This value for the mean number of tumors occurring in genetic carriers may be used to estimate the mutation rate for each mutation. The germinal and somatic rates for the first, and the somatic rate for the second, mutation, are approximately equal. The germinal mutation may arise in some instances from a delayed mutation.

Most cancers have many chromosomal abnormalities, both in number and in structure, whereas some show only a single aberration. In the era before molecular biology, cancer researchers, studying both human and animal cancers, proposed that a small number of events was needed for carcinogenesis. Evidence from the recent molecular era also indicates that cancers can arise from small numbers of events that affect common cell birth and death processes.

This article extends our previous quantitative analysis of the relationship between the dynamics of the primary structure of DNA and mutagenesis associated with single-strand lesions to an analysis of the production and processing of endogenous doublestrand breaks (EDSBs) and to their implications for oncogenesis. We estimate that in normal human cells Ϸ1% of single-strand lesions are converted to Ϸ50 EDSBs per cell per cell cycle. This number is similar to that for EDSBs produced by 1.5-2.0 Gy of sparsely ionizing radiation. Although EDSBs are usually repaired with high fidelity, errors in their repair contribute significantly to the rate of cancer in humans. The doubling dose for induced DSBs is similar to doubling doses for mutation and for the induction of carcinomas by ionizing radiation. We conclude that rates of production of EDSBs and of ensuing spontaneous mitotic recombination events can account for a substantial fraction of the earliest oncogenic events in human carcinomas.

A model for carcinogenesis is presented that provides a framework for understanding the roles of "spontaneous" events, hereditary factors, and environmental agents in human carcinogenesis and for interpreting experimental carcinogenesis. This model incorporates two features: a) transition of target stem cells into cancer cells via an intermediate stage in two irreversible steps, and b) growth and differentiation of normal target and intermediate cells. Cast in mathematical terms, the model can be fitted to age-specific incidence data on human cancers of both children and adults and can illuminate the relative importance of agents that affect transition rates, tissue growth, and tissue differentiation. This is illustrated by application of the model to a) the epidemiology of lung cancer with emphasis on the role of cigarette smoking and b) the epidemiology of breast cancer with emphasis on the roles of hormones, radiation, and hereditary. The nature of the two events and of the intermediate stage is considered in light of hereditary conditions that predispose to cancer in humans. The modes of action of radiation and chemicals in carcinogenesis are discussed, as are predictions based on the model and amenable to experimental verification.

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