Bellouquid A., Delitala M. Mathematical Modeling of Complex Biological Systems: A Kinetic Theory Approach Modeling and Simulation in Science, Engineering and Technology

Birkhäuser, 2006. — 194 p.The scientific community is aware that the great scientific revolution of this century will be the mathematical formalization, by methods of applied mathematics, of complex biological systems. A fascinating prospect is that biological sciences will finally be supported by rigorous investigation methods and tools, similar to what happened in the past two centuries in the case of mechanical and physical sciences.
It is not an easy task, considering that new mathematical methods may be needed to deal with the inner complexity of biological systems which exhibit features and behaviors very different from those of inert matter.
Microscopic entities in biology, say cells in a multicellular system, are characterized by biological functions and the ability to organize their dynamics and interactions with other cells. Indeed, cells organize their dynamics according to the above functions, while classical particles follow deterministic laws of Newtonian mechanics. Cells have a life according to a cell cycle which ends up with a programmed death. The dialogue among cells can modify their behavior. The activity of cells includes proliferation and/or destructive events which may, in some cases, result in dangerously reproductive events. Finally, a cellular system may move far from equilibrium in physical situations where classical particles generally show a tendency toward equilibrium.
An additional source of complexity is that biological systems always need a multiscale approach. Specifically, the dynamics of a cell, including its life, are ruled by sub-cellular entities, while most of the phenomena can be effectively observed only at the macroscopic scale.
This book deals with the modelling of complex multicellular systems by a mathematical approach which is related to mathematical kinetic theory. Applications refer to the mathematical description of the immune competition with special attention to the interactions between tumor and immune cells.On the Modelling of Complex Biological Systems
Mathematical Frameworks of the Generalized Kinetic (Boltzmann) Theory
Modelling the Immune Competition and Applications
On the Cauchy Problem
Simulations, Biological Interpretations, and Further Modelling Perspectives
Models with Space Structure and the Derivation of Macroscopic Equations
Critical Analysis and Forward Perspectives
Basic Tools of Mathematical Kinetic Theory