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.
