Beyond the Cancer Slope Factor: Broad Application of Bayesian and Probabilistic Approaches for Cancer Dose-Response Assessment

Abstract

Traditional cancer slope factors derived from linear low-dose extrapolation give little consideration to uncertainties in dose-response model choice, interspecies extrapolation, and human variability. As noted previously by the National Academies, probabilistic methods can address these limitations, but have only been demonstrated in a few case studies. Here, we applied probabilistic approaches for Bayesian Model Averaging (BMA), interspecies extrapolation, and human variability distributions to 255 animal cancer bioassay datasets previously used by governmental agencies. We then derived predictions for both population cancer incidence and individual cancer risk. For model uncertainty, we found that lower confidence limits from BMA and from U.S. Environmental Protection Agency (EPA)’s Benchmark Dose Software (BMDS) correlated highly, with 86% differing by <10-fold. Incorporating other uncertainties and human variability, the lower confidence limits of the probabilistic risk-specific dose (RSD) at 10⁻⁶ population incidence were typically 3− to 30-fold lower than traditional slope factors. However, in a small (<7%) number of cases of highly non-linear experimental dose-response, the probabilistic RSDs were >10-fold less stringent. Probabilistic RSDs were also protective of individual risks of 10⁻⁴ in >99% of the population. We conclude that implementing Bayesian and probabilistic methods provides a more scientifically rigorous basis for cancer dose-response assessment and thereby improves overall cancer risk characterization.

1. Introduction

The cancer slope factor (CSF), also known as the oral slope factor (OSF), is a toxicity value defined as an estimate approximating an 95% upper bound of the lifetime increased cancer risk for an incremental increase in lifetime average daily dose (U.S. EPA, 2005), usually expressed in units of risk of cancer per daily exposure to an agent in mg/kg-day. Dose-response relationships for carcinogenic responses are considered stochastic “non-threshold,” meaning that any exposure level may still have the possibility of producing a carcinogenic response. Once a CSF is derived, the combination of CSF and exposure scenarios, such as intake rates of a contaminated medium, exposure frequency and duration, enables the calculation of cancer risk for exposure for a lifetime, facilitating risk assessment and risk management decision-making (U.S. EPA (Environmental Protection Agency), 1989).

The traditional CSF assumes linear low-dose extrapolation and is derived by identifying a human equivalent dose (HED) point of departure (POD) at a 10% extra risk of cancer, and then taking the quotient 0.1/HED. The HED is derived from benchmark dose (BMD) modeling of experimental data, taking the lower confidence limit of benchmark dose (BMDL), and converting from animal doses using a dosimetric adjustment factor (DAF). For the benchmark response (BMR) in determining the BMDL, the response level of 10% extra risk of cancer is recommended as the default value appropriate for typical rodent bioassay in U.S. EPA’s 2005 Guidelines for Carcinogen Risk Assessment (U.S. EPA, 2005). In contrast to the derivation of the reference dose (RfD), which is an estimate of a daily oral exposure that is likely to be without risk of non-cancer deleterious effects, the traditional method for deriving the CSF generally neglects the uncertainties in addressing animal-to-human difference (UF_A) and human variability (UF_H) (U.S. EPA, 2002).

There have been growing concerns about the limitations of traditional approaches for CSF calculation using these simple assumptions and neglecting uncertainties and human variability in its extrapolation methods (National Research Council, 2009). On the other hand, others have criticized the current approach to linear extrapolation as overestimating cancer risk (Calabrese, 2022; Li, 2020). The absence of addressing these uncertainties is based on the assumptions that they are negligible and/or that linear extrapolation is sufficiently conservative to neglect them. However, the actual degree of uncertainties is not sufficiently known or quantified except in a few case studies (Chiu & Slob, 2015; Rheinberger, 2021; Slob et al., 2014). As a result, there have been intense, but largely theoretical, debates about the appropriateness of making adjustments to traditional cancer risk assessment practices to address population variability (Bogen, 2014; Finkel, 2014; Varshavsky et al., 2023).

Implementation of a comprehensive framework suggested by the World Health Organization/ International Programme on Chemical Safety (WHO/IPCS) and deployment of distributions of the uncertainty factors enable the quantification of uncertainties in both cancer and non-cancer endpoints (Chiu & Slob 2015; WHO/IPCS 2018). As previously demonstrated for non-cancer effects (Chiu et al. 2018), risk quantification at the individual level can be supported by employing the concept of the HD_M^I, which is defined as the human dose associated with an effect of magnitude M with population incidence I. For cancer, which is a stochastic endpoint, the magnitude of effect M at the individual level is the individual probability, or risk, of cancer (WHO/IPCS, 2018). The population incidence I in this case represents the more sensitive subpopulation, which will have a greater individual risk at a given dose as compared to the more resistant subpopulation. Thus, for an individual risk estimate, the HD_M^I has been suggested by WHO/IPCS to provide the dose-response curves addressing human variability. For instance, HD₀₁⁰¹ of an agent means the human dose at which adverse effects of magnitude M ≥ 1% will occur with an incidence no more than I = 1% in the population.

For cancer risk assessment, however, there is also interest in deriving the overall population risk, which integrates over the variable population susceptibilities to derive the dose associated with an overall population risk of cancer. This can be expressed as a RSD, which is the dose corresponding to a specified upper bound estimate for population incidence, similar to how the CSF is currently used. For example, for an RSD₀₆ corresponding to 10⁻⁶ population incidence, in a population of one million who are exposed at the RSD₀₆ of an agent for a lifetime, on average there would be no more than one extra individual (above background) experiencing cancer at 95% confidence.

Thus, although previous case studies exist as to how the CSF could be replaced by a probabilistic version, broad application of a probabilistic cancer dose-response approach has not yet been attempted (Chiu & Slob, 2015; Evans et al., 2001). Here, we developed a standardized probabilistic cancer dose-response workflow and applied it to a large database of over 250 animal bioassay dose-response datasets that have previously been used to derive traditional CSF values. This large number of datasets enables us to address several long-standing questions with respect to cancer dose-response assessment: (1) Is traditional low dose linear extrapolation over- or under-protective of cancer risk; (2) What are the implications of assessing population versus individual cancer risk when incorporating human variability in susceptibility; and (3) What are the largest contributors to uncertainties in cancer risk estimation?

2. Materials and Methods

2.1. Data Collection and Cleaning

Figure 1 illustrates the overall workflow from the processes of data collection to data analysis. All animal bioassay data were from health assessments (or references therein) by California Environmental Protection Agency (CalEPA) and U.S. EPA (including Health Effects Assessment Summary Tables [HEAST], Integrated Risk Information System [IRIS], and Provisional Peer-Reviewed Toxicity Values [PPRTV]). These were previously collected into two datasets, one (Wignall et al., 2014) containing the data of toxicity values, dose, number of animals, and number of animals with incidence, and another (Wignall et al., 2018) containing toxicity values and the detailed study information such as species, strains, sex, etc. Relevant data were extracted from these databases, which contained information from different types of toxicity values such as reference concentration, reference dose, inhalation unit risk, and CSF. Relevant information from Wignall et al. (2014) and Wignall et al. (2018) were merged based on their Chemical Abstracts Service (CAS) number, data source, reference study, organ, effect, number of animals, and number of animals with observed adverse effect. After merging, toxicity values from human epidemiological data were excluded because this analysis focused on animal bioassay data. Data containing less than 3 dose-response data points (including controls), missing response data, and derived from routes other than oral administration were also excluded. Also, datasets that have the same study information with the same dose-response data were considered as duplicates, although they may have been coded slightly differently or were collected from different governmental agencies. After cleaning, 255 datasets remained for further investigation.

Figure 1.

Overview of the workflow to derive probabilistic CSF for estimating population and individual risk and characterizing uncertainty contributions. First, cancer dose-response animal study data from different databases are merged and cleaned (left panels). Then, Bayesian benchmark dose modeling and probabilistic inter-/intra-species extrapolations were conducted, with comparison with traditional approaches (middle panels). Finally, the results were used to estimate individual risk, in terms of HD_M^I, and population incidence, in terms of risk-specific dose (RSD), and to characterize contributions to uncertainty.

2.2. Bayesian Modeling of Dose-response Relationship

Using these cancer animal bioassay dose-response data, we used a BMA approach to determine the BMD, fitting 8 different dichotomous models—quantal linear, logistic, probit, Weibull, multistage 2, log-logistic, log-probit, and dichotomous hill implemented in the Bayesian benchmark dose modeling (BBMD) system (Shao & Shapiro, 2018). BMA and other ensemble prediction methods are widely used in various fields, such as ecological, agricultural, and toxicological science (Bailer et al., 2005; Bazrafshan et al., 2022; Massoud et al., 2020; Shao & Gift, 2014). The BBMD system uses Markov Chain Monte Carlo (MCMC) sampling with 5,000 iterations, considering the posterior weights of each model to extract the posterior distribution of the model-averaged BMD. The model weights were calculated based on posterior likelihood as a statistic for cross-model comparison. Posterior samples of model parameters were used to calculate average posterior model weights for the models included. We also extracted the best fitting (the highest weight) model’s BMD for comparison. Additionally, we also used the BMDS from U.S. EPA (2012) to derive a BMD and BMDL with the same dose-response data points for comparison with the BMA BMD. In contrast to BBMD, BMDS selects only the best fitting model that has the lowest Akaike information criterion (AIC) to derive the BMD. In all cases, when applying linear extrapolation, we used a BMR = 10%.

For interspecies extrapolation, application of a DAF is the default approach recommended by U.S. EPA, specifically the use of allometric scaling by body weight to the ¾ power to address physiological differences (U.S. EPA, 2011). Based on the species, strain, and sex of each animal model, default average body weights were assigned based on recommended values from U.S. EPA (1988). For probabilistic analyses, the fixed allometric scaling power was replaced by a distribution as previously described (Chiu et al., 2018; Chiu & Slob, 2015; WHO/IPCS, 2018). Additionally, for probabilistic analyses, previously derived distributions for the uncertainties from inter-species differences (after allometric scaling) and human variability were also applied to the BMA BMD values to implement the probabilistic extrapolations to derive probabilistic CSF. See Table 1 for the uncertainty distributions for allometric scaling and other uncertainties.

Table 1.

Summary of Parameter Distributions Recommended by WHO/IPCS (2018) for Probabilistic Extrapolation.

Uncertainty Parameter	Distribution	Source
DAF = (BWa/BWh)1 – α	α ~ Norm(0.70, 0.0243)	Bokkers and Slob (2007)
UFA	LogNorm(1, 1.95)	Bokkers and Slob (2007)
σH	LogNorm(0.746, 1.5935)	Hattis and Lynch (2007)

Notes: DAF: Dosimetric Adjustment Factor; BWa: experimental animal body weight; BWh: default human body weight (70 kg); Norm(m,sd): normal distribution with mean m and standard deviation sd; LogNorm(GM,GSD): Lognormal distribution with geometric mean GM and geometric standard deviation GSD; UF_A: Animal to human uncertainty factors; σ_H: natural log-scale standard deviation for human variability. Distribution were adopted from (Bokkers & Slob, 2007) and (Hattis & Lynch, 2006). See Section 2.3 for calculation details.

2.3. Population and Individual Risk Estimation

Two different risk estimates are of interest for cancer risk assessment: population risk and individual risk. Population risk is a fraction of population experiencing cancers responding to the chemical agents and individual risk is a possibility of observing cancers for each individual in response to lifetime exposure to the agents. The traditional CSF does not distinguish between the two, essentially assuming that everyone has the same individual risk, which is therefore equal to the population risk. This limitation has been noted by the National Academies (National Research Council, 2009). However, in the probabilistic framework developed by WHO/IPCS, which includes variability in susceptibility across the population, these two risk estimates are not the same (Chiu & Slob, 2015; WHO/IPCS, 2018). The process for calculating both individual and population risk estimates was previously described in detail by Chiu and Slob (2015), and briefly reviewed here.

Because population risk is based on integrating over the population’s individual risks, we first consider individual risk. Individual risk estimation is characterized by the human dose associated with an effect of magnitude M and population incidence I (HD_M^I), where for cancer endpoints (assumed to be stochastic) the magnitude M is an individual’s probability (expressed as “extra risk”) of experiencing cancer (above background). M is equivalent to the BMR used for BMD calculation. The HD_M^I is calculated in a manner conceptually similar to the RfD, except taking the BMA-BMD uncertainty distribution for a benchmark response extra risk of M, multiplying by the uncertainty distribution for the DAF, and dividing by the uncertainty distributions for remaining inter-species differences (UF_A) and for intra-species variability (UF_H,I).

$$\text{H}{{\text{D}}_{\text{M}}}^{\text{I}}=\text{BMA}-\text{BM}{\text{D}}_{\text{M}}\times \text{DAF}÷\left(\text{U}{\text{F}}_{\text{A}}\times \text{U}{\text{F}}_{\text{H},\text{I}}\right)$$ (1)

Here, UF_H,I = exp(|z_I| × σ_H), where z_I is the z-score for the Ith percentile, σ_H is the natural log-standard deviation associated with human variability (assumed to be lognormally distributed), and the absolute value is used to ensure UF_H,I > 1. The uncertainty distributions for the BMA-BMD utilize the posterior samples from the MCMC simulation, and the other uncertainty distributions are the same as previously implemented for non-cancer RfDs (Chiu et al., 2018). We investigated the human dose at which a I = 1% of the population shows an effect of magnitude M or greater for the adverse effect considered, for extra risk values M = 10⁻², 10⁻⁴, and 10⁻⁶. An incidence I = 1% was chosen to represent the sensitive population and the extra risk values were selected based on typical benchmarks for severe non-cancer endpoints (M = 10⁻²) or cancer risk management (acceptable risk range of 10⁻⁴ to 10⁻⁶). These values of M are lower than customary for BMD calculations because we are using BMA instead of linear extrapolation for calculating lower risk levels. Calculations were performed using Monte Carlo sampling of 2,000 uncertainty parameter samples.

For population risk, the individual risk estimates must be integrated across the population. Here we utilize the concept of the RSD, described by the National Academies (National Research Council, 2009) as the dose causing a specific level of risk in the population. For cancer, this is often a one-in-a-million extra risk (ER = 10⁻⁶). The RSD_ER for a particular ER can only be derived implicitly, as described by Chiu and Slob (2015), using the following formula:

$$\text{ER}\left(\text{RS}{\text{D}}_{\text{ER}}\right)=\int \mathrm{\varphi }\left(\text{z}\right)f_{\text{A}}\left[\text{RS}{\text{D}}_{\text{ER}}\times \text{exp}\left(\text{z}\times {\text{σ}}_{\text{H}}\right)\times \text{U}{\text{F}}_{\text{A}}÷\text{DAF}\right]dz$$ (2)

Here, ϕ(z) is the standard normal distribution density for a z-score of z, f_A is the experimental animal dose-response function, and the other symbols are the same as for individual risk. For uncertainty distributions, f_A is sampled from the posterior distribution from the Bayesian BMD analysis, which includes weights for the different dichotomous models as well distributions for model parameters. RSD values were calculated at three different risk levels (10⁻⁴, 10⁻⁵, and 10⁻⁶ risk), denoted RSD₀₄, RSD₀₅, and RSD₀₆ respectively. All calculations were performed using Monte Carlo sampling, utilizing 10,000 random individuals for integrating equation (2) and 2,000 uncertainty parameter samples. These probabilistic RSDs using BMA-BMDs and probabilistic extrapolations were compared with traditional RSDs derived using linear extrapolation from BMDS and BMA-BMDs.

We investigated further whether any common “classes” of datasets could be identified using hierarchical clustering, examining correlations between the posterior weights of 8 models, the CIs of BMA-BMDs and probabilistic RSDs, the ratios of BBMD to BMDS BMDLs, and the ratio of the probabilistic to traditional RSDs. These values were standardized using scale function in R and illustrated using heatmaps using the gplots R package.

Finally, it is important to understand the most important sources of uncertainty in the resulting risk estimates. The overall variance is composed of contributions from each step in the derivation—BMD model selection, BMD parameters, DAF, interspecies toxicokinetics and toxicodynamics, and intraspecies variability. The eta-squared statistic is a standard measure of contributions to variation and was calculated using lsr R package. We also stratified chemicals by the size of the overall confidence interval (CI) to examine whether contributions varied depending on the overall uncertainty.

2.4. Data Processing and Reproducibility

All data analysis, modeling, and visualizations are performed using R (version 4.1.2) and RStudio (version 2022.12.0+353) under the operating system of Windows 11. U.S. EPA Benchmark Dose Software version 3.2.01 was used. All raw data and codes for functions and data visualization to check reproducibility are available in GitHub repository (https://github.com/Suji-Jang/Probabilistic-Cancer-D-R). More details are also provided in the Supplementary Materials.

3. Results

After data curation and cleaning, there were 255 datasets with dose-response data and background study information, consisting of 159 unique chemicals (see Supplemental Table 1). Figure 1 shows that 82 out of all 367 datasets after merging had only 2 data points and were excluded. A plurality of datasets (38.4%) contain 3 data points and the datasets containing more than 4 data points are only 10% of the overall database. Interesting, the organs that were most commonly identified were liver (45.9%), followed by stomach/forestomach (7.5%), bladder (5.1%), gastrointestinal (3.5%), and “systemic” tumors (3.5%), with the remaining being other tumor types.

Figure 2 shows the results of comparing Bayesian and traditional BMDS BMDLs. Figure 2A illustrates that BMDS BMDLs are generally located below the median of the BMA BMDs and correlate with the BMA BMDLs. Interestingly, the BMDS BMDL values have greater variation, with a range from 5.41×10⁻⁷ to 3,180, as compared to the BMA BMDL values with a range from 1.07×10⁻⁵ to 1,056, showing that considering model selection uncertainty reduces overall variance. The scatter plot (Figure 2B) compares the BMDLs derived from traditional BMDS and BMA-BMDs, demonstrating that log₁₀ BMDLs have a high correlation (r² = 0.61). The traditional BMDL is typically within an order of magnitude of the Bayesian model averaged BMDL in either direction, but with substantial outliers, as shown in Figure 2C. The same analysis was applied to the BMDL values derived from the selection of a model that has the highest weight (HW BMDLs) for comparison with BMA BMDL values (Supplemental Figure 1). All HW BMDLs are located within the range of 5^th percentile-median of BMA BMDLs, with a range from 8.82×10⁻⁶ to 1,034, more similar to BMA BMDLs rather than BMDS BMDLs. Their log₁₀-transformed values also show a greater correlation with BMA BMDL values, with r² of 0.92. This makes sense because BMDS usually selects the model with the lowest AIC, which is another measure of model fit. See Supplemental Materials for BMDL values for each chemical estimated using BMDS, HW-BMD, and BMA-BMD (Supplemental Tables 2, 3, and 4, respectively).

Figure 2.

Comparison of benchmark dose derived from the BBMD system and the U.S. EPA Benchmark Dose Software (BMDS). (A) 90% CI of Bayesian model averaged (BMA) BMD (purple bar), the median of BMA BMD (purple circle), and the lower 95^th confidence bound of benchmark dose (BMDL) from BMDS (green triangle). (B) Scatterplot of BMA BMDL versus BMDS BMDL. Black solid line denotes equality. (C) Histogram of the ratio of BMDS BMDL to BMA BMDL. Black dashed line denotes equality.

We also investigated how probabilistic population risk estimates, in particular the RSD₀₆ corresonding to the one-in-a-million extra risk, differ from traditional methods using linear extrapolation (Figure 3). BMA BMDLs were converted to RSDs using both traditional and probabilistic methods, and BMDS BMDLs were converted to RSDs using only traditional linear extrapolation. Figure 3A shows 90% CI of probabilistic RSD₀₆ compared to linearly extrapolated RSD₀₆ derived from BMA and BMDS BMDLs. The lower 95% confidence bound of probabilistic RSD values range from 1.84×10⁻¹¹ to 1.07, with a greater variance compared to linearly extrapolated MA RSD values, ranging from 1.07×10⁻¹⁰ to 0.003, or BMDS RSD values, ranging from 5.41×10⁻¹² to 0.005. Almost all RSDs using linear extrapolation methods are located between the 5^th percentile and median of probabilistic RSDs, indicating that traditional linear extrapolation is conservative, but tends to be less protective than the 5^th percentile of probabilistically derived RSDs.

Figure 3.

Comparison of risk-specific dose for a one-in-a-million extra risk (RSD₀₆) using Bayesian model averaged benchmark dose and probabilistic extrapolation (BMA-BMD + Prob.), using BMA-BMD and linear extrapolation (BMA-BMD + Linear), using the U.S. EPA Benchmark Dose Software (BMDS) and linear extrapolation (BMDS + Linear). (A) 90% CI of BMA-BMD + Prob. RSD₀₆ (green bar), the median of BMA-BMD + Prob. RSD₀₆ (green square), BMA-BMD + Linear RSD₀₆ (purple circle), and BMDS + Linear RSD₀₆ (blue triangle). (B), (C) Scatterplots comparing the 5^th percentile of BMA-BMD + Prob. RSD₀₆ with BMA-BMD + Linear RSD₀₆ and BMDS + Linear RSD₀₆, respectively. Black solid line denotes equality. (D), (E) Histograms of the ratio BMA-BMD + Linear RSD₀₆ to BMA-BMD + Prob. RSD₀₆ and ratio BMDS + Linear RSD₀₆ to BMA-BMD + Prob. RSD₀₆. Black dashed line denotes equality.

The 5^th percentile of the probabilistic RSD was compared to the two different derivations of traditional, linearly-extrapolated RSDs in Figure 3B–C. Probabilistic RSD values are less than 87.1% of MA RSD values and 88.2% of BMDS RSD values using linear extrapolation. In Figure 2, on average BMDS and BBMD give similar results for BMD, but the probabilistic RSD tends to be lower due to human variability. Population risk is derived by the more sensitive tail of the human variability distribution. As shown in Figure 3B and 3C, log₁₀-transformed BMDS RSD values showed significant correlation (r² = 0.40) with the probabilistic RSDs, and traditional linearly extrapolated MA RSD values showed similar correlation (r² = 0.46). Figure 3D and 3E also show that the ratio of linearly extrapolated RSD values to probabilistic RSD values are generally greater than 1, with median (IQR) ratios of 8.2− (3.5− to 30−) fold and 10.6− (4.3 to 27−) fold for BMDS- and BMA-based values, respectively. In both cases, there are heavy tails, with the 5^th–95^th percentile ranges of 0.05− to 240-fold and 0.12− to 150-fold. RSD values for each chemical derived from linear extrapolation using BMDS, linear extrapolation using HW-BMD, and both linear and probabilistic extrapolation using BMA-BMD, are contained in Supplemental Tables 2, 3, and 4, respectively.

Figure 4 compares the results of probabilistically-derived population- and individual-based risk estimates. The three rows (4A-C, 4D-F, and 4G-I) correspond to results of using individual risks HD_M^I for I = 1^st percentile sensitive population and M of 0.01, 10⁻⁴, and 10⁻⁶ extra risk, respectively. The first column (4A, 4D, 4G) shows the results of calculating the confidence interval for population incidence (equation (2)) at dose levels corresponding to the lower confidence bound on the HD_M^I. In Figure 4A and 4D, the medians of estimated population incidences are typically located close to 10⁻⁴ and 10⁻⁶ respectively, corresponding approximately to the multiplication of I = 0.01 and M = 0.01 and 10⁻⁴ respectively. Median estimates for the data presented in Figure 4G are close to 10⁻⁸ although they are not shown. However, the upper confidence bounds on population incidence extend quite a bit higher. These results suggest that setting exposure limits based on individual risk protecting 99% of the population may not be protective at the population level at 95% confidence. This is due to population risk being driven by the more sensitive tail of the human variability distribution. HD_M^I values derived from probabilistic extrapolation using BMA-BMD for each dataset are contained in Supplemental Table 5.

Figure 4.

Comparison of population risk estimates, in terms of probabilistic risk-specific dose (upper 95% confidence bound) for a one-in-a-million extra risk (Prob. RSD₀₆), and individual risk estimates, in terms of human dose associated with an effect of magnitude M and population incidence I = 1% (HD_M^1%) at varying levels of M. (A), (D), (G) 90% CI and median of population incidence for I = 1% at M = 0.01, 10⁻⁴, and 10⁻⁶, respectively. (B), (E), (H) Scatterplots comparing the lower 5^th percentile confidence bound of the Prob. RSD₀₆ with the HD₀₁⁰¹, HD_1E-4⁰¹, and HD_1E-6⁰¹, respectively. Black solid line denotes equality. (C), (F), (I) Histograms of the ratio HD₀₁⁰¹, HD_1E-4⁰¹, and HD_1E-6⁰¹ to the lower 5^th percentile confidence bound of the Prob. RSD₀₆. Black dashed line denotes equality.

Scatterplots in Figure 4B, 4E, and 4H summarize the correlations of probabilistic RSD₀₆ with HD₀₁⁰¹, HD_1E-4⁰¹ HD_1E-6⁰¹. In Figure 4B and 4E, HD_M^I at I = 1% with M = 10⁻² and 10⁻⁴ are typically greater than probabilistic RSD₀₆, and HD_M^I at I = 1% with M = 10⁻⁶ are highly correlated with and mostly similar to probabilistic RSD₀₆ (r²=0.48). Clusters of values are shifted down as magnitude decreases. Interestingly, there are ten datasets for which the HD_M^I remains substantially less protective than the RSD₀₆. These mostly have flat-shaped dose-response data and/or only one non-zero dose-response point, resulting in an extremely wide confidence interval of population incidence, as shown at the rightmost confidence bounds in Figure 4A, 4D, and 4G; likewise, they are the top-left outliers in Figure 4E and 4H. These ten datasets (which are different from the BMDL outliers in Figure 2C) have HD_M^I values about 20,000-fold less stringent than the probabilistic RSD₀₆ as shown in Figure 4H. This is because the individual risk dose-response curves for these datasets are much steeper than the population dose-response curves. Histograms of the ratio of HD_M^I and probabilistic RSD₀₆ in Figure 4C, 4F, and 4I support these results. On average, the ratio gets closer to 1 when M decreases, but the ten outliers remain. At a magnitude M =10⁻⁶ the HD_1E-6⁰¹ a generally similar to probabilistic RSD₀₆ with most differences less than 10-fold. Alternative choice of M and I are illustrated in Supplemental Figure 2. The increase in M and decrease in I typically makes the HD_M^I values greater. When M = 10⁻⁵ and I = 0.1%, the HD_M^I values showed the greatest correspondence with RSD₀₆.

Datasets clustered into 6 groups based on the hierarchical clustering dendrogram and examples of each group are shown in Figure 5. Group I datasets have relatively high values for the Dichotomous Hill model weight, the ratio of BBMD to BMDS BMDLs, and the BMA-BMD CI. Typical dose-response data for this group is a “saturated” (plateau) response even at the lowest dose above controls. Group II also have elevated Dichotomous Hill weight, but the ratios derived from BMD are not as high as Group I and the ratios derived from RSD are relatively low compared to other groups. This is because the dose-response data for this group, while “saturated” at higher doses, had at least one lower dose group with a low response. Moreover, for Groups I and II, traditional linear extrapolation tends to be more stringent than, and often outside the confidence interval of, the results of probabilistic analyses. For Group III, the weight of Log-logistic model in Group III was higher, and all the ratios were lower; these datasets tended to have an intermediate response at the lowest dose above controls. Groups IV-VI tend to have multiple models with similar weights, and all have more gradually increasing dose-response data. The main differences are subtle variations in the shape of the dose response data: Group V tend to look more “linear” (higher weights on Quantal-linear, Weibull, Multistage 2, Log-logistic, and Log-probit), Group VI tends to look most “non-linear” (higher weights on Weibull and Log-probit), and Group IV more intermediate, with many different models with slightly higher or lower weights. Supplemental Figures 3 and 4 show the correlations among 8 different model weights and ratios of BMD and RSD values and corroborates the results in Figure 5. For the ratio of BBMD BMDLs to BMDS BMDLs and that of BMD CI, both tend to have fairly strong positive correlation with the weight of Dichotomous Hill model and weak negative correlation with the others. By contrast, the ratio of probabilistic RSD to traditional RSD is strongly negatively correlated with Dichotomous Hill model and strongly positively correlated with Probit and Logistic models. The ratio of RSD CI is also fairly negatively correlated with the weight of Dichotomous Hill model and positively correlated with both Probit and Logistic models, with significant positive correlation with the ratio of probabilistic RSD to traditional RSD. Furthermore, the similarities between the models are also shown, for example, weights of the Probit and Logistic, and Weibull and Log-probit models have strong positive correlations.

Figure 5.

Clustered heatmap for the weights of models, the ratio of BMDL from BBMD to BMDL from the BMDS, the ratio of the 5^th and 95^th percentile of BMA-BMD, the ratio of probabilistic RSD₀₆ to RSD₀₆ derived from the BMDS and linear extrapolation, and the ratio of the 5^th and 95^th percentile of probabilistic RSD₀₆. The examples from each cluster, the top graph depicts the 90% CIs of dose-response curve (yellow band), the dose-response data, and approximate benchmark dose lower confidence bound (BMDL), at which 10% increase in response corresponds with the upper 95% confidence bound (black open triangle). Second through fourth graphs depict predicted the 90% confidence bounds (bands) and median estimate (solid line) of the human dose-response curves for the median (I = 50%) individual risk estimate (purple band), 1^st percentile individual risk estimate (blue band), and population risk estimate (green band). In each of these three lower graphs, traditional estimate of cancer risk linearly extrapolated from the BMDL (black dot) is also shown (black dashed line).

In Figure 6, contributions of different factors to the overall uncertainty in RSD₀₆ are shown stratified by the width of the 90% confidence interval. The RSDs for most datasets have at least 5 orders of magnitude of uncertainty width, with the largest contribution of uncertainty from BMD model selection followed by BMD parameter sampling. For these datasets, the largest contributions to overall uncertainty were mainly composed of BMD model and BMD parameters uncertainties; uncertainties from DAF and animal-to-human extrapolation are generally negligible. Interestingly, the datasets that have only 2 orders of magnitude uncertainty width have their largest contribution of uncertainty from uncertainties in human variability. These generally correspond to datasets with high weight on the dichotomous Hill model.

Figure 6.

Uncertainties with the largest contribution to the overall uncertainty in RSD₀₆, stratified by 90% uncertainty width in log10 (orders of magnitude) units. BMD: benchmark dose; DAF: Dosimetric Adjustment Factor; UF_A: animal to uncertainty; σ_H: log-scale standard deviation for human variability.

4. Discussion

There have been persistent concerns that the traditional approaches for cancer dose-response assessment have significant limitations: the assumption of linearity in low-dose extrapolation, the neglect of uncertainties in model choice and inter-species differences, the lack of incorporation of human variability, absence of determining the contribution of uncertainty, and the lack of distinction between the concepts of individual and population cancer risk. The impacts of these limitations have not been systematically evaluated, potentially leading to inaccuracies in cancer dose-response assessment. To address these limitations, numerous novel methodologies have been introduced over time, but none have been broadly applied and compared to traditional approaches across a large number of datasets. Our goal was to fill this gap, applying a standardized probabilistic approach using over 250 datasets from experimental animal studies to better understand the broad impact of existing assumptions.

It is often difficult to extrapolate from a dose-response curve to a range of low doses for providing more quantitative risk information for risk management decision-making. Linear and/or nonlinear extrapolation can be selected accounting for the chemical agent’s mode of action (MOA). Developing a toxicodynamic model for extrapolation is considered the preferred approach, though the uncertainty of any such model should be characterized (U.S. EPA, 2005). However, in the vast majority of current cancer risk assessments, the MOA is not ascertained, and linear extrapolation is used as the default approach. Thus, the first part of our probabilistic approach is an alternative to a single one-size-fits-all linear low-dose extrapolation approach – instead, we utilized BMA, implementing low-dose extrapolation with multiple dose-response models, each weighted by its posterior probability (Shao and Shapiro, 2018).

Additionally, unlike non-cancer dose-response assessment (U.S. EPA, 2002), the uncertainty coming from animal-to-human extrapolation and intra-species variability is usually omitted in cancer dose-response assessment. The U.S. EPA Guidelines for Carcinogen Risk Assessment (2005) does not require quantifying uncertainties beyond the parameter uncertainties associated with BMD modeling, implying that inter- and intra-species extrapolation are considered insignificant in cancer dose-response assessment. Our probabilistic analysis approach addresses this issue by utilizing the uncertainty and variability distributions previously only implemented for non-cancer effects (Chiu et al., 2018). Another advantage of the probabilistic approach is that uncertainty distribution can be better understood by analyzing the relative contribution of each uncertainty to overall uncertainty. Identifying the magnitude of uncertainties increases the transparency of a probabilistic CSF, enables the quantitative definition of uncertainties, facilitates the prioritization of research, and eventually aids risk management decisions.

Furthermore, the probabilistic framework suggested by WHO/IPCS (2018) and Chiu and Slob (2015) includes a way to distinguish between individual and population risk. We estimated both for each dataset, calculating a risk-specific dose for population risk (RSD₀₆ for one in a million risk) and the HD_M^I (where M is individual risk of cancer) for individual risk.

With this standardized approach, we were able to address three long-standing questions about regarding cancer risk assessment:

1. How protective is traditional linear low-dose extrapolation? In short, the answer is “it depends.” We found that in the majority of cases, traditional low-dose extrapolation is conservative, but less protective of population risk than the probabilistic approach, with the probabilistic RSD₀₆ typically 3− to 30− fold more stringent (Figure 3E). However, in a small number of cases where the dose-response data are highly non-linear (Groups I and II in Figure 5), linear extrapolation is over-protective, sometimes by many orders of magnitude. Thus, the degree of conservatism of linear extrapolation cannot be absolutely generalized, and probabilistic methods provide a more case-specific approach to characterizing low-dose-extrapolation uncertainty. Probabilistic approaches overall provide a more transparent and consistent basis for risk management, since the degree of protection (individual or population risk level) and statistical confidence level (e.g., 95% confidence) are made explicit.

2. Given human variability in susceptibility, is using population or individual risk more protective? Our results suggest that population risk tends to be more protective than individual risk. We found that in general, dose levels set based on protecting population risk are protective of individual risk, but the converse is not always true. As shown in Figures 4C and 4F a dose level protective of population risk of 10⁻⁶ at 95% confidence is protective of individual risk of less than 10⁻⁴ in >99% of the population. By contrast, as shown in Figure 4G, even protecting 99% of the population to 10⁻⁶ individual risk at 95% confidence, for a number of datasets, the population risk exceeds 10⁻⁴. In sum, protecting population risk at the level 10⁻⁶ is protective of individual risk at the level 10⁻⁴ in >99% of the population. Thus, HD_M^I should not be used alone, and population risk calculations are always recommended.

3. What are the largest sources of uncertainty in cancer risk assessment? Perhaps unsurprisingly, we found that in the vast majority of cases, uncertainty in model choice was the greatest contributor to uncertainties in cancer risk estimates (see Figure 6). Similarly, for non-cancer effects, we previously found that in the absence of BMD modeling, the point of departure was the largest source of uncertainty. Importantly, uncertainties in population RSD₀₆ values were almost all greater than four orders of magnitude, which contrasts highly with the results of probabilistic analyses for non-cancer effects, which typically had only two orders of magnitude of uncertainty (Chiu et al., 2018). Additionally, in the subset of datasets where uncertainties were narrower (<1000 fold), there are also many cases where the largest source of uncertainty was uncertainty in human variability, similar to findings for non-cancer (Chiu et al., 2018). Thus, in general, low dose-extrapolation of the shape of the dose-response curve remains the greatest source of uncertainty, therefore, the importance of probabilistic approaches to BMD modeling and parameter estimation should be highlighted. However, for datasets where the dose-response curve is better characterized, human variability is the greatest uncertainty.

This study has some important limitations. First, chemical-specific information such as physiologically based pharmacokinetic (PBPK) modeling, which is for predicting the absorption, distribution, metabolism and excretion of chemical substances, or MOA studies, which refers to the means by which an agent achieves its toxic effect or action, is not included in the derivation of a CSF. Additionally, this analysis does not include specific sources of variability in susceptibility, such as gender or life-stage (Barton et al., 2005; Li & Xiong, 2023; Zeise et al., 2013). This was intentional, as this study is designed to be a generalized and comprehensive approach for most chemicals. Future work can account for these more detailed characteristics of chemicals when there is sufficient information. However, it should be recalled that the very low risk levels deemed important for risk management are impossible to verify empirically, even for ionizing radiation, for which there are more carcinogenicity data as compared to any other agent (Crouch, 1986). Thus, it is unlikely that even biologically-based modeling will be able to resolve the low dose extrapolation issue, as has been previously pointed out (Crump, Chen, et al., 2010; Crump, Chiu, et al., 2010). Moreover, it has been previously shown that model averaging approaches that incorporate model uncertainty provide a more accurate prediction of low dose risks than the traditional approach of linear extrapolation from a POD (Wheeler & Bailer, 2007, 2013).

Additionally, the uncertainty distributions used for inter-species extrapolation and intra-species variability are based on historical data on non-cancer effects, and thus assumed to also apply to cancer. However, because there are very few data on either inter-species extrapolation or human variability in susceptibility to carcinogens, it is challenging to address this limitation. Nonetheless, the impact of this limitation may be limited, as we have shown that in most cases by far, the greatest source of uncertainty is BMD model choice rather than these other sources of uncertainty. Also, the largest contribution of BMD model uncertainty indicates that low-dose extrapolation is the biggest uncertainty source, and this is one of the most challenging uncertainties to address. Although biologically based dose-response models have been suggested for estimating low-dose risks, they still exhibit remaining uncertainty regarding the mechanisms of toxicity in low doses (Crump, Chen, et al., 2010). Another option is to identify precursor effects using non-cancer RfD instead of cancer dose-response; however, it does not provide estimates for cancer population incidence or individual risk. Also, as this option relies on variability distributions to protect sensitive subpopulations, similar extrapolation issues still exist in the tail of the distribution at low incidence I, despite the use of log-normal distributions in this study (Crump, Chiu, et al., 2010; Finkel, 2014; Varshavsky et al., 2023). Finkel (2013) has suggested independently that the population risk estimates have to be adjusted upward by roughly a factor of 7-fold to address human variability, which is comparable to our results of 8− to 10-fold differences in the upper 95% confidence bound (Finkel, 2013).

Finally, as with all risk assessments based on cancer bioassays, this framework is still not able to fully quantify the severity of cancer considering organ types and effects. For example, pancreatic cancer, for which the relative survival rate is 11.5%, is not the same as thyroid cancer, for which the relative survival rate is 98.4% (Siegel et al., 2023). However, because site concordance between experimental animals and humans is not expected as a general rule, this limitation cannot be addressed without human epidemiologic data.

5. Conclusion

We previously demonstrated the feasibility of Bayesian and probabilistic approaches and applied them in noncancer risk assessment for 1,400 endpoints obtained from the governmental agencies (Chiu et al., 2018; Chiu & Slob, 2015). Here, this study verifies our previous conclusions and extends the application of these methods to cancer dose-response animal study data, while also incorporating Bayesian model averaging. In particular, we have implemented a Bayesian and probabilistic approach for cancer dose-response assessment that enables a more statistical, transparent, and informative estimation of population and individual cancer risk that addresses uncertainties of model selection, inter-species extrapolation, and intra-species variability. Using over 250 cancer datasets from cancer bioassays used in government cancer assessments, we have demonstrated the feasibility and applicability of the approach for broad application across chemicals. The results derived from the Bayesian and probabilistic approach go “beyond the CSF” and instead utilize two complementary concepts: a population risk-specific dose (RSD) and the individual protective human dose HD_M^I. Future work remains to implement more refined and extended applications of the probabilistic approaches, such as incorporation of chemical-specific pharmacokinetic and MOA information, as well as application to agents causing cancer by inhalation. Overall, our results support a conclusion that combining Bayesian and probabilistic methods provides a more scientifically rigorous basis for cancer dose-response assessment and therefore can be used to improve overall cancer risk characterization.