Mathematical Analysis and Numerical Simulations for a System Modeling Acid-Mediated Tumor Cell Invasion

Abstract

This work is concerned with the mathematical analysis of a model proposed by Gatenby and Gawlinski (1996) in order to support the hypothesis that tumor-induced alteration of microenvironmental pH may provide a simple but comprehensive mechanism to explain cancer invasion. We give an intuitive proof for the existence of a solution under general initial conditions upon using an iterative approach. Numerical simulations are also performed, which endorse the predictions of the model when compared with experimentally observed qualitative facts.

A stochastic multiscale model for acid mediated cancer invasion

“…To ensure an upper bound for C 0 , consider equation (46b) and use the following auxiliary function (like in [28]):…”

“…Further developments of Gatenby & Gawlinski's model involve both vascular and avascular growth of multicellular tumor spheroids, assuming rotational symmetry, for which existence and qualitative properties of the solutions were investigated [36]. In [28] the model in [8] for acid-mediated tumor invasion was reconsidered, wherein crowding effects (due to competition with cancer cells) in the growth of normal cells was taken into account. The global existence of a unique solution was proved via an iteration argument.…”

Mathematical Analysis and Numerical Simulations for the Cancer Tissue Invasion Model

“…al. [5] and the well posedness of an extended model, which has been developed to include the crowding e¤ects in the growth of normal cells, has been shown in [6]. Then the model has been extended to include the intracellular dynamics and it has been shown that the new multiscale model has a global unique solution [7].…”

A multiscale model for acid-mediated tumor invasion: Therapy approaches

“…Most of them reconsider the model in [5] and possibly extend it, e.g., to account for vascular and avascular growth of multicellular tumor spheroids [6]. The settings involve reaction-diffusion equations for the densities of cancer cells and of normal cells, coupled with an equation for the concentration of extracellular protons, see also [7,8]. All of these models, however, are set on the macroscopic scale of cell populations, whereas the microscopic, subcellular level is known to significantly influence (and even control) the macroscale behavior, e.g., by the intracellular proton dynamics, as mentioned above.…”