Mathematical Framework for Health Risk Assessment

Transcription

1 Mathematical Framework for Health Risk Assessment

2 Health Risk Assessment Does a substance pose a health hazard and, if so how is it characterized? A multi-step process Risk Characterization MSOffice1 Hazard Identification Dose- Response Assessment Exposure Assessment

3 投影片 2 MSOffice1, 2006/8/11

4 A General Regulatory Framework Chemical information chemical and physical properties: chemical formula, boiling and melting points, etc. use and occurrences. Toxicokinetic information absorption, distribution, elimination, metabolism physiologically based toxicokinetic model Hazard characterization Dose-response assessment Risk characterization.

5 Hazard Characterization Human studies, epidemiology and case reports Animal Studies: Pre-chronic (< 90 days), chronic cancer studies, Reproductive/developmental studies Other studies: Acute toxicity, mixture, mode of action, genotoxicity Synthesis of human, animal, and other evidences for cancer and non-cancer characterization. Susceptible populations: child, woman

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7 An Example of RA of a Pesticide Five historical studies on rats or mouse: four studies either concluded no clear evidence or not carcinogenic in the animals tested. One study concluded that there was ``equivocal evidence. Based on the historical data and 1986 EPA guideline this compound was classified as not classifiable (inadequate studies). Three new animal studies for risk assessment: male and female mouse and female rat

8 Toxicity Observed in Male Mice ppm Toxicity 0 adequate 100 adequate 800 adequate plasma ChE 24% inhibition RBC ChE 44% 8000 excessive decrease BW 14% plasma 24%, RBC 90%*, brain 23% excessive decrease BW 20% plasma 95%*, RBC 92%*, brain 37%* The survival rates of the five groups were about equal, at 78, the numbers alive at 78 weeks are 50, 51, 48, 54, 50. The mean BW were 34, 34, 34, 30, 28 gm.

9 Male Mice Liver Tumors from Compound X Tumor 0 ppm Adenomas 4/54 8/54 7/55 p= Carcinomas 0/54 4/54 2/55 p= Combined 4/54 10/54 9/55 p= /55 49/ /55 0/ /55 49/ Historical control range of the testing facility adenomas: 14 to 22% carcinomas: 0 t0 6.4%

10 Weight of Evidence Five descriptions for summarizing weight of evidence (EPA, 1999 draft Cancer Guidelines) 1. Carcinogenic to humans 2. Likely to be carcinogenic to humans 3. Suggestive evidence of carcinogenic, but not sufficient to assess human carcinogenic 4. Data are inadequate for an assessment of human carcinogenic potential 5. Not likely to be carcinogenic to humans

11 Proposed Classification Three studies on the compound and one on a structurally related compound : i. occurrence of liver tumors in male and female mice and in female rats at excessive dose. ii. the presence of a few rare tumors in rats. iii. the evidence for mutagenicity is not supportive. iv. The structurally related compound is not carcinogenic. The proposed classification having suggestive evidence of carcinogenicity but not sufficient to assess human carcinogenic potential.

12 Dose-Response Assessment Selection of principal studies from the available data for use to represent human risk selection of studies and critical endpoints Dose-response modeling (BMD approach) dichotomous models continuous models Adjust for a toxicological equivalency across species cancer and non-cancer endpoints inter- and intra-species adjustment factors. Others less than life time exposure, etc

13 Benchmark Dose (BMD) Approach Response (Risk) BMR 0 BMDL BMD Dose EPA's BMDS package (

14 (Traditional) Cancer versus Non-Cancer Modeling Cancer: Model chance events, but assume no inter-individual variation in susceptibility Non-Cancer: Model inter-individual susceptibility, but no role for chance events

15 BMDL: Point of Departure (POD) The BMDL is used as a starting point of departure (POD) to determine human safe exposure level. Two distinct approaches are followed: Linear extrapolation approach for cancer effects: a reference dose for a small risk, e.g.,10-5, is estimated as RfD = BMDL(10%) x Uncertainty factors approach to non-cancer effects: the RfD is calculated by dividing BMDL by several UFs RfD = BMDL/(UF1 x UF2 x UF3 x ) Interspecies uncertainty and intraspecies uncertainty.

16 BMD Approach for Cancer Endpoints Dose(mg/kg/day) Tumor Response 5/50 7/50 13/50 20/50 The generalized multistage dose response model: P(d)=1- exp { - θ 0 - θ 1 d - θ 2 d θ k d k }, θ i > 0. BMDL calculations: 1. Select a mathematical model to fit the animal DR data. 2. Specify BMR level. (extra risk: 10%) 3. Use the MLE method to estimate BMD (13.4 mg/kg) 4. Calculate the BMDL (preferably LR method) (7.4 mg/kg) 5. Linear extrapolation: RfD = 7.4 x 10-4 (mg/kg)

17 BMD Approach: Reproductive Effect Five groups of pregnant female CD-1 mice are exposed to DEHP with dose levels:0, 44, 91,191, and 292 mg/(kg BW) (Kodell et al. 1991, RA). Weibull dose response model (Chen & Kodell,1988, JASA): P(d) = 1 - exp (- θ 0 θ 1 d w ), w > 0. The Weibual model was fitted (BMD software: nested model). The BMDL at BMR = 1% is 23.0 mg/kg RfD = BMDL/(UF1 x UF2 x UF3 x ) = 23/(10 x 10) = 2.3 mg/kg.

18 Hypothesis Example: Non-Cancer Inhalation study 12h/day, 7 d/week for 12 weeks Dose (ppm) n mean s.d Hill Model: y = a + b dose /(k + dose) US EPA's (Benchmark Software, BMD = 156 ppm at BMR =1 std of control mean BMDL (95% confidence) = 122 ppm

19 Hypothesis Example: Rfc The BMDL of 122 ppm (430 mg/m 3 ) BMDL ADI = 430 x 12/24 = 215 mg/m 3 Rfc = BMDL/(Uncertainty Factors) UFs: interspecies, intraspecies, subchronic to chronic, susceptible group Rfc = 215/(UF1 x UF2 x. x UFk) Rfc = 215/(10 x 10) = 2.15 mg/m 3

20 Summary: Current Approach Human Safe Exposure Dose (RfD) = Benchmark Dose Lcf (BMDL) Human risk characterization (uncertainty) Factors 1. choose the most relevant (animal) datasets to represent human risk 2. fit a dose-response model to the data 3. adjust for human risk characterization factors

21 Adjustment for Human Risk Characterization: Estimating uncertainty factors

22 BMDL for a Prospective Human Interspecies: Animal D a Average human D h D a : BMDL estimated from a fitted dose response function T a : The random variable representing the ratio of human-toanimal sensitivity over all chemicals E(D a /T a ): BMD for a prospective human exposed to an average chemical D h : BMDL for average humans: the 95% LC limit on E(D a /T a )- account for an uncertainty in sensitivity to the tested chemical between the tested animals and humans). D h is the dose such that P[(D a /T a ) > D h ] = 0.95 => D h = D a /T 95, where T 95 is the 95th percentile of T a

23 Subpopulation BMDL Intraspecies: Average human D h Subpopulation D s (p) T h : The random variable representing the ratio of human-tohuman sensitivity to the tested chemical D h : The 95% lower confidence limit BMDL for average humans D s (p): The BMDL for the subpopulation with the sensitivity the 100p-th percentile. D s (p) = D h /T h,100p targets to protect the p fraction of population Combined Inters- and intrapecies: D a D s (p) D s (p) = D a /(T a,95 x T h,100p )

24 Shifted-lognormal Distribution The shifted-lognormal distribution is used to model ratios of interspecies and intraspecies uncertainty (T a and T h ) and the distribution of exposure (T g ): T = 10 μ+zσ + τ <=> log 10 (T - τ) ~ N(μ, σ 2 ) τ is the lower bound of T; when τ = 0, T is a lognormal LN: τ=1, μ=0, σ= T 50 T 95

25 Hierarchical model T a has a shifted LN with pdf - f a (t a μ a, σ a, τ a ) T h has a prior shifted LN - f h (t h μ h =c, σ h, τ h ) conditional on T a =t a, T h has a shifted LN - f h (t h μ h =log(t a )+c, σ h, τ h ) Bayesian Hierarchical model for pdf of T s : )=,,,, ( a a a h h s t s f τ σ μ τ σ + = a a dt a a a a t a f h h c a t h t h f h τ τ σ μ τ σ μ ),, ( ),, ) log( ( Human to Human Animal to Human

26 Human Extrapolated Dose Lower 100p% confidence limit on human dose: Calculate by D a /T s,100p, where T s,100p is the 100p th percentile of the unconditional human susceptibility distribution Instead of D a /(T a,100p T h,100p ) or [D a /(10 10)] In general, T s,100p can be expected to be smaller than T a,100p x T h,100p

27 The 90%, 95%, and 99% percentiles of distribution of human-to-human variability ratio from the conventional product and hierarchical model approaches. Conventional Hierarchical T h (T 50, T 95, τ) T 90 T 95 T 99 T 90 T 95 T 99 H1:(1, 10, 0) H2:(3.16, 10, 1) H3:(2, 15, 1) T a : Shifted-LN τ=1, μ=0, σ=0.580, T 50 = 2, T 95 =10 T h1 : Shifted-LN τ=0, μ=0, σ=0.608, T 50 = 1, T 95 =10 T h2 : Shifted-LN τ=1, μ=0.335, σ=0.377, T 50 = 3.16 T 95 =10 T h3 : Shifted-LN τ=1, μ=0, σ=0.697, T 50 = 2, T 95 =15

28 Proportion of Population at Risk: Non-cancer Effects RfD: the dose at which the risk of an adverse effect is presumed to be zero (or negligible). With RfD = D s (p), the proportion p of the population is expected not to have an adverse effect. The proportion (1-p) would have a risk of an adverse effect. The dose response function for the proportion of the population at risk can be constructed P[D s (p)] = 1-p.

29 Hypothesis Example Inhalation study 12h/day, 7 d/week for 12 weeks Dose (ppm) n mean s.d Hill Model: y = a + b dose /(k + dose) US EPA's (Benchmark Software, BMD = 156 ppm at BMR =1 std of control mean BMDL (95% confidence) = 122 ppm

30 Hypothesis Example: Rfc The BMDL of 122 ppm (430 mg/m 3 ) BMDL ADI = 430 x 12/24 = 215 mg/m 3 Rfc = BMDL/(Uncertainty Factors) UFs: interspecies, intraspecies, subchronic to chronic, susceptible group Rfc = 215/(UF1 x UF2 x. x UFk) Rfc = 215/(10 x 10) = 2.15 mg/m 3

31 Proportion of Population at Risk DR function for at risk sub-population: P(D s (p)) = 1-p T a : S-LN (T 50 =2,T 95 =10, τ a =0) (μ a =0,σ a =0.580) T h : S-LN (T 50 =3.16,T 95 =10, τ h =1) (μ=0.335, σ=0.377) Conventional: D s (p) = D a /(T a,95 T h,100p ) = 21.5/T h,100p p=0.95: D s (p=0.95) = 21.5/10 = 2.15 => 5% at risk p=0.90: D s (p=0.90) = 21.5/7.6 = 2.83 => 10% at risk Hierarchical: D s (p) = D a /(T s,100p ) = 215/T s,100p p=0.95: D s (p=0.95) = 215/38 = 5.66 => 5% at risk p=0.90: D s (p=0.90) = 215/24 = 8.96 => 10% at risk

32 Risks from Population Exposure Distribution of Exposure: The amount of exposure to individuals d is a random variable: the probability of the exposure the amount of exposures In general, exposure profiles often can be approximately by a skewed distribution such as a lognormal. Given the exposure density function g(d μ g,σ g ) and the dose response function that individuals will have an adverse P(d), the expected population risk is

33 Example: Reproductive Effect Five groups of pregnant female CD-1 mice are exposed to DEHP with dose levels:0, 44, 91,191, and 292 mg/(kg BW). Weibull dose response model: P(d) = 1 - exp (- θ 0 θ 1 d w ), w > 0. The Weibual model was fitted (BMD software: nested model). The BMDL at BMR = 1% is 23.0 mg/(kg BW) Interspecies: Animal BMDL D a => human D s

34 RfD for DEHP Reproductive Effect The BMDL at BMR = 1% is 23.0 mg/kg Convential: T a S-LN(T 50 = 2,T 95 =10, τ = 1) => D h = D a /10 = 2.3 mg/kg T h S-LN(T 50 = 1,T 95 =10, τ = 0)=>D s = D h /10 = 0.23 mg/kg Hierarchical: D s = D a /(T s,95 = 34) = 2.3/34 = 0.68 mg/kg For a population exposure g(d μ g,σ g ): The fraction of the population above the exposure dose above d is P(T g > d) = f The expected proportion of population at risk is

35 The expected proportion of the population at risk from an exposure distribution T g (-0.24, 0.426,0) a for the two subpopulations: T h (0, 0.608, 0) and T h (0.335, 0.377, 1) T g (-2.4, 0.426,0) Expected Proportion at Risk dose f = G(T g > d) T h (0, 0.608, 0) T h (0.335, 0.377, 1) x x x x x x x x x x 10-5 a. T 50 =0.004 mg/kg and T 95 = mg/kg

36 Hierarchical Models for Dose Response Assessment: link the pharmacokinetic (PK) and pharmacodynamic (PD) component RfD = BMDL/(UF1 x UF2 x.. x UFk)

37 Administration and Internal Doses Fit a mathematical model to D-R data: P(D)=1- exp {-q 0 -q 1 E(d D) q k E[(d D)] k }. D is administered (external) dose, d is internal dose E(d D) is a function between d and D E(d D) is estimated at each D by a PK analysis: AUC (area under the blood concentration-time curve) Cmax (maximum blood concentration) of parent chemical or metabolite The estimate d is then simply substituted for D.

38 Hierarchical DR model PK model: h(d D) is pdf of the internal dose d for administered dose D. PD model: P(d) is DR model for development of a tumor. P ( tumor D) = P + (1 P ) P ( tumor x) h( x D) dx PD Model PK Model

39 Example M1- PK: Michaelis-Menten, PD: Weibull PK: h(d D) ~ Normal [μ=2d/(10+d), σ=0.2μ] f(d D)={1/[σ (2π)]}exp{-½[(d- μ)/σ] 2 } PD: Weibull: 1-exp(-βd k ) M2- PK: Linear, PD: 2-Stage PK: h(d D) ~ Normal [μ=0.1d, σ=0.2μ] PD: two-stage: 1-exp(-β 0 -β 1 d-β 2 d 2 ) Fit hierarchical model using nonlinear least squares with numerical integration.

40 Hierarchical PK/PD Model Fit D Tumor (x/n) p = x/n Weibull 2-stage 0 5/ / / / MM-Weibull: BMD 10 = 13.91; BMDL= 6.86 mg/kg Linear-2stage: BMD 10 = 12.64; BMDL= 8.22 EPA BMD 2-stage: BMD 10 = 13.4; BMDL= 7.4

41 Effects of PK Modeling on BMDs and BMDLs PD: Weibull: 1-exp(-βd k ) M1: σ = 0.2μ PK: h(d D) ~ Normal [μ=2d/(10+d), σ=0.2μ] h(d D)={1/[σ (2π)]} exp{-½[(d- μ)/σ] 2 } PD: Weibull: 1-exp(-βd k ) M2: σ = 0, d = 2D/(10+D) PK-PD: Weibull: 1-exp{-β[2D/(10+D)] k } M3: σ = 0, d = D PK-PD: Weibull: 1-exp(-βd k )

42 BMD and BMDL Estimates PK (MM) BMD 01 BMDL 01 BMD 10 BMDL 10 M1: σ=0.2μ M2: σ=0 (mean) M3: None PK analysis on internal dose can reduce uncertainty in BMD estimation by improving the estimate of mean risk But, the complete distribution of internal dose does not appear to affect the characterization of uncertainty.

43 ( Exposure Dose-Response ) (Animal Average Human Sensitive Human) + = p h a T h a a a a a a h h a h h h dt dt t f c t t f dx x D f k tumor x g 100 ),, ( ), ) log( ( ),, ( ),, ( 0 τ τ τ σ μ τ σ μ σ μ β Overall Summary

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45 Uncertainty Analysis: distributions of BMD from the MM- Weibull model at BMR = 1 and 10% and distributions of excess risk at D = 10 and 20, based on 1,000 replicates. 16 BMD BMD Percent 8 Percent th Percentile = 0.95 Median = th Percentile = Median = Excess Risk Dose= Excess Risk Dose=20 Percent 12 8 Percent Median = th Percentile = Median = th Percentile = 0.271

46 Dose response function P(d) for the proportion of population at risk from the conventional product (--- o ---) and hierarchical model ( ), T a (2, 10. 1) a and T h (3.16, 10, 1) b RfD a: μ=0, σ=0.580, τ=1; b: μ=0.335, σ=0.377, τ=1