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

Abstract

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.analytic semigroup, cell proliferation, degradation enzyme, diffusion, extracellular matrix, fractional power, haptotaxis, tumor cells,

Global Existence and Boundedness of Solutions to a Model of Chemotaxis

“…The solutions exhibit "numerical blow-up". species and a bound chemoattractant, do not exhibit blow-up behavior ( [7], [8], [9], [36]). In chemotaxis models these two spatial movements -diffusion away from concentrations and chemotactic attraction toward concentrations -are directly confronted.…”

A Spatial Model of Tumor Growth with Cell Age, Cell Size, and Mutation of Cell Phenotypes

“…There is an extensive literature of spatial models of tumor growth that have included many of these processes in both continuum and discrete frameworks. Our objective here is to continue the study of a continuum model of tumor growth first proposed in [5] (see also [3] and [4]), and later developed in [6], [7], [8], [18], and [19]. The salient feature of this model is the use of cell age to track passage of individual cells through the cell cycle.…”

“…In this paper we extend the model in [6], [7] to include cell size as a structure variable analogous to cell age, which allows consideration of individual cell growth and division as a component of the spatial movement of the total tumor mass. Additionally, we include transition of cells to and from quiescence as a constraint on tumor growth limited by availability of oxygen and other nutrients.…”

“…The salient feature of this model is the use of cell age to track passage of individual cells through the cell cycle. In [6], [7] a basic theory of existence, uniqueness, positivity, and regularity of solutions was established for the system of nonlinear partial differential equations of the model. A major difficulty in the analysis was to treat the term corresponding to haptotaxis, the directed movement of tumor cells up-gradient of the environmental extracellular matrix, as mediated by a degradation enzyme produced by the tumor cells.…”

Glenn F.Webb: A Career in Mathematics

“…As a result of this visit, Glenn not only learned to speak fractured Italian, but more importantly he began an ongoing love affair with Italy and was introduced to the powerful Italian analysis and partial differential equations research community, and had the opportunity to meet a large number of people including Giuseppe da Prato, Ennio de Georgi, Mimmo Iannelli, Vincenzo Capasso, Allessandra Lunardi, Eugenio Sinestrari, and Gabriella di Blasio. Toward the end of his visit he met Rosanna Villella-Bressan, beginning his ongoing collaboration and friendship with her and Janet Dyson [50,89,93,[103][104][105][106]108,[116][117][121][122][123][138][139][140]. About a year before his Italian sojourn Glenn began to consider functional differential equations with discrete or distributed delay.…”