An Exact Analysis of the Multistage Model Explaining Dose‐Response Concavity

The traditional multistage (MS) model of carcinogenesis implies several empirically testable properties for dose‐response functions. These include convex (linear or upward‐curving) cumulative hazards as a function of dose; symmetric effects on lifetime tumor probability of transition rates at different stages; cumulative hazard functions that increase without bound as stage‐specific transition rates increase without bound; and identical tumor probabilities for individuals with identical parameters and exposures. However, for at least some chemicals, cumulative hazards are not convex functions of dose. This paper shows that none of these predicted properties is implied by the mechanistic assumptions 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 model with only two stages is presented, both for the usual case where a carcinogen acts on both stages simultaneously, and also for idealized initiation‐promotion experiments in which one stage at a time is affected. The exact two‐stage model successfully fits bioassay data for chemicals (e.g., 1,3‐butadiene) with concave cumulative hazard functions 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 contribution is to show that properties of the traditional MS model family that appear to be inconsistent with empirical data for some chemicals can be explained easily if an exact, rather than an approximate model, is used. 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 performed.