Are two mutations sufficient to cause cancer? Some generalizations of the two-mutation model of carcinogenesis of Moolgavkar, Venzon, and Knudson, and of the multistage model of Armitage and Doll

Some generalizations of the two-mutation carcinogenesis model of Moolgavkar, Venzon, and Knudson (to allow for an arbitrary number of mutational stages) and of the multistage model of Armitage and Doll are shown to have the property that, at least in the case when the parameters of the model are eventually constant, the excess relative and absolute risks following changes in any of the parameters will eventually tend to zero. It is also shown that when the parameters governing the processes of cell division, death, or additional mutation at the penultimate stage are subject to perturbations, there are relatively large fluctuations in the hazard function for carcinogenesis for either model, which start almost as soon as the parameters are changed. For this reason it appears that without some extra stochastic ''stage'' appended (such as might be provided by consideration of the process of development of a malignant clone from a single malignant cell) the two-mutation model is not well able to describe the pattern of excess risk for solid cancers that is often seen after exposure to ionizing radiation, although leukemia may be better fitted by the two-mutation model in this respect. An examination of the results of perturbing various of the parameters for models that require three or more mutations provides indications that these models are easier to reconcile with the results from a body of epidemiological data relating to solid cancers.