Issue Math. Model. Nat. Phenom. Volume 3, Number 7, 2008 Special issue dedicated to Glenn Webb Page(s) 90 - 114 DOI 10.1051/mmnp:2008043 Published online 23 October 2008

Math. Model. Nat. Phenom. Vol. 3, No. 7, 2008, pp. 90-114
DOI: 10.1051/mmnp:2008043

Reaction-Difusion Model of Early Carcinogenesis: The Effects of Influx of Mutated Cells

Anna Marciniak-Czochra¹ and Marek Kimmel<sup>2, 3

1</sup>  Institute of Applied Mathematics, University of Heidelberg, 69120 Heidelberg, Germany
²  Department of Statistics, Rice University, Houston, TX 77005, USA
³  Systems Engineering Group Silesian University of Technology, 44-100 Gliwice, Poland

anna.marciniak@iwr.uni-heidelberg.de

Published online: 23 October 2008

Abstract
In this paper we explore a new model of field carcinogenesis, inspired by lung cancer precursor lesions, which includes dynamics of a spatially distributed population of pre-cancerous cells c(t, x), constantly supplied by an influx of mutated normal cells. Cell proliferation is controlled by growth factor molecules bound to cells, b(t, x). Free growth factor molecules g(t, x) are produced by precancerous cells and may diffuse before they become bound to other cells. The purpose of modelling is to investigate the existence of solutions, which correspond to formation of multiple spatially isolated lesions of pre-cancerous cells or, mathematically, to stable spike solutions. These multiple lesions are consistent with the field theory of carcinogenesis. In a previous model published by these authors, the influx of mutated cells was equal to zero, = 0, which corresponded to a single pre-malignant colony of cells. In that model, stable patterns appeared only if some of the growth factor was supplied from outside, arguably, a biologically tenuous hypothesis. In the present model, when > 0, that hypothesis is no more required, which makes this model more realistic. We present a range of results, both mathematical and computational, which taken together allow understanding the dynamics of this model. The equilibrium solutions in the current model result from the balance between new premalignant colonies being initiated and the old ones dying out.

Mathematics Subject Classification. 35K57, 35B35, 35B40, 92C37, 92C50

Key words: cancer modelling -- cooperation -- reaction-diffusion equations -- pattern formation -- spike solutions

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