierarchical Models for hi l M d l f Quantifying Uncertainty in Quantifying Uncertainty in Human Health Risk/Safety Assessment Ralph L. Kodell, Ph.D. Department of Biostatistics University of Arkansas for Medical Sciences Little Rock, AR 1
Risk/Safety Assessment: A M l i A Multi-step Process P Risk/Safety Characterization Dose-Response Hazard Exposure Assessment Identification Assessment 2
Outline Outline • Background on Human Risk/Safety Assessment • Exposure-to-Dose Response Exposure-to-Dose Response – PK/PD relationship via hierarchical model – Benchmark dose estimation (distributions) – How uncertainty can be reduced by PK information • Dose Response-to-Risk/Safety Characterization – Inter-species and intra-species uncertainties I t i d i t i t i ti – BMD conversion via hierarchical model • Summary and Conclusions • Summary and Conclusions • Challenges and Needs – Model uncertainty y 3
Uncertainty Analysis y y • Issue: There are many uncertainties in getting from Hazard and Dose-Response Assessment p in experimental (animal) settings to Exposure and Risk/Safety Characterization for human settings settings • Challenge: How to properly reflect these • Challenge: How to properly reflect these uncertainties • Today’s Talk: How Hierarchical Probabilistic Models can help to characterize and manage these uncertainties 4
Usual Approach to Exposure Setting: T Two-Step Process S P • Human Exposure (Risk) = Animal-Derived Benchmark Dose (Risk) Animal → Average Human → Sensitive Human Exposure → Dose-Response ( ) (Dose-Response → Risk/Safety Characterization) (D R Ri k/S f t Ch t i ti ) 5
Dose-Response Modeling for BMD Estimation: Illustration E ti ti Ill t ti D n #tumors Observed Predicted 0 50 5 0.10 0.096 10 10 50 50 7 7 0 14 0.14 0.157 0 157 20 50 13 0.26 0.239 40 40 50 50 20 20 0 40 0.40 0 407 0.407 • Weibull model: P(D)= α +(1 α )[1 exp( β D γ )] • Weibull model: P(D)= α +(1- α )[1-exp(- β D γ )] • P(D)=0.096 + 0.904 [1-exp(-0.0035D 1.30 )] • Goodness-of-fit p-value = 0.61 6
Weibull Model with 0.95 Confidence Level 0.6 Weibull BMD Lower Bound BMD Lower Bound 0.5 0.4 ed Fraction Affecte 0.3 0.2 0.1 BMDL BMDL BMD BMD 0 0 5 10 15 20 25 30 35 40 dose 11:59 10/03 2007 0.71 2.25 7
Exposure → Dose-Response • Context: Dose-response analysis for cancer –Fit a mathematical model to D-R data: Prob ( tumor | D ) = F ( D ) | D ) b ( t F ( D ) P –D is administered (external) dose D is administered (external) dose • Generally acknowledged that PK information on internal dose (d) should be information on internal dose (d) should be incorporated whenever possible – e.g., d = mean AUC in tissue or blood AUC i ti bl d d 8
PK/PD Hidden Structure PK/PD Hidden Structure • However, most often there is no formal However, most often there is no formal attempt to separate the hidden Pharmacokinetic (PK) and ( ) Pharmacodynamic (PD) components of F that might explain the transformation of an external exposure into the development of t l i t th d l t f a tumor –e.g., F(D), F(d): multistage, probit, F(D) F(d) lti t bit Weibull 9
Hierarchical Model Hierarchical Model • The most natural way to link the PK and The most natural way to link the PK and PD components of a dose-response model is via a hierarchical model is via a hierarchical model ∞ ∫ ∫ = + − ( | ) ( 1 ) ( ) ( ) P tumor D P P g tumor x f x D dx 0 0 0 0 PD PK Model Model Background Risk 10
How to implement the model How to implement the model • PK: Experiment, e.g., rats, n animals/D – Calculate mean and s.d. of d ≡ AUC C l l t d d f d AUC – Assume normal distribution for f( d |D) • Simple PK: variability in internal dose • Complex PK: variability + parameter uncertainty • PD: Mechanism/Mode of Action? – e.g., two-stage clonal growth model for cancer g g g – OR, multistage, probit, Weibull • Numerical integration to fit hierarchical model Numerical integration to fit hierarchical model 11
Example Example • PK analysis – f(d|D) ~ Normal [ μ =(2D/(10+D) σ =0 2 μ ] f(d|D) Normal [ μ (2D/(10+D), σ 0.2 μ ] – f(d|D)={1/[ σ √ (2 π )]}exp{-½[(d- μ )/ σ ] 2 } • PD model – g(tumor|d): Weibull model – g(tumor|d)=1-exp(- β d k ) ( β d k ) (t |d) 1 • Fit hierarchical model using nonlinear least g squares with numerical integration (e.g., SAS NLIN) ) 12
Results Results D n #tumors proportion PK/PD fit 0 50 5 0.10 0.098 10 10 50 50 7 7 0 14 0.14 0 145 0.145 20 50 13 0.26 0.256 40 40 50 50 20 20 0.40 0 40 0 402 0.402 • μ =(2D/(10+D), σ =0.2 μ (from PK analysis) (2D/(10 D) 0 2 (f PK l i ) • β =0.0406, k=4.65 (from fit to tumor data) 13
Benchmark Doses Benchmark Doses • Can get BMD on scale of external Can get BMD on scale of external (administered) dose – Fix the parameters at estimated values Fix the parameters at estimated values – Let the desired BMD, e.g., BMD 10 , be the “parameter” of interest parameter of interest – Set BMR (0.10) = [P(tumor|D)-P 0 ]/[1-P 0 ] • Estimated BMD 10 is 13.91 (SAS NLIN) 14
Uncertainty Analysis Uncertainty Analysis • Can simulate a complete distribution of Can simulate a complete distribution of BMD 100BMR for any BMR using Monte Carlo bootstrap re-sampling of the tumor Carlo bootstrap re sampling of the tumor data. • Similarly, can simulate a distribution of excess risks for any D i k f D 15
BMD01 16 BMD10 16 10 12 12 Percent Percent 8 8 4 4 0 0 0.0 4.8 9.6 14.4 19.2 24.0 28.8 0 4 8 12 16 20 24 28 32 5th Percentile = 5th Percentile = Median = Median = 6.86 0.95 4.94 14.45 Use 5 th percentile as 95% U 5 th til 95% BMD05 16 BMDL 100BMR 12 nt Perce Useful for managing risk: 8 BMDL 10 = 6.86 4 BMDL 01 = 0.95 BMDL = 0 95 0 0 3 6 9 12 15 18 21 24 27 30 33 5th Percentile = Median = 16 3.55 9.64
Reduced Uncertainty in BMDs Reduced Uncertainty in BMDs PK (f) PD (g) BMR BMDL(05) Mic-Men Mi M Weibull W ib ll 0 01 0.01 0 97 0.97 (mean only) 0.10 6.29 Mic-Men Mic-Men Weibull Weibull 0.01 0 01 0 95 0.95 (distribution) 0.10 6.86 None Weibull 0.01 0.09 0.10 4.80 • Nonlinear PK info can reduce the spread of distributions of BMDs (reduce the data uncertainty). But, mean internal dose seems sufficient. , 17
Why the Mean Seems Sufficient Why the Mean Seems Sufficient ∞ ∫ ∫ = + − ( | ) ( 1 ) ( ) ( ) P tumor D P P g tumor x f x D dx 0 0 0 ∞ ∫ ∫ − − = [ [ ( ( | | ) ) ] ] /( /( 1 1 ) ) ( ( ) ) ( ( ) ) P P tumor tumor D D P P P P g g tumor tumor x x f f x x D D dx dx 0 0 0 ≅ [ ( ) ] [ ( )] E g tumor d D g tumor E d D f f 18
Comparison of Variation from Hierarchical M d l Model with Ordinary Binomial Variation ith O di Bi i l V i ti D N Mean SD Bin. SD 10 100 0.1432 0.0466 0.0495 20 100 0.2450 0.0620 0.0620 40 40 100 100 0 4066 0.4066 0 0659 0.0659 0 0695 0.0695 • Model: Hierarchical model with P0=0.098, g: Weibull (0.0406, 4.65), f: N(2D/(10+D), 0.4D/(10+D)) ( , ), ( ( ), ( )) • Mean: average of N generated tumor proportions • SD: observed std dev of N generated tumor proportions • Bin. SD: std dev calculated by [p(1-p)/50] 1/2 , where p=observed mean and 50 is number of animals/group 19
Combining PK and PD Results OSHA M OSHA: Methylene Chloride 1997 h l Chl id 199 • Internal dose • Risk estimate I t l d Ri k ti t from PK analysis from PD model • Mean d • MLE excess risk • UCL on excess risk UCL i k • UCL on d • MLE excess risk • UCL on excess risk 20
Usual Approach to Exposure Setting: T Two-Step Process S P • Human Exposure = Animal-Derived NOAEL or Benchmark Dose Animal → Average Human → Sensitive Human Exposure → Dose-Response ) ( (Dose-Response → Risk/Safety Characterization) (D R Ri k/S f t Ch t i ti ) 21
Dose-Response → Ri k Ch Risk Characterization i i • Inter-species extrapolation: Inter species extrapolation: – Animal → Human – Location extrapolation, from susceptibility of Location extrapolation from susceptibility of test animal to center (mean), μ H , of human susceptibility distribution susceptibility distribution – Uncertainty is due to a lack of knowledge about μ H , because of the variability among chemicals in their differential effects on test animals and humans 22
Dose-Response → Risk Characterization (cont.) Ri k Ch i i ( ) • Intra-species extrapolation: – Human → Human – Scale extrapolation, from the center, μ H , of the S l t l ti f th t f th human susceptibility distribution to an extreme tail area t t il – Uncertainty is due to the inherent inter- i di id individual variability in human sensitivity l i bilit i h iti it 23
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