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Attractors for the Navier-Stokes-Cahn-Hilliard system
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An oxygen driven proliferative-to-invasive transition of glioma cells: An analytical study
Università di Modena e Reggio Emilia, Dipartimento di Scienze Fisiche, Informatiche e Matematiche, Via Campi 213/B, I-41125 Modena, Italy |
Our aim in this paper is to analyze a model of glioma where oxygen drives cancer diffusion and proliferation. We prove the global well-posedness of the analytical problem and that, in the longtime, the illness does not disappear. Besides, the tumor dynamics increase the oxygen levels.
Addendum: "This research has been performed within the framework of the grant MIUR-PRIN 2020F3NCPX “Mathematics for industry 4.0 (Math4I4)”." is added under Fund Project. We apologize for any inconvenience this may cause.
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Interaction between astrocytes and neurons studied using a mathematical model of compartmentalized energy metabolism, J. Cereb. Blood Flow Metab., 25 (2005), 1476-1490.
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Analysis of a model for tumor growth and lactate exchanges in a glioma, Discrete Contin. Dyn. Syst. Ser. S, 14 (2021), 2729-2749.
doi: 10.3934/dcdss.2020457. |
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M. Conti, S. Gatti and A. Miranville, Mathematical analysis of a model for proliterative-to-invasive transition of hypoxic glioma cells, Nonlinear Anal., 189 (2019), 111572, 17 pp.
doi: 10.1016/j.na.2019.111572. |
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H. Gomez,
Quantitative analysis of the proliferative-to-invasive transition of hypoxic glioma cells, Integr. Biol., 9 (2017), 257-262.
doi: 10.1039/C6IB00208K. |
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P. Hartman, Ordinary Differential Equation, Corrected Reprint of the Second (1982) Edition, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002.
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L. Li,
On a coupled Cahn-Hilliard/Cahn-Hilliard model for the proliferative-to-invasive transition of hypoxic glioma cells, Commun. Pure Appl. Anal., 20 (2021), 1545-1557.
doi: 10.3934/cpaa.2021032. |
| [8] |
L. Li, L. Cherfils, A. Miranville and R. Guillevin,
A Cahn-Hilliard model with a proliferation term for the proliferative-to-invasive transition of hypoxic glioma cells, Commun. Math. Sci., 19 (2021), 1509-1532.
doi: 10.4310/CMS.2021.v19.n6.a3. |
| [9] |
L. Li, A. Miranville and R. Guillevin,
Cahn-Hilliard models for glial cells, Appl. Math. Optim., 84 (2021), 1821-1842.
doi: 10.1007/s00245-020-09696-x. |
| [10] |
L. Li, A. Miranville and R. Guillevin,
A coupled Cahn-Hilliard model for the proliferative-to-invasive transition of hypoxic glioma cells, Quart. Appl. Math., 79 (2021), 383-394.
doi: 10.1090/qam/1585. |
| [11] |
B. Mendoza-Juez, A. Martínez-González, G. F. Calvo and V. M. Peréz-García,
A mathematical model for the glucose-lactate metabolism of in vitro cancer cells, Bull. Math. Biol., 74 (2012), 1125-1142.
doi: 10.1007/s11538-011-9711-z. |
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A. Miranville, E. Rocca and G. Schimperna,
On the long time behavior of a tumor growth model, J. Differential Equations, 267 (2019), 2616-2642.
doi: 10.1016/j.jde.2019.03.028. |
| [13] |
B. Muz, P. de la Puente and F. Azab ande A. K. Azab,
The role of hypoxia in cancer progression, angiogenesis, metastasis, and resistance to therapy, Hypoxia, 3 (2015), 83-92.
doi: 10.2147/HP.S93413. |
show all references
References:
| [1] |
A. Aubert and R. Costalat,
Interaction between astrocytes and neurons studied using a mathematical model of compartmentalized energy metabolism, J. Cereb. Blood Flow Metab., 25 (2005), 1476-1490.
doi: 10.1038/sj.jcbfm.9600144. |
| [2] |
A. Aubert, R. Costalat, P. Magistretti, J. Pierre and L. Pellerin,
Brain lactate kinetics: Modeling evidence for neuronal lactate uptake upon activation, Proc. National Acad. Sci. USA, 102 (2005), 16448-16453.
doi: 10.1073/pnas.0505427102. |
| [3] |
L. Cherfils, S. Gatti, A. Miranville and R. Guillevin,
Analysis of a model for tumor growth and lactate exchanges in a glioma, Discrete Contin. Dyn. Syst. Ser. S, 14 (2021), 2729-2749.
doi: 10.3934/dcdss.2020457. |
| [4] |
M. Conti, S. Gatti and A. Miranville, Mathematical analysis of a model for proliterative-to-invasive transition of hypoxic glioma cells, Nonlinear Anal., 189 (2019), 111572, 17 pp.
doi: 10.1016/j.na.2019.111572. |
| [5] |
H. Gomez,
Quantitative analysis of the proliferative-to-invasive transition of hypoxic glioma cells, Integr. Biol., 9 (2017), 257-262.
doi: 10.1039/C6IB00208K. |
| [6] |
P. Hartman, Ordinary Differential Equation, Corrected Reprint of the Second (1982) Edition, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002.
doi: 10.1137/1.9780898719222. |
| [7] |
L. Li,
On a coupled Cahn-Hilliard/Cahn-Hilliard model for the proliferative-to-invasive transition of hypoxic glioma cells, Commun. Pure Appl. Anal., 20 (2021), 1545-1557.
doi: 10.3934/cpaa.2021032. |
| [8] |
L. Li, L. Cherfils, A. Miranville and R. Guillevin,
A Cahn-Hilliard model with a proliferation term for the proliferative-to-invasive transition of hypoxic glioma cells, Commun. Math. Sci., 19 (2021), 1509-1532.
doi: 10.4310/CMS.2021.v19.n6.a3. |
| [9] |
L. Li, A. Miranville and R. Guillevin,
Cahn-Hilliard models for glial cells, Appl. Math. Optim., 84 (2021), 1821-1842.
doi: 10.1007/s00245-020-09696-x. |
| [10] |
L. Li, A. Miranville and R. Guillevin,
A coupled Cahn-Hilliard model for the proliferative-to-invasive transition of hypoxic glioma cells, Quart. Appl. Math., 79 (2021), 383-394.
doi: 10.1090/qam/1585. |
| [11] |
B. Mendoza-Juez, A. Martínez-González, G. F. Calvo and V. M. Peréz-García,
A mathematical model for the glucose-lactate metabolism of in vitro cancer cells, Bull. Math. Biol., 74 (2012), 1125-1142.
doi: 10.1007/s11538-011-9711-z. |
| [12] |
A. Miranville, E. Rocca and G. Schimperna,
On the long time behavior of a tumor growth model, J. Differential Equations, 267 (2019), 2616-2642.
doi: 10.1016/j.jde.2019.03.028. |
| [13] |
B. Muz, P. de la Puente and F. Azab ande A. K. Azab,
The role of hypoxia in cancer progression, angiogenesis, metastasis, and resistance to therapy, Hypoxia, 3 (2015), 83-92.
doi: 10.2147/HP.S93413. |
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