August  2022, 15(8): 2233-2248. doi: 10.3934/dcdss.2022002

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

*Corresponding author: Stefania Gatti

The author is a member of GNAMPA (Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM(Istituto Nazionale di AltaMatematica)
Dedicated to Maurizio Grasselli on the occasion of his 60th birthday

Received  October 2021 Published  August 2022 Early access  January 2022

Fund Project: This research has been performed within the framework of the grant MIUR-PRIN 2020F3NCPX “Mathematics for industry 4.0 (Math4I4)”

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.

Citation: Stefania Gatti. An oxygen driven proliferative-to-invasive transition of glioma cells: An analytical study. Discrete and Continuous Dynamical Systems - S, 2022, 15 (8) : 2233-2248. doi: 10.3934/dcdss.2022002
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. AubertR. CostalatP. MagistrettiJ. 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. CherfilsS. GattiA. 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. LiL. CherfilsA. 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. LiA. 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. LiA. 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-JuezA. Martínez-GonzálezG. 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. MiranvilleE. 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. MuzP. 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. AubertR. CostalatP. MagistrettiJ. 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. CherfilsS. GattiA. 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. LiL. CherfilsA. 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. LiA. 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. LiA. 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-JuezA. Martínez-GonzálezG. 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. MiranvilleE. 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. MuzP. 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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