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.

SUMMARY The method of constructing confidence regions based on the generalised likelihood ratio statistic is well known for parameter vectors. A similar construction of a confidence interval for a single entry of a vector can be implemented by repeatedly maximising over the other parameters. We present an algorithm for finding these confidence interval endpoints that requires less computation. It employs a modified Newton‐Raphson iteration to solve a system of equations that defines the endpoints.

We use general multistage models to fit the age-specific incidence of colorectal cancers in the Surveillance, Epidemiology, and End Results registry, which covers Ϸ10% of the U.S. population, while simultaneously adjusting for birth cohort and calendar year effects. The incidence of colorectal cancers in the Surveillance, Epidemiology, and End Results registry is most consistent with a model positing two rare events followed by a high-frequency event in the conversion of a normal stem cell into an initiated cell that expands clonally to give rise to an adenomatous polyp. Only one more rare event appears to be necessary for malignant transformation. The two rare events involved in initiation are interpreted to represent the homozygous loss of adenomatous polyposis coli gene function. The subsequent transition of a preinitiated stem cell into an initiated cell capable of clonal expansion via symmetric division is predicted to occur with a frequency too high for a mutational event but may reflect a positional effect in colonic crypts. Our results suggest it is not necessary to invoke genomic instability to explain colorectal cancer incidence rates in human populations. Temporal trends in the incidence of colon cancer appear to be dominated by calendar year effects. The model also predicts that interventions, such as administration of nonsteroidal anti-inflammatory drugs, designed to decrease the growth rate of adenomatous polyps, are very efficient at lowering colon cancer risk substantially, even when begun later in life. By contrast, interventions that decrease the rate of mutations at the adenomatous polyposis coli locus are much less effective in reducing the risk of colon cancer.T he first attempts to formulate a quantitative description of carcinogenesis reflecting essential biological processes on the pathway from a normal cell to a cancer cell go back almost half a century (1). Perhaps the best known model is due to Armitage and Doll (2), who noticed that the age-specific incidence of many carcinomas appeared to increase approximately with power of age, which could be related to the number of rate-limiting steps involved in the formation of a malignant tumor. However, it was also realized that a two-stage model with clonal expansion of intermediate cell populations could generate similar age-specific incidence curves (3). These considerations, combined with the idea of recessive oncogenesis first formulated by Knudson (4), led to the two-stage clonal expansion (TSCE) model, which explicitly incorporates clonal expansion as a stochastic process during carcinogenesis (5-7).Recent studies of the genetic profiles of various tumors suggest the involvement of several genes during tumorigenesis. A case in point is colorectal cancer, perhaps the best studied cancer in terms of the putative sequence of genetic events in its pathogenesis (8-12). Over the past 10 years, an impressive number of studies have been carried out identifying several molecular pathways involved in the development of colorectal cancer (see ref...