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February  2004, 4(1): 59-80. doi: 10.3934/dcdsb.2004.4.59

From a class of kinetic models to the macroscopic equations for multicellular systems in biology

N. Bellomo 1, and  A. Bellouquid 1,

1. 

Department of Mathematics, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy, Italy

Received  January 2003 Revised  July 2003 Published  November 2003

This paper deals with the development of an asymptotic theory for large systems of interacting cells in a vertebrate. Macroscopic diffusion and evolution equations are derived from the microscopic behavior represented by a class of nonlinear kinetic equations obtained as a generalization of the Boltzmann equation in mathematical biology. The analysis shows how the time-scaling plays a crucial role in the derivation of different type of equations. The application, developed in the second part of the paper refers to a model of progressing tumor cells in competition with the immune system. The asymptotic analysis is addressed to derive the mathematical framework of macroscopic equations to describe the evolution of solid tumors in "vivo" or in "vitro" environments.
Keywords: Statistical mechanics for multicellular systems, asymptotic theory, kinetic theory, from mesoscopic to macroscopic..
Mathematics Subject Classification: P93 B-F, 92C3.
Citation: N. Bellomo, A. Bellouquid. From a class of kinetic models to the macroscopic equations for multicellular systems in biology. Discrete and Continuous Dynamical Systems - B, 2004, 4 (1) : 59-80. doi: 10.3934/dcdsb.2004.4.59
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