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

R. Bertolusso; M. Kimmel

Mathematical Modelling of Natural Phenomena (2012)

  • Volume: 7, Issue: 1, page 245-260
  • ISSN: 0973-5348

Abstract

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We consider the early carcinogenesis model originally proposed as a deterministic reaction-diffusion system. The model has been conceived to explore the spatial effects stemming from growth regulation of pre-cancerous cells by diffusing growth factor molecules. The model exhibited Turing instability producing transient spatial spikes in cell density, which might be considered a model counterpart of emerging foci of malignant cells. However, the process of diffusion of growth factor molecules is by its nature a stochastic random walk. An interesting question emerges to what extent the dynamics of the deterministic diffusion model approximates the stochastic process generated by the model. We address this question using simulations with a new software tool called sbioPN (spatial biological Petri Nets). The conclusion is that whereas single-realization dynamics of the stochastic 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 degree of similarity depends on model parameters. Interestingly, despite the differences, typical realizations of the stochastic process include spikes of cell density, which however are spread more uniformly and are less dependent of initial conditions than those produced by the reaction-diffusion system.

How to cite

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Bertolusso, R., and Kimmel, M.. "Modeling Spatial Effects in Early Carcinogenesis : Stochastic Versus Deterministic Reaction-Diffusion Systems." Mathematical Modelling of Natural Phenomena 7.1 (2012): 245-260. <http://eudml.org/doc/222405>.

@article{Bertolusso2012,
abstract = {We consider the early carcinogenesis model originally proposed as a deterministic reaction-diffusion system. The model has been conceived to explore the spatial effects stemming from growth regulation of pre-cancerous cells by diffusing growth factor molecules. The model exhibited Turing instability producing transient spatial spikes in cell density, which might be considered a model counterpart of emerging foci of malignant cells. However, the process of diffusion of growth factor molecules is by its nature a stochastic random walk. An interesting question emerges to what extent the dynamics of the deterministic diffusion model approximates the stochastic process generated by the model. We address this question using simulations with a new software tool called sbioPN (spatial biological Petri Nets). The conclusion is that whereas single-realization dynamics of the stochastic 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 degree of similarity depends on model parameters. Interestingly, despite the differences, typical realizations of the stochastic process include spikes of cell density, which however are spread more uniformly and are less dependent of initial conditions than those produced by the reaction-diffusion system.},
author = {Bertolusso, R., Kimmel, M.},
journal = {Mathematical Modelling of Natural Phenomena},
keywords = {cancer modelling; deterministic; stochastic; reaction-diffusion equations; pattern formation; spike solutions},
language = {eng},
month = {1},
number = {1},
pages = {245-260},
publisher = {EDP Sciences},
title = {Modeling Spatial Effects in Early Carcinogenesis : Stochastic Versus Deterministic Reaction-Diffusion Systems},
url = {http://eudml.org/doc/222405},
volume = {7},
year = {2012},
}

TY - JOUR
AU - Bertolusso, R.
AU - Kimmel, M.
TI - Modeling Spatial Effects in Early Carcinogenesis : Stochastic Versus Deterministic Reaction-Diffusion Systems
JO - Mathematical Modelling of Natural Phenomena
DA - 2012/1//
PB - EDP Sciences
VL - 7
IS - 1
SP - 245
EP - 260
AB - We consider the early carcinogenesis model originally proposed as a deterministic reaction-diffusion system. The model has been conceived to explore the spatial effects stemming from growth regulation of pre-cancerous cells by diffusing growth factor molecules. The model exhibited Turing instability producing transient spatial spikes in cell density, which might be considered a model counterpart of emerging foci of malignant cells. However, the process of diffusion of growth factor molecules is by its nature a stochastic random walk. An interesting question emerges to what extent the dynamics of the deterministic diffusion model approximates the stochastic process generated by the model. We address this question using simulations with a new software tool called sbioPN (spatial biological Petri Nets). The conclusion is that whereas single-realization dynamics of the stochastic 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 degree of similarity depends on model parameters. Interestingly, despite the differences, typical realizations of the stochastic process include spikes of cell density, which however are spread more uniformly and are less dependent of initial conditions than those produced by the reaction-diffusion system.
LA - eng
KW - cancer modelling; deterministic; stochastic; reaction-diffusion equations; pattern formation; spike solutions
UR - http://eudml.org/doc/222405
ER -

References

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  8. R. Erban, S. J. Chapman, P. Maini. A practical guide to stochastic simulations of reaction-diffusion processes. ArXiv e-prints, (2007), April.  
  9. S. A. Isaacson, C. S. Peskin. Incorporating diffusion in complex geometries into stochastic chemical kinetics simulations. SIAM J. Scientific Computing, 28 (2006), No. 1, 47–74.  
  10. A. Slepoy, A. P. Thompson, S. J. Plimpton. A constant-time kinetic monte carlo algorithm for simulation of large biochemical reaction networks. J. Chem. Phys., 128 (2008), May, 205101.  
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