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Mathematical Methods for Cancer Evolution (Paperback, 1st ed. 2017)
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Mathematical Methods for Cancer Evolution (Paperback, 1st ed. 2017)
Series: Lecture Notes on Mathematical Modelling in the Life Sciences
Expected to ship within 10 - 15 working days
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The purpose of this monograph is to describe recent developments in
mathematical modeling and mathematical analysis of certain problems
arising from cell biology. Cancer cells and their growth via
several stages are of particular interest. To describe these
events, multi-scale models are applied, involving continuously
distributed environment variables and several components related to
particles. Hybrid simulations are also carried out, using
discretization of environment variables and the Monte Carlo method
for the principal particle variables. Rigorous mathematical
foundations are the bases of these tools.The monograph is composed
of four chapters. The first three chapters are concerned with
modeling, while the last one is devoted to mathematical analysis.
The first chapter deals with molecular dynamics occurring at the
early stage of cancer invasion. A pathway network model based on a
biological scenario is constructed, and then its mathematical
structures are determined. In the second chapter mathematical
modeling is introduced, overviewing several biological insights,
using partial differential equations. Transport and gradient are
the main factors, and several models are introduced including the
Keller-Segel systems. The third chapter treats the method of
averaging to model the movement of particles, based on mean field
theories, employing deterministic and stochastic approaches. Then
appropriate parameters for stochastic simulations are examined. The
segment model is finally proposed as an application. In the fourth
chapter, thermodynamic features of these models and how these
structures are applied in mathematical analysis are examined, that
is, negative chemotaxis, parabolic systems with non-local term
accounting for chemical reactions, mass-conservative
reaction-diffusion systems, and competitive systems of chemotaxis.
The monograph concludes with the method of the weak scaling limit
applied to the Smoluchowski-Poisson equation.
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