A mixed model of carcinogenesis with applications to retinoblastoma

Comparison of multi-stage dose-response mixture models, with applications

This article concerns the analysis of a stochastic model that we propose for the population that generates a response (response measure) to the dose with the multi-stage model. The parameter uncertainty is dealt with via random dose and random size of the population at risk. The response measure is modeled by a random sum of mixed Bernoulli random variables with arbitrary distribution for the mixing parameters. Some extensions of the model are defined by functionals of the infection probability, fulfilling some convexity properties. We analyze the response by stochastic comparisons under different stochastic relations on the random dosages and the random sizes of the population at risk; or on the random infection rates. We provide stochastic exact bounds of the mixture model for the response, using inequalities and the positive quadrant dependence. Numerical bounds of the response by a dose having a scalar value or having an exponential or uniform distributions are obtained. Some conclusions are derived: the lower estimation of the response measure in the increasing convex order sense by replacing the dosages by their means; effects of the variation of the dose on the magnitude of the probability distribution of the response; effects of parameter correlation on the degree of variability of the response to any random dose; the low-dose region assessment; and also, the classical multi-stage model is compared versus the mixture model featuring independence and versus that with positive quadrant dependence.

Analysis of retinoblastoma age incidence data using a fully stochastic cancer model

Retinoblastoma (RB) is an important ocular malignancy of childhood. It has been commonly accepted for some time that knockout of the two alleles of the RB1 gene is the principal molecular target associated with the occurrence of RB. In this article, we examine the validity of the two-hit theory for RB by comparing the fit of a stochastic model with two or more mutational stages. Unlike many such models, our model assumes a fully stochastic stem cell compartment, which is crucial to its behavior. Models are fitted to a population-based dataset comprising 1,553 cases of RB for the period 1962-2000 in Great Britain (England, Scotland and Wales). The population incidence of RB is best described by a fully stochastic model with two stages, although models with a deterministic stem cell compartment yield equivalent fit; models with three or more stages fit much less well. The results strongly suggest that knockout of the two alleles of the RB1 gene is necessary and may be largely sufficient for the development of RB, in support of Knudson's two-hit hypothesis.

Heterogeneity in multistage carcinogenesis and mixture modeling

Carcinogenesis is commonly described as a multistage process, in which stem cells are transformed into cancer cells via a series of mutations. In this article, we consider extensions of the multistage carcinogenesis model by mixture modeling. This approach allows us to describe population heterogeneity in a biologically meaningful way. We focus on finite mixture models, for which we prove identifiability. These models are applied to human lung cancer data from several birth cohorts. Maximum likelihood estimation does not perform well in this application due to the heavy censoring in our data. We thus use analytic graduation instead. Very good fits are achieved for models that combine a small high risk group with a large group that is quasi immune.