On a reaction-cross-diffusion system modeling the growth of glioblastoma
Burger M, Friele P, Pietschmann JF (2020)
Publication Type: Journal article
Publication year: 2020
Journal
Book Volume: 80
Pages Range: 160-182
Journal Issue: 1
DOI: 10.1137/18M1194559
Abstract
We investigate a recently proposed cross-diffusion system modeling the growth of gliobastoma taking into account size exclusion in both the migration and the proliferation process. In addition to degenerate nonlinear cross-diffusion, the model includes reaction terms as in the Fisher-Kolmogorov equation and linear ones modeling transition between states of proliferation and migration. We discuss the mathematical structure of the system and provide a complete existence analysis in spatial dimension one. The proof is based on exploiting partial entropy dissipation techniques and fully implicit time discretizations. In order to prove the existence of the latter, appropriate variational and fixed-point techniques are used, together with suitable a priori estimates. Moreover, we review the existence of traveling waves and their relation to potential growth or decay of glioblastoma. Finally, we provide extensive numerical studies in one and two spatial dimensions, including the effect of anisotropic diffusions as found in neural tissues.
Authors with CRIS profile
Involved external institutions
Westfälische Wilhelms-Universität (WWU) Münster
Germany (DE)
Technische Universität Chemnitz
Germany (DE)
Germany (DE)
Technische Universität Chemnitz
Germany (DE)
How to cite
APA:
Burger, M., Friele, P., & Pietschmann, J.F. (2020). On a reaction-cross-diffusion system modeling the growth of glioblastoma. SIAM Journal on Applied Mathematics, 80(1), 160-182. https://doi.org/10.1137/18M1194559
MLA:
Burger, Martin, Patricia Friele, and Jan Frederik Pietschmann. "On a reaction-cross-diffusion system modeling the growth of glioblastoma." SIAM Journal on Applied Mathematics 80.1 (2020): 160-182.
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