Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-27T23:09:39.712Z Has data issue: false hasContentIssue false
  • English
  • Français

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]
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Marciniak-Czochra, A., Kimmel, M.. Reaction-difusion model of early carcinogenesis : The effects of influx of mutated cells. Mathematical Modelling of Natural Phenomena, 3 (2008), No. 7, 90–114. CrossRefGoogle Scholar
Marciniak-Czochra, A., Kimmel, M.. Dynamics of growth and signaling along linear and surface structures in very early tumors. Computational & Mathematical Methods in Medicine, 7 (2006), No. 2/3, 189–213. CrossRefGoogle Scholar
Marciniak-Czochra, A., Kimmel, M.. Modelling of early lung cancer progression : Influence of growth factor production and cooperation between partially transformed cells. Math. Mod. Meth. Appl. Sci., 17S (2007), 1693–1719. CrossRefGoogle Scholar
Marciniak-Czochra, A., Kimmel, M.. Reaction–diffusion approach to modeling of the spread of early tumors along linear or tubular structures. Journal of Theoretical Biology, 244 (2006), No. 3, 375–387. CrossRefGoogle ScholarPubMed
Marciniak-Czochra, A., Ptashnyk, M.. Derivation of a macroscopic receptor-based model using homogenization techniques. SIAM J. Math. Anal., 40 (2008), No. 1, 215–237. CrossRefGoogle Scholar
A. Marciniak-Czochra, G. Karch, K. Suzuki. Unstable patterns in reaction-diffusion model of early carcinogenesis. arXiv :1104.3592v1, (2011).
R. Bertolusso. Computational models of signaling processes in cells with applications : Influence of stochastic and spatial effects. PhD thesis (2011), Rice University, Houston, TX.
R. Erban, S. J. Chapman, P. Maini. A practical guide to stochastic simulations of reaction-diffusion processes. ArXiv e-prints, (2007), April.
Isaacson, S. A., Peskin, C. S.. Incorporating diffusion in complex geometries into stochastic chemical kinetics simulations. SIAM J. Scientific Computing, 28 (2006), No. 1, 47–74. CrossRefGoogle Scholar
Slepoy, A., Thompson, A. P., Plimpton, S. J.. A constant-time kinetic monte carlo algorithm for simulation of large biochemical reaction networks. J. Chem. Phys., 128 (2008), May, 205101. CrossRefGoogle Scholar
Paulsson, J., Berg, O. G., Ehrenberg, M.. Stochastic focusing : fluctuation-enhanced sensitivity of intracellular regulation. Proc. Natl. Acad. Sci. U.S.A., 97 (2000), June, 7148–53. CrossRefGoogle ScholarPubMed