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Mathematical methods for cancer evolution. (English) Zbl 1397.92005
Lecture Notes on Mathematical Modelling in the Life Sciences. Singapore: Springer (ISBN 978-981-10-3670-5/pbk; 978-981-10-3671-2/ebook). viii, 144 p. (2017).
This book deals with the mathematical analysis of selected models of reaction-diffusion systems. The book is intended to both graduate as well as advanced undergraduate students in applied mathematics. The book assumes familiarity of PDE theory. A working knowledge of cell biological insights is somewhat essential. In detail, the book is divided into four chapters. Chapter 1 (Molecular dynamics) explains how to build a system of ordinary differential equations (ODEs) for modeling the pathway network of 12 species, at the early stage of cancer invasion, that results from the attachment and detachment of the molecules MMP2, TIMP2 and MT1-MMP. The solution is graphically represented by time-course simulation. Chapter 2 (Amounting the balance) offers an overview of some selected models to the invadopodia, using partial differential equations. Namely, it underlies (1) the Keller-Segel systems that account for the density of the cellular slime mold and the chemical concentrations; (2) the Smoluchowski-ODE systems that account for the existence probability and the control species density; and (3) the Chaplain-Anderson system that accounts for the ECM density, the MMP concentration and the enzyme concentration. Moreover, a free boundary problem is described for the motion of the plasma membrane. This chapter ends by covering a simplified Smoluchowski-Poisson system under diffusion coefficient equal to 1, and the stationary solution is given. In Chapter 3 (Averaging particle movements), the Smoluchowski equation is derived under constant diffusion coefficient, by means of mean field approximations, employing deterministic and stochastic approaches. Regarding each F-actin as a segment, diffusion equations to the averaged probability are derived for either no polymerization or the polymerization of F-actins between G-actins being taken into account. Chapter 4 (Mathematical analysis) closes the book with regularity and existence results. This chapter begins by proving the existence of extension global-in-time classical solution, for the case of one-dimensional space, of the Smoluchowski-ODE system with negative chemotaxis. Asymptotic convergence properties for parabolic systems with non-local term accounting for chemical reactions, and for mass-conservative reaction-diffusion systems are studied. In the case of two-dimensional space, the blowup set of classical solutions to parabolic-elliptic systems for two chemotactic cell types is studied. This chapter ends by applying the method of the weak scaling limit to the simplified Smoluchowski-Poisson system, defined in 2D space, introduced in Section 2.4. The selected list of literature covers a vast area of applications that goes from the biological scenario of cell invasion until transport of contaminants in geological formations. Unfortunately, the first reference is not conveniently quoted.
##### MSC:
 92-02 Research exposition (monographs, survey articles) pertaining to biology 92C50 Medical applications (general) 92C37 Cell biology 35Q92 PDEs in connection with biology, chemistry and other natural sciences 35K57 Reaction-diffusion equations 92C17 Cell movement (chemotaxis, etc.)
##### Keywords:
diffusion; chemotaxis; Smoluchowski equation
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