An Age and Spatially Structured Model of Tumor Invasion with Haptotaxis II

An analysis of a model of tumor growth into surrounding tissue is continued from an earlier treatment, in which the global existence of unique solutions to the model was established. The model consists of a system of nonlinear partial differential equations for the population densities of tumor cells, extracellular matrix macromolecules, oxygen concentration, and extracellular matrix degradative enzyme concentration. The spatial growth of the tumor involves the directed movement of tumor cells toward the extracellular matrix through haptotaxis. Cell age is used to track progression of cells through the cell cycle. Regularity, positivity, and global bounds of the solutions of the model are proved.

[1]  Dirk Horstmann,et al.  F ¨ Ur Mathematik in Den Naturwissenschaften Leipzig from 1970 until Present: the Keller-segel Model in Chemotaxis and Its Consequences from 1970 until Present: the Keller-segel Model in Chemotaxis and Its Consequences , 2022 .

[2]  Glenn F. Webb,et al.  Population Models Structured by Age, Size, and Spatial Position , 2008 .

[3]  Alissa M. Weaver,et al.  Tumor Morphology and Phenotypic Evolution Driven by Selective Pressure from the Microenvironment , 2006, Cell.

[4]  G. Webb.  Exponential representation of solutions to an abstract semi-linear differential equation. , 1977 .

[5]  H. Stewart.  Generation of analytic semigroups by strongly elliptic operators , 1974 .

[6]  Alexander R. A. Anderson,et al.  Computational Methods and Results for Structured Multiscale Models of Tumor Invasion , 2005, Multiscale Model. Simul..

[7]  Amnon Pazy,et al.  Semigroups of Linear Operators and Applications to Partial Differential Equations , 1992, Applied Mathematical Sciences.

[8]  G. Webb.  Theory of Nonlinear Age-Dependent Population Dynamics , 1985 .

[9]  Christoph Walker.  Global well-posedness of a haptotaxis model with spatial and age structure , 2007, Differential and Integral Equations.

[10]  H. Amann.  Dual semigroups and second order linear elliptic boundary value problems , 1983 .

[11]  H. Bruce Stewart,et al.  Generation of analytic semigroups by strongly elliptic operators under general boundary conditions , 1980 .

[12]  Glenn F Webb,et al.  A fully continuous individual-based model of tumor cell evolution. , 2008, Comptes rendus biologies.

[13]  G. F. Webb,et al.  An age and spatially structured model of tumor invasion with haptotaxis , 2007 .

[14]  A. Anderson,et al.  A hybrid mathematical model of solid tumour invasion: the importance of cell adhesion. , 2005, Mathematical medicine and biology : a journal of the IMA.

[15]  Christoph Walker,et al.  Global Existence of Classical Solutions for a Haptotaxis Model , 2007, SIAM J. Math. Anal..