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

Mathematical Modelling of Natural Phenomena (2012)

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

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topBertolusso, 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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- A. Marciniak-Czochra, M. Kimmel. 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.
- A. Marciniak-Czochra, M. Ptashnyk. Derivation of a macroscopic receptor-based model using homogenization techniques. SIAM J. Math. Anal., 40 (2008), No. 1, 215–237. Zbl1178.35050
- A. Marciniak-Czochra, G. Karch, K. Suzuki. Unstable patterns in reaction-diffusion model of early carcinogenesis. arXiv :1104.3592v1, (2011). Zbl1295.35072
- 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.
- 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. Zbl1109.65008
- 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.
- J. Paulsson, O. G. Berg, M. Ehrenberg. Stochastic focusing : fluctuation-enhanced sensitivity of intracellular regulation. Proc. Natl. Acad. Sci. U.S.A., 97 (2000), June, 7148–53.

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