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[…] Fundamental principles. Principle 1. Individual and population-level dose response. The starting point of this framework is that a conceptual distinction exists between effects on the individual and effects on the population. In particular, the effect of exposure at the level of the individual is the “magnitude” of a measure of toxicological effect. The result of a fixed exposure in a population will be different magnitudes of effect in different individuals in that population. Therefore, for a particular magnitude of effect, the result in the population is expressed as an “incidence.” In the present framework, the magnitude of change needs to be ordinally related to severity—so a greater magnitude constitutes a more severe effect. For instance, a body weight (BW) decrease of 20% is more severe than a BW decrease of 10%, and a moderate liver lesion is more severe than a mild liver lesion. Thus, for a monotonic dose response in an individual, it may be imagined that a higher exposure will, for any given endpoint, lead to more severe effects. In a population, increasing exposure levels will result in simultaneously increasing both incidence and severity: more and more individuals will suffer from more and more severe effects.For convenience, we establish the notation whereby human dose or exposure is denoted HD, the magnitude of effect is denoted by M, and incidence is denoted I. Because M is assumed to have an ordinal relationship with severity, incidence can be characterized as the incidence of effects of magnitude equal or greater than M, denoted I≥M. Because it is customary to discuss “incidence” in terms of effects that may be of concern, we use the simpler notation HDMI as shorthand for HD(I≥M). In addition, we use an asterisk (*) to indicate fixed or target values, such as a “critical effect size” (M*), target human dose (HD*), or target incidence level (I*).Given these definitions, the output of a human dose–response assessment is concerned with the quantitative relationships among HD, M, and I, along with their uncertainty. We focus on the most common type of output, which is developing a human health–based exposure limit (some other types of outputs are discussed in Supplemental Material, “Performing a population assessment”). In our notation, this means estimating a target human dose, HD*, which is regarded as a function of a two-dimensional protection goal: the target level of incidence (I*) and the specified level of effect magnitude (M*), both of which may be selected based on risk management considerations. Specifically, it is the dose at which only a small fraction of the population (low incidence of I*) will experience effects ≥ M*, which can be writtenHD* = HD(I*≥M*) = HDM*I*. [1]For instance, one could write HD(1%≥ 10%BW) = HD1001 for the dose at which only 1% of the population has > 10% change in BW.Principle 2. Continuous parameters underlying all observed dose–response endpoints. The second element of this framework is that observed dose–response relationships for all endpoints can, at the individual level, be recast as relating to an underlying continuous measure of response. Obviously, this principle applies to endpoints that are directly observed as continuous data. In that case, the observed dose response for the average responses as a function of dose may be imagined to reflect the dose response in an individual animal (namely, the average animal), even though an individual’s dose response is not directly observable in most toxicological studies. Endpoints that are generally measured as quantal response rates in a study population require some additional discussion. Below we discuss two options, briefly indicated as “deterministic” and as “stochastic” quantal endpoints ().Many quantal endpoints, such as in histopathology, are ordinally scored (e.g., minimal, mild, moderate, severe). Such endpoints can be considered as gradually increasing in magnitude at the individual level, but are reported in “bins,” or severity categories rather than as a continuous measure. In this way, the reported incidences can be thought of as relating to a single category (or a limited number of categories) of severity, whereas for other severities the incidences are not reported. In fact, any continuous dataset can be “quantized” and transformed into such a quantal (or ordinal) dataset by setting one (or more) cut points, resulting in incidences Y associated with each cut point. For instance, changes in hematocrit from the mean level in the controls can be separated into < 5% and ≥ 5%, and the fraction of animals below and above the 5% cut point can treated as quantal data. In this case, the effective dose (ED) for a 5% change in the continuous hematocrit data (i.e., ED05) is equal to the ED for a 50% quantal response, EDY
= 50%, as illustrated in . This concept of quantal endpoints has previously been discussed by .Therefore, when quantal data can be viewed as reflecting the incidence of a continuous effect above or below a “determined” cut point equal to M*, then the endpoint is referred to as a “deterministic quantal endpoint.” This is generally appropriate for effects that can occur in different degrees of severity. Furthermore, for the purposes of dose–response data analysis, the ED50 from the quantal response data would be used to estimate the EDM* corresponding to M* of the underlying continuous data. When the available dose–response data report the incidences related to various severity categories, then one of them may be chosen as being minimally adverse. When they report only the incidences related to a single severity category, this severity may be more than minimally adverse, in which case additional uncertainty arises in estimating the dose for a minimally adverse level of severity.However, not all quantal effects may be derived from applying a cut point to an underlying continuous variable. Some effects appear to have discrete outcomes, without an underlying, gradually increasing level of severity. An example of such an endpoint is malformations, which often do not show different degrees of severity: It is there or it is not. Cancer may be considered another example, because a particular tumor is present or not (ignoring observational practicalities). For such endpoints, an alternative interpretation of the dose response is possible: The observed incidences at each dose are considered as resulting from a “stochastic” process, where the observation that an individual animal has a tumor or not is analogous to drawing a lottery ticket, with probability equal to the expected incidence at that dose (and time of observation). That is, given all relevant circumstances for the particular individual (such as genetic make-up or experimental conditions), the effect is not fully determined, but rather any particular animal may be (un)lucky or not. If it were possible to perform a study in which all animals were identical, and identically treated (except the dose, but without dosing errors), then the quantal dose response would estimate the “individual probability of effect.” In this case, the observed incidence Y is treated as an estimate of the underlying individual probability of effect M, so M* would correspond to an incidence Y* = M*, as depicted in . This concept of quantal endpoints has previously been discussed by .Therefore, when quantal data are assumed to reflect the individual probability of an outcome as a result of a stochastic process, the endpoint is referred to as a “stochastic quantal endpoint.” In reality, there are always small differences between animals (including the experimental conditions), which will have some impact on the dose response. However, this additional impact is generally not separately distinguishable from the dose–response data. This interindividual variability might be assumed to be relatively small in the study population and hence ignored. If so, the observed quantal dose response approximates the individual probability of effect as a function of dose.Whether the “deterministic” or “stochastic” interpretation of quantal endpoints is correct remains uncertain, because even for endpoints such as cancer or malformations, it might be the case that the effect in an individual subject is evoked deterministically as soon as a given internal dose in that individual is reached. For risk assessment, the distinction between the two interpretations is important for the following reason: In the deterministic interpretation, the animal dose–response curve reflects the experimental variation and errors in the animal study, and therefore its shape (e.g., slope) would not be relevant information for predicting risks in humans. However, in the stochastic interpretation, the dose–response curve may be regarded as a model for the human individual probability of effect, and therefore its shape would be relevant as information for human risks.Principle 3. Selecting a basis for inference: the “effect metric.” The third fundamental element in this framework is that inferences are made on the basis of a selected “effect metric” that defines “toxicological equivalent” magnitudes of effect. This effect metric should reflect the effect size in such a way that it applies across species (or populations) as well as across individuals within a species (or population). Changes of the same magnitude in this metric are considered to reflect equal toxicologically induced changes.In addition, the magnitude of effect should also be ordinally related to severity at the level of the individual. For a continuous endpoint, severity increases with an increase in the percent change of a continuous endpoint (e.g., from 5 to 7% decrease in hematocrit). For a deterministic quantal endpoint, the severity is related to the category of effect (e.g., from “mild” to more severe liver lesions). For a stochastic quantal endpoint, severity is related to the probability of experiencing the effect (e.g., from 1 to 2% individual probability of cancer).“Equipotent doses” are defined as doses that elicit the same size of effect metric. Thus, individuals with the same equipotent doses (at all effect sizes) are defined as equally sensitive to the chemical for the endpoint.Note that it is assumed that the effect has previously been determined to be an appropriate basis for making inferences about human health effects—for example, that the effects observed in the test animal are relevant to humans. In this context, “relevance” needs to be determined only in the qualitative sense: Could a similar effect occur in humans, or not? When the answer is “yes,” quantitative differences, including large differences that are expected based on MOA considerations, should be addressed explicitly and quantitatively, taking uncertainty into account as well (e.g., using a probabilistic CSAF or DDEF; see Supplemental Material, “Chemical-specific/data-derived toxicokinetics or toxicodynamics”).The use of an effect metric does not necessarily imply that a given change is equally adverse in all individuals (or species). For instance, a 5% decrease in hematocrit may be considered as a toxicologically equivalent effect metric in all individuals, but be adverse in persons with anemia and nonadverse in healthy persons. Finally, it should be noted that further inferences are possible from the toxicologically equivalent effect metric to other measures of health effect, such as if an adverse outcome pathway can quantify the linkage between a change in effect metric and the likelihood of an adverse health outcome. For instance, if the effect metric is a percent change in serum cholesterol, given an adverse outcome pathway linking serum cholesterol changes to cardiovascular disease, one might aim to estimate the risk of fatal myocardial infarction in a specific population. Because variability in baseline serum cholesterol levels and other relevant risk factors (e.g., blood pressure, C-reactive protein) may differ across different populations (e.g., geographic regions, socioeconomic groups, lifestages), analyses of such “downstream effects” would necessarily be specific to the population(s) being assessed, even if the relationship between exposure and the effect metric is assumed to be the same across populations. Such analyses may also be useful for socioeconomic analyses because a fixed magnitude of effect may have different cost implications across human subpopulations (e.g., modifying insulin for diabetics vs. nondiabetics). This is discussed further in Supplemental Material, “Extrapolation to downstream health endpoints and adverse outcome pathways” and Figure S1.Principle 4. Making adjustments while accounting for variability and uncertainty. The final fundamental element in this framework is that dose–response assessment involves making inferences about the human population of interest for risk assessment (the “target population”) based on information obtained from a scientific study (the “study population”). In the usual deterministic approach, these inferences are accomplished using the “uncertainty factors” to address (potential) differences due to differing species, human variability, suboptimal study conditions, and so on. However, these factors are often mixtures of multiple elements that need to be clearly specified in a probabilistic framework. Specifically, making inferences between the “study” and “target” populations involves making adjustments from the study to the target populations while accounting for variability and uncertainty:Adjustments are needed to correct for differences between the study and target populations in order to make inferences as to the potential health effects in the population of interest, with the relevant exposure conditions. For example, on average across chemicals, the dose in milligrams per day eliciting the same effect differs between species due to differences in body size. The usual (implicit) adjustment is to divide the dose by BW, which is also intended to normalize across individual subjects in the (study or target) population. But data increasingly support the idea that the dose in milligrams per kilogram BW may need additional adjustment by an allometric scaling factor to achieve equivalent effects (e.g., ; ; ; ; ). Further, it might be known that, for any particular chemical, there are specific differences in toxicokinetic or toxicodynamic properties, which, for instance, make it plausible that one species would be more sensitive than others (e.g., resulting in a CSAF or DDEF). As another example, for some classes of effects, the expected relationship between a benchmark dose (BMD) and duration of exposure might be reflected by Haber’s law (toxicity depends on the product of concentration and exposure time), which may be used to adjust the BMD to other exposure durations. Usually, differences in study population/conditions and the target population/conditions can be better characterized (i.e., its uncertainty reduced) with additional data or analysis, and some can even be eliminated by conducting new studies that require fewer adjustments (e.g., conducting a chronic study to replace a subchronic study).“Variability” refers to intrinsic heterogeneity about a central tendency, usually between the individuals in the “target” population. For example, different individuals (humans) will exhibit different sensitivity to toxic effects from the same exposure due to various sources of variability (e.g., genetics, lifestyle, health status). Additional data or analysis can make an estimate of human variability more precise, but the variability itself cannot be eliminated.“Uncertainty” refers to a lack of knowledge that could, in principle, be reduced with additional data or analysis. Uncertainty can relate to the degree of adjustment (e.g., the exact allometric power with which to adjust for BW differences) but also to the degree of variability (e.g., how much more sensitive is the 95% individual relative to the median individual). As another example, because toxicity studies have finite numbers of individuals per dose group, the BMD is uncertain. This uncertainty can, in principle, be reduced by performing a larger or better designed study. Similarly, “missing studies” represent an uncertainty that can be quantified by meta-analyses comparing the overall differences between study types and capture that in a distribution (e.g., ). In some cases, observed variability among chemicals, in general, can be used to inform the uncertainty in an adjustment factor for a specific chemical. For instance, observed variability among chemicals in the dose ratio between subchronic and chronic studies for the same effect translates into uncertainty in the subchronic/chronic difference for a specific chemical for which no such data are available (e.g., ).As shows, all typical uncertainty factors include an uncertainty component, and an adjustment component, except for the intraspecies factor, where the adjustment component is replaced by a variability component.Prototypical approach implementing a unified probabilistic framework. The principles described above underlying a unified probabilistic framework can be applied to any type of study or endpoint that has dose–response information, but here we address the most common case of using animal toxicology data. The primary assumption is that the candidate critical endpoint(s) from an animal toxicology study is relevant in the sense that similar effects might be expected to occur in humans (uncertainty in the qualitative cross-species concordance is not addressed in this framework). Additional assumptions are as follows:The toxicity data are from a study conducted in an (inbred) laboratory animal strain, with the purpose of mimicking what might happen in a typical human being. Intrastudy variability reflects experimental errors (e.g., dosing errors, imperfectly controlled experimental conditions) and remaining differences (genetic, or otherwise) among animals. This is treated as statistical uncertainty in estimating a POD, which is supposed to mimic an equipotent dose in a typical human being.In the effect range of interest, the continuous dose–response relationships are monotonic and parallel on a log-dose scale across species and across individuals within a species, so that the values (distributions) for any adjustments, variability, or uncertainties are independent of the selected critical effect size M*. found evidence consistent with this assumption.The basic steps of the procedure under these assumptions are as follows (see also and ):Select a toxicologically equivalent effect metric and an associated critical effect size (M*), and conduct a BMD analysis with benchmark response (BMR) = M* () to derive the uncertainty distribution for the dose corresponding to M* in the animal (ADM*).Apply probabilistic interspecies and other adjustments to ADM* to derive the uncertainty distribution for the dose corresponding to M* in the median human (HDM*).Select a human variability distribution (e.g., log-normal), setting the median to HDM* with an uncertainty distribution as obtained in step 2. The measure of dispersion of this human variability distribution [such as geometric standard deviation; GSD = exp(σH)] has an uncertainty distribution, reflecting that we are uncertain about the degree of variability among individuals. From this (uncertain) human variability distribution, we derive an (uncertain) human variability factor HVI* for the ratio between the quantile corresponding to a selected target incidence (I*) value and the median, so that HDM*I* = HDM* × HVI*.This output is an estimate of the HDM*I* in the form of an uncertainty distribution, and any given level of confidence may be chosen for deriving an exposure limit (e.g., a “probabilistic RfD”), by taking the associated lower percentile of the uncertainty distribution of HDM*I*. Alternatively, the full uncertainty distribution can be combined with exposure information to inform risk management decisions. Details of each step are described below along with Monte Carlo (MC) procedures for the overall calculation.Step 1: Estimating the animal dose corresponding to the critical effect size for the selected toxicologically equivalent effect metric. The purpose of this step is to establish the uncertainty distribution for ADM*, the animal dose associated with a specified effect size M* (= BMR) based on a specified toxicologically equivalent effect metric.The key issue in defining the effect metric is how to address baseline differences across species or individuals in order to make changes “comparable.” For instance, a decrease of 10 g in rat body weight does not compare to a 10-g change in human body weight. For most (continuous) parameters, a percent change would be the obvious effect metric, being the only measure that may be defined as representing an equal effect size among different species and individuals (with different background responses). Note that an equal effect size does not imply that it will always be equally adverse in different species/individuals (such as a 5% decrease in hematocrit in anemic vs. nonanemic persons). Severity categories in histopathological lesions appear to directly apply as a measure of equivalent effect magnitude. However, for endpoints measuring an increase in individual probability of effect, the question of how to correct for the background risk is not easily answered. Various measures are being used, such as additional, extra, or relative risk, which all correct for background in a different way. It remains unclear, however, which of these measures reflects an equivalent measure of risk (if any), in particular when background risks among species (populations) differ greatly.After having chosen the effect metric, one also needs to specify a critical effect size—the magnitude of effect size M* of interest, as defined by the problem formulation and risk management context for the assessment. The term “critical” here should be understood in a wide sense, that is, it is a selected value (or even a range of values) that forms a starting point for doing the probabilistic calculations. Often, the problem formulation suggests that the critical effect size should reflect the effect size that is considered to be “minimally adverse” biologically. However, current toxicological knowledge does not allow one to unequivocally define minimally adverse effect sizes for all potentially critical endpoints. Further, a given effect size might not be minimally adverse in one species or individual, while it is minimally adverse in another (e.g., hematocrit and anemia, discussed above). As a practical limitation, the choice of M may be restricted by the available data. For instance, the reported data may relate to discrete values of M only (e.g., specific severity categories of lesions, as in histopathological data). Moreover, the lower the value of M, the less precise the estimates of the associated doses will be. For biologically defined M*s, one might aim to specify study designs that are likely to achieve “adequate” statistical precision for dose estimates related to that value M*. However, even then, the study design needed to achieve that goal may be impractical (e.g., unrealistic number of animals needed). If so, one may decide to use a statistically based M* (i.e., the lowest value of M* that achieves the desired level of statistical precision) as a surrogate. Such statistically based M*s could reflect levels of effect that are larger than minimally adverse levels, and this can be regarded as an additional source of uncertainty or addressed by setting a more stringent protection goal in terms of incidence. Typical examples of effect metrics and critical effect sizes are shown in , along with the BMD approach implied, by treating all endpoints as fundamentally continuous.The result of the dose–response analysis is an uncertainty distribution for ADM*, the animal dose corresponding to M*. Approaches to establishing the uncertainty distribution include a) translating the BMD confidence limits obtained by BMD software into a distribution, b) parametric bootstrapping [; implemented in the R package (version 3.2.2; ) PROAST ()], or c) Bayesian analysis (). It should be noted that fitting a single dose–response model may not fully capture the uncertainties in the dose–response data. Therefore, instead of deriving a BMD distribution from a single model, various models should be fitted to address model uncertainty. These model-specific distributions may be simply pooled in a single distribution (e.g., ), or one may apply “model averaging,” for which various approaches have been proposed (; ; ). In addition, if different dose–response datasets are available for the same endpoint, they could be combined in a joint dose–response analysis, with study as a covariate in the analysis, that is, some of the parameters of the dose–response model are study specific, and others are not ().Step 2: Adjustments due to interspecies differences and study conditions. The purpose of this step is to establish an uncertainty distribution for the “typical” human dose associated with a specified magnitude of effect and endpoint, and with specified exposure conditions. This step combines with the results of step 1. The “typical” human is defined as the median person of the population. This interspecies step involves addressing three separate aspects:A dosimetric adjustment factor (DAF) for generic physiological differences (e.g., body size differences for oral dose; respiratory tract differences for inhalation exposures) between the test animal and (median) human, along with uncertainty in the expected adjustmentAnimal-to-human uncertainties (AHU) due to potential chemical-specific toxicokinetic or toxicodynamic differences between the test animal and humans, resulting in differences in sensitivity for a given chemicalOther uncertainties (OU) due to specific study conditions that differ from the target exposure conditions (e.g., exposure duration, or exposure pattern).The result of this step is an uncertainty distribution for the human dose at which 50% of the human population has effects greater than (or equal to) M*:HD(0.5≥M*) = ADM* × DAF / (AHU × OU). [2]Each of the adjustments is described in more detail below.Dosimetric adjustments. It is increasingly evident that generic differences in physiology (e.g., body size) across species can be accounted for by multiplying the animal dose by a DAF, or equivalently, by dividing by an “assessment” factor (AF) accounting for interspecies body size differences (AFinter-bs).For oral exposures, scaling doses by a fractional power of BW has been found to better account for interspecies differences in body size than scaling by BW alone. Because oral doses are usually expressed as milligrams per kilogram BW, a correction factor is needed to convert the doses in milligrams per kilogram into allometrically scaled doses. Thus, the DAF and AFinter-bs are given byDAForal = (animal BW/human BW)1 – α [3]AFinter-bs(oral) = (human BW/animal BW)1 – α, [4]where α is the allometric power. This power is not exactly known, and can be represented by a distribution (e.g., normal). Because this adjustment is meant to extrapolate from the test animal to the median human, the average (median) animal BW in the study and the median human BW in the target (sub)population should ideally be used (). If these are not available, then standard values can be used (e.g., ), with an uncertainty that is probably negligible compared with the uncertainty in the allometric power (although the BW uncertainty could be included in the assessment).For inhalation exposures, different types of DAFs have been derived for particles (regional deposited dose ratio, or RDDR) and gases (regional gas dose ratio, or RGDR) (). Based on interspecies information about respiratory tract geometries and air flow rates, the inhalation DAFs differ depending on whether the effects of interest are regional or systemic. For example, for effects in the upper airways, DAFs are based on the surface areas of relevant regions of the respiratory tract and the inhalation minute-volume. For systemic effects that involve transport by blood, DAFs utilize information on species differences (if any) in blood-air and blood-tissue partition coefficients. As with the oral DAFs, these are meant to extrapolate between the (median) test animal and the median human. Standard values, rather than statistically based medians or values specific to the study, are usually employed, but clearly these are uncertain as well. Thus, one could define the uncertainty in the DAF (or in the analogous AFinter-bs(inhalation)) by assuming log-normal residual uncertainty:DAFinhalation = (RDDR or RGDR) × exp(εDAF) [5]AFinter-bs(inhalation) = (RDDR or RGDR)–1 × exp(εDAF), [6]where εDAF is normally distributed with a standard deviation of σDAF. The value of σDAF might be based on propagating the uncertainties in the parameters occurring in the calculations predicting the RDDR or RGDR or based on expert judgment.Chemical-specific toxicokinetic or toxicodynamic differences. Test animals and humans differ not only generically (e.g., in body size) but also in compound-specific toxicokinetics or toxicodynamics. Although on average across chemicals, the DAF is intended to neither under- nor overestimate the interspecies differences, the actual interspecies difference for any particular chemical is unknown in the absence of chemical-specific data. This uncertainty is addressed by subsequently dividing by a distribution for animal-to-human uncertainty (AHU), reflecting the additional differences in sensitivity between animal and median human beyond those addressed by the DAF (i.e., the toxicokinetic/dynamic differences specifically related to the chemical considered). For instance, assuming a log-normal uncertainty, one could defineAHU = exp(εAHU), [7]where εAHU is normally distributed with a standard deviation of σAHU. With chemical-specific toxicokinetic or toxicodynamic data, a CSAF or DDEF may be developed, resulting inAHU = (CSAF or DDEF) × exp(εAHU), [8]where the standard deviation of εAHU would normally be smaller than that of the default value related to Equation 6, as discussed in Supplemental Material, “Chemical-specific/data-derived toxicokinetics or toxicodynamics.”Additional study-specific adjustments. Depending on the situation (e.g., experimental setup of a critical study, toxicity database), additional issues may need to be addressed to infer the equipotent dose in the median human under the target conditions. Those additional adjustments and their associated uncertainties that are specific to the study (or endpoint) are addressed in step 2 as well. The purpose of these adjustments is to account for the “other uncertainties” (OU) in characterizing the uncertainty distribution for the median human dose associated with a specified magnitude of effect, based on a specified study and endpoint. Examples of additional uncertainties include the following:The human hazard to be assessed relates to a different duration of exposure than that in the study. For instance, when the effect was in a subchronic rather than chronic study, the animal dose for the selected magnitude of effect might have been smaller in a chronic study. Based on historical data, one can estimate the empirical distribution for the ratio of chronic to subchronic dose (e.g., using equipotent doses from studies of both durations across many chemicals). Or, in specific situations a dose–time relationship (e.g., cumulative dose = constant, analogous to Haber’s law) could be postulated, along with a distribution reflecting the uncertainty in how accurately the relationship holds.The human hazard to be assessed relates to a different route of exposure than that in the study, such as inhalation versus oral. Again, both an empirical (e.g., ratio of inhalation to oral equipotent doses), theoretical (e.g., based on total intake or absorbed dose), or model-based [e.g., physiologically based pharmacokinetic (PBPK) model] approach can be used, along with a distribution reflecting the uncertainty in how accurately the assumed relationship is believed to hold.The hazard is being assessed for a different exposure pattern than that in the study, such as continuous exposure in humans versus daily bolus exposure in the test animal. In this case, it is common to make assumptions about the dose–time relationship, such as peak or cumulative dose, as the basis for adjustment. If multiple assumptions are plausible, the uncertainty among the different options can be characterized through a distribution. For instance, when there is uncertainty whether a given peak exposure would be equivalent to a three times lower or a three times higher equivalent continuous dose as compared with Haber’s rule, this could be reflected by taking those values as the lower 5th and upper 95th percentiles of the equivalent dose distribution for constant exposure.Note that in this step uncertainties are are with respect to the same magnitude of the same effect (endpoint). Uncertainties with respect to possibly different effects due to missing studies, even if they are at a similar level of severity, are not addressed here. This additional uncertainty is best addressed after completing steps 1–3, which are all related to the specific effect under consideration. For a discussion of some of these additional uncertainties, see Supplemental Material, “Cross-study/endpoint uncertainties.”Step 3: Accounting for human interindividual variability in sensitivity. The aim of this step is to take into account differences in sensitivity across individuals in the population. For an exposure limit, for example, the result would be the uncertainty distribution for the dose associated with a specified endpoint and magnitude of effect (M*) for a “sensitive” individual, defined in terms of a percentile or incidence in the population (I*). To make these inferences, a population distribution representing the variation in equipotent doses among individuals needs to be specified. Because there are usually limited data as to the magnitude of this variation, this uncertainty needs to be taken into account as well.Assuming a log-normal distribution for human variability, with standard deviation σH on a log-scale, the relationship between M*, the incidence of effects I≥M*, and human dose HD is given byI≥M*(HD) = Φ[{ln HD – ln HD(0.5≥M*)}/σH], [9]where Φ is the standard normal cumulative distribution. A similar relationship can be derived for any other assumed human variability distribution. For an exposure limit, one selects a target incidence value I*≥M* and solves for dose D. Given that the median of the distribution HD(0.5≥M*) was calculated in step 2, this can be calculated by multiplying the median by the ratio between the I* quantile of the variability distribution and its median, denoted the human variability factor HVI*. For a log-normal distributionHVI* = exp{zI* σH}, [10]where zI* is the normal z-score corresponding to a quantile I*≥M*. For instance, at a 5% incidence, z5% = –1.64; at a 1% incidence, z1% = –2.33. Combining Equations 2 and 10, the resulting equation isHD(I*≥M*) = ADM* × DAF × HVI*/(AHU × OU). [11]For discussion of recent analyses of human variability data, see . For instance, Hattis and colleagues (; ) estimated equipotent doses in a number of individuals, and calculated the standard deviations σH of the log-transformed equipotent doses, representing the variability in sensitivity among individuals. Then, they fitted a log-normal distribution to these standard deviations established for different chemicals (studies). They separated the available data into toxicokinetic and toxicodynamic factors, and estimated the uncertainty in the overall human variability as a combination of toxicokinetic and toxicodynamic variability. In this way, a default uncertainty distribution for intraspecies variation may be defined ().For some effects, we might suspect larger differences in sensitivity than others, or it might be known that the particular target subpopulation is highly sensitive for the agent considered. Or, we might be more uncertain for some effects than for others, for instance, for effects that did not occur in the database underlying the default distribution. In such cases, one may decide to deviate from the default distribution in the appropriate direction. If compound- and endpoint-specific toxicokinetic or toxicodynamic data are available, these may be used to define a case-specific human variability distribution, with case-specific uncertainty about that distribution (see Supplemental Material, “Chemical-specific/data-derived toxicokinetics or toxicodynamics”).MC (Monte Carlo) calculation of HDMI. Keeping variability and uncertainty distinct in the calculation of HDMI requires a hierarchical approach to implementation. In addition, because the individual distributions cannot be combined in closed-form (particularly incorporating uncertainty in the extent of human variability), an MC simulation approach is necessary (see , for an “approximate probabilistic approach” that can be implemented in a spreadsheet without MC simulation). Specifically, at each MC iteration, all the steps addressing uncertainty are done first, followed by the steps evaluating variability:Evaluating uncertainty: Simultaneously draw MC samples [j] from ADM*, DAF, AHU, OU, and σH. Obtaining MC samples from ADM* is not a standard output from U.S. Environmental Protection Agency (EPA) Benchmark Dose Software (BMDS) (), but can be generated with PROAST using the bootstrap method (). Bayesian methods offer another approach to generating such samples, and software such as **WinBUGS** (version 1.4.3; ), JAGS (version 4.0.0; ), or **Stan** (version 2.8.0; ) can be used.Evaluating variability: Combine (ADM*[j] × DAF[j])/(AHU[j] × OU[j]) to obtain one sample of the “median” human dose HD(0.5≥M*)[j]. Next, given the target incidence I*, evaluate one sample of the human variability factor HVI*[j] = exp{zI* σH[j]}. Combining these results is one MC sample of the human target dose associated with a particular incidence I* and magnitude of effect M*: HDM*I*[j] = HD(0.5≥M*)[j] × HVI*[j].The result after many samples is the uncertainty distribution for HDMI.For the illustrative datasets discussed below, this procedure was used with 107 MC samples for uncertainty (j) (for ADM*, 103 bootstrap samples were resampled with replacement). Datasets and computer code are available in Supplemental Material, Table S1). An example of this approach for an exposure limit was provided by who used the term “ICED” (individual critical effect dose) rather than HD.Calculating population incidence for stochastic quantal endpoints. In the deterministic interpretation of quantal endpoints, the calculated incidence directly represents the expected incidence in the overall population. However, in the stochastic interpretation, the calculated incidence relates to a single individual’s probability M of experiencing the quantal endpoints (such as a tumor). For this reason, the HDMI for stochastic and deterministic quantal endpoints cannot be directly compared. To make such comparison possible, for the stochastic interpretation, the expected incidence in the overall population needs to be calculated by integrating all possible values of M (). The calculation is simplified by the preceding assumption that the underlying continuous dose–response relationships are “monotonic and parallel on a log-dose scale across species and across individuals within a species.” Specifically, let the animal dose–response function be represented byMA(AD) = f(AD, θ), [12]where MA is the magnitude of effect, AD is the animal dose, and f is the dose–response function with parameters θ. Based on “step 2,” the median human has the same magnitude of response as the animal [i.e., MA = MH,I
> 50%] when the human dose HD = AD × DAF/(AHU × OU). Rearranging so that AD = HD × AHU × OU/DAF, the dose–response function for the median human will beMH,I
> 50%(HD) = f(HD × AHU × OU/DAF, θ), [13]with the same model parameters θ. From “step 3,” the equipotent dose across human individuals is distributed log-normally with log-transformed standard deviation σH. Therefore, the magnitude of effect for a particular percentile of the population with z-score z, will beMH,z(HD) = f(exp[z × σH] × HD × AHU × OU / DAF, θ). [14]For a log-normally distributed population of equipotent doses, z has a normal distribution. Therefore, the population arithmetic mean of MH will be equal to the expected value of MH,z over a normally distributed z:

## Pipeline specifications

Software tools | WinBUGS, Stan |
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Applications | Drug design, Miscellaneous |

Organisms | Homo sapiens |

Diseases | Neoplasms, Drug-Related Side Effects and Adverse Reactions |