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Heavy-tailed distributions in branching process models of secondary cancerous tumors

Part of: Markov processes

Published online by Cambridge University Press:  01 February 2019

Philip A. Ernst ,
Marek Kimmel ,
Monika Kurpas  and
Quan Zhou
Philip A. Ernst*
Affiliation:
Rice University
Marek Kimmel*
Affiliation:
Rice University and Silesian University of Technology
Monika Kurpas*
Affiliation:
Silesian University of Technology
Quan Zhou*
Affiliation:
Rice University
*
Department of Statistics, Rice University, MS-138, 6100 Main Street, Houston, TX 77005, USA.
Departments of Statistics and Bioengineering, Rice University, MS-138, 6100 Main Street, Houston, TX 77005, USA. Email address: kimmel@stat.rice.edu
Systems Engineering Group, Silesian University of Technology, Akademicka 16, 44-100, Glinice, Poland.
Department of Statistics, Rice University, MS-138, 6100 Main Street, Houston, TX 77005, USA.
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Abstract

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Recent progress in microdissection and in DNA sequencing has facilitated the subsampling of multi-focal cancers in organs such as the liver in several hundred spots, helping to determine the pattern of mutations in each of these spots. This has led to the construction of genealogies of the primary, secondary, tertiary, and so forth, foci of the tumor. These studies have led to diverse conclusions concerning the Darwinian (selective) or neutral evolution in cancer. Mathematical models of the development of multi-focal tumors have been devised to support these claims. We offer a model for the development of a multi-focal tumor: it is a mathematically rigorous refinement of a model of Ling et al. (2015). Guided by numerical studies and simulations, we show that the rigorous model, in the form of an infinite-type branching process, displays distributions of tumor size which have heavy tails and moments that become infinite in finite time. To demonstrate these points, we obtain bounds on the tails of the distributions of the process and an infinite series expansion for the first moments. In addition to its inherent mathematical interest, the model is corroborated by recent literature on apparent super-exponential growth in cancer metastases.

Keywords

Branching processmutationcancer cellheterogeneityheavy tailYule‒Simon distributioninfinite moment

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J85: Applications of branching processes
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue A: Branching and Applied Probability , December 2018 , pp. 99 - 114
Copyright
Copyright © Applied Probability Trust 2018 

References

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