Monotonic Bayesian Semiparametric Benchmark Dose Analysis

Quantitative risk assessment proceeds by first estimating a dose‐response model and then inverting this model to estimate the dose that corresponds to some prespecified level of response. The parametric form of the dose‐response model often plays a large role in determining this dose. Consequently, the choice of the proper model is a major source of uncertainty when estimating such endpoints. While methods exist that attempt to incorporate the uncertainty by forming an estimate based upon all models considered, such methods may fail when the true model is on the edge of the space of models considered and cannot be formed from a weighted sum of constituent models. We propose a semiparametric model for dose‐response data as well as deriving a dose estimate associated with a particular response. In this model formulation, the only restriction on the model form is that it is monotonic. We use this model to estimate the dose‐response curve from a long‐term cancer bioassay, as well as compare this to methods currently used to account for model uncertainty. A small simulation study is conducted showing that the method is superior to model averaging when estimating exposure that arises from a quantal‐linear dose‐response mechanism, and is similar to these methods when investigating nonlinear dose‐response patterns.     

Empirical Scaling of Single Oral Lethal Doses Across Mammalian Species Based on a Large Database

The scaling of administered doses to achieve equal degrees of toxic effect in different species has been relatively poorly examined for noncancer toxicity, either empirically or theoretically. We investigate empirical patterns in the correspondence of single oral dose LD, values across several mammalian species for a large number of chemicals based on data reported in the RTECSQ database maintained by the National Institute for Occupational Safety and Health. We find a good correspondence of LD, values across species when the dose levels are expressed in terms of mgadministered per kg of body mass. Our findings contrast with earlier analyses that support scaling doses by the 3/4-power of body mass to achieve equal subacute toxicity of antineoplastic agents. We suggest that, especially for severe toxicity, single- and repeated-dosing regimes may have different cross-species scaling properties, as they may depend on standing levels of defenses and rate of regeneration of defenses, respectively.     

New Approaches in Risk Assessment for Carcinogens

Methods are needed to improve the ability of biomonitoring and epidemiological studies to identify potential carcinogenic hazards and to quantify human risk. The limitations of pharmacokinetic models can be mitigated by the direct measurement of molecular markers of biologically effective dose of carcinogen. Parallel animal and human studies are recommended as a means of validating these markers.     

The Fallacy of Ranking Possible Carcinogen Hazards Using the TD50

Ames et al. have proposed a new model for evaluating carcinogenic hazards in the environment. They advocate ranking possible carcinogens on the basis of the TD50, the estimated dose at which 50% of the test animals would get tumors, and extrapolating that ranking to all other doses. We argue that implicit in this methodology is a simplistic and inappropriate statistical model. All carcinogens are assumed to act similarly and to have dose-response curves of the same shape that differ only in the value of one parameter. We show by counterexample that the rank order of cancer potencies for two chemicals can change over a reasonable range of doses. Ames et al.'s use of these TD50 ranks to compare the hazards from low level exposures to contaminants in our food and environment is wholly inappropriate and inaccurate. Their dismissal of public health concern for environmental exposures, in general, based on these comparisons, is not supported by the data.     

Implications of Developmental Toxicity Study Design for Quantitative Risk Assessment

Standard experimental designs for conducting developmental toxicity studies typically include three- or four-dose levels in addition to a control group. Some researchers have suggested that designs with more exposure groups would improve dose-response characterization and risk estimation. Such proposals have not, however, been supported by the results of simulation studies, which instead back the use of fewer dose levels. This discrepancy is partly due to using a known dose–response pattern to generate data, making model choice obvious. While the carcinogenicity literature has explored implications of different study designs, little attention has been given to the role of design in developmental toxicity risk assessment (or noncancer toxicology in general). In this research, we explore the implications of various experimental designs for developmental toxicity by resampling data from a large study of 2,4,5-trichlorophenoxyacetic acid in mice. We compare the properties of benchmark dose (BMD) estimation for different design strategies by randomly selecting animals within particular dose groups from the entire 2,4,5-T database of over 77,000 birth outcomes to create smaller "pseudo-studies" that are representative of standard bioassay sample sizes. Our results show that experimental designs which include more dose levels have advantages in terms of risk characterization and estimation.     

Benchmark Dose Analysis via Nonparametric Regression Modeling

Estimation of benchmark doses (BMDs) in quantitative risk assessment traditionally is based upon parametric dose‐response modeling. It is a well‐known concern, however, that if the chosen parametric model is uncertain and/or misspecified, inaccurate and possibly unsafe low‐dose inferences can result. We describe a nonparametric approach for estimating BMDs with quantal‐response data based on an isotonic regression method, and also study use of corresponding, nonparametric, bootstrap‐based confidence limits for the BMD. We explore the confidence limits’ small‐sample properties via a simulation study, and illustrate the calculations with an example from cancer risk assessment. It is seen that this nonparametric approach can provide a useful alternative for BMD estimation when faced with the problem of parametric model uncertainty.     

Optimal Experimental Design Strategies for Detecting Hormesis

Hormesis is a widely observed phenomenon in many branches of life sciences, ranging from toxicology studies to agronomy, with obvious public health and risk assessment implications. We address optimal experimental design strategies for determining the presence of hormesis in a controlled environment using the recently proposed Hunt‐Bowman model. We propose alternative models that have an implicit hormetic threshold, discuss their advantages over current models, and construct and study properties of optimal designs for (i) estimating model parameters, (ii) estimating the threshold dose, and (iii) testing for the presence of hormesis. We also determine maximin optimal designs that maximize the minimum of the design efficiencies when we have multiple design criteria or there is model uncertainty where we have a few plausible models of interest. We apply these optimal design strategies to a teratology study and show that the proposed designs outperform the implemented design by a wide margin for many situations.     

Risk Assessment of Nongenotoxic Carcinogens Based upon Cell Proliferation/Death Rates in Rodents

Increased cell proliferation increases the opportunity for transformations of normal cells to malignant cells via intermediate cells. Nongenotoxic cytotoxic carcinogens that increase cell proliferation rates to replace necrotic cells are likely to have a threshold dose for cytotoxicity below which necrosis and hence, carcinogenesis do not occur. Thus, low dose cancer risk estimates based upon nonthreshold, linear extrapolation are inappropriate for this situation. However, a threshold dose is questionable if a nongenotoxic carcinogen acts via a cell receptor. Also, a nongenotoxic carcinogen that increases the cell proliferation rate, via the cell division rate and/or cell removal rate by apoptosis, by augmenting an existing endogenous mechanism is not likely to have a threshold dose. Whether or not a threshold dose exists for nongenotoxic carcinogens, it is of interest to study the relationship between lifetime tumor incidence and the cell proliferation rate. The Moolgavkar–Venzon–Knudson biologically based stochastic two-stage clonal expansion model is used to describe a carcinogenic process. Because the variability in cell proliferation rates among animals often makes it impossible to detect changes of less than 20% in the rate, it is shown that small changes in the cell proliferation rate, that may be obscured by the background noise in rates, can produce large changes in the lifetime tumor incidence as calculated from the Moolgavkar–Venzon–Knudson model. That is, dose response curves for cell proliferation and tumor incidence do not necessarily mimic each other. This makes the use of no observed effect levels (NOELs) for cell proliferation rates often inadmissible for establishing acceptable daily intakes (ADIs) of nongenotoxic carcinogens. In those cases where low dose linearity is not likely, a potential alternative to a NOEL is a benchmark dose corresponding to a small increase in the cell proliferation rate, e. g., 1%, to which appropriate safety (uncertainty) factors can be applied to arrive at an ADI.     

Estimating Upper Confidence Limits for Extra Risk in Quantal Multistage Models

Multistage models are frequently applied in carcinogenic risk assessment. In their simplest form, these models relate the probability of tumor presence to some measure of dose. These models are then used to project the excess risk of tumor occurrence at doses frequently well below the lowest experimental dose. Upper confidence limits on the excess risk associated with exposures at these doses are then determined. A likelihood-based method is commonly used to determine these limits. We compare this method to two computationally intensive "bootstrap" methods for determining the 95% upper confidence limit on extra risk. The coverage probabilities and bias of likelihood-based and bootstrap estimates are examined in a simulation study of carcinogenicity experiments. The coverage probabilities of the nonparametric bootstrap method fell below 95% more frequently and by wider margins than the better-performing parametric bootstrap and likelihood-based methods. The relative bias of all estimators are seen to be affected by the amount of curvature in the true underlying dose-response function. In general, the likelihood-based method has the best coverage probability properties while the parametric bootstrap is less biased and less variable than the likelihood-based method. Ultimately, neither method is entirely satisfactory for highly curved dose-response patterns.     

Comparing Experimental Designs for Benchmark Dose Calculations for Continuous Endpoints

The BMD (benchmark dose) method that is used in risk assessment of chemical compounds was introduced by Crump (1984) and is based on dose-response modeling. To take uncertainty in the data and model fitting into account, the lower confidence bound of the BMD estimate (BMDL) is suggested to be used as a point of departure in health risk assessments. In this article, we study how to design optimum experiments for applying the BMD method for continuous data. We exemplify our approach by considering the class of Hill models. The main aim is to study whether an increased number of dose groups and at the same time a decreased number of animals in each dose group improves conditions for estimating the benchmark dose. Since Hill models are nonlinear, the optimum design depends on the values of the unknown parameters. That is why we consider Bayesian designs and assume that the parameter vector has a prior distribution. A natural design criterion is to minimize the expected variance of the BMD estimator. We present an example where we calculate the value of the design criterion for several designs and try to find out how the number of dose groups, the number of animals in the dose groups, and the choice of doses affects this value for different Hill curves. It follows from our calculations that to avoid the risk of unfavorable dose placements, it is good to use designs with more than four dose groups. We can also conclude that any additional information about the expected dose-response curve, e.g., information obtained from studies made in the past, should be taken into account when planning a study because it can improve the design.     

Risk/Benefit Assessments of Human Diseases: Optimum Dose for Intervention

A simple procedure is proposed in order to quantify the tradeoff between a loss suffered from an illness due to exposure to a microbial pathogen and a loss due to a toxic effect, perhaps a different illness, induced by a disinfectant employed to reduce the microbial exposure. Estimates of these two types of risk as a function of disinfectant dose and their associated relative losses provide information for the estimation of the optimum dose of disinfectant that minimizes the total expected loss. The estimates of the optimum dose and expected relative total loss were similar regardless of whether the beta-Poisson, log-logistic, or extreme value function was used to model the risk of illness due to exposure to a microbial pathogen. This is because the optimum dose of the disinfectant and resultant expected minimum loss depend upon the estimated slope (first derivative) of the models at low levels of risk, which appear to be similar for these three models at low levels of risk. Similarly, the choice among these three models does not appear critical for estimating the slope at low levels of risk for the toxic effect induced by the use of a disinfectant. For the proposed procedure to estimate the optimum disinfectant dose, it is not necessary to have absolute values for the losses due to microbial-induced or disinfectant-induced illness, but only relative losses are required. All aspects of the problem are amenable to sensitivity analyses. The issue of risk/benefit tradeoffs, more appropriately called risk/risk tradeoffs, does not appear to be an insurmountable problem.     

Characterizing Dose-Response I: Critical Assessment of the Benchmark Dose Concept

We present a critical assessment of the benchmark dose (BMD) method introduced by Crump⁽¹⁾ as an alternative method for setting a characteristic dose level for toxicant risk assessment. The no-observed-adverse-effect-level (NOAEL) method has been criticized because it does not use all of the data and because the characteristic dose level obtained depends on the dose levels and the statistical precision (sample sizes) of the study design. Defining the BMD in terms of a confidence bound on a point estimate results in a characteristic dose that also varies with the statistical precision and still depends on the study dose levels.⁽²⁾ Indiscriminate choice of benchmark response level may result in a BMD that reflects little about the dose-response behavior available from using all of the data. Another concern is that the definition of the BMD for the quantal response case is different for the continuous response case. Specifically, defining the BMD for continuous data using a ratio of increased effect divided by the background response results in an arbitrary dependence on the natural background for the endpoint being studied, making comparison among endpoints less meaningful and standards more arbitrary. We define a modified benchmark dose as a point estimate using the ratio of increased effect divided by the full adverse response range which enables consistent placement of the benchmark response level and provides a BMD with a more consistent relationship to the dose-response curve shape.     

On the Correlation Coefficient Between the TD50 and the MTD

The existence of correlation between the carcinogenic potency and the maximum tolerated dose has been the subject of many investigations in recent years. Several attempts have been made to quantify this correlation in different bioassay experiments. By using some distributional assumptions, Krewski et al .⁽¹⁾ derive an analytic expression for the coefficient of correlation between the carcinogenic potency TD₅₀ and the maximum tolerated dose. Here, we discuss the deviation that may result in using their analytical expression. By taking a more general approach we derive an expression for the correlation coefficient which includes the result of Krewski et al .⁽¹⁾ as a special case, and show that their expression may overestimate the correlation in some instances and yet underestimate the correlation in other instances. The proposed method is illustrated by application to a real dataset.     

Peak Exposures in Epidemiologic Studies and Cancer Risks: Considerations for Regulatory Risk Assessment

We review approaches for characterizing “peak” exposures in epidemiologic studies and methods for incorporating peak exposure metrics in dose–response assessments that contribute to risk assessment. The focus was on potential etiologic relations between environmental chemical exposures and cancer risks. We searched the epidemiologic literature on environmental chemicals classified as carcinogens in which cancer risks were described in relation to “peak” exposures. These articles were evaluated to identify some of the challenges associated with defining and describing cancer risks in relation to peak exposures. We found that definitions of peak exposure varied considerably across studies. Of nine chemical agents included in our review of peak exposure, six had epidemiologic data used by the U.S. Environmental Protection Agency (US EPA) in dose–response assessments to derive inhalation unit risk values. These were benzene, formaldehyde, styrene, trichloroethylene, acrylonitrile, and ethylene oxide. All derived unit risks relied on cumulative exposure for dose–response estimation and none, to our knowledge, considered peak exposure metrics. This is not surprising, given the historical linear no‐threshold default model (generally based on cumulative exposure) used in regulatory risk assessments. With newly proposed US EPA rule language, fuller consideration of alternative exposure and dose–response metrics will be supported. “Peak” exposure has not been consistently defined and rarely has been evaluated in epidemiologic studies of cancer risks. We recommend developing uniform definitions of “peak” exposure to facilitate fuller evaluation of dose response for environmental chemicals and cancer risks, especially where mechanistic understanding indicates that the dose response is unlikely linear and that short‐term high‐intensity exposures increase risk.     

A Malformation Incidence Dose-Response Model Incorporating Fetal Weight and/or Litter Size as Covariates

A dose-response model is often fit to bioassay data to provide a mathematical relationship between the incidence of a developmental malformation and dose of a toxicant. To utilize the interrelations among the fetal weight, incidence of malformation and number of the live fetuses, a conditional Gaussian regression chain model is proposed to model the dose-response function for developmental malformation incidence using the litter size and/or the fetal weight as covariates. The litter size is modeled as a function of dose, the fetal weight is modeled as a function of dose conditional on the litter size, and the malformation incidence is modeled as a function of dose conditional on both the litter size and the fetal weight, which itself is also conditional on the litter size. Data from a developmental experiment conducted at the National Center for Toxicological Research to investigate the growth stunting and increased incidence of cleft palate induced by Dexamethasone (DEX) exposure in rats was used as an illustration.     

Model Uncertainty and Model Averaging in the Estimation of Infectious Doses for Microbial Pathogens

Food‐borne infection is caused by intake of foods or beverages contaminated with microbial pathogens. Dose‐response modeling is used to estimate exposure levels of pathogens associated with specific risks of infection or illness. When a single dose‐response model is used and confidence limits on infectious doses are calculated, only data uncertainty is captured. We propose a method to estimate the lower confidence limit on an infectious dose by including model uncertainty and separating it from data uncertainty. The infectious dose is estimated by a weighted average of effective dose estimates from a set of dose‐response models via a Kullback information criterion. The confidence interval for the infectious dose is constructed by the delta method, where data uncertainty is addressed by a bootstrap method. To evaluate the actual coverage probabilities of the lower confidence limit, a Monte Carlo simulation study is conducted under sublinear, linear, and superlinear dose‐response shapes that can be commonly found in real data sets. Our model‐averaging method achieves coverage close to nominal in almost all cases, thus providing a useful and efficient tool for accurate calculation of lower confidence limits on infectious doses.     

Influence of Distribution of Animals between Dose Groups on Estimated Benchmark Dose and Animal Distress for Quantal Responses

Increasingly, dose‐response data are being evaluated with the benchmark dose (BMD) approach rather than by the less precise no‐observed‐adverse‐effect‐level (NOAEL) approach. However, the basis for designing animal experiments, using equally sized dose groups, is still primed for the NOAEL approach. The major objective here was to assess the impact of using dose groups of unequal size on both the quality of the BMD and overall animal distress. We examined study designs with a total number of 200 animals distributed in four dose groups employing quantal data generated by Monte Carlo simulations. Placing more animals at doses close to the targeted BMD provided an estimate of BMD that was slightly better than the standard design with equally sized dose groups. In situations involving a clear dose‐response, this translates into fewer animals receiving high doses and thus less overall animal distress. Accordingly, in connection with risk and safety assessment, animal distress can potentially be reduced by distributing the animals appropriately between dose groups without decreasing the quality of the information obtained.     

Genome rearrangements with partially ordered chromosomes

Genomic maps often do not specify the order within some groups of two or more markers. The synthesis of a master map from several sources introduces additional order ambiguity due to markers missing from some sources. We represent each chromosome as a partial order, summarized by a directed acyclic graph (DAG), to account for poor resolution and missing data. The genome rearrangement problem is then to infer a minimum number of translocations and reversals for transforming a set of linearizations, one for each chromosomal DAG in the genome of one species, to linearizations of the DAGs of another species. We augment each DAG to a directed graph (DG) in which all possible linearizations are embedded. The chromosomal DGs representing two genomes are combined to produce a single bicoloured graph. From this we extract a maximal decomposition into alternating coloured cycles, determining an optimal sequence of rearrangements. We test this approach on simulated partially ordered genomes and on marker data from maize and sorghum chromosomal maps.     

An Exact Analysis of the Multistage Model Explaining Dose-Response Concavity

The traditional multistage (MS) model of carcinogenesis implies several empirically testable properties for dose-response functions. These include convex (linear or upward-curving) cumulative hazards as a function of dose; symmetric effects on lifetime tumor probability of transition rates at different stages; cumulative hazard functions that increase without bound as stage-specific transition rates increase without bound; and identical tumor probabilities for individuals with identical parameters and exposures. However, for at least some chemicals, cumulative hazards are not convex functions of dose. This paper shows that none of these predicted properties is implied by the mechanistic assumptions of the MS model itself. Instead, they arise from the simplifying "rare-tumor" approximations made in the usual mathematical analysis of the model. An alternative exact probabilistic analysis of the MS model with only two stages is presented, both for the usual case where a carcinogen acts on both stages simultaneously, and also for idealized initiation-promotion experiments in which one stage at a time is affected. The exact two-stage model successfully fits bioassay data for chemicals (e.g., 1,3-butadiene) with concave cumulative hazard functions that are not well-described by the traditional MS model. Qualitative properties of the exact two-stage model are described and illustrated by least-squares fits to several real datasets. The major contribution is to show that properties of the traditional MS model family that appear to be inconsistent with empirical data for some chemicals can be explained easily if an exact, rather than an approximate model, is used. This suggests that it may be worth using the exact model in cases where tumor rates are not negligible (e.g., in which they exceed 10%). This includes the majority of bioassay experiments currently being performed.     

Loss/Gain Framing,Dose, and Reactance: A Message Experiment

Whether a loss or gain frame has a persuasive advantage in communicating health risks is a matter of ongoing debate. Findings reported in the literature are mixed, suggesting that framing effects are likely complex and may be influenced by a combination of factors. This study examined reactance as a mediator and dose as a moderator of loss/gain framing effects. Adults (N = 1,039) read framed messages about the health consequences of physical (in)activity in varying message doses (i.e., number of framed statements). Compared to loss frames, gain frames generated more threat to freedom and reactance. Dosage exerted significant influence at the extremes; the one‐dose messages invoked less intentions to exercise compared to the four‐dose messages. Planned contrasts revealed significant frame × dose interactions. Notably, the one‐dose gain‐framed messages triggered significantly more freedom threat and less intentions to engage in physical activity—a situation that changed when the information was loss‐framed or when the dosage was increased.