In this work we analyse a PDE-ODE problem modelling the evolution of a Glioblastoma, which includes chemotaxis term directed to vasculature. First, we obtain some a priori estimates for the (possible) solutions of the model. In particular, under some conditions on the parameters, we obtain that the system does not develop blow-up at finite time. In addition, we design a fully discrete finite element scheme for the model which preserves some pointwise estimates of the continuous problem. Later, we make an adimensional study in order to reduce the number of parameters. Finally, we detect the main parameters determining different width of the ring formed by proliferative and necrotic cells and different regular/irregular behaviour of the tumor surface.
Keywords: Glioblastoma, chemotaxis, PDE-ODE system, numerical scheme
@article{M2AN_2022__56_2_407_0,
author = {Fern\'andez-Romero, Antonio and Guill\'en-Gonz\'alez, Francisco and Su\'arez, Antonio},
title = {A {Glioblastoma} {PDE-ODE} model including chemotaxis and vasculature},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {407--431},
year = {2022},
publisher = {EDP-Sciences},
volume = {56},
number = {2},
doi = {10.1051/m2an/2022012},
mrnumber = {4382752},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2022012/}
}
TY - JOUR AU - Fernández-Romero, Antonio AU - Guillén-González, Francisco AU - Suárez, Antonio TI - A Glioblastoma PDE-ODE model including chemotaxis and vasculature JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2022 SP - 407 EP - 431 VL - 56 IS - 2 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2022012/ DO - 10.1051/m2an/2022012 LA - en ID - M2AN_2022__56_2_407_0 ER -
%0 Journal Article %A Fernández-Romero, Antonio %A Guillén-González, Francisco %A Suárez, Antonio %T A Glioblastoma PDE-ODE model including chemotaxis and vasculature %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2022 %P 407-431 %V 56 %N 2 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2022012/ %R 10.1051/m2an/2022012 %G en %F M2AN_2022__56_2_407_0
Fernández-Romero, Antonio; Guillén-González, Francisco; Suárez, Antonio. A Glioblastoma PDE-ODE model including chemotaxis and vasculature. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 56 (2022) no. 2, pp. 407-431. doi: 10.1051/m2an/2022012
[1] , , , , , , and , The biology and mathematical modelling of glioma invasion: A review. J. R. Soc. Interface 14 (2017) 20170490.
[2] , An application of the invariance principle to reaction-diffusion equations. J. Differ. Equ. 33 (1979) 201–225. | MR | Zbl
[3] and , Continuous and discrete mathematical models of tumor-induced angiogenesis. Bull. Math. Biol. 60 (1998) 857–899. | Zbl
[4] , , , , , , , , , and , From patient-specific mathematical neuro-oncology to precision medicine. Front. Oncol. 3 (2013) 62.
[5] , , and , Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues. Math. Models Methods Appl. Sci. 25 (2015) 1663–1763. | MR
[6] , Mathematical modelling of angiogenesis. J. Neuro-Oncol. 50 (2000) 37–51.
[7] and , Maximum principle and uniform convergence for the finite element method. Comput. Methods Appl. Mech. Eng. 2 (1973) 17–31. | MR | Zbl
[8] , and , Global solutions of some chemotaxis and angiogenesis systems in high space dimensions. Math. Models Methods Appl. Sci. 72 (2004) 1–28. | MR | Zbl
[9] and , Existence of solutions and optimal control for a model of tissue invasion by solid tumours. J. Math. Anal. App. 421 (2015) 842–877. | MR | Zbl
[10] and , Mathematical modeling of tumor growth and treatment. Curr. Pharm. Des. 20 (2014) 4934–4940.
[11] , and , Determining parameters giving different growths of a new Glioblastoma differential model. Preprint (2021). | arXiv
[12] , and , Theoretical and numerical analysis for a hybrid tumor model with diffusion depending on vasculature. J. Math. Anal. App. 503 (2021) 29. | MR
[13] , Partial Differential Equations. Holt, Reinhart and Winston, New York (1969). | MR | Zbl
[14] and , Stability of solutions of chemotaxis equations in reinforced random walks. J. Math. Anal. App. 272 (2002) 138–163. | MR | Zbl
[15] and , A users guide to PDE models for chemotaxis. J. Math. Biol. 58 (2009) 183–217. | MR | Zbl
[16] , From 1970 until present: The Keller–Segel model in chemotaxis and its consequences. Jahresber. Dtsch. Math.-Ver. 1052003 (1970) 103–165. | MR | Zbl
[17] , and , A Brownian dynamics tumor progression simulator with application to glioblastoma. Converg. Sci. Phys. Oncol. 4 (2018) 015001.
[18] and , Convergence to traveling waves of a singular PDE-ODE hybrid chemotaxis system in the half space. J. Differ. Equ. 268 (2020) 6940–6970. | MR
[19] and , Asymptotic behavior of global solutions to a model of cell invasion. Math. Models Methods Appl. Sci. 20 (2010) 1721–1758. | MR | Zbl
[20] and , Boundedness of solutions of a haptotaxis model. Math. Models Methods. Appl. Sci. 20 (2010) 449–476. | MR | Zbl | DOI
[21] , , and , Hypoxic cell waves around necrotic cores in glioblastoma: A mathematical model and its therapeutical implications. Bull. Math. Biol. 74 (2012) 2875–2896. | MR | Zbl | DOI
[22] , , , , and , Combined therapies of antithrombotics and antioxidants delay in silico brain tumour progression. Math. Med. Biol. 32 (2015) 239–262. | MR | DOI
[23] , , , and , Prognostic models based on imaging findings in glioblastoma: Human versus Machine. Sci. Rep. 9 (2019) 5982. | DOI
[24] and , On a parabolic-ODE system of chemotaxis. Disc. Cont. Dyn. Syst. Ser. S 13 (2020) 279–292. | MR
[25] , and , A note on a periodic parabolic-ODE chemotaxis system. Appl. Math. Lett. 106 (2020) 106351. | MR | DOI
[26] , , , , , , , and , CBTRUS statistical report: Primary brain and central nervous system tumors diagnosed in the united states in 2007–2011. Neuro-Oncol. 16 (2014) iv1–iv63. | DOI
[27] , , , , , , , , , and , Glioblastoma: Does the pretreatment geometry matter? A postcontrast T1 MRI-based study. Eur. Radiol. 27 (2017) 163–169. | DOI
[28] , , , , , , , , , and , Tumor surface regularity at MR imaging predicts survival and response to surgery in patients with glioblastoma. Radiology 288 (2018) 218–225. | DOI
[29] , , , , , , , , , and , Morphological MRI-based features provide pretreatment survival prediction in glioblastoma. Eur. Radiol. 29 (2019) 1968–1977. | DOI
[30] , and , Tumor width on T1-weighted MRI images of glioblastoma as a prognostic biomarker: A mathematical model. Math. Model. Nat. Phenom. 15 (2020) 19. | MR | DOI
[31] , PDE models for chemotactic movements: Parabolic, hyperbolic and kinetic. Appl. Math. 49 (2004) 539–564. | MR | Zbl | DOI
[32] , , , , , and , Clinical implications of in silico mathematical modeling for glioblastoma: A critical review. J. Neuro-Oncol. 136 (2018) 1–11. | DOI
[33] and , A system of reaction diffusion equations arising in the theory of reinforced random walks. SIAM J. Appl. Math. 57 (1997) 683–730. | MR | Zbl | DOI
[34] , The derivation of chemotaxis equations as limit dynamics of moderately interacting stochastic many-particle systems. SIAM J. Appl. Math. 61 (2000) 183–212. | MR | Zbl | DOI
[35] and , Aggregation, blowup, and collapse: The ABC’s of taxis in reinforced random walks. SIAM J. Appl. Math. 57 (1997) 1044–1081. | MR | Zbl | DOI
[36] and , A density-dependent chemotaxis-haptotaxis system modeling cancer invasion. J. Math. Anal. App. 367 (2010) 612–624. | MR | Zbl | DOI
[37] and , A combined chemotaxis-haptotaxis system: The role of logistic source. SIAM J. Appl. Math. 41 (2009) 1533–1558. | MR | Zbl | DOI
[38] and , A chemotaxis-haptotaxis model: The roles of nonlinear diffusion and logistic source. SIAM J. Appl. Math. 43 (2012) 685–704. | MR | Zbl | DOI
[39] , , , , , and , Radiotherapy planning for glioblastoma based on a tumor growth model: Improving target volume delineation. Phys. Med. Biol. 59 (2014) 747–770. | DOI
Cité par Sources :
