Additive and Multiplicative Relative Risk in the Two‐Stage Clonal Expansion Model of Carcinogenesis

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Abstract
The effects of exposure to two carcinogens are explored within the context of the two-stage clonal expansion model of carcinogenesis. This biologically based model provides a useful framework for the quantitative description of carcinogenesis, and for defining carcinogenic agents that act as initiators, promoters, and completers. This paper addresses the combined effects of simultaneous lifetime exposure to two carcinogens as well as nonoverlapping partial lifetime exposure to each agent. Whereas the age-specific relative risk for exposure to two initiators or two completers is additive, a multiplicative relative risk model holds for exposure to an initiator and a completer, or to a promoter and a completer. Exposure to two promoters yields supra-multiplicative relative risk. Exposure to an initiator and promoter leads to multiplicative and supra-multiplicative relative risks for simultaneous lifetime and nonoverlapping partial lifetime exposures, respectively. Although departures from the additive relative risk model may thus occur at moderate to high doses, conditions are identified under which additivity will provide a good approximation to the joint risk at low doses. The methods of analysis used in this paper can also be used to determine the joint effects of exposure to two carcinogens which may affect more than one stage (initiation, promotion, completion) of the process of carcinogenesis. In general, the joint effects of exposure to such agents depends on the relative magnitude of the effects on individual stages.