The multistage model, introduced by Armitage and Doll, was very successful at describing many features of cancer development. Doll and Peto noted a significant departure below the prediction of the model and suggested that this could be due to undercounting of cases at older ages, or to the 'biology of extreme old age.' Moolgavkar pointed out that it could also be due to the approximation used. The recent observation that cancer incidence falls rapidly above age 80 has stimulated new modelling investigations, such as the Pompei-Wilson beta model (which does reproduce the rapid fall). In the present paper, we argue that Moolgavkar's criticisms, while mathematically correct, do not affect the conclusions, particularly the constancy of the number of stages across different cancer registries (Cook, Doll and Fellingham. 1969: A mathematical model for the age distribution of cancer in man. International Journal of Cancer 4, 93-112). We discuss several exact solutions, compare them with the most recent data, and prove rigorously that the standard Armitage-Doll multistage model can never reproduce the sharp turnaround in cancer incidence at old age seen in the data. We discuss in detail multistage processes which have a property observed in many laboratory studies, namely that some stages progress much faster than the others. We verify mathematically the intuition that sufficiently fast stages do not appreciably affect the incidence rate of cancer, and discuss implications of this fact for cancer treatment strategies. We also show that the simplest possible modification of the Armitage-Doll model to incorporate cellular senescence just leads to the Pompei-Wilson beta model.
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