## Mathematical Modeling of Complex Biological Systems: A Kinetic Theory ApproachContentsandScienti?cAims The scienti?c community is aware that the great scienti?c 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 ?nally be supported by rigorous investigation me- ods 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 maybeneededtodealwiththeinnercomplexityofbiologicalsystemswhich exhibit features and behaviors very di?erent 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 eq- librium 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. Speci?cally, the dynamics of a cell, including its life, are ruled by sub-cellular entities, while most of the phenomena can be e?ectively observed only at the macroscopic scale. |

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40 t Fig abnormal cells analyzed antibody antigen apoptosis application assumption asymptotic behavior asymptotic limit Bellouquid biological functions biological interactions biological interpretations Boltzmann equation cell populations cellular characterized collision complex biological systems computational conservative continuum mechanics corresponding cytokine deals decreases deﬁned Delitala dendritic cell depletion of abnormal developed differential equations discrete distribution function dv dv dynamics endothelial cells evolution equations immune cells immune competition immune response immune system increases inhibit immune cells initial value problem integrating Lemma lymphocytes macrophages macroscopic equations mathematical framework mathematical kinetic theory mathematical model methods microscopic interactions models proposed Moreover multicellular systems number of cells obtained ordinary differential equations parameters particles particular pathogen phenomenological proliferating or destructive proliferation and destruction Proof proposed in Chapter qualitative analysis refers satisﬁes Section simulations solution space spatially homogeneous specific models statistical statistical mechanics suitable test cell Theorem tion tumor cells velocity yields zero

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Page i - USA wpaepmnas . epm . ornl . gov KR Rajagopal (Multiphase Flows) Department of Mechanical Engineering Texas A&M University College Station, TX 77843, USA KRaj agopaltaengr . tamu . edu Y.

Page 1 - The work was repugnant to me, chiefly from my not being able to see any meaning in the early steps in algebra. This impatience was very foolish, and in after years I have deeply regretted that I did not proceed far enough at least to understand something of the great leading principles of mathematics, for men thus endowed seem to have an extra sense.

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