Multistage model interpretation of additive and multiplicative carcinogenic effects

Under the assumption of multistage carcinogenesis, a multiplicative carcinogenic effect would be produced by the action of different carcinogens in a mixture on different stages of the carcinogenic process. An additive effect would be produced by the effect of different carcinogens on the same stage. A mathematical argument for these hypotheses is presented here.     

A Cellular Dynamics Model of Experimental Bladder Cancer: Analysis of the Effect of Sodium Saccharin in the Rat

To make the methodology of risk assessment more consistent with the realities of biological processes, a computer-based model of the carcinogenic process may be used. A previously developed probabilistic model, which is based on a two-stage theory of carcinogenesis, represents urinary bladder carcinogenesis at the cellular level with emphasis on quantification of cell dynamics: cell mitotic rates, cell loss and birth rates, and irreversible cellular transitions from normal to initiated to transformed states are explicitly accounted for. Analyses demonstrate the sensitivity of tumor incidence to the timing and magnitude of changes to these cellular variables. It is demonstrated that response in rats following administration of nongenotoxic compounds, such as sodium saccharin, can be explained entirely on the basis of cytotoxicity and consequent hyperplasia alone.     

Carcinogenic Risk Assessment with Time-Dependent Exposure Patterns

Previous applications of carcinogenic risk assessment using mathematical models of carcinogenesis have focused largely on the case where the level of exposure remains constant over time. In many situations, however, the dose of the carcinogen varies with time. In this paper, we discuss both the classical Armitage-Doll multistage model and the Moolgavkar-Venzon-Knudson two-stage birth-death-mutation model with time-dependent dosing regimens. Bounds on the degree of underestimation of risk that can occur through the use of a simple time-weighted average dose are derived by means of comparison with an equivalent constant dose corresponding to the actual risk under the time-dependent dosing regimen.     

Estimation of a Benchmark Dose in the Presence or Absence of Hormesis Using Posterior Averaging

U.S. Environment Protection Agency benchmark doses for dichotomous cancer responses are often estimated using a multistage model based on a monotonic dose‐response assumption. To account for model uncertainty in the estimation process, several model averaging methods have been proposed for risk assessment. In this article, we extend the usual parameter space in the multistage model for monotonicity to allow for the possibility of a hormetic dose‐response relationship. Bayesian model averaging is used to estimate the benchmark dose and to provide posterior probabilities for monotonicity versus hormesis. Simulation studies show that the newly proposed method provides robust point and interval estimation of a benchmark dose in the presence or absence of hormesis. We also apply the method to two data sets on carcinogenic response of rats to 2,3,7,8‐tetrachlorodibenzo‐p‐dioxin.     

Hormesis Without Cell Killing

Stochastic two-stage clonal expansion (TSCE) models of carcinogenesis offer the following clear theoretical explanation for U-shaped cancer dose-response relations. Low doses that kill initiated (premalignant) cells thereby create a protective effect. At higher doses, this effect is overwhelmed by an increase in the net number of initiated cells. The sum of these two effects, from cell killing and cell proliferation, gives a U-shaped or J-shaped dose-response relation. This article shows that exposures that do not kill, repair, or decrease cell populations, but that only hasten transitions that lead to cancer, can also generate U-shaped and J-shaped dose-response relations in a competing-risk (modified TSCE) framework where exposures disproportionately hasten transitions into carcinogenic pathways with relatively long times to tumor. Quantitative modeling of the competing effects of more transitions toward carcinogenesis (risk increasing) and a higher proportion of transitions into the slower pathway (risk reducing) shows that a J-shaped dose-response relation can occur even if exposure increases the number of initiated cells at every positive dose level. This suggests a possible new explanation for hormetic dose-response relations in response to carcinogenic exposures that do not have protective (cell-killing) effects. In addition, the examples presented emphasize the role of time in hormesis: exposures that monotonically increase risks at younger ages may nonetheless produce a U-shaped or J-shaped dose-response relation for lifetime risk of cancer.     

A Trichloroethylene Risk Assessment Using a Monte Carlo Analysis of Parameter Uncertainty in Conjunction with Physiologically-Based Pharmacokinetic Modeling

A Monte Carlo simulation is incorporated into a risk assessment for trichloroethylene (TCE) using physiologically-based pharmacokinetic (PBPK) modeling coupled with the linearized multistage model to derive human carcinogenic risk extrapolations. The Monte Carlo technique incorporates physiological parameter variability to produce a statistically derived range of risk estimates which quantifies specific uncertainties associated with PBPK risk assessment approaches. Both inhalation and ingestion exposure routes are addressed. Simulated exposure scenarios were consistent with those used by the Environmental Protection Agency (EPA) in their TCE risk assessment. Mean values of physiological parameters were gathered from the literature for both mice (carcinogenic bioassay subjects) and for humans. Realistic physiological value distributions were assumed using existing data on variability. Mouse cancer bioassay data were correlated to total TCE metabolized and area-under-the-curve (blood concentration) trichloroacetic acid (TCA) as determined by a mouse PBPK model. These internal dose metrics were used in a linearized multistage model analysis to determine dose metric values corresponding to 10^-6 lifetime excess cancer risk. Using a human PBPK model, these metabolized doses were then extrapolated to equivalent human exposures (inhalation and ingestion). The Monte Carlo iterations with varying mouse and human physiological parameters produced a range of human exposure concentrations producing a 10^-6 risk.     

Symmetry, Identifiability, and Prediction Uncertainties in Multistage Clonal Expansion (MSCE) Models of Carcinogenesis

Many models of exposure-related carcinogenesis, including traditional linearized multistage models and more recent two-stage clonal expansion (TSCE) models, belong to a family of models in which cells progress between successive stages-possibly undergoing proliferation at some stages-at rates that may depend (usually linearly) on biologically effective doses. Biologically effective doses, in turn, may depend nonlinearly on administered doses, due to PBPK nonlinearities. This article provides an exact mathematical analysis of the expected number of cells in the last ("malignant") stage of such a "multistage clonal expansion" (MSCE) model as a function of dose rate and age. The solution displays symmetries such that several distinct sets of parameter values provide identical fits to all epidemiological data, make identical predictions about the effects on risk of changes in exposure levels or timing, and yet make significantly different predictions about the effects on risk of changes in the composition of exposure that affect the pharmacodynamic dose-response relation. Several different predictions for the effects of such an intervention (such as reducing carcinogenic constituents of an exposure) that acts on only one or a few stages of the carcinogenic process may be equally consistent with all preintervention epidemiological data. This is an example of nonunique identifiability of model parameters and predictions from data. The new results on nonunique model identifiability presented here show that the effects of an intervention on changing age-specific cancer risks in an MSCE model can be either large or small, but that which is the case cannot be predicted from preintervention epidemiological data and knowledge of biological effects of the intervention alone. Rather, biological data that identify which rate parameters hold for which specific stages are required to obtain unambiguous predictions. From epidemiological data alone, only a set of equally likely alternative predictions can be made for the effects on risk of such interventions.     

An Improved Approximation to the Exact Solution of the Two-Stage Clonal Growth Model of Cancer

Multistage clonal growth models are of interest for cancer risk assessment because they can explicitly incorporate data on cell replication. Both approximate and exact formulations of the two stage growth model have been described. The exact solution considers the conditional probability of tumors arising in previously tumor-free animals; the approximate solution estimates total probability of tumor formation. The exact solution is much more computationally intensive when time-dependent cell growth parameters are included. The approximate solution deviates from the exact solution at high incidences and probabilities of tumor. This report describes a computationally tractable,'improved approximation'to the exact solution. Our improved approximation includes a correction term to adjust the unconditional expectation of intermediate cells based on the time history of formation of intermediate cells by mutation of normal cells (recruitment) or by cell division in the intermediate cell population (expansion). The improved approximation provided a much better match to the exact solution than the approximate solution for a wide range of parameter values. The correction term also appears to provide insight into the biological factors that contribute to the variance of the expectation for the number of intermediate cells over time.     

Could Removing Arsenic from Tobacco Smoke Significantly Reduce Smoker Risks of Lung Cancer?

If a specific biological mechanism could be determined by which a carcinogen increases lung cancer risk, how might this knowledge be used to improve risk assessment? To explore this issue, we assume (perhaps incorrectly) that arsenic in cigarette smoke increases lung cancer risk by hypermethylating the promoter region of gene p16INK4a, leading to a more rapid entry of altered (initiated) cells into a clonal expansion phase. The potential impact on lung cancer of removing arsenic is then quantified using a three‐stage version of a multistage clonal expansion (MSCE) model. This refines the usual two‐stage clonal expansion (TSCE) model of carcinogenesis by resolving its intermediate or “initiated” cell compartment into two subcompartments, representing experimentally observed “patch” and “field” cells. This refinement allows p16 methylation effects to be represented as speeding transitions of cells from the patch state to the clonally expanding field state. Given these assumptions, removing arsenic might greatly reduce the number of nonsmall cell lung cancer cells (NSCLCs) produced in smokers, by up to two‐thirds, depending on the fraction (between 0 and 1) of the smoking‐induced increase in the patch‐to‐field transition rate prevented if arsenic were removed. At present, this fraction is unknown (and could be as low as zero), but the possibility that it could be high (close to 1) cannot be ruled out without further data.     

On the Exact Hazard and Survival Functions of the MVK Stochastic Carcinogenesis Model

The MVK two-stage carcinogenesis model is one of the most widely accepted mechanistic models in carcinogenesis modeling. However, due to a perceived difficulty in obtaining analytic solutions for the hazard and survival functions, approximations and numerical methods have been used to calculate these two fundamental quantities. This paper focuses on a special case of the homogeneous MVK model where the number of normal cells is constant. The probability generating function (pgf) for the number of tumor cells is derived, and the exact analytic solutions to the hazard and survival functions are obtained from the pgf.     

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.     

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.     

Pharmacokinetic Principles for Dose-Rate Extrapolation of Carcinogenic Risk from Genetically Active Agents

Neither experimental animal exposures nor real-life human exposures are delivered at a constant level over a full lifetime. Although there are strong theoretical reasons why all pharmacokinetic processes must "go linear" at the limit of low dose rates, fluctuations in dose rate may produce nonlinearities that either increase or decrease actual risks relative to what would be expected for constant lifetime exposure. This paper discusses quantitative theory and specific examples for a number of processes that can be expected to give rise to pharmacokinetic nonlinearities at high dose rates–including transport processes (e.g., renal tubular secretion), activating and detoxifying metabolism, DNA repair, and enhancement of cell replication following gross toxicity in target tissues. At the extreme, full saturation of a detoxification or DNA repair process has the potential to create as much as a dose² dependence of risk on dose delivered in a single burst, and if more than one detoxification step becomes fully saturated, this can be compounded. Effects via changes in cell replication rates, which appear likely to be largely responsible for the steep upward turning curve of formaldehyde carcinogenesis in rats, can be even more profound over a relatively narrow range of dosage. General suggestions are made for experimental methods to detect nonlinearities arising from the various sources in premarket screening programs.     

Cancer Risk Assessment with Intermittent Exposure

Applications of methods for carcinogenic risk assessment often focus on estimating lifetime cancer risk. With intermittent or time-dependent exposures, lifetime risk is often approximated on the basis of a lifetime average daily dose (LADD). In this article, we show that there exists a lifetime equivalent constant dose (LECD) which leads to the same lifetime risk as the actual time-dependent exposure pattern. The ratio C = LECD/LADD then provides a measure of accuracy of risk estimates based on the LADD, as well as a basis for correcting such estimates. Theoretical results derived under the classical multistage model and the two-stage birth-death-mutation model suggest that the maximum value of C, which represents the factor by which the LADD may lead to underestimates of risk, will often lie in the range of 2- to 5-fold. The practical application of these results is illustrated in the case of astronauts subjected to relatively short-term exposure to volatile organics in a closed space station environment, and in the case of the ingestion of pesticide residues in food where consumption patterns vary with age.     

Uncertainties in Biologically-Based Modeling of Formaldehyde-Induced Respiratory Cancer Risk: Identification of Key Issues

In a series of articles and a health-risk assessment report, scientists at the CIIT Hamner Institutes developed a model (CIIT model) for estimating respiratory cancer risk due to inhaled formaldehyde within a conceptual framework incorporating extensive mechanistic information and advanced computational methods at the toxicokinetic and toxicodynamic levels. Several regulatory bodies have utilized predictions from this model; on the other hand, upon detailed evaluation the California EPA has decided against doing so. In this article, we study the CIIT model to identify key biological and statistical uncertainties that need careful evaluation if such two-stage clonal expansion models are to be used for extrapolation of cancer risk from animal bioassays to human exposure. Broadly, these issues pertain to the use and interpretation of experimental labeling index and tumor data, the evaluation and biological interpretation of estimated parameters, and uncertainties in model specification, in particular that of initiated cells. We also identify key uncertainties in the scale-up of the CIIT model to humans, focusing on assumptions underlying model parameters for cell replication rates and formaldehyde-induced mutation. We discuss uncertainties in identifying parameter values in the model used to estimate and extrapolate DNA protein cross-link levels. The authors of the CIIT modeling endeavor characterized their human risk estimates as "conservative in the face of modeling uncertainties." The uncertainties discussed in this article indicate that such a claim is premature.     

Carcinogenic Mixtures

Human populations are generally exposed simultaneously to a number of toxicants present in the environment, including complex mixtures of unknown and variable origin. While scientific methods for evaluating the potential carcinogenic risks of pure compounds are relatively well established, methods for assessing the risks of complex mixtures are somewhat less developed. This article provides a report of a recent workshop on carcinogenic mixtures sponsored by the Committee on Toxicology of the U.S. National Research Council, in which toxicological, epidemiological, and statistical approaches to carcinogenic risk assessment for mixtures were discussed. Complex mixtures, such as diesel emissions and tobacco smoke, have been shown to have carcinogenic potential. Bioassay-directed fractionation based on short-term screening test for genotoxicity has also been used in identifying carcinogenic components of mixtures. Both toxicological and epidemiological studies have identified clear interactions between chemical carcinogens, including synergistic effects at moderate to high doses. To date, laboratory studies have demonstrated over 900 interactions involving nearly 200 chemical carcinogens. At lower doses, theoretical arguments suggest that risks may be near additive. Thus, additivity at low doses has been invoked as as a working hypothesis by regulatory authorities in the absence of evidence to the contrary. Future studies of the joint effects of carcinogenic agents may serve to elucidate the mechanisms by which interactions occur at higher doses.     

随机多阶段分销网络设计模型

为了更合理的设计分销网络,本文提出了一种随机多阶段的联合选址-库存模型。在该模型中,不仅考虑了经济规模和分摊效益的影响。同时通过情景规划,考虑了在多阶段的分销网络设计中,对未来市场环境的不确定性。该模型的目标是使整个战略周期内的总期望成本(包括库存、运输、选址成本与损失的收益)最小。本文将该模型建立成为了一个非线性的整数规划模型,同时提出了一种基于拉格朗日松弛的求解算法。最后,本文使用该算法求解了三组不同规模的算例,得到的计算结果证明了拉格朗日算法是求解该模型的有效算法。     

Additive and Multiplicative Models and Multistage Carcinogenesis Theory

In light of the Armitage-Doll multistage carcinogenesis theory, this paper examines the assumption that an additive relative risk relationship is indicative of two carcinogens that affect the same stage in the cancer process. We present formulas to compute excess cancer risks for a variety of patterns for limited exposure durations to two carcinogens that affect the first and penultimate stages; and using an index of synergy proposed by Thomas (1982), we find a number of these patterns to produce additive, or nearly additive, relative risk relationships. The consistent feature of these patterns is that the two exposure periods are of short duration and occur close together.     

Characterizing Dose-Response Relationships in Multiple Cancer Bioassays

In the evaluation of chemical compounds for carcinogenic risk, regulatory agencies such as the U.S. Environmental Protection Agency and National Toxicology Program (NTP) have traditionally fit a dose-response model to data from rodent bioassays, and then used the fitted model to estimate a Virtually Safe Dose or the dose corresponding to a very small increase (usually 10(-6)) in risk over background. Much recent interest has been directed at incorporating additional scientific information regarding the properties of the specific chemical under investigation into the risk assessment process, including biological mechanisms of cancer induction, metabolic pathways, and chemical structure and activity. Despite the fact that regulatory agencies are currently poised to allow use of nonlinear dose-response models based on the concept of an underlying threshold for nongenotoxic chemicals, there have been few attempts to investigate the overall relationship between the shape of dose-response curves and mutagenicity. Using data from an historical database of NTP cancer bioassays, the authors conducted a repeated-measures Analysis of the estimated shape from fitting extended Weibull dose-response curves. It was concluded that genotoxic chemicals have dose-response curves that are closer to linear than those for nongenotoxic chemicals, though on average, both types of compounds have dose-response curves that are convex and the effect of genotoxicity is small.