Probabilistic risk assessment's use of trees and distributions to reflect uncertainty and variability and to overcome the limitations of default assumptions

Pergamon

Environment

International, Vol: 25, No. 617, pp. 755-772.1999 Copyright 01999 Ekcvicr Science Ltd Printed in the USA. All rights resewed 0160-4120/99/S-see front matter

PI1SO160-4120(99)00053-7

PROBABILISTIC RISK ASSESSMENT’S USE OF TREES AND DISTRIBUTIONS TO REFLECT UNCERTAINTY AND VARIABILITY AND TO OVERCOME THE LIMITATIONS OF DEFAULT ASSUMPTIONS R.L. Sielken, Jr. and C. Valdez-Flores Sielken, Inc., 3833 Texas Avenue, Suite 230, Bryan, TX 77802, USA

EI

9903-149 M (Received 30 March 1999; accepted 25 May 1999)

Probabilistic risk assessment is an emerging approach to exposure assessment and quantitative cancer and non-cancer risk characterizations. The approach is easily extended to othertypes ofrisks and outcomes. A tree, like a decision tree or probability tree, encourages the evaluation of not only the default assumptions but also alternatives to those defaults, and reflects the uncertainty in the current state of knowledge. Trees are used in both the characterization of the dose received by individuals in a potential exposure situation and the characterization of the dose-response relationship for a specified response of concern. Probability distributions are used to reflect the variability in exposure, dose, and dose-response relationships among individuals and over time within individuals. Distributions incorporating variabilities, uncertainties, subjective probabilities, and expert judgements are used to characterize the probabilities of observing an individual in a population with a specified dose from exposure, with a specified probability of a certain adverse health effect for a designated dose, and with a specified probability of a certain adverse health effect (i.e., a specified risk). Some suggestions are given on how a risk manager can incorporate adistributional risk characterization into decision making. Some discussion is included concerning sensitivity analyses and path analyses. The major finding is methodology to explicitly incorporate variability, uncertainty, and alternatives to defaults into exposure, dose-response, and risk characterizations. 01999 Elsevier Science Ltd

INTRODUCTION Quantitative characterizations of exposure, doseresponse, and the risk of cancer or noncancer health effects have often been dominated by default assumptions and have failed to adequately reflect uncertainty and variability (NRC 1993; NRC 1994). Exposure characterizations frequently focus only on assumed, maximally exposed subpopulations rather than actual or sampled people. Exposure characterizations also frequently assume oversimplified exposure equations or models, which fail to realistically reflect

lifetime exposures that vary over time (from hour to hour, day to day, and year to year). They also assume extreme values, rather than likelihood distributions reflecting uncertainty and individual variability (NRC 1983; USEPA 1992). Dose-response characterizations that are based on the use of default models and a default assumption of lowdose linearity reflect several false premises (Sielken 1985; ILSI 1995; Sielken and Stevenson 1994; Sielken and Stevenson 1997; BELLE 1997; Crump et al. 1997). 75.5

R.L. Sielken, Jr. and C. Valdez-Flores

756

In many cases, linearity at low doses would not be expected based on the interaction between the multiple components in the biological processes. For example, for cancer, the two-stage growth models (Moolgavkar and Venzon 1979; Moolgavkar and Knudson 198 l), involving multiple mutations and cell birth and death rates, provide one means of exploring these interactions. In addition, if carcinogenesis reflects the imbalance between invading substances and defense mechanisms, then the cancer probability depends on, not only how much the substance increases or decreases the number of invaders, but also on how much the substance increases or decreases the number of defenders or their efficiency (Stevenson et al. 1994; Sielken et al. 1995; Sielken and Stevenson 1998). Dose-response characterizations use default assumptions about the dose-response model, dose scales, pharmacokinetics, pharmacodynamics, individual variability, interspecies extrapolation, extrapolations among exposure routes, the feasible range of dose-response model parameter values, and the relevance to humans of different responses in animals. Default quantitative risk characterizations are often based on only a single number for exposure and a single potency value to characterize the dose-response relationships. Furthermore, risk is often described solely in terms of an mcrease in the frequency of an adverse health response by one specific age rather than reff ecting a shortening in the response-free period. The limitations of default assumptions and the challenges to the appropriateness of such assumptions have stimulated the development of new quantitative risk assessment methodologies initially for cancer effects and more recently for noncancer health effects (HERA 1996a). The U.S. National Academy of Sciences and many others have noted the need for quantitative health risk assessment methodologies that go beyond a simple screening analysis based on upper bounds on risk (NRC 1993; 1994). The Academy recommended adoption of methodologies providing a higher-tier analysis based on realistic estimates of risk reflecting more of the available information. In addition, cost-benefit analyses and other contributors to risk management are more effective when they are based on realistic estimates of risk and accurate characterizations of uncertainty that reflect the state of knowledge about a substance’s risks and the variability of those risks among individuals in the population. The need for greater realism and more accuracy calls for a more comprehensive treatment of the available information. The challenges to the current risk assessment methodology

in the U.S.A. and the needs for a more comprehensive approach have stimulated the development of probabilistic risk assessment (Holland and Sielken 1993; Evans et al. 1993, 1994, 1995; Sielken et al. 1995; Presidential/Congressional Commission 1997; HERA 1998). Probabilistic risk assessment has also been stimulated by the passage of the Food Quality Protection Act (FQPA 1996) and the required aggregate and cumulative risk assessments (ILSI 1998a, 1998b; USEPA 1998b, 1998~). There is an opportunity to expand probabilistic risk assessments to include a weight-of-evidence based on distributional risk characterization. This would explicitly incorporate all ofthe relevant alternatives for each of the components in the exposure and dose-response assessments and their probabilities, rather than emphasizing worst-case values, default assumptions, and only one alternative for each of the factors in the risk assessment (NRC 1996). The distributional characterization could include the distributions of the dose from exposure, the added cancer risk at a dose, the benchmark dose, and uncertainty factors. This could include model and parameter uncertainty as well as variability. EXPOSURE

CHARACTERIZATION

Exposure generally refers to the frequency or length of time an individual comes into contact with an agent and the amount, concentration, or intensity of the agent during that period. The duration and level of exposure are components in the determination of the dose from exposure or dose. While exposure usually refers to the external situation, dose usually refers more to the internal situation for an individual and can be more complex. Dose can refer to intake (the amount of the agent that an individual takes in), delivered dose (the amount of agent that reaches a specific tissue), or biologically effective dose (the net amount of a specific biological activity or consequence stemming from the delivered dose). Exposure characterization often encompasses the characterization of the dose from exposure. Uncertainty

and variability

In most cases, dose cannot be conveniently observed or measured directly, and there are multiple equations, models, or other means of estimating it. In such cases, if these alternatives imply different quantitative values, then there is uncertainty about the dose. There is a specific value for the dose, but that value is not known with certainty (NRC 1994 (Chapter 9)).

Probabilistic risk assessment’s use of trees

Alternative Methods of Dose Calculation Weights Explicitly Indicating the Uncertainty in the Current State of Knowledge IO

Dose (mglkgday)

or

Chamctertzation1 of a Pammeter

of a Pammeter Characterization2 of a Pammeter wt - 0.5

Characterization1 of a Parameter

1

i

wt = 0.7x0.6 - 0.56

wt = 0.7x0.2 - 0.14

Dose (mglkg-day) ioi wt = 0.3~0.5 - 0.15 wt = 0.3~05 = 0.15 -Weight Weight-of-Evidence Based Distributional Characterization of Dose

1

10

2

For example,

the most probable

Dose (mgfkg-day)

dose is 1 mg /kg-day

which has a relative likelihood of 56%. Fig. 1. Simple trees explicitly indicating the results of alternative calculations of dose from exposure and explicitly reflecting the uncertainty in the current state of knowledge about dose.

In most cases, the dose is not the same for every individual in the population being evaluated. Variability refers to the different values of the dose for different individuals. Variability can exist regardless of whether thedoseiscertainoruncertain(NRC 1994(Chapter 10)). Although uncertainty is often used to describe both uncertainty and variability, the two are different. A decision maker can spend resources to reduce uncertainty without affecting variability, and he or she can focus on a subpopulation or even a specific individual and reduce variability without affecting uncertainty. Uncertainty

If there are alternative methods of calculating the dose from exposure, then the calculation should be done explicitly using each of the alternatives. The resulting calculations can be presented in a tree. It provides a structure for identifying the alternative methods of calculation, and displaying the results of each calculation method. In addition, a tree can incorporate the relative amount of scientific support (subjective probability) for using one method of calculation vs.

another and ultimately a means of combining the information. For example, it may be desired to determine the dose in mg of chemical per kg of body weight per day (mg/kg-d) from inhalation exposure to ambient air containing a specified chemical. There may be four alternative methods of making this calculation. There may be disagreement in the relevant scientific community concerning which alternative should be used. Rather than making the calculation using only one alternative (on the basis of a default assumption or policy), probabilistic risk assessment can make the calculation using all four alternatives and explicitly present the results of each. In this simple example, the corresponding tree is simple, as shown in the top tree in Fig. 1. In many cases, alternative methods of calculation can be decomposed to reveal explicitly the different factors involved in the calculation and the alternatives for each factor, as illustrated in the bottom tree. In addition, a tree allows for the explicit assignment of a weight on each alternative for each factor in the calculation. When the weights correspond to the

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probability of the alternatives under the current state of knowledge, the weights are a means for explicitly reflecting the uncertainty in the current state of knowledge (Graham 1993; Evans et al. 1993, 1994, 1995; Sielken 1993; ILSI 1995). The weights also allow the distributional characterization of dose. The key feature of trees is that they explicitly identify: 1) the structure of the alternative methods of calculation; 2) the quantitative results of the alternative methods of calculation; 3) the amount of certainty or degree of uncertainty about each alternative; and 4) the combined distributional characterization reflecting all of the relevant alternatives and the current state of knowledge (uncertainty) about each alternative. Variability

Different individuals may have different values for the component parameters in an exposure equation or model. Probability distributions indicating the frequencies of different values for the component parameters for different individuals in the population can be used to capture this variability (AIHC 1994; USEPA 1997, 1998a; Presidential/Congressional Commission 1997). For example, if the dose from exposure via inhalation of ambient air is calculated according to the equation: (mg/kg-d) = [Chemical Concentration in Ambient Air (mg/m’)] [Inhalation Rate (m3/h)] [Ambient Air Exposure Time (h/d)] [Exposure Frequency (d/y)] [Exposure Duration (y)] [ 1/Averaging Time (d)] [l/Body Weight (kg)],

Dose

then the variability among individuals in their values for the components in this equation can be reflected in distributions and combined using Monte Carlo simulation software (e.g., DistGEN 1995, Crystal Ball 1996, @Risk 1996), or other suitable software (Metzger et al. 1998), as depicted in Fig. 2 (HERA 1996b). Inter-individual variability: Subpopulations and populations

For the public, legislators, agencies, and other risk managers to make the best use of an exposure characterization and to judge its relevance and appropriateness, they need to know the answers to these and other questions. Whose exposures are being characterized? Do the exposure values being incorporated into the

R.L. Sielken, Jr. and C. Valdez-Flores

exposure characterization correspond to real people? Does each individual in the population have the same exposure? How different are the exposures in one subpopulation compared to the exposures in different subpopulations and the combined population? How many people are there in the population and each subpopulation? What is the frequency distribution of exposures in the entire population, and what are the frequency distributions of exposures in different component subpopulations? (NRC 1983, 1994; Presidential/ Congressional Commission 1997). Exposure characterizations can describe actually exposed individuals instead of assuming maximally exposed individuals. Exposure characterizations can also describe the number of people with specific exposures in a population where different individuals can have different exposures. Probability distributions can be used to characterize the frequency of different exposures in subpopulations of interest as well as the whole population (Sielken and Trenary 1995). Distributional exposure characterizations eliminate the need to assume that every person in a population has the same exposure, and provide much more information than the information provided by upper bounds on exposures. Figure 3 provides an example in which probability distributions rather than default constants are used to characterize exposure parameter values for subpopulations and populations and to estimate the relative likelihood of different exposures and the variability of exposure from individual to individual. Some of the numerical consequences of using distributions as opposed to default constant values are shown in Table 1. The estimated distributions of exposure parameter values inferentially correspond to actual populations and subpopulations, rather than hypothetical “maximally” exposed individuals. Separate distributional characterizations are developed and presented for each subpopulation and combined population. The differences between subpopulations and populations are preserved and made available for possible incorporation into the decision making. lntraindividual variability: Variability over time

Exposure characterization does not necessarily require the value of a component in the exposure equation or model remain the same at all times or for all ages. For example, an individual’s lifetime exposure duration (LED) in hours per lifetime can be determined by summing the number of exposure hours (HPD, hours per day) in a day, over all days (DPY, days per year) in

759

Probabilistic risk assessment’s use of trees

Probability

l.L

L

Chemical Specific Parameters:

General Parameters:

Pathway Specific Parameters:

Chemical Concentration in Ambient Air

Body Weight

Inhalation Rats

Exposure Frequency ( days I year )

Exposure Time ( hours I day ) k-

Exposure Duration ( years )

Averaging Time ( days ) I Fig. 2. Probability distributions reflecting the inter-individual variability in the values of the component parameters in the equation or model for calculating dose and the resulting probability distribution for dose from exposure (the distributions are shown as histograms for simplicity’s sake)

Population

of Recreational

Visitors

I 309 728

Number in Population: Visitation Years per Lifetfme (YPL):

LN (7.87, 22.47)

Visitation Days per Year (DPY):

LN (3.47, 4.57)

Visitation Hours per Day (HPD):

Angler Vbiton

Number in Subpopulations:

602

Neighborhood General Visiton

Regional General Visiton

9 649

296 101

Organized Sportr ViSitOf

3 376

Visitation Yeers per Lifetime (YPL):

LN (7.82, 23.41)

LN (8.92, 21.93)

Visitation Days per Year (DPY):

LN (11.95. 13.75)

LN (29.87, 30.83)

LN (2.91, 1.94)

LN (10.80, 10.49)

Visitation Houra per Day (HPD):

LN (3.33, 1.58)

LN (2.10, 1.95)

LN (2.10, 1.95)

LN (2.46, 3.03)

LN (7.82, 23.41)

LN (7.82, 23.41)

* LN (7.87,22.47) denotes a lognormal distribution with mean 7.67 and standard deviation 22.47 Fig. 3. Characteristics of the population and subpopulations in a site-specific evaluation of exposure duration (lognormal distributions fit to Monte Carlo simulated values).

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R.L. Sielken, Jr. and C. Valdez-Flares

Table 1. Quantitative aspects of using probability distributions for exposure components rather than upper bounds or fixed default values.

Subpopulations Anglers

Populations

;,;:h;yhood

E;te;;l

Visitors

Visitors

Organized s orts V!sitors

&Me;

and

Visitors

Anglers and All Visitors

95th Percentile of Lifetime Exposure Duration (HPD~DPYxYPL),~ (Hours Per Day x Days Per Year x Years Per Lifetime = Hours Per Lifetime)

1206.0

184.7

1607.0

747.5

199.0

201.2

HPDg5= 3.6 DPY,,=10.7 YPL,,=29.4

HPD,,= 3.7 DPY,,=10.9 YPL,,=29.4

95th Percentile of Individual HPD, DPY, and YPL Distributions HPDys= 6.3 DPY,,=35.5 YPL,,=30.0

6709.5

HPD,,= 5.6 DPY,,=84.1 YPL,,=26.6

HPDyS= 5.6 DPY,,= 6.6 YPL,,=30.0

12 527.5

1108.8

Probability( 0.0042

0.0028

HPDg5= 7.5 DPY,,=29.2 YPL,,=30.0

6570.0

HPDxDPYxYPL 0.0032

1 HPD9,xDPY,,xYPL,,

1132.5

1185.7

0.0040

0.0037

5.7

5.9

)

0.0025

HPD9,xDPY,,xYPL, / (HPDxDPYxYPL),, The Product of the 95th Percentiles for HPD, DPY, and YPL Divided by the 95th Percentile of HPDxDPYxYPL

5.6

7.8

6.0

8.8

HPDxDPYxYPL Derived from Fixed Default Values (No Data) 8x52~25 = 10400

8x52~25 = 10 400

8x52~25 = 10 400

8x52~25 = 10 400

8x52~25 = 10400

8x52~25 = 10400

HPDxDPYxYPL Derived from Fixed Values Divided by (HPD~DPYxYPL)~~ Derived from Data

I 8.6

6.5

52.3

13.9

56.3

51.7

I

which the individual is exposed in a year, over all years (YPL, years per lifetime) in which there is an opportunity for exposure. Mathematically,

exposure day in year y. A commonly tion to the summation in (1) is

used approxima-

LED = YPL x DPY x HPD YPL DPY, LED = c 1 HPD,, y=l d=l

(1)

where DPY, is the number of exposure days in year y and HPD,, is the number of exposure hours on the d-th

(2)

However, Eq. 1 is more realistic in that it does not assume that the exposure duration was the same for every day in which there was some exposure, nor does it assume that the number of exposure days is the same for every year. The principal impact of using Eq. 2

Probabilistic risk assessment’s use of trees

761

--qQ&yt

Default Calculation

Added Cancer Risk at 1 ppm (Hypothetical Numbers) Fig. 4. A hypothetical tree decomposing the cancer dose-response assessment into component factors and providing a structure for explicitly considering the multiple alternatives for each dose-response factor. Note: ppm = mg/kg.

instead of Eq. 1 is, roughly, that Eq. 2 exaggerates the lengths of the tails in the distribution of LED (that is, underestimates the lower percentiles and overestimates the upper percentiles). Accounting for variability over time leads to more realistic exposure characterization (Sielken and Trenary 1995). DOSE-RESPONSE

CHARACTERIZATION

Probabilistic risk assessment can use a tree to decompose the dose-response assessment into its component factors and to provide a structure for explicitly considering the uncertainty associated with the alternatives for each factor. A hypothetical example of such a tree in the context of a cancer dose-response assessment is shown in Fig. 4. In this tree, there are three factors - dose scale, dose-response model, and data set. The first factor is represented as having two alternatives or branches -the dose scale being a delivered dose based on physiologically based pharmacokinetic (PBPK) modeling and the dose scale being a default representation of intake without any PBPK modeling. For each alternative for the first factor, there are two alternatives for the second factor, the dose-response model. The first alternative for the dose-response

model is denoted in Fig. 4 by M, and could represent a specific time-to-response model (e.g., a two-stage growth model including the dependence of the response frequency on age or the length of time since exposure began) while the second alternative M, might be a default model (e.g., a multistage quanta1 dose-response model). For each alternative for the first two factors (the dose scale and the dose-response model), there are two alternatives in Fig. 4 for the third factor, the data set. The alternative data sets are denoted by D, and D, and could correspond to either epidemiological or animal bioassay data. Although two alternatives are shown for each of the three factors, there could be more than three factors, several alternatives for a factor, and different numbers of alternatives for different factors. In a tree such as that in Fig. 4, a combination of one alternative for each factor is a “path” through the tree and represents one method of dose-response analysis (one method of quantifying the added cancer risk at the specified dose). The default practice of considering only one default possibility for each factor results in only one path being explicitly evaluated and the uncertainty in the dose-response relationship not being characterized. In Fig. 4, that one path corresponds to

762

R.L. Sielken, Jr. and C. Valdez-Flores

Dose Scales

0.002

Added Cancer Risk at 1 ppm

Fig. 5. An example of a tree’s distributional characterization of the added cancer risk at 1 ppm indicating not only upper and lower bounds but also the probabilities of all possible values based on all of the available information and the current state of knowledge. Note: ppm = mg/kg.

the dose scale being “Default Intake”, the dose-response model being “MZ”, and the data set being “D2”. On the other hand, by explicitly identifying the potency corresponding to each path through the tree (as shown in Fig. 4), probabilistic risk assessment provides a true representation of the current state of knowledge and the uncertainty. The amount of scientific data and support for different alternatives may be different. Thus, the branches in the tree are weighted accordingly. At the most general level, the weights correspond to the relevant community of scientific experts and their current state of knowledge. In sensitivity analyses, the weights may correspond to individual experts or perspectives. The weights are numbers greater than or equal to zero and less than or equal to one. The weights on the branches emanating from each node in the tree sum to one. For example, in Fig. 5, the weights on the two alternative dose scales are 0.8 and 0.2. These weights sum to 1.0 and correspond to the PBPK-based delivered dose scale having four times greater support than using default intake. The weights on the alter-

natives for the second factor (the dose-response model) may be different for different choices of the dose scale. For example, the weights for M, and M, are 0.9 and 0.1, respectively, when the dose scale is delivered dose and are a different pair of values (0.5 and 0.5, respectively) when default intake is the dose scale. Intuitively, the weights reflect relative likelihood, expert judgement, scientific support, or “plausibility” (Graham 1993; Evans et al. 1993; Evans et al. 1994; Evans et al. 1995; Sielken 1993; ILSI 1995). Mathematically, the weights are conditional subjective probabilities. Thus, the probability for the numerical value or distribution of values for the added cancer risk at the end of a path in the tree is the product of the weights on the branches contained on that path. Thus, the weight on the added cancer risk at 1 mg/kg from the far left path in Fig. 5 is 0.216 (because 0.8x0.9x0.3=0.216), and the weight on the added cancer risk at 1 mg/kg from the far right path is 0.07 (because 0.2x0.5x0.7=0.07). The weights on the added cancer risks at 1 mg/kg at the end of each path corresponding to each method of analysis can be combined to determine the combined

Probabilistic risk assessment’s use of trees

probability on each value of the added cancer risk at 1 mg/kg that has some support in the relevant scientific community. For example, in Fig. 5 the weights on the added cancer risks at 1 mg/kg being 0.00001 sum to 0.528. The result is a probability distribution. The most probable added cancer risk at 1 mg/kg is 0.00001 with probability 0.528. The range (0.000005 to 0.002) of possible values and, perhaps more importantly, the probabilities of the different values in that range are shown. Uncertainty

Trees, such as the tree in Fig. 5, explicitly incorporate and disclose the uncertainty in the current state of knowledge about dose-response relationships. The weights in the tree are an explicit quantitative indication of “model” uncertainty and an indication of the relative certainty about different alternatives. Of course, as the current state of knowledge changes, the weights and possibly the structure of the tree can be updated using Bayesian rules or other methods (Brand and Small 1995). Although the outcome at the bottom of each path through the tree in a cancer risk assessment could be a function (or distribution of functions) describing the added cancer probability as a function of dose, the outcome is usually the added cancer probability at a specified dose. Thus, there are usually several trees one for each dose of interest or one for each dose in a specified set of doses. When there are separate trees for separate doses, the weights in the trees may change with the specified dose level if the current states of knowledge about different dose levels are different. Computer software to develop probability distributions of added cancer risks at specified doses using the probabilistic risk assessment methodology is available (UNRAVEL 1995). Frequently, the tree for the dose-response relationship involves more factors and alternatives than in the relatively simple examples in Figs. 4 and 5 (Evans et al. 1993; Evans et al. 1994; Evans et al. 1995; Sielken 1993 and 1995; ILSI 1995). For example, the preliminary tree for the cancer dose-response relationship for butadiene contains the following factors and alternatives (Sielken et al. 1996): Factor 1: Human target organ Alternative 1: Lung Alternative 2: Hematopoietic system Alternative 3: Any tissue

763

Factor 2: Carcinogenic mechanism Alternative 1: Genotoxicity Alternative 2: Nongenotoxicity Alternative 3: Genotoxicity and nongenotoxicity Factor 3: Dose Scale: Metabolite Alternative 1: Butadiene Alternative 2: Monoepoxide Alternative 3: Diepoxide Alternative 4: Combination of mono-epoxide and diepoxide Factor 4: Dose Scale: PBPK model for experimental animal Alternative 1: Model 1 Alternative 2: Model 2 Alternative 3: Model 3 Alternative 4: Model 4 Factor 5: Dose Scale: Relevant measure of delivered dose Alternative 1: Area under the curve (AUC) Alternative 2: Area under the curve above a critical level Alternative 3: Duration above a critical level Alternative 4: Peak concentration Factor 6: Dose Scale: Tissue where agent is quantified Alternative 1: Lung Alternative 2: Liver Alternative 3: Blood Alternative 4: Heart Factor 7: Dose-response model: Shape Alternative 1: Linear at low doses Alternative 2: Sublinear at low doses Factor 8: Experimental data: study Alternative 1: NTP II mouse study Alternative 2: Rat study Factor 9: Experimental data: Definition of response (tissue and severity) Alternative 1: Combination of all statistically significantly increased responses plus rare responses Multiple alternatives: A specific tissue with a statistically significantly increased response rate

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Factor 10: Experimental data: Time-to-response data Alternative 1: Not included Alternative 2: Included Factor 11: Interspecies extrapolation of pharmacokinetic differences Alternative 1: Human PBPK determined by Method 1 Alternative 2: Human PBPK determined by Method 2 Factor 12: Interspecies extrapolation of pharmacodynamic differences Alternative 1: Human pharmacodynamic differences from animal determined by Method 1: No differences Alternative 2: Human pharmacodynamic differences from animal determined by Method 2: (body weight)3’4 The use of epidemiological data to affect the weights in a tree, to expand it, or to form the basis of a separate tree is discussed elsewhere (Crouch 1996a; Crouch 1996b; Sielken 1995). “Parameter” uncertainty associated with the calculation for a path can be incorporated by determining a distributional characterization of this outcome not only for the case where the outcome is the “best estimate,” such as the maximum likelihood estimate, but also for the case where the outcome includes the uncertainty associated with variability in observing data, such as would be reflected if the outcome were the bootstrap distribution of maximum likelihood estimates (Crouch 1996a; Sielken 1993). Variability Although uncertainty in the dose-response assessment is incorporated through a tree’s outcomes and the weights which reflect the probabilities of the alternatives in the tree, interindividual variability can be incorporated through the alternatives themselves. Alternatives can include distributions reflecting interindividual variability. For example, an alternative dose scale used in the dose-response models could include not only the external dose from exposure but also a distribution of background doses, distributions of pharma-. cokinetic parameters in the determination of a distribution of delivered doses, and distributions of susceptibilities involved in determining biologically effective

doses from delivered doses. Another example of an alternative incorporating interindividual variability would be a distribution characterization of the interspecies differences in pharmacodynamic sensitivity to a given dose. Noncancer dose-response characterization Figures 4 and 5 provide a simple example of the tree and dose-response characterization for a cancer health effect where the target attribute is the added cancer risk at a specified dose. The target attribute may be different for other effects. For example, for noncancer health effects a currently, widely discussed target attribute is the reference dose (analogous to the reference concentration, tolerable intake, or acceptable daily intake) which is defined by the U.S. Environmental Protection Agency as “an estimate (with uncertainty spanning perhaps an order of magnitude) of a daily exposure to the human population (including sensitive subgroups) that is likely to be without appreciable risk of deleterious effects during a lifetime” (Barnes and Dourson 1988). The corresponding reference dose is calculated using an equation such as Reference

Dose = [Indicator Dose] / [Uncertainty

Factor]

in which the “Indicator Dose” is a No Observed Adverse Effect Level (NOAEL), a Lowest Observed Adverse Effect Level (LOAEL), or a Benchmark Dose (BMD) and the “Uncertainty Factor” is either a single uncertainty or safety factor or product of such factors (e.g., factors for interindividual or intraspecies variation, interspecies differences, extrapolating from a LOAEL to a NOAEL, extrapolating from a subchronic result to a chronic result, and data base completeness or other modifying factors). A simple hypothetical example of a tree indicating the available alternatives for determining the indicator dose, the available altematives for the uncertainty factor, and the resulting calculated reference dose is shown in Fig. 6 which is analogous to Fig. 4 in a cancer dose-response assessment. A more realistic example tree would include multiple factors for multiple uncertainty factors and distributions for at least some of the alternatives for the uncertainty factor values (Baird et al. 1996; Swartout et al. 1998). Of course, the indicator doses could also be distributions instead of single values (Sielken and Valdez-Flores 1996; Slob and Pieters 1998).

Probabilistic risk assessment’s

use of trees

765

Health Effects

Reference

Dose (mg/kg-day)

Fig. 6. Explicit evaluation and presentation of the calculated reference doses associated with all of the relevant combinations of indicator and uncertainty factor values in a noncancer dose-response assessment.

Weights would be added to Fig. 6 to reflect the current state of knowledge and attendant uncertainty about the noncancer health effects, and a distributional characterization of the probabilities of different values for the reference dose derived (analogous to Fig. 5). Computer software to develop probability distributions of reference doses using the probabilistic risk assessment methodology is available (UNRAVELN 1994). RISK CHARACTERIZATION

Distributional characterizations of exposure can be combined with distributional characterizations of the dose-response relationships to produce distributional characterizations of risk. Such distributional risk characterizations account for the uncertainty and variability in exposure and the dose-response relationships (NRC 1993). For example, for cancer risks, there are usually separate distributional characterizations of the added cancer risk at different dose levels (or for different ranges of doses) because the cancer dose-response relationship is seldom linear for all doses of interest. For instance, Fig. 7 shows three different distributional characterizations for the cancer potency where

Cancer Potency at a Specified Dose = [Added Cancer Risk at the Specified /Specified Dose].

Dose]

In Fig. 7, there is one distributional characterization of cancer potency when the dose is very low (between dose A and dose B), one when the dose is somewhat higher (between dose B and dose C), and one for larger doses (between dose B and dose C). Using Monte Carlo simulation software, these cancer potency distributions can be combined with the distribution of the dose from exposure to determine the distribution of Added Cancer Risk = [Dose from Exposure] x [Cancer Potency when the Dose is in the Range containing the Dose from Exposure 1. For noncancer health effects, the distributions of the dose from exposure and the reference dose along with their associated uncertainty and variability can be combined to determine the probabilities of different values of the Hazard Index = [Dose from Exposure] / [Reference

Dose 1.

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Dose from Exposure ( mg/kg-day

)

Probabili

Probability

I

ADDED RISK per mgl kgday

ADDED RISK per mslkaday from EXPOSUkE in the-range from B to C

when the DOSE from EXPOSURE in the range from A to B

Q ADDED RISK per mglkgday from EXPOSURE In the range from C to D

Added Risk per mglkgday x D&se from Exposure RISK CHARACTERIZATIQN

Probability of Increased Risk

_I

Increased Risk

None Expected

Combining Distributional Evaluations of Dose-Response and Dose from Exposure when a Chemical has Different Cancer Potencies in Different Dose Ranges Fig. 7. Determining the probabilities of different risks from distributional characterizations incorporating the uncertainty and variability in the dose from exposure and the cancer potency in different dose ranges.

DISTRIBUTIONAL CHARACTERIZATIONS DECISION MAKING

IN

The probabilistic risk assessment approach can be applied to characterize different types of outcomes. For example, the outcomes at the bottom of the tree could be the dose from exposure, the added cancer risk or potency at a specified dose, an indication of the amount of time without a certain carcinogenic response at a specified dose (i.e., the mean response free period), the dose (e.g., the ED,,,) corresponding to an increase of 0.10 in the probability of a specified carcinogenic response, or a reference dose for noncancer health effects. Furthermore, the distributional characterizations of the dose from exposure and another outcome can be combined using Monte Carlo simulation software to provide a distributional characterization of risk. Here, “risk” could be added cancer risk or a margin of exposure (such as the dose with a specified decrease in the me addition an response free period, the ED,,, or the NOAEL divided by the dose from exposure). The discussion in this section uses the terminology that would be appropriate if the outcome under con-

sideration were added cancer risk; however, an analogous discussion applies for other types of outcomes. Steps in implementing an expanded probabilistic risk assessment The first step in developing a weight-of-evidence based distributional characterization of, for example, added cancer risk is the identification of the tree structure. This includes identifying the major factors and the plausible alternatives or branches for each factor (e.g., one factor is the relevant chemical upon which the dose scale should be based and the alternatives could be the parent compound and its metabolites or combinations of these). The second step is to completely specify the details of the alternatives in the tree. For example, if the factor is the relevant chemical and the alternative is a specific metabolite, then this step could involve the specification of the tissue in which the metabolite was to be measured and how the measurement was to be done or modeled. If it were modeled, then the values of the parameters in that model need to be specified.

Probabilistic risk assessment’s use of trees

Frequently, during this step, additional alternatives or factors are added to the tree. The third step is to quantify the outcome (e.g., the added cancer risk) for each path through the tree. Each path corresponds to a combination of one alternative for each of the factors in the tree. Hence, each path corresponds to one analytical method. The third step includes all of the calculations or Monte Carlo simulations involved in modeling as well as the incorporation of parameter uncertainty and human variability. The fourth step is to determine the frequency of different values for the outcome in the tree and, by including or excluding different alternatives, to perform sensitivity analyses quantifying the relative impacts of the different alternatives for the different factors. The fifth step is to assign the alternatives for each factor on each path weights indicating the probabilities of the alternatives under the current state of knowledge and then to determine the probability distribution ofthe outcome. The sixth step is to perform sensitivity analyses to determine the impacts of the weights on the distributional characterization of the outcome. Problems with some current risk characterizations Instead of describing the relative likelihoods of different levels of risk, the current quantitative risk assessment procedures used by most federal and state agencies in the U.S.A. characterize the quantitative risk solely in terms of an upper bound that is calculated on the basis of a single set of default assumptions and statistical approximations. Default characterizations can be misleading when they fail to include all of the available, relevant information. These characterizations also do not encourage additional research and data collection because they may not reflect the additional information, even if it were obtained. Current default characterizations frequently fail to differentiate between the magnitudes of risk associated with different substances. Hence, such characterizations can result in the misallocation of limited health protection resources. These default characterizations contain assumptions and policy decisions that masquerade as scientific evidence and fail to make the use of judgements in the quantitative characterization of isk explicit. This failure makes it more difficult to critically evaluate these characterizations, identify important information needs,

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prioritize research objectives, route research funding, and incorporate new information. Default characterizations hinder the risk management process by failing to answer several important questions that could be answered if the current state of knowledge were fully reflected in the risk characterization. When an agency risk characterization only provides an upper bound on risk, the following are among the unanswered questions that could be better answered using the probabilistic risk assessment: How likely is the risk to be near zero? How likely is the risk to be a specific value, say l/100 000, between zero and the upper bound? How likely is the risk to be near the regulatory upper bound? How likely is the risk to be greater than that upper bound? Advantages of distributional risk characterizations An advantage of a distributional characterization of risk is that it explicitly quantifies the probability of the different possible risk levels. Current regulatory characterizations are usually limited to a single number represented as a plausible upper bound on risk. Rather than having the risk characterization being limited to an implicit range from zero to an upper bound, the distributional characterization indicates the relative likelihood of each risk value in that range. In addition, the distributional characterization indicates the probability of being in any specified range (e.g., less than l/l 000000, less than l/10000, and between l/10000 and l/1000). Distributional characterizations of risk also clearly indicate the most likely risk, the median risk, the mean risk, and any desired percentile of the risk distribution. Single number characterizations do not indicate this information. Distributional characterizations explicitly indicate the magnitude of the uncertainty about the risk associated with the uncertainty in the current state of knowledge. The range of risks between the 95th and 5th percentiles and the range between the 75th and 25th percentiles of the distributional characterization of risk are just two examples of the explicit quantitative indication of the magnitude of the uncertainty possible using a distributional characterization. The shape of the risk distribution in a distributional characterization can also provide useful information that is not contained in single number, upper bound characterization. For example, the distribution may be highly skewed and indicate that most of the likelihood is near zero risk with the remaining likelihood spread

R.L. Sielken, Jr. and C. Valdez-Flores

768

Information for Risk Managers Evaluating Alternative Remediation Strategies Confidence that a Soil Concentration Limit Satisfies Specified Health Protection Objectives

I

I 10’

I

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I 102 I B

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t

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I

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DIRECT and INDIRECT HEALTH BENEFIT

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Fig. 8. Illustration using distributional characterizations to quantify and compare the different probabilities of different risk management alternatives.

out thinly over several orders of magnitude. The distribution may contain more than one mode and indicate that there are sensitive subpopulations or altematives with substantial support in the relevant scientific community that differ markedly in their risk implications. Current default characterizations often give similar indications of the upper bounds on risks for two different substances because the impact of the default assumptions tends to dominate the impact of the substance specific data. This makes it often very difficult to differentiate between the risks for two different substances. Distributional characterizations provide much more information that can help the risk manager differentiate between the risks for two different substances. Even if, for example, the 95th percentiles in the distributional characterizations of two different substances were similar, the risks for the two substances could still be differentiated on the basis oftheir 50th percentiles, the range between the 95th and 5th percentiles, etc.

a

desired effect associated with

Facilitating risk management decision making Distributional characterizations can be helpful for the risk manager who must evaluate multiple alternative actions in the face of uncertainty (Graham 1993; Finley and Paustenbach 1994; Thompson and Graham 1996; Morgan 1998). For example, as suggested in Fig. 8, a risk manager might be trying to evaluate multiple remediation strategies, each with its own resultant soil concentration limit or upper bound. The usual type of single number characterization based on one set of default assumptions and an upper bound on risk would result in a single upper limit on the soil concentration. With respect to this “bright line” characterization of admissible soil concentrations, if aremediation strategy resulted in soil concentrations less than that limit, then that strategy would be admissible; otherwise, it would be no longer considered. However, a soil concentration exceeding a conservative lower bound on the maximum allowable soil concentration does not mean that the soil concentration necessarily exceeds a more likely value

Probabilistic risk assessment’s use of trees

169

Concentration Levels: The Cost I Benefit Relationship The Cost of Increasing the Confidence that the Added Risk is Less Than Particular Levels Probablllty

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Fig. 9. Example using cumulative distributions to quantify the changes in the probability of a desired effect associated with different risk management decisions.

ofthe maximum allowable soil concentration. Distributional characterizations indicate the probability of different possible values for the maximum allowable soil concentration and, hence, indicate the probability of the soil concentration resulting from a specific remediation strategy exceeding the maximum allowable soil concentration. The distributional characterization in Fig. 8 indicates the confidence the risk manager should have that the resulting soil concentration is below the maximum allowable soil concentration. This is approximately 30%, 87%, 97%, and 100% for remediation strategies A, B, C, and D, respectively. Thus, distributions make the level of uncertainty associated with each alternative action quantitative, complete, and explicit. Having an explicit, quantitative measure ofthe cost (likelihood of exceeding the maximum allowable soil concentration) of an action (here, a remediation strategy) enhances cost-benefit analyses. In Fig. 8 the risk management decision might come down to whether the gain (from 87% to 99%) in confidence about the soil concentration being less than its maximum allowable limit associated with using remediation strategy C instead of B is worth the reduction in

total direct and indirect health benefits. Without the distributional characterization, the relative costs of B and C would not have been available to theriskmanager. Figure 9 depicts the fact that decreasing a limit, such as the limit on a chemical’s concentration in soil, does not decrease the added risk from “unsafe” (above a specified level) to “safe” (below a specified level), but rather merely decreases the probability ofthe added risk being “unsafe.” Thus, the real choice for the risk manager is not a “safe” or “unsafe” level but rather how much he or she is willing to spend to decrease the probabilityofbeing”unsafe.“Distributional characterizations help make this choice by quantifying the probabilities, whereas single number characterizations based on one set of default assumptions and an upper bound do not. Distributions give risk managers the opportunity to make more informed decisions and facilitate decision making under budgetary restrictions. Distributional characterizations can also improve risk communication by incorporating all of the relevant information and different viewpoints and by providing sensitivity analyses of the quantitative impacts of different alternatives (Morgan 1998).

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SENSITIVITY

R.L. Sielken, Jr. and C. Valdez-Flores

ANALYSES

Both qualitative and quantitative sensitivity analyses are an important part of risk characterization. Quantitative sensitivity analyses can quantify the impacts on the risk characterization ofthe different alternatives for the component factors (Evans et al. 1993; Evans et al. 1994; Evans et al. 1995; Sielken 1993). This provides a quantitative perspective on the “robustness” of the risk characterization with respect to these alternatives and helps prioritize and direct research and resource allocation. Tree analyses which utilize weights to reflect the probabilities of alternatives provide a distributional characterization that is, itself, the ultimate sensitivity. analysis. This follows because the tree analysis combines the quantitative impacts of all of the relevant alternatives and gives a quantitative characterization in the form of a distribution of the probabilities of the possible outcomes and reflects the uncertainty in the current state of knowledge. The characterization ofthe current state of knowledge can reflect variability in expert judgement and uncertainty in assessing or expressing expert judgement (Graham 1993; Evans et al. 1993; Evans et al.1994; Evans et al. 1995; Sielken 1993). By using computer software that can quickly re-evaluate the probability distribution for different sets of weights and weights that are themselves distributions, the distributional characterizations of dose, potency, or risk can include the variability among different experts and the uncertainty in determining and characterizing the current state of knowledge. Each combination of one alternative (branch) for each factor (level) through a tree is a path. Path analyses can identify the paths with the larger or smaller outcomes or the paths with the outcome distributions that have the larger or smaller means, medians, modes, percentiles, ranges, and other statistics. Path analyses can also identify the frequency of a specific alternative for a factor among the paths with the larger or smaller outcomes (Sielken 1993). Tree, sensitivity, and path analyses can all contribute to a quantitative perspective on the robustness of the distributional characterization with respect to the alternatives and uncertainties in calculating doses from exposure, cancer potency, noncancer reference doses, cancer risks, noncancer hazard indices, and margins of exposure. DISCUSSION

By incorporating butions, probabilistic

tree-based analyses and distririsk assessment can integrate the

available information and reflect the current state of knowledge including expert judgements, uncertainty, and variability. A tree explicitly identifies and incorporates the uncertainty (model uncertainty) associated with the existence of multiple alternatives for component factors in the assessment. Weights reflect the probabilities of different alternatives under the current state of knowledge, and a distributional characterization combines the outcomes of the multiple paths through the tree. Distributions are used to reflect variability. The variability (parameter uncertainty) within the alternatives for the component factors in the assessment is represented by distributions. Also, the variability among experts and the uncertainty in assessing or expressing expertjudgements are represented by distributions (distributions of weights). The expansion of probabilistic risk assessment developed in this paper can be applied to several types of outcomes (Budnitz et al. 1998; Cooke and Jager 1998; Moses 1998). For example, an outcome could be the dose resulting from exposure. The outcome could refer to the dose-response relationship. For cancer responses, this outcome could be the ED,, or other risk specific doses (RSDs) or the dose corresponding to a specified decrease in the mean response-free period. For noncancer responses, the outcome characterizing the doseresponse relationship could be the no-observedadverse-effect level (NOAEL), the lowest-observedadverse-effect level (LOAEL), the Benchmark Dose (BMD), or these doses divided by uncertainty factors (i.e., a reference dose (RfD)). The distributional characterizations of the outcomes for dose and the doseresponse relationship can be combined to provide a distributional characterization for “risk.” For cancer health effects, “risk” might be the added cancer risk, the margin of exposure, or the reduction in the mean response-free period. For noncancer health effects, the “risk” might be the hazard index, the percentage of the reference dose, or the margin of exposure. REFERENCES AIHC (American Industrial Health Council). Exposure factors sourcebook. Washington, DC: AIHC; 1994. @Risk Release 3.5d. Risk Analysis & Modeling. Newfield, NY: Palisade Corporation; 1996. Baird, S.J.S.; Cohen, J.T.; Graham, J.D.; Shlyakhter, A.I.; Evans, J.S. Noncancer risk assessment: a probabilistic alternative to current practice. Hum. Ecol. Risk Assess. 2: 103-129; 1996. Barnes, D. G.; Dourson, M. Reference Dose (RfD): Description and Use in Health Risk Assessments; Regul. Toxicol. Pharmacol. 8: 471-486; 1988.

Probabilistic risk assessment’s use of trees

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