AN EXACT ANALYSIS OF THE MULTISTAGE MODEL EXPLAINING DOSE-RESPONSE CONCAVITY

The traditional multistage (MS) model of carcinogenesis implies severa l empirically testable properties for dose-response functions. These i nclude convex (linear or upward-curving) cumulative hazards as a funct ion of dose; symmetric effects on lifetime tumor probability of transi tion rates at different stages; cumulative hazard functions that incre ase without bound as stage-specific transition rates increase without bound; and identical tumor probability for individuals with identical parameters and exposures. However, for at least some chemicals, cumula tive hazards are not convex functions of dose. This paper shows that n one of these predicted properties is implied by the mechanistic assump tions of the MS model itself. Instead, they arise from the simplifying ''rare-tumor'' approximations made in the usual mathematical analysis of the model. An alternative exact probabilistic analysis of the MS m odel with only two stages is presented, both for the usual case where a carcinogen acts on both stages simultaneously, and also for idealize d initiation-promotion experiments in which one stage at a time is aff ected. The exact two-stage model successfully fits bioassay data for c hemicals (e.g., 1,3-butadiene) with concave cumulative hazard function s that are not well-described by the traditional MS model. Qualitative properties of the exact two-stage model are described and illustrated by least-squares fits to several real datasets. The major contributio n is to show that properties of the traditional MS model family that a ppear to be inconsistent with empirical data for some chemicals can be explained easily if an exact, rather than an approximate model, is us ed. This suggests that it may be worth using the exact model in cases where tumor rates are not negligible (e.g., in which they exceed 10%). This includes the majority of bioassay experiments currently being pe rformed.