## Parameter Identifiability and Redundancy in a General Class of Stochastic Carcinogenesis Models

### Background

Heidenreich *et al.* (*Risk Anal* 1997 **17** 391–399) considered parameter identifiability in the context of the two-mutation cancer model and demonstrated that combinations of all but two of the model parameters are identifiable. We consider the problem of identifiability in the recently developed carcinogenesis models of Little and Wright (*Math Biosci* 2003 **183** 111–134) and Little *et al.* (*J Theoret Biol* 2008 **254** 229–238). These models, which incorporate genomic instability, generalize a large number of other quasi-biological cancer models, in particular those of Armitage and Doll (*Br J Cancer* 1954 **8** 1–12), the two-mutation model (Moolgavkar *et al. Math Biosci* 1979 **47** 55–77), the generalized multistage model of Little (*Biometrics* 1995 **51** 1278–1291), and a recently developed cancer model of Nowak *et al.* (*PNAS* 2002 **99** 16226–16231).

### Methodology/Principal Findings

We show that in the simpler model proposed by Little and Wright (*Math Biosci* 2003 **183** 111–134) the number of identifiable combinations of parameters is at most two less than the number of biological parameters, thereby generalizing previous results of Heidenreich *et al.* (*Risk Anal* 1997 **17** 391–399) for the two-mutation model. For the more general model of Little *et al.* (*J Theoret Biol* 2008 **254** 229–238) the number of identifiable combinations of parameters is at most less than the number of biological parameters, where is the number of destabilization types, thereby also generalizing all these results. Numerical evaluations suggest that these bounds are sharp. We also identify particular combinations of identifiable parameters.

### Conclusions/Significance

We have shown that the previous results on parameter identifiability can be generalized to much larger classes of quasi-biological carcinogenesis model, and also identify particular combinations of identifiable parameters. These results are of theoretical interest, but also of practical significance to anyone attempting to estimate parameters for this large class of cancer models.