Biologically Based Models for Cancer Risk Assessment: A Cautionary Note¹

Abstract

Biologically based models of carcinogenesis are becoming increasingly popular for risk assessment.(1-3) The multistage model, proposed by Armitage and Doll" to explain the observation that the age-specific incidence rates of many cancers increase with a power of age, views the process of carcinogenesis as the progressive deterioration of a normal cell through a sequence of intermediate stages to malignancy. When a couple of crucial approximations are made, the Armitage-Doll model generates an incidence function that increases with a power of age. These approximations have been examined in detail in a previous publicatiod5) and will not be reviewed here. Suffice it to say that the approximations are unlikely to hold in animal experiments in which, typically, a large proportion of animals develops tumors. For risk assessment the multistage model is often used in the form P ( d , t ) =1-exp[-g ( d ) H ( t ) ]where P(d,t) is the probability of a carcinogenic response by time r, g ( d ) is a polynomial in dose, and H ( r ) is a power of time. This formulation depends crucially on the adequacy of the approximations alluded to above, and may, thus, be inappropriate in precisely those situations in which it is most widely used, viz., in the analysis of animal experimental data.More recently Thorslund et ~l . (~) have proposed that risk assessment be based on a two-stage model for carcinogenesis developed by Moolgavkar and 'Research supported by USPHS grant CA-39949 from NIH. Venzod6) and shown to be consistent with a large body of epidemiologic and experimental data by Moolgavkar and budson.(') In the original mathematical development of the model@) the parameters of the model were assumed to be independent of time. Thorslund et present an expression for the incidence function generated by the model when the parameters are made time dependent. This allows the use of the model for the analysis of carcinogenesis experiments with time-dependent dosing patterns. Thorslund etdo not present a derivation of the expression for the incidence function. We would like to point out that the derivation depends upon the adequacy of an approximation. As in the case of the Armitage-Doll model, this approximation is unlikely to be adequate when dealing with animal experiments in which the probability of tumor is high. Specifically, let Y( t) and Z ( t ) represent the number of intermediate (premalignant) and malignant cells at time r, and let p 2 ( r ) represent the second event (mutation) rate. Then the incidence function predicted by the model is where E is the expectation. When the probability of tumor is small, it is reasonable to make the approximation E [ Y ( t ) l Z ( t ) =o] = E [ Y (~) ]That is, the conditional expectation may be replaced to a good approximation by the unconditional expectation of the random variable Y ( t ) . This approximation is essential for the expression presented by 5 0272-4332/SS/0300-05$06.CCl/101988 Society tor Risk Analysis

Radon-induced lung cancer in smokers and non-smokers: risk implications using a two-mutation carcinogenesis model

“…The simplified calculation may differ from the exact solution especially for tumour prevalences higher than 60% [15,16,22]. In order to avoid miscalculations as a result of the simplified calculation, in this paper the exact calculation as developed and used by Moolgavkar et al and Luebeck et al [23,24] has been used with a time unit of 1 month.…”

“…This can be corrected using the exact calculation [15,17,21]. Moolgavkar and co-workers [15,16,22] have shown that for high tumour prevalences, deviations from Poissonian can be expected. However, for prevalences of 60% or lower, the deviations turn out to be small, and also in the analyses presented in this paper, the deviations between the two methods are negligible.…”

“…Later on, Knudson and Moolgavkar [12,13] derived a two-stage model with clonal expansion to explain human cancer incidence in general and inherited forms of childhood cancer in particular, thus providing a strong biological basis for their model [14]. They [11,13,15,16] developed a two-stage model with clonal expansion and applied it to different epidemiological data including the Colorado miners' study [17]. They conclude that smoking and radon are the major factors influencing lung cancer incidence and showed that combined analysis of the British doctors' study done by Doll and Peto [7] and of the Colorado miners' data led to a coherent description.…”

Using cell replication data in mathematical modeling in carcinogenesis.

“…We would not support this approach because the approximation is likely to be inadequate and could lead to false inferences from the model. This has been observed and discussed for other multistage models in the context of tumor incidence modeling (32,33,34), and it is unlikely that approximations for this class of models will be any different. Furthermore, it would be far better to match the mathematical development to the biology.…”

Biological Bases for Cancer Dose-Response Extrapolation Procedures

“…[As Moolgavkar and Dewanji pointed out (17), this equation is valid only for I(t) < 0.2; at higher incidence a more nearly exact formulation must be used.] In this formulation, the synergistic effects of initiated cell mitotic rate on mutation probability become apparent.…”