Stability of stationary solutions for the glioma growth equations with radial or axial symmetries

Abstract

We investigate a class of nonlinear time‐partial differential equations describing the growth of glioma cells. The main results show sufficient conditions for the stability of stationary solutions for these kind of equations. More precisely, we study different spatial variables involving radial or axial symmetries. In addition, we also numerically simulate the system based on three distinct scenarios by considering symmetry across all spatial variables. The numerical results confirm the presence of possible stable states.

An Overview of Mathematical Modelling in Cancer Research: Fractional Calculus as Modelling Tool

“…Moreover, these models have a propensity to integrate well-established conservation laws into their modelling assumptions. This has thus enabled the modelling of phenomena such as vascular growth processes through the transport equation [59], the tumour's metastatic spreading adjoined to the possibility of drug resistance from treatment [60] or the evolution of cell population density across tissues by means of diffusion equations [61]. Noteworthy initial contributions on this end have been the Greenspan and proliferation-invasion models (both approached in [32]) with other relevant developments seen in [18,62].…”

On the Stability of Stationary States in Diffusion Models in Biology and Humanities

“…It is obvious that a vector function with coordinates u 1 = 0, q = 0 is an unstable stationary solution of system (26). However, for system (22), the same vector function may also turn out to be a stable stationary solution if the following condition is satisfied:…”

“…In the work [25], in order to prove the sufficiency of conditions ( 24), ( 25), the authors had to use a weighted version of the Steklov-Poincare-Friedrichs inequality because of the convective term −v∂q/∂x in the second equation. Another weighted variant is used in the work [26], where it is shown that the condition…”

Fractional mathematical oncology: cancer-related dynamics under an interdisciplinary viewpoint

“…In a surrogate approach, the diffusion equation can be used to study the dynamic of cell population density across tissues POLOVINKINA et al, 2021). In those studies, one may consider different combinations of population heterogeneity, possibly including stem and regular tumor cells, dead cells, healthy cells and even lymphocytes or similar (ADAM; MAGGELAKIS, 1990;PHAM et al, 2012;WONG et al, 2015).…”

“…With their relative simplicity, ODE-based approaches enable analytical solutions and have conveniences that motivate their use until today (WODARZ; KO-MAROVA, 2014;BENZEKRY et al, 2014;HARTUNG et al, 2014). On the other hand, PDEs can model tumor growth into surrounding tissue (POLOVINKINA et al, 2021). Some models describe tumors as a fluid or mixture via transport equations (BYRNE;PREZIOSI, 2003), while others employ transport phenomena to model metastatic processes and beyond (HARTUNG et al, 2014;XU;VILANOVA;GOMEZ, 2016).…”