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Theoretical and numerical analysis for a hybrid tumor model with diffusion depending on vasculature. (English) Zbl 1475.35352

Summary: In this work we analyse a PDE-ODE problem modelling the evolution of a Glioblastoma, which includes an anisotropic nonlinear diffusion term with a diffusion velocity increasing with respect to vasculature. First, we prove the existence of global in time weak-strong solutions using a regularization technique via an artificial diffusion in the ODE-system and a fixed point argument. In addition, stability results of the critical points are given under some constraints on parameters. Finally, we design a fully discrete finite element scheme for the model which preserves the pointwise and energy estimates of the continuous problem.

MSC:

35Q92 PDEs in connection with biology, chemistry and other natural sciences
92C37 Cell biology
92C35 Physiological flow
35K10 Second-order parabolic equations
35D30 Weak solutions to PDEs
35D35 Strong solutions to PDEs
35B45 A priori estimates in context of PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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References:

[1] Amann, H., Highly degenerate quasilinear parabolic systems, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4), 18, 135-166 (1991) · Zbl 0738.35029
[2] Amann, H., Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, (Schmeisser, Hans-Jürgen; Triebel, Hans, Function Spaces, Differential Operators and Nonlinear Analysis (1993), Vieweg+Teubner Verlag: Vieweg+Teubner Verlag Wiesbaden), 9-126 · Zbl 0810.35037
[3] Baldock, A. L.; Ahn, S.; Rockne, R.; Johnston, S.; Neal, M.; Corwin, D.; Clark-Swanson, K.; Sterin, G.; Trister, A. D.; Malone, H.; Ebiana, V.; Sonabend, A. M.; Mrugala, M.; Rockhill, J. K.; Silbergeld, D. L.; Lai, A.; Cloughesy, T.; McKhann, G. M.; Bruce, J. N.; Rostomily, R. C.; Canoll, P.; Swanson, K. R., Patient-specific metrics of invasiveness reveal significant prognostic benefit of resection in a predictable subset of gliomas, PLoS ONE, 9, 1-10 (2014)
[4] Bitsouni, V.; Chaplain, M. A.J.; Eftimie, R., Mathematical modelling of cancer invasion: the multiple roles of TGF-β pathway on tumour proliferation and cell adhesion, Math. Models Methods Appl. Sci., 27, 1929-1962 (2017) · Zbl 1375.35564
[5] Ciarlet, P.; Raviart, P.-A., Maximum principle and uniform convergence for the finite element method, CMAME, 2, 17-31 (1973) · Zbl 0251.65069
[6] Cruz, E.; Negreanu, M.; Tello, J. I., Asymptotic behavior and global existence of solutions to a two-species chemotaxis system with two chemicals, Z. Angew. Math. Phys., 69, 20 (2018) · Zbl 1400.35025
[7] de Araujo, A. L.; de Magalhães, P. M., Existence of solutions and optimal control for a model of tissue invasion by solid tumours, J. Math. Anal. Appl., 421, 842-877 (2015) · Zbl 1323.92038
[8] Elliott, C. M.; Stuart, A. M., The global dynamics of discrete semilinear parabolic equations, SIAM J. Numer. Anal., 30, 1622-1663 (1993) · Zbl 0792.65066
[9] Evans, L. C., Partial Differential Equations, Graduate Studies in Mathematics (1998), American Mathematical Society · Zbl 0902.35002
[10] Faragó, I.; Karátson, J.; Korotov, S., Discrete maximum principles for nonlinear parabolic pde systems, IMA J. Numer. Anal., 32, 1541-1573 (2012) · Zbl 1258.65088
[11] Fernández-Romero, A.; Guillén-González, F.; Suárez, A., Theoretical analysis for a pde-ode system related to a glioblastoma tumor with vasculature, Z. Angew. Math. Phys., 72, 97 (2020) · Zbl 1464.35360
[12] A. Fernández-Romero, F. Guillén-González, A. Suárez, Determining parameters giving different growths of a new Glioblastoma differential model, (2021) 15, submitted.
[13] Guillén-González, F.; Gutiérrez-Santacreu, J., From a cell model with active motion to a Hele-Shaw-like system: a numerical approach, Numer. Math., 143, 107-137 (2019) · Zbl 1419.92004
[14] Klank, R. L.; Rosenfeld, S. S.; Odde, D. J., A Brownian dynamics tumor progression simulator with application to glioblastoma, Converg. Sci. Phys. Oncol., 4, 16 (2018)
[15] Kubo, A.; Tello, J. I., Mathematical analysis of a model of chemotaxis with competition terms, Differ. Integral Equ., 29, 441-454 (2016) · Zbl 1363.35041
[16] Lou, Y.; Tao, Y.; Winkler, M., Approaching the ideal free distribution in two-species competition models with fitness-dependent dispersal, SIAM J. Numer. Anal., 46, 1228-1262 (2014) · Zbl 1295.35098
[17] Martínez-González, A.; Durán-Prado, M.; Calvo, G. F.; Alcaín, F. J.; Pérez-Romasanta, L. A.; Pérez-García, V. M., Combined therapies of antithrombotics and antioxidants delay in silico brain tumour progression, Math. Med. Biol., 32, 239-262 (2015) · Zbl 1325.92048
[18] Martínez-González, A.; Calvo, G. F.; Pérez-Romasanta, L. A.; Pérez-García, V. M., Hypoxic cell waves around necrotic cores in glioblastoma: a biomathematical model and its therapeutic implications, Bull. Math. Biol., 74, 2875-2896 (2012) · Zbl 1264.92028
[19] Molina, D.; Pérez-Beteta, J.; Martínez-González, A.; Sepúlveda, J. M.; Peralta, S.; Gil-Gil, M. J.; Reynes, G.; Herrero, A.; Peñas, R. D.L.; Luque, R.; Capellades, J.; Balaña, C.; Pérez-García, V. M., Geometrical measures obtained from pretreatment postcontrast t1 weighted mris predict survival benefits from bevacizumab in glioblastoma patients, PLoS ONE, 11, 1-16 (2016)
[20] Nie, Y.-Y.; Thomée, V., A lumped mass finite-element method with quadrature for a non-linear parabolic problem, IMA J. Numer. Anal., 5, 371-396 (1985) · Zbl 0591.65079
[21] Pang, P. Y.H.; Wang, Y., Global boundedness of solutions to a chemotaxis-haptotaxis model with tissue remodeling, Math. Models Methods Appl. Sci., 28, 2211-2235 (2018) · Zbl 1416.35052
[22] Pérez-Beteta, J.; Belmonte-Beitia, J.; Pérez-García, V. M., Tumor width on t1-weighted mri images of glioblastoma as a prognostic biomarker: a mathematical model, Math. Model. Nat. Phenom., 15, 10 (2020) · Zbl 1467.92108
[23] Pérez-Beteta, J.; Martínez-González, A.; Molina, D.; Amo-Salas, M.; Luque, B.; Arregui, E.; Calvo, M.; Borrás, J. M.; López, C.; Claramonte, M.; Barcia, J. A.; Iglesias, L.; Avecillas, J.; Albillo, D.; Navarro, M.; Villanueva, J. M.; Paniagua, J. C.; Martino, J.; Velásquez, C.; Asenjo, B.; Benavides, M.; Herruzo, I.; del Carmen Delgado, M.; del Valle, A.; Falkov, A.; Schucht, P.; Arana, E.; Pérez-Romasanta, L.; Pérez-García, V. M., Glioblastoma: does the pre-treatment geometry matter? A postcontrast T1 MRI-based study, Eur. Radiol., 27, 1096-1104 (2016)
[24] Pérez-Beteta, J.; Molina-García, D.; Ortiz-Alhambra, J. A.; Fernández-Romero, A.; Luque, B.; Arregui, E.; Calvo, M.; Borrás, J. M.; Melédez, B.; Rodríguez de Lope, Á.; Moreno de la Presa, R.; Iglesias Bayo, L.; Barcia, J. A.; Martino, J.; Velásquez, C.; Asenjo, B.; Benavides, M.; Herruzo, I.; Revert, A.; Arana, E.; Pérez-García, V. M., Tumor surface regularity at MR imaging predicts survival and response to surgery in patients with glioblastoma, Radiology, 288, 218-225 (2018)
[25] Tello, J. I.; Wrzosek, D., Inter-species competition and chemorepulsion, J. Math. Anal. Appl., 459, 1233-1250 (2018) · Zbl 1381.92087
[26] Thomée, V., On positivity preservation in some finite element methods for the heat equation, Int. J. Numer. Math. Appl., 13-24 (2015) · Zbl 1360.65243
[27] Thomée, V.; Wahlbin, L. B., On the existence of maximum principles in parabolic finite element equations, Math. Comput., 77, 11-19 (2008) · Zbl 1128.65085
[28] Winkler, M.; Lou, Y., Advantage and disadvantage of dispersal in two-species competition models, CSIAM Trans. Appl. Math., 1, 86-103 (2020)
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