Convergence and Numerical Solution of a Model for Tumor Growth

In this paper, we show the application of the meshless numerical method called “Generalized Finite Diference Method” (GFDM) for solving a model for tumor growth with nutrient density, extracellular matrix and matrix degrading enzymes, [recently proposed by Li and Hu]. We derive the discretization of the parabolic–hyperbolic–parabolic–elliptic system by means of the explicit formulae of the GFDM. We provide a theoretical proof of the convergence of the spatial–temporal scheme to the continuous solution and we show several examples over regular and irregular distribution of points. This shows the feasibility of the method for solving this nonlinear model appearing in Biology and Medicine in complicated and realistic domains.

[1]  Liliana Borges de Menezes,et al.  Tumor microenvironment components: Allies of cancer progression. , 2019, Pathology, research and practice.

[2]  Mark A. J. Chaplain,et al.  Mathematical modelling of cancer invasion of tissue: dynamic heterogeneity , 2006, Networks Heterog. Media.

[3]  Luis Gavete,et al.  Influence of several factors in the generalized finite difference method , 2001 .

[4]  Mihaela Negreanu,et al.  Solving a reaction-diffusion system with chemotaxis and non-local terms using Generalized Finite Difference Method. Study of the convergence , 2021, J. Comput. Appl. Math..

[5]  Bei Hu,et al.  A parabolic–hyperbolic system modeling the growth of a tumor , 2018, Journal of Differential Equations.

[6]  A. M. Vargas,et al.  On the numerical solution to a parabolic-elliptic system with chemotactic and periodic terms using Generalized Finite Differences , 2020, Engineering Analysis with Boundary Elements.

[7]  A. E. del Río Hernández,et al.  Role of Extracellular Matrix in Development and Cancer Progression , 2018, International journal of molecular sciences.

[8]  Luis Gavete,et al.  Solving second order non-linear elliptic partial differential equations using generalized finite difference method , 2017, J. Comput. Appl. Math..

[9]  P. Cirri,et al.  Nutritional Exchanges Within Tumor Microenvironment: Impact for Cancer Aggressiveness , 2020, Frontiers in Oncology.

[10]  A.C. Albuquerque-Ferreira,et al.  The generalized finite difference method with third- and fourth-order approximations and treatment of ill-conditioned stars , 2021 .

[11]  David Levin,et al.  The approximation power of moving least-squares , 1998, Math. Comput..

[12]  A. Marciniak-Czochra,et al.  Mathematical modelling of the influence of heat shock proteins on cancer invasion of tissue , 2009, Journal of mathematical biology.