Commun. Comput. Phys., 11 (2012), pp. 1-18. |
A Bistable Field Model of Cancer Dynamics C. Cherubini ^{1*}, A. Gizzi ^{2}, M. Bertolaso ^{3}, V. Tambone ^{3}, S. Filippi ^{1} 1 Nonlinear Physics and Mathematical Modeling Lab, University Campus Bio-Medico, I-00128, Via A. del Portillo 21, Rome, Italy; and International Center for Relativistic Astrophysics-I.C.R.A., University of Rome "La Sapienza", I-00185 Rome, Italy.2 Nonlinear Physics and Mathematical Modeling Lab, University Campus Bio-Medico, I-00128, Via A. del Portillo 21, Rome, Italy; and Alberto Sordi Foundation-Research Institute on Aging, I-00128 Rome, Italy. 3 Institute of Philosophy of Scientific and Technological Activity, University Campus Bio-Medico, I-00128, Via A. del Portillo 21, Rome, Italy. Received 27 July 2010; Accepted (in revised version) 22 February 2011 Available online 5 September 2011 doi:10.4208/cicp.270710.220211a Abstract Cancer spread is a dynamical process occurring not only in time but also in space which, for solid tumors at least, can be modeled quantitatively by reaction and diffusion equations with a bistable behavior: tumor cell colonization happens in a portion of tissue and propagates, but in some cases the process is stopped. Such a cancer proliferation/extintion dynamics is obtained in many mathematical models as a limit of complicated interacting biological fields. In this article we present a very basic model of cancer proliferation adopting the bistable equation for a single tumor cell dynamics. The reaction-diffusion theory is numerically and analytically studied and then extended in order to take into account dispersal effects in cancer progression in analogy with ecological models based on the porous medium equation. Possible implications of this approach for explanation and prediction of tumor development on the lines of existing studies on brain cancer progression are discussed. The potential role of continuum models in connecting the two predominant interpretative theories about cancer, once formalized in appropriate mathematical terms, is discussed. AMS subject classifications: 92B05, 35K59PACS: 87.19.xj, 87.23.Cc, 02.60.-x Key words: Reaction-diffusion equations, cancer modeling, finite elements method. *Corresponding author. Email: c.cherubini@unicampus.it (C. Cherubini), a.gizzi@unicampus.it (A. Gizzi), m.bertolaso@unicampus.it (M. Bertolaso), v.tambone@unicampus.it (V. Tambone), s.filippi@unicampus.it (S. Filippi) |