INTRODUCTION

In recent years, it has become clear that carcinogenesis is a multistep process that requires deregulation of cellular growth. Cell growth and differentiation are normally under genetic regulation, so it may be assumed that the critical events in carcinogenesis involve genetic damage and inappropriate genetic expression (Weinberg, 1988; 1989). Mathematical models based on those biologic considerations can be simple or complex depending on assumptions about the number and nature of the events required to transform normal cells into cancer cells and about the sequence of events.

The simplest model judged to be consistent with the data available—a model that assumes two critical stages—was selected for evaluation by the Committee on Risk Assessment Methodology (CRAM). The two-stage model has been proposed as an improvement over currently used models for estimating carcinogenic risks to health, because it incorporates biologic considerations, notably cell population kinetics. The principal purposes of this CRAM study were to assess the scientific basis of the two-stage model of carcinogenesis and to evaluate the possible applications of the two-stage model to health risk assessment.

As part of the information-gathering process, the committee held a workshop on November 8, 1990, with presentations by the originators and proponents of the two-stage model and by invited discussants. A workshop summary appears in Appendix A. Postworkshop discussions were also held with the workshop speakers and with representatives of the federal liaison group to clarify the issues. However, this report and its recommendations were prepared solely by the committee.

BIOLOGIC CONSIDERATIONS

Clinicians and pathologists have long recognized that cancer formation in humans is often preceded by a series of preneoplastic changes. Confirmation of the multistage nature of certain human cancers has been obtained by studies of the role of changes in oncogenes and suppressor genes in human colon cancer (Hollstein et al., 1991). Similar observations have been made on laboratory animals that were exposed to carcinogens experimentally (Barbacid, 1987; Balmain and Brown, 1988). Morphologic or histopathologic studies do not always lend themselves well, however, to conclusions as to the biologic potential or ultimate fate of individual precancerous lesions. Some uncertainty exists about the identification of particular lesions as part of a neoplastic process, their place in the pathologic sequence, the inevitability of their progression to the next stage, and the rate of transition when they progress. For risk assessment, it is important to be able to distinguish lesions that are reversible from lesions that will irreversibly lead to neoplastic disease. The frequency with which most preneoplastic lesions pass from stage to stage appears to be low. In such model systems as the production of hepatic tumors in rats that are given known hepatic carcinogens, one can typically produce around thousands of biochemically altered cell foci per liver, which will be followed by the appearance of several adenomas and then by one or two hepatocarcinomas (Moolgavkar et al., 1990a; Cohen and Ellwein, 1991; Luebeck et al., 1991). In examples of that sort, almost all the early lesions do not progress to cancer, but remain the same or regress; thus cancer is a rare biologic outcome. Nevertheless, the consistent association of altered cell foci with later cancer formation has prognostic value and may help in developing preventive measures.

Underlying the structural stages are molecular events, or steps, that define the beginning and end of each stage. As noted before, cancer involves a disturbance of cell growth and cell growth is under genetic regulation, so genetic damage is likely to be important in carcinogenesis.

Target genes include those related to cell division and proliferation (pro-to-oncogenes) or those which cause cells to stop dividing (anti-oncogenes or tumor-suppressor genes). The available evidence strongly supports the general concept that the cells of some cancers in humans and laboratory animals contain activated or mutated oncogenes and, in some cancer cells, tumor-suppressor genes are inactive or missing (Weinberg, 1988; 1989). Under active investigation are the extent to which those genetic events are necessary and sufficient to result in cancer and whether the sequence of genetic events is important if more than one genetic event is necessary. Other possible target genes, such as the genes that contribute to cell division cycles or the genes that affect the microenvironment in which developing cancer cell clones might be inhibited or selectively enhanced, have received less attention. It is generally assumed, too, that cell proliferation in general increases the probability of inheritance of random mutations by somatic cells, thus contributing indirectly to the carcinogenic process (Cohen et al., 1991).

The concept of two-stage models emerged from Knudson's studies of heritable childhood cancers (Moolgavkar and Knudson, 1981), and was an extension of the work of Armitage and Doll (1957). For retinoblastoma in particular, the relation of tumor incidence to age suggested that one event is necessary in the somatic cells of hereditary carriers and two events are necessary in nonhereditary carriers. Molecular genetic analyses of cells from affected children have revealed that the critical event can be the loss or inactivation of both alleles of the retinoblastoma tumor-suppressor gene (RB1) (Gaillie et al., 1990). The developing retina might contain three types of cells: normal retinoblasts with two normal RB1 alleles, intermediate retinoblasts with one altered or lost RB1 gene, and retinoblasts with both RB1 genes altered. In the herediary form in which one parentally acquired allele is altered, the probability of retinoblastoma is increased, because all the developing retinoblasts have an abnormal RB1 gene and are at risk of a second event. That three or four tumors develop in the typical gene carrier suggests that the second event is not very common. The process is limited, as a child ages, by differentiation of the entire embryonal retinoblast pool into adult nondividing retinal cells.

Other cancers appear more complicated. For example, mutations in both RB1 and p53 suppressor genes are thought to be involved, with a third presumed suppressor gene, in small-cell lung tumors (Takahashi et al., 1989). Data on colon cancer suggest five or six critical events (Vogelstein et al., 1988; Goyette et al., 1992). These examples suggest that more complex models might be required. Conceptually, the two-stage model could be extended to any number of stages.

The biologic basis of carcinogenesis is still incompletely understood. In spite of recent rapid advances at the molecular level, many of the events described cannot yet be demonstrated to be essential for the pathogenesis of cancer. Some might be incidental phenomena with no causal relationship to the carcinogenic process. Research utilizing dose-response modeling can provide insights into which events are necessary and sufficient to produce cancer by demonstrating which mechanistic assumptions are consistent with the dose-response data.

THE TWO-STAGE MODEL

The two-stage model developed by Moolgavkar, Venzon, and Knudson (Moolgavkar and Venzon, 1979; Moolgavkar and Knudson, 1981) postulates two critical events in carcinogenesis that are specific, irreversible, and hereditary (at the cell level). The model supposes three cell compartments: normal stem cells, intermediate cells that have been altered by one genetic event, and malignant cells that have been altered by two genetic events. The size of each compartment is affected by cell birth, death, and differentiation processes and by the rates of transition between cell compartments.

The model is consistent with current concepts regarding the roles of inactivated tumor-suppressor genes and activated oncogenes in carcinogenesis. It explicitly accounts for many processes considered important in carcinogenesis, including cell division, mutation, differentiation, and death and the clonal expansion of populations of cells. Although the various carcinogenic processes might have more than two steps, a major assumption is that each of them can be described as consisting of two critical, genomic events: the first is assumed to give a small growth advantage through partial abrogation of growth control, and the second is assumed to lead to total abrogation of growth control. Among the other assumptions are that a cancer arises from a single cell, that transformations of stem cells are independent events, that each transformed cell will become a tumor, and that the time required to develop from a single transformed cell into a tumor is constant.

The mathematical aspects of two-stage model development and application have been described by Moolgavkar, Cohen, and Portier and their associates (Greenfield et al., 1984; Portier, 1987; Ellwein and Cohen, 1988; Moolgavkar, 1988; Moolgavkar and Luebeck, 1990; Portier and Edler, 1990; Tan, 1991). The model permits computation of both the rate at which tumors form (incidence function) and the probability of tumor formation with respect to time. Both stochastic and deterministic forms of the model have been described.

In the schematic representation of the model (Figure 1) as described by Moolgavkar and Knudson (1981), C₀, C₁, C₂, and D represent stem cells, intermediate cells, malignant cells, and differentiated or dead cells. A normal stem cell can divide into two stem cells, die, or be transformed by mutation into an altered intermediate cell. An intermediate cell similarly divides into two intermediate cells, dies, or becomes transformed into a fully malignant cell. α₁ is the rate at which cells divide (normal cells at α₁, and intermediate cells cells at α₂), β₁ the rate at which they die, and µ₁ the rate at which they are transformed.

FIGURE 1

Two-stage model paradigm. Source: Moolgavkar and Knudson, 1981.

That formulation assumes that normal cells behave independently, which implies that they either die out (generally early in life, which would result in death of the subject) or grow exponentially throughout life. Neither alternative is realistic, so the normal stem cell population is generally modeled as growing deterministically, with intermediate cells arising from normal cells in a nonhomogeneous Poisson process with intensity function X(s)v(s), where X(s) is the (deterministic) size of the normal stem cell population at age s and v(s) is the rate at which stem cells are converted into intermediate cells. With that form of the model, closed-form mathematical expressions can be obtained for the probability of forming a malignant cell by time t, P(t), and the associated instantaneous-hazard function h(t), as long as the parameters are time-independent. By integrating Expression 24 of Moolgavkar et al. (1988), one obtains

and

where

and

X(s) and (s) are not separately identifiable, and only their product can be estimated. Intermediate cells behave independently and with exponentially distributed lifespans. Consequently, clones of intermediate cells either die out or increase exponentially in size; there is no provision for growth regulation. Those assumptions are unrealistic, and alternatives have been proposed (Moolgavkar and Luebeck, 1990); however, implementation of more realistic alternatives greatly complicates the mathematical analysis and may not be necessary in providing good estimates of h(t).

The time between the occurrence of the first malignant cell and a clinically detectable cancer or death is generally modeled as a constant (Moolgavkar and Luebeck, 1990). A different assumption could easily be made, but doing so is likely to make the resulting mathematics intractable.

Before the model can be used in risk assessment, the effect of does must be incorporated. That is generally accomplished by treating model parameters as functions of instantaneous dose (Thorslund et al., 1987; Moolgavkar and Luebeck, 1990). Dose can be incorporated into the model by introducing a dose-effect relationship into the transition rate from normal cells to intermediate cells (µ₁), into the transition rate from intermediate cells to malignant cells (µ₂ ), or into the growth rate of clones of intermediate cells (α₂ -β₂). If the dose rate changes over time, then the corresponding parameter that dose affects is time-dependent and the solutions presented earlier do not apply. Explicit formulas for obtaining solutions when the parameters are piece-wise constant are also found in Moolgavkar and Luebeck (1990). Quinn (1989) and Moolgavkar and Luebeck (1990) showed how to obtain numerical solutions with the method of characteristics when the parameters are generally time-dependent.

One of the most important applications of dose-response models in risk assessment is to predict increased risk from exposure to low doses of a chemical. Typically, increased cancer risks on the order of 1/100 cannot be accurately measured in either a standard animal bioassay or an epidemiologic study due to limitations of sample size, yet increased risks in human populations of this magnitude, and even smaller, are of concern.

Small increased risks from low exposures are often estimated by fitting a dose-response model to data collected at higher exposures. The form assumed for the dose-response model is of critical importance to the resulting risk estimate (NRC, 1983). Regulatory agencies have frequently applied models that assume the increased risk is linearly related to exposure (i.e., the increased risk is proportional to the amount of exposure), at least at low exposures. However, it is frequently the case that nonlinear models will fit the data equally well and predict much lower risks at low exposures. The most extreme case of a nonlinear model is a threshold model, which assumes that there is a critical exposure (i.e., a threshold) below which the risk is not increased.

As is the case with simpler descriptive models, the manner in which the effect of dose is modeled will be the determining factor in the predictions of the two-stage model at low doses. If at least one of the transition rates is assumed to vary linearly with dose at low doses and the background incidence of cancer is not zero, then the probability of cancer will vary linearly with dose at low doses, although the low dose linear slope could differ appreciably from that predicted from high dose data. However, if all the dose-related rates are assumed to vary nonlinearly with dose at low doses or to exhibit a threshold dose below which the rate is not affected by dose, then the probability of cancer will likewise vary nonlinearly with dose at low doses or exhibit a threshold below which dose cannot cause cancer, respectively. Consequently, the manner in which dose is introduced into the two-stage model is a critical assumption for risk assessment.

Depending on how the model parameters are selected, there might be a number of parameters to estimate. For example, in their application of the two-stage model to data on 1,797 rats exposed to radon, Moolgavkar and Luebeck (1990) assumed that the number of normal cells was a constant and a clone of malignant cells of any size could be identified as a tumor at necropsy. They also assumed that the rate at which intermediate cells divided (2) was 10 times per week—the measured cell division rate for adenomas in rat lungs. Despite those simplifying assumptions, six parameters still had to be estimated by fitting the model to the cancer bioassay data. Although those data constituted an extraordinarily large data set, the precision with which some of the parameters could be estimated from the tumor data was low.

Application of a six-parameter model illustrates one of the key advantages of the two-stage model, as well as one of its disadvantages. Unlike the parameters of descriptive models (e.g., statistical models not derived from underlying biologic mechanisms), the parameters of the two-stage model are required to relate to actual biologic phenomena. Thus, Moolgavkar and Luebeck made the assumption that intermediate cells are equivalent to adenoma cells. This assumption implies that adenomas progress to carcinomas and is open to investigation. Generally, the two-stage model is more useful than mainly descriptive models for testing mechanistic hypotheses of this type, because several models can be developed based on alternative biologic hypotheses, which can then be tested on the basis of goodness-of-fit. Descriptive models are developed by fitting them to data, and hypotheses regarding underlying biologic mechanisms generally cannot be tested on the basis of fit.

A disadvantage of the two-stage model illustrated by the analysis of Moolgavkar and Luebeck is the potentially large number of model parameters to be estimated. To estimate as many as six parameters reliably might require large numbers of animals exposed to various dose patterns and with serial sacrifices. Thus, most examples of application of the two-stage model have used large data sets—e.g., the ED₀₁ study of 2-AAF (Cohen and Ellwein, 1990) and the study involving 1,797 animals exposed to radon (Moolgavkar and Luebeck, 1990). Such extensive data sets are available for only a few chemicals.

An alternative to estimating large numbers of parameters from tumor bioassay data is estimating specific parameters with data from other sources. Moolgavkar and Luebeck's estimation of the cell division rate of intermediate cells on the basis of the cell division rate in adenomas illustrates the approach. Although the approach is potentially quite useful, the knowledge needed for its general application is not yet available. In general, it requires an understanding of the steps in carcinogenesis and identification of cell types produced in the progression from normal to malignant cells, measures of the proliferation rates of intermediate cells and the rates of transformation of cells from one-stage to another, and whenever these quantities are dose-related, measures of the response of the parameters to dose. The latter measures are critical in determining the dose-response relationship and consequently assessing risk and estimating potency. Ideally, one would be able to measure the dose-response relationship accurately at doses to which humans are likely to be exposed (which might be much lower than the doses that produced measurable tumor responses in a standard animal bioassay); otherwise, one must assume a functional form for the relationship, which can introduce large uncertainties.

Environmental agents, as well as interindividual genetic differences in their metabolism, can affect tumor incidence through their effects on either mutation rates or the kinetics of cell division and differentiation, or both (Nebert, 1989; 1991a,b; Nebert et al., 1991, 1993 and references reviewed therein). Numerous studies in both mice and humans have demonstrated striking genetic differences in benzo[a]pyrene-induced tumor initiation, in cigarette smoke-induced tumor initiation (and probably tumor promotion), and in dioxin-induced toxicity (and possibly tumor promotion). A mutagenic substance can increase the intermediate-cell population (be an initiator) and can also cause conversion to malignancy (be a complete carcinogen). If only the kinetics of cell division and differentiation are affected by a substance, two general outcomes would be possible: in one, cell division and differentiation would be increased equally, resulting in increased mutation rates related to the increased cell division rate; in the second, the rate of cell division would be increased disproportionately to the rate of differentiation, and the increase would result in greater numbers of cells at risk of mutation.

APPLICATIONS OF THE TWO-STAGE MODEL TO ANIMAL DATA

Applications of the two-stage model have been few, because of limited data availability. Standard two year chronic carcinogenicity bioassays are not designed to provide information on the contribution of cell proliferation to tumor rates, and there are few data on the time-and dose-response effects of agents on cell proliferation. As a result, models that have been developed have generally relied on indirect measures of cell proliferation, such as increases in organ weights, or on measures of cell proliferation performed in independent experiments with protocols that provide less than ideal data for modeling purposes. Generating adequate data for the characterization of dose-response relationships for cell proliferation rates and of their contribution to tumor rates is a critical need.

The best-known examples of applications of the two-stage model to animal data are those for 2-acetylaminofluorene (2-AAF) in the mouse liver and bladder (Cohen and Ellwein, 1990), for saccharin in the rat bladder (Ellwein and Cohen, 1988), for radon in the rat lung (Moolgavkar et al., 1990b), and for N-nitrosomorpholine (NNM) in the rat liver (Moolgavkar et al., 1990a). In the case of 2-AAF, a genotoxic agent, liver tumor rates are consistent with its effect on the rates of transition between cell stages and have a linear dose-response relationship, as does the rate of DNA adduct formation. In the bladder, a linear rate of DNA adduct formation is observed as well, but the tumor rate is consistent with a nonlinear increase in the rate of cell proliferation at high doses and with an effect of dose on rates of transition between cell stages. Saccharin, a nongenotoxic agent, appears to induce bladder cancer as a result of toxicity-induced regenerative hyperplasia, and its dose-response model is thus based on an effect on the cell growth rate function and not on transition rates. The model that was developed for radon is consistent with a primary effect on the rate of first transition between cell stages, a less-pronounced effect on the rate of second transition, and an increase in the proliferation rate of intermediate cells. Analysis of the data on liver foci development associated with NNM indicates that it is a strong initiator that has a primary effect on the rate of transition to intermediate cells (as detected by foci formation) and that it has a weak promoting effect as well (as determined by its effect on the rate of foci proliferation).

The modeling approaches of Cohen and Ellwein and of Moolgavkar differ. Moolgavkar applies standard statistical methods (e.g., maximum likelihood) that have well-understood statistical properties, to a closed-form solution of the two-stage model. Uncertainty in model parameters and goodness-of-fit of the model to the experimental data can be investigated using standard statistical methods. However, the procedure requires closed-form solutions for the two-stage model, which are only available for a few special cases (although more general cases can be approximated quite closely by the cases for which solutions are available). When a simulation approach is used or a closed-form solution does not exist, assessing both the model's goodness-of-fit and the uncertainty in parameter estimates is important to understand the applicability of the model for calculating risk. Sensitivity analyses that explore alternative model assumptions and parameter values should be carried out to support such an assessment. Formal statistical techniques should be used whenever possible.

On the other hand, the Cohen and Ellwein approach involves specifying values for each model parameter. A computer simulation is then used to compute realizations of the subsequent tumor response. Parameter values are varied until the realizations conform to the actual data. This procedure can be applied more generally than that of Moolgavkar because a closed-form solution is not required. However, since there are no clear criteria for selecting trial values for the parameters or for determining adequacy of fit, the most appropriate parameters may not be found and results may not be reproducible. Moreover, the method does not readily lend itself to assessing goodness-of-fit and expressing uncertainty in parameter estimates.

Several other applications of the two-stage model have been attempted. Two examples are described here: one for the genotoxic agent benzo[a]pyrene (B[a]P) (Clement Associates, 1988) and one for the nongenotoxic agent chlordane (Thorslund and Charnley, 1988). Both applications involve an approximation to the two-stage model that gives good results only when the probability of a tumor is not too near unity (Moolgavkar et al., 1988). The mathematical expressions defining this approximate model are much less complicated than those required for the exact model; however, the approximation will be poor in some circumstances.

These examples are presented only to illustrate particular points and do not represent complete risk analyses. The committee believes that a complete risk analysis based on a two-stage model should include statistical confidence-intervals for model parameters and estimates of excess risk, which would permit determination of the ranges of risk that are consistent with the data and the particular form of the two-stage model being employed.

• Benzo[a]pyrene. In an inhalation study, respiratory tract tumors were induced in hamsters exposed daily to a B[a]P/sodium chloride aerosol throughout their lives. These occurred in the nasal cavity, larynx, and trachea (Thyssen et al., 1981). Doses and tumor rates are shown in Table 1. B[a]P is assumed to increase the rates of transition between cell stages and (in the absence of information to the contrary) the B[a]P-induced rates of transition between normal and intermediate and between intermediate and cancerous cells are assumed to be equally likely. It is further assumed that these transition rates are linear functions of dose (which is likely at low doses) and that the growth rates of normal and preneoplastic cells are independent of exposure. Under these assumptions, the probability of tumor development at time t as a result of exposure to level x of B[a]P (or any other genotoxic agent for which the above assumptions are applicable) is:

TABLE 1

Benzo[a]pyrene Bioassay Data: Observed and Predicted Tumor Rates.

where M is the background tumor rate parameter, S is the exposure-dependent transition rate between cell stages, G is the exposure-independent growth rate of intermediate cells, and t is the time (or age) at which risk is evaluated. In this case, the level x at the target tissue is assumed to be directly proportional to the administered dose.

The bioassay data in Table 1 can be used to estimate the exposure-induced relative transition rates as well as a background (spontaneous) transition rate, using Equation 8. Fitting the equation to the data by the method of maximum likelihood and using the average survival time as the length of observation yields the following equation for determining the cancer risk for each dose level:

As Table 1 indicates, the number of tumors predicted by this equation is very similar to that observed. However, because none of the animals at the lowest dose developed tumors, the data are also consistent with a threshold-type dose-response.

If survival had not been affected by exposure, Equation 8 could have been reduced to the simple quadratic form:

Equation 10 has the mathematical form of EPA's multistage model, but it is further constrained by the limitation to the first and second powers of x, and it has only two free parameters to determine the three coefficients in the exponent.

If 96.4 weeks is used as the average survival time for the control (unexposed) group, the following time-independent lifetime-risk relationship is obtained with the simplified form of the equation:

With this simplified form and an assumption that at low doses the product of the linear term of the equation and the dose is a close approximation of the estimated cancer risk, the linear term may be expressed as

To extrapolate that potency value to a human cancer risk estimate for lung cancer and B[a]P, the experimental exposure periods (4.5 hours/day, 7 days/week, for 10 weeks, then 3 hours/day thereafter) can be converted to an average 24 hour exposure period to yield a human cancer potency value of

That value can be compared with the cancer potency value of 1.7 (mg/m³)^-1 calculated using the linearized multistage modeling procedure by EPA (1980) based on the same bioassay data. This approximately order-of-magnitude difference is due, at least in part, to the use of the 95% upper confidence limit by EPA instead of the maximum likelihood estimate.

• Chlordane. The termiticide chlordane produces liver tumors in CD-1 mice (IRDC, 1973), promotes the incidence of liver tumors initiated by dimethylnitrosamine in B6C3F₁ mice (Williams and Numoto, 1984), and is only a weak mutagen (Cavender et al., 1986). As a result, its tumorigenicity has been attributed to its tumor-promoting, not-initiating, ability. A nongenotoxic mechanism of action may be proposed that involves hepatocellular mitogenesis, on the basis of observation of proliferative activity in the livers of exposed animals (IRDC, 1973). In the context of the two-stage model, chlordane's tumorigenicity could be proposed to result from its ability to increase the birth rate of intermediate cells while having no direct effect on the rates of transition between cell stages. The probability that a tumor will develop by time t after constant exposure to chlordane at dose x under those assumptions can be expressed as

where M = M₀M₁ and, again, the approximate form of the two-stage model has been applied. Using this model, the growth rate of intermediate cells, G(x), may be expressed in the form

where G(0) is the normal intermediate-cell growth rate, G(∞) is the upper bound on the chlordane-induced intermediate-cell growth rate, and R(x) is the fraction of the maximal increase in the intermediate-cell growth rate that is induced by a constant exposure to chlordane at dose x.

The functional form chosen for R(x) was that of a bounded log-logistic function on the basis of the following logic: at low doses, the function is bounded by the background number of intermediate cells; at high doses, it is reasonable to assume that proliferation, and therefore the number of intermediate cells, reaches a plateau, because it cannot increase indefinitely. That logic has not been examined carefully or validated because of the difficulty inherent in identifying intermediate cells and their kinetics. The log-logistic form assumed for the growth-rate function is

where chlordane is assumed to elicit its proliferative effect as a result of binding at a cellular receptor of some kind, I is proportional to its binding constant, and S is the average number of receptors in (or on) the affected cells.

Equation 14 can be fitted to the bioassay data for chlordane shown in Table 2 by using the log-logistic form of the growth-rate function described above and making a number of assumptions regarding the needed parameter estimates (see Table 3):

TABLE 2

Chlordane Bioassay Data: Observed and Predicted Tumor Rates.

TABLE 3

Parameter Estimates for Chlordane Dose-Response Model.

• Use human age-specific liver cancer death rates to estimate G(0).
• Use the background rate of spontaneous liver tumors in CD-1 mice to estimate M in male mice and the maximum-likelihood method and total tumor response to estimate M in female mice.
• Fix S at 4 (based on analogy to other logistic responses).
• Estimate G(∞) and I by equating the observed tumor rates at the two highest chlordane doses to the parametric form of the model and solving the resulting two nonlinear equations with two unknowns.

The predicted tumor rates are quite similar to the observed rates (Table 2) using Equation 16 and the parameter estimates specified in Table 3. The human cancer dose-response model developed on the basis of mouse data for chlordane yields the estimates of cancer risk shown in Table 4. These predictions are at least 3 orders of magnitude less than those obtained using the EPA's linearized multistage procedure, also shown.

TABLE 4

Estimates of Chlordane's Human Cancer Risk.

To further explore the utility of two-stage models in risk assessment, the committee has conducted additional calculations involving the chlordane example. In the chlordane analysis conducted by Thorslund and Charnley, it was assumed that the two-stage model is appropriate for chlordane. It was further assumed that chlordane increases the rate of division of intermediate cells, but otherwise does not affect tumor rates. The latter is a critical assumption, because, even if the two-stage framework is appropriate for chlordane, the weak dose-response relationship of chlordane could be considerably different from that predicted by the model if chlordane also affects some other step in the carcinogenesis process (e.g., if chlordane also had some effect on the rate of transition from normal to intermediate cells).

Even if all those assumptions are correct, the critical assumption regarding how chlordane affects the growth rate of intermediate cells still requires validation. Having no data for determining that growth rate, Thorslund and Charnley assumed that it had a particular mathematical form, expressed as G(x), where x is the dose of chlordane. To explore the sensitivity of the risk assessment results to this assumption, the committee experimented with other functional forms to determine whether other forms might also describe the data but yield different risk assessment results. One functional form considered by the committee, which varies only slightly from that of Thorslund and Charnley but that also describes the data was

That expression was substituted for the formula for cell-proliferation rate assumed by Thorslund and Charnley, which was

All other parameter values used by Thorslund and Charnley were retained (M = 2.8622 × 10^-6, t = 78 weeks).

Table 5 shows that the fit of Equation 17 to the chlordane data on male mice is comparable with that of the model used by Thorslund and Charnley. However, as Table 6 indicates, the predictions of the two models differ sharply at doses below the experimental range, which is generally the range of interest for environmental regulation. The risk estimates calculated with the committee's exploratory model are higher than those obtained with the model of Thorslund and Charnley by a factor of about 39 for a chlordane dose of 1 ppm, about 24,000 for a dose of 0.1 ppm, and about 23,000,000 for a dose of 0.01 ppm. Those large differences are due solely to differences in the assumed cell-proliferation rate G(x). Thus, distinguishing between these risk estimates would require distinguishing between the underlying cell-proliferation rates determined by the two alternative expressions for G(x).

TABLE 5

Chlordane Data: Fit of Alternative Approximate Two-Stage Models.

TABLE 6

Low Dose Cancer Risk Estimates for Chlordane Derived from Two-Stage Models.

Table 6 compares the values of the cell-proliferation rates from the Thorslund and Charnley model with those from the alternative model at both the experimental doses and the lower doses for which additional risk was estimated. The predictions of cell-proliferation rates agree closely both at the experimental doses and at lower doses. Extremely small differences in cell-proliferation rate can result in large differences in additional risk. At doses of 0.1 and 0.01 ppm, the two cell-proliferation rates differ only in the sixth decimal place, whereas the resulting extra risks differ by factors of about 4 and 7 orders of magnitude, respectively. Thus, tiny changes in the cell-proliferation rate can make enormous differences in the resulting risk estimates. Given the variation that is normal in biologic systems, it is highly unlikely that such small differences in cell-proliferation rate could ever be accurately distinguished. An additional source of uncertainty could be introduced by assuming that as a weak mutagen, chlordane could have an effect on the transition rates in addition to the cell growth rates. This assumption could alter the risk estimates even more.

In the committee's discussions of these results, it was suggested that they might be due to the use of the approximation to the two-stage model and that the exact form of the model might not exhibit such instabilities. To explore that issue, the exact form of the two-stage model was fitted to the chlordane data on male rats. As in the application of the approximate model, chlordane was assumed to affect the division and death of intermediate cells. The following specific parameter values were used in the fitting:

As before, x is the dose of chlordane. The two specific forms of G(x) (Expressions 17 and 18) applied to the approximate solution were also applied here in connection with the exact solution. The resulting exact solutions are virtually indistinguishable from the corresponding approximate solutions. Table 7 shows that both exact models fit the data on male mice almost exactly, just as the approximate models do. Table 8 shows virtually the same risks at low doses for the exact models as shown in Table 6 for the approximate models.

TABLE 7

Chlordane Data: Fit of Alternative Exact Two-Stage Models.

TABLE 8

Low Dose Cancer Risk Estimates for Chlordane Derived from Exact Two-Stage Models.

The two exact and two approximate dose-responses are depicted graphically in Figures 2 and 3. Figure 2 shows that the exact expression for the probability of response, P(x), agrees closely with the approximate solution throughout the complete range of exposures when both are based on the same expression for G(x). It also shows that the two expressions for G(x) provide comparable response probabilities at the experimental exposure levels (0 ppm, 5 ppm, 25 ppm, and 50 ppm) and at all exposure levels below 5 ppm. Figure 3, which is the same as Figure 2 except that log scales are used and the vertical axis is the additional probability of response induced by exposure [P(x) - P(0)], shows that the exact solution for additional probability also agrees closely with the corresponding approximate solution over a wide exposure range, including very low exposures. It also shows that the two expressions for G(x) provide similar results for additional risk at high exposures but very different values at low exposures.

FIGURE 2

Comparison of exact and approximate two-stage models applied to chlordane data on male mice.

FIGURE 3

Comparison of exact and approximate two-stage models at low exposures applied to chlordane data on male mice.

Thus, the exact models produce results in this case that are virtually indistinguishable from those produced by the approximate models that use the same cell-proliferation rate function G(x). The two expressions for G(x) provide very similar response probabilities at the experimental exposure levels (and therefore are indistinguishable based on the experimental data) but predict divergent estimates of additional risk at low exposures. Consequently, there is virtually no difference between the exact and approximate two-stage solutions in this case, and the exact model is subject to the same instabilities as the approximate model.

This type of instability is likely to be the rule, rather than the exception. A general model for the probability of cancer arising from a dose x can be written as

where β₁,…,β_k are parameters that are unaffected by dose and G(x) is the parameter that is affected by dose. In the chlordane example, G(x) represented cell-proliferation rate, although for the purposes of the current argument it could be any biologic parameter that is affected by dose. At low doses, the probability is approximately

Thus, the dose-response relationship will behave at low doses like a linear function of G(x). That implies that the dose-response relationship for cancer at low doses will mimic that of the parameter that dose affects. The Red Book (NRC, 1983) showed that different dose-response curves for P(x) could be obtained that fit data in the observable range but yield results for incremental risk (above background) that differ by many orders of magnitude in the low dose range. The same arguments apply to G(x) and therefore, through Equation 20 relating P(x) to G(x), to P(x) again.

Those considerations suggest steps that are critical in using a two-stage model (or any other biologically based model) for low dose extrapolation. Identifying the biologic steps that lead to cancer and determining which ones are affected by the chemical insult are the first steps in the process. Another is the specification of the dose-response relationship for the parameter G(x) (cell-proliferation rate, mutation rate, etc.) or parameters that are affected by the chemical insult. Regardless of how detailed and reliable the model is otherwise, if it does not specify a mathematical form for G(x), the quantitative predictions of the model at low doses are essentially arbitrary. More precisely, given a specific two-stage model that fits a given set of dose-response data adequately, the function G can be adjusted so that the adjusted model describes the data equally well but corresponds to estimates of additional cancer risk over the background risk that differ from estimates based on the original model by arbitrarily large factors at any dose below the lowest experimental dose.

Prescribing with confidence the mathematical form of the dose-response relationship for any particular biologic parameter that depends on dose is likely to be difficult. Kopp and Portier (1989) found that when the approximate form of the model fails to characterize accurately the cumulative distribution function of the time to tumor onset, bias may result in the estimates of the remaining parameters. The critical need in applying biologically based models will be for data on the response of the model parameters that are affected by dose. While the committee encourages the use of formal statistical methods, the application of such methods to estimate model parameters from bioassay data does not resolve uncertainty about the relationship of these parameters to dose at low doses. Bioassay data such as those for benzo[a]pyrene in Table 1 and chlordane in Table 2 do not provide a basis for determining the shape of a dose-response relationship at low doses. These data sets would be consistent with a model that predicted zero incremental cancer risk at the lowest positive dose. These data sets are also consistent with the predictions of incremental risk of 2% to 6% over background at the lowest positive dose, as shown in Tables 1 and 2. The shape of the dose-response relationship will be determined by assumptions about how the parameters in the model depend upon dose, supplemented by direct measurements of cell kinetics to the extent that such measurements are available. As the chlordane example illustrates, alternative functional forms that fit the data in the experimental range can lead to widely differing estimates of risk in the low dose range. Narrowing the uncertainty in the low dose range will require improved mechanistic understanding of how exposure to low doses of a toxicant affects the kinetics of cell transformation and proliferation.

DISCUSSION

CONCLUSIONS AND RECOMMENDATIONS

Mechanistic understanding of toxicity has strong implications for improvement in the development of low dose extrapolation for the regulation of chemical substances. Currently, low dose extrapolation uses a multistage model with data developed from human occupational exposures or from whole animal bioassays (Anderson et al., 1983). The newer two-stage model examined in this report attempts to use data related more to mechanisms of toxicity. Its potential utility (and the gathering of the data needed to use it) derives from the recent rapid development of biologic investigative techniques.

Understanding of and data on cell birth and death are required for the development and use of the two-stage model, but they do not exist for most chemicals. A better mechanistic understanding must be developed, if modeling efforts are to take advantage of cell birth and death data. If the mechanism of a toxic effect is not understood, inappropriate dose-response data are likely to be used in the extrapolation process, which could then produce an incorrect result. The committee recommends that when critical assumptions about mechanisms of toxicity are made, they must be clearly stated.

The two-stage model can be used as a basis for decision-making, if there is sufficient mechanistic understanding and if a sufficient data base is available. The committee recognizes that regulators face the question of how to determine when such understanding, data, and models are sufficient and appropriate; no hard and fast rules can be given. Complicating the issue is that there is a continuum in the extent of mechanistic understanding and data on any chemical. The risk management context need also be considered.

Scientific work on the two-stage model of carcinogenesis has proceeded sufficiently for it to be clear that its further development should be strongly encouraged. Regulatory agencies might review decisions and standards on materials with economic or public health importance to see whether enough data are available or can be rapidly collected to permit application of the two-stage model for additional perspective. The judgment of scientists as to whether sufficient data are available or could be collected might be helpful to regulatory agencies before they decide to apply this newer model in risk assessment. Experience in conducting such reviews will probably lead to a set of criteria for determining when the two-stage model should be used. The proposed reviews should be conducted on only a narrowly limited number of materials. And they must not be allowed to substitute for or interfere with the prompt regulation of or setting of standards for materials currently or soon to be under examination.

The committee recommends exploratory applications of the two-stage model along with its testing and validation. A first stage in the testing requires mechanistic understanding and the gathering of sufficient data to permit its use. Comparative information on humans must be developed as a part of the validation process. The committee also recommends that statistical confidence-interval methods, sensitivity analyses and related quantitative methods as appropriate be applied to determine the extent that the data are consistent with other mathematical representations and ranges of risk. Although the committee recognizes that the simulation approach to model fitting can have very important uses, particularly for exploratory data analysis and when no closed-form solution of the two-stage model is available, the committee recommends that whenever the model is applied in formal risk assessment, formal statistical methods (e.g., maximum likelihood) should be employed.