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


Book Volume: 80

Pages Range: 160-182

Journal Issue: 1

DOI: 10.1137/18M1194559


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.

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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://dx.doi.org/10.1137/18M1194559


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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