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Modeling Spatial Effects in Early Carcinogenesis : StochasticVersus Deterministic Reaction-Diffusion Systems

Published online by Cambridge University Press:  25 January 2012

R. Bertolusso  and
M. Kimmel
R. Bertolusso
Affiliation:
Department of Statistics, Rice University, 6100 Main Street, MS138, Houston, TX 77005, USA
M. Kimmel*
Affiliation:
Department of Statistics, Rice University, 6100 Main Street, MS138, Houston, TX 77005, USA Systems Engineering Group, Silesian University of Technology, 44-100 Gliwice, Poland
*
⋆Corresponding author. E-mail: [email protected]
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Abstract

We consider the early carcinogenesis model originally proposed as a deterministicreaction-diffusion system. The model has been conceived to explore the spatial effectsstemming from growth regulation of pre-cancerous cells by diffusing growth factormolecules. The model exhibited Turing instability producing transient spatial spikes incell density, which might be considered a model counterpart of emerging foci of malignantcells. However, the process of diffusion of growth factor molecules is by its nature astochastic random walk. An interesting question emerges to what extent the dynamics of thedeterministic diffusion model approximates the stochastic process generated by the model.We address this question using simulations with a new software tool called sbioPN (spatialbiological Petri Nets). The conclusion is that whereas single-realization dynamics of thestochastic process is very different from the behavior of the reaction diffusion system,it is becoming more similar when averaged over a large number of realizations. The degreeof similarity depends on model parameters. Interestingly, despite the differences, typicalrealizations of the stochastic process include spikes of cell density, which however arespread more uniformly and are less dependent of initial conditions than those produced bythe reaction-diffusion system.

Keywords

cancer modellingdeterministicstochasticreaction-diffusion equationspattern formationspike solutions
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 7 , Issue 1: Cancer modeling , 2012 , pp. 245 - 260
Copyright
© EDP Sciences, 2012

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