Identifiability of parameters in the Yakovlev-Polig model of carcinogenesis

The paper discusses the problem of identifiability for two versions of a two-stage model of carcinogenesis recently introduced by Yakovlev and Polig. In this model, cell killing is allowed to compete with tumor promotion. In the first version of the Yakovlev-Polig model, which is referred to as Model 1, cell killing starts immediately after a carcinogen is administered. In the second version, called Model 2, it is assumed that a cell may be killed only after the process of initiation has been completed. The two versions of the Yakovlev-Polig model suggest explicit formulas for the distribution of time to tumor onset (that is, appearance of the first malignant clonogenic cell) counted from the initial moment of the exposure to a carcinogen. A model of carcinogenesis is identifiable if the set of all model parameters is uniquely determined by the distribution of time to tumor onset. It is shown that, under a natural necessary condition of overlap of supports of the dose-rate function h and the promotion time distributions from a family F, Model 1 is identifiable in the family F for many practically important functions h. In particular, this is the case for a simple model of spontaneous carcinogenesis (h = 1) and for a class of piecewise constant dose-rate functions h with arbitrary family F. Also, this holds for the family of gamma distributions and h supported on an interval and non-vanishing in the interior of this interval. More restrictions need to be imposed on the dose-rate function and the family of promotion time distributions for Model 2 to be identifiable. In particular, for h = 1, Model 2 turns out to be non-identifiable even in the family of gamma distributions.