The statistical analyses of whole-animal bioassay data have employed over the years a number of mathematical models in an attempt to relate experimental data to the human situation, espe- cially for the purposes of quantitating human risk insofar as is possible. As Gaylor and Shapiro (1979) have pointed out, “There is no choice but to extrapolate.” This means, in essence, that because of the insensitivity of epidemiologic studies and the number and quantity of actual and potential carcinogens in our environment, one must make every attempt possible to relate data from bioassay studies to the human condition, especially the potential risk to the public. Most of these mathematical models have as a basic tenet the assumption that carcinogenic agents lack a threshold, act irreversibly, and have effects that are additive. Equations for some of the more commonly used models are given in Table 13.10. None of these models can prove or disprove the existence of a threshold of response, and none can be completely verified on the basis of biological argument; however, the models have been useful in data evaluation and are presently being used by some federal agencies in extrapolating experimental data to the human risk situa- tion. One of the most commonly used techniques is the log-probit model. In the earlier use of this model, the procedure was to regard every agent as carcinogenic. On this assumption, one must determine some “safe” dosage level at which the risk calculated would not exceed some very small level such as 1 in 100,000,000 or 10–8.
The linear multistage model, first proposed by Armitage and Doll (1954), incorporates the idea of multiple steps into a statistical approach for risk analysis. This multistage model (Figure 13.15) incorporates one aspect of the pathogenesis of neoplastic development, that of multiple stages, but cell cycle-dependent processes, the dynamics of cell kinetics, birth rate, and death
rate are not considered. Furthermore, the transition from one stage to the next is considered irre- versible. Despite these deficiencies, the linearized multistage model is one of the most com- monly utilized models at the present time. At a low dose, the multistage model is used to fit the observed tumor incidence data to a polynomial of the dose as noted in Table 13.10. The linear multistage model is not appropriate for estimating low-dose carcinogenic potency for many chemicals. In most cases, the dose response of high doses of testing differs substantially from the considerably lower doses for exposure. Pharmacokinetic and pharmacodynamic models pro- vide information that can help bridge the gap between the high dose and low dose scenarios (Anderson, 1989). A second problem is associated with extrapolation of lifetime exposure of animals to the MTD of a compound to the less than lifetime exposure common for humans. This problem has been addressed by the EPA through the use of the Weibull model (Hanes and Wedel, 1985), which assumes that risk is greater when encountered at a younger age, and, once exposure occurs, risk continues to accrue despite the cessation of exposure. However, observa- tions in humans and experimental animals have demonstrated that in many cases risk decreases after exposure ceases, as would be true if the agent were a promoting agent.
More recently, biomathematical modeling of cancer risk assessment has been used in an attempt to relate such models more closely to the biological characteristics of the pathogenesis of neoplasia. The best known of these biologically based models is that described originally by Moolgavkar, Venzon, and Knudson, termed the MVK model (Moolgavkar, 1986). This model, which is depicted in Figure 13.15, reproduces quite well the multistage characteristics of neo- plastic development with µ1, the rate at which normal cells are converted to “intermediate” cells (initiated cells), and µ2, the rate at which intermediate cells are converted to neoplastic (N) cells.
These rates model the rates of initiation and progression in multistage carcinogenesis, while the stage of promotion represents the expansion of the intermediate cell population, which is a function of α2, the rate of division of “intermediate cells,” and β2, the rate of differentiation and/or death of intermediate cells. Other factors in the model that are also true in biology are the rate of replication and cell death of normal or stem cells. While this model originally was developed to explain certain epidemiological characteristics of breast cancer incidence and mortality in humans (Moolgavkar, 1986), it has found potential application in a variety of multi- stage models including that of rat liver (Luebeck et al., 1991). Application of the model to risk assessment problems has not found wide use, but this may change in the next few years (Ander- son et al., 1992). In addition, integration of biological data, including pharmacokinetic and pharmacodynamic parameters, should aid in the development of a more biologically based risk assessment model.
Figure 13.15 The Armitage and Doll (upper) and MKV (lower) models of multistage carcinogenesis. In the former, the number of stages is unspecified (Tk), and the transition between them is irreversible. In the MKV model, the fates of stem cells (S) and intermediate (I) cells are death (D) or proliferation. Rarely, I cells undergo µ2 to malignancy (M). The rates of replication (α2) and apoptosis (β2) for I cells are indi- cated, and similar rates for S cells are implied. µ1 and µ2 are the rates of the first genetic event (initiation) and the second genetic event (progression). (Adapted from Pitot and Dragan, 1996, with permission of the authors and publishers.)