A mathematical solution to Peto\u2019s paradox using Polya\u2019s urn model: implications for the aetiology of cancer in general

Anastasio Salazar-Bañuelos

doi:10.1007/s12064-019-00290-6

Abstract

Ageing is the leading risk factor for the emergence of cancer in humans. Accumulation of pro-carcinogenic events throughout life is believed to explain this observation; however, the lack of direct correlation between the number of cells in an organism and cancer incidence, known as Peto’s Paradox, is at odds with this assumption. Finding the events responsible for this discrepancy can unveil mechanisms with potential uses in prevention and treatment of cancer in humans. On the other hand, the immune system is important in preventing the development of clinically relevant tumours by maintaining a fine equilibrium between reactive and suppressive lymphocyte clones. It is suggested here that the loss of this equilibrium is what ultimately leads to increased risk of cancer and to propose a mechanism for the changes in clonal proportions based on decreased proliferative capacity of lymphocyte clones as a natural phenomenon of ageing. This mechanism, being a function of the number of cells, provides an explanation for Peto’s Paradox.

Highlights

Carcinogenesis, multistep models and Peto’s paradoxThe aetiology of cancer in humans has been the subject of intense study since the beginning of medicine as a scientific discipline
As demonstrated by the simulations, the probability of the variation in the proportion of balls is inversely proportional to the number of the initial balls, and if we consider that the proliferation of the lymphocytes is stochastic in nature (Cosette et al 2015; Zilman et al 2010) following the Pólya’s urn model, we will expect an increased probability of variation in the final proportion of clones if the number of progenitor cells decreases for any reason
This process can be described as the recursive proliferation process of lymphocyte clones creating an Eigenvalue for the proportion of clones where the probability of change is in inverse proportion to the number of cells recursively producing the lymphocytes

Summary

Introduction

Carcinogenesis, multistep models and Peto’s paradoxThe aetiology of cancer in humans has been the subject of intense study since the beginning of medicine as a scientific discipline. One of the more convincing theories on how this transformation occurs is the Multistage Model of carcinogenesis. This model was originally proposed more than 50 years ago by analysing the epidemiological data of cancer in human populations (Nordling 1953). It proposes that the pure mutation of a cell is not enough to explain the development of clinically relevant tumours, and that more than. Most hypotheses that attempt to explain Peto’s paradox are centred in mechanisms directly related to the mutated tissue, e.g. by decreasing the mutagenic susceptibility of cells, decreasing the growth rate, self-tumour control, etc. We consider here that Peto’s paradox comes as a result of considering the development of clinically relevant tumours, exclusively as an intrinsic process of the mutated tissue, and not considering the significant participation of extrinsic factors such as the immune system

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