Hostname: page-component-76fb5796d-skm99 Total loading time: 0 Render date: 2024-04-27T04:55:57.398Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

[1]Abramowitz, M. and Stegun, I. A. (1964).Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables(Nat. Bureau Stand. Appl. Math. Ser. 55).US Government Printing Office,Washington, DC.Google Scholar
[2]Arino, O. and Kimmel, M. (1993).Comparison of approaches to modeling of cell population dynamics.SIAM J. Appl. Math. 53,1480–1504.Google Scholar
[3]Baratchart, E. et al. (2015).Computational modelling of metastasis development in renal cell carcinoma.PLoS Comput. Biol. 11, 23pp.Google Scholar
[4] Cerone, P. (2007).Special functions: approximations and bounds.Appl. Anal. Discrete Math. 1,72–91.Google Scholar
[5]Iwata, K., Kawasaki, K. and Shigesada, N. (2000).A dynamical model for the growth and size distribution of multiple metastatic tumors.J. Theoret. Biol. 203,177–186.Google Scholar
[6]Kimmel, M. and Axelrod, D. (2015).Branching Processes in Biology.Springer,New York.Google Scholar
[7]Ling, S. et al. (2015).Extremely high genetic diversity in a single tumor points to prevalence of non-Darwinian cell evolution.Proc. Nat. Acad. Sci. USA 112, E6496‒E6505.Google Scholar
[8]Metz, J. A. and Diekmann, O. (eds) (2014).The Dynamics of Physiologically Structured Populations(Lecture Notes Biomath. 68).Springer,Berlin.Google Scholar
[9]Simon, H. A. (1955).On a class of skew distribution functions.Biometrika 42,425–440.Google Scholar
[10]Tao, Y. et al. (2015). Further genetic diversification in multiple tumors and an evolutionary perspective on therapeutics. Preprint. Available at https://www.biorxiv.org/content/early/2015/08/25/025429.Google Scholar
[11]Yule, G. U. (1925).A mathematical theory of evolution, based on the conclusions of Dr. J. C. Willis, F.R.S.Phil. Trans. Royal Soc. London B 213,21–87.Google Scholar