Open Access
Issue |
ESAIM: M2AN
Volume 56, Number 2, March-April 2022
|
|
---|---|---|
Page(s) | 407 - 431 | |
DOI | https://doi.org/10.1051/m2an/2022012 | |
Published online | 18 February 2022 |
- J.C.L. Alfonso, K. Talkenberger, M. Seifert, B. Klink, A. Hawkins-Daarud, K.R. Swanson, H. Hatzikirou and A. Deutsch, The biology and mathematical modelling of glioma invasion: A review. J. R. Soc. Interface 14 (2017) 20170490. [CrossRef] [PubMed] [Google Scholar]
- N.D. Alikakos, An application of the invariance principle to reaction-diffusion equations. J. Differ. Equ. 33 (1979) 201–225. [CrossRef] [Google Scholar]
- A.R.A. Anderson and M.A.J. Chaplain, Continuous and discrete mathematical models of tumor-induced angiogenesis. Bull. Math. Biol. 60 (1998) 857–899. [CrossRef] [Google Scholar]
- A. Baldock, R. Rockne, A. Boone, M. Neal, C. Bridge, L. Guyman, M. Mrugala, J. Rockhill, K.R. Swanson, A.D. Trister and A. Hawkins-Daarud, From patient-specific mathematical neuro-oncology to precision medicine. Front. Oncol. 3 (2013) 62. [CrossRef] [Google Scholar]
- N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues. Math. Models Methods Appl. Sci. 25 (2015) 1663–1763. [CrossRef] [MathSciNet] [Google Scholar]
- M.A.J. Chaplain, Mathematical modelling of angiogenesis. J. Neuro-Oncol. 50 (2000) 37–51. [CrossRef] [Google Scholar]
- P. Ciarlet and P.-A. Raviart, Maximum principle and uniform convergence for the finite element method. Comput. Methods Appl. Mech. Eng. 2 (1973) 17–31. [CrossRef] [Google Scholar]
- L. Corrias, B. Perthame and H. Zaag, Global solutions of some chemotaxis and angiogenesis systems in high space dimensions. Math. Models Methods Appl. Sci. 72 (2004) 1–28. [Google Scholar]
- A.L. de Araujo and P.M. de Magalhães, Existence of solutions and optimal control for a model of tissue invasion by solid tumours. J. Math. Anal. App. 421 (2015) 842–877. [CrossRef] [Google Scholar]
- H. Enderling and M.A.J. Chaplain, Mathematical modeling of tumor growth and treatment. Curr. Pharm. Des. 20 (2014) 4934–4940. [CrossRef] [Google Scholar]
- A. Fernández-Romero, F. Guillén-González and A. Suárez, Determining parameters giving different growths of a new Glioblastoma differential model. Preprint arXiv:2104.04560 (2021). [Google Scholar]
- A. Fernández-Romero, F. Guillén-González and A. Suárez, Theoretical and numerical analysis for a hybrid tumor model with diffusion depending on vasculature. J. Math. Anal. App. 503 (2021) 29. [Google Scholar]
- A. Friedman, Partial Differential Equations. Holt, Reinhart and Winston, New York (1969). [Google Scholar]
- A. Friedman and J.I. Tello, Stability of solutions of chemotaxis equations in reinforced random walks. J. Math. Anal. App. 272 (2002) 138–163. [CrossRef] [Google Scholar]
- T. Hillen and K.J. Painter, A users guide to PDE models for chemotaxis. J. Math. Biol. 58 (2009) 183–217. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
- D. Horstmann, From 1970 until present: The Keller–Segel model in chemotaxis and its consequences. Jahresber. Dtsch. Math.-Ver. 1052003 (1970) 103–165. [Google Scholar]
- R.L. Klank, S.S. Rosenfeld and D.J. Odde, A Brownian dynamics tumor progression simulator with application to glioblastoma. Converg. Sci. Phys. Oncol. 4 (2018) 015001. [CrossRef] [Google Scholar]
- J. Li and Z. Wang, Convergence to traveling waves of a singular PDE-ODE hybrid chemotaxis system in the half space. J. Differ. Equ. 268 (2020) 6940–6970. [CrossRef] [Google Scholar]
- G. Litcanu and C. Morales-Rodrigo, Asymptotic behavior of global solutions to a model of cell invasion. Math. Models Methods Appl. Sci. 20 (2010) 1721–1758. [CrossRef] [MathSciNet] [Google Scholar]
- A. Marciniak-Czochra and M. Ptashnyk, Boundedness of solutions of a haptotaxis model. Math. Models Methods. Appl. Sci. 20 (2010) 449–476. [CrossRef] [MathSciNet] [Google Scholar]
- A. Martínez-González, G.F. Calvo, L.A. Pérez-Romasanta and V.M. Pérez-García, Hypoxic cell waves around necrotic cores in glioblastoma: A mathematical model and its therapeutical implications. Bull. Math. Biol. 74 (2012) 2875–2896. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
- A. Martínez-González, M. Durán-Prado, G.F. Calvo, F.J. Alcaín, L.A. Pérez-Romasanta and V.M. Pérez-García, Combined therapies of antithrombotics and antioxidants delay in silico brain tumour progression. Math. Med. Biol. 32 (2015) 239–262. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
- D. Molina, L. Vera-Ramírez, J. Pérez-Beteta, E. Arana and V.M. Pérez-García, Prognostic models based on imaging findings in glioblastoma: Human versus Machine. Sci. Rep. 9 (2019) 5982. [CrossRef] [Google Scholar]
- M. Negreanu and J.I. Tello, On a parabolic-ODE system of chemotaxis. Disc. Cont. Dyn. Syst. Ser. S 13 (2020) 279–292. [Google Scholar]
- M. Negreanu, J.I. Tello and A.M. Vargas, A note on a periodic parabolic-ODE chemotaxis system. Appl. Math. Lett. 106 (2020) 106351. [CrossRef] [MathSciNet] [Google Scholar]
- Q.T. Ostrom, H. Gittleman, P. Liao, C. Rouse, Y. Chen, J. Dowling, Y. Wolinsky, C. Kruchko and J. Barnholtz-Sloan, CBTRUS statistical report: Primary brain and central nervous system tumors diagnosed in the united states in 2007–2011. Neuro-Oncol. 16 (2014) iv1–iv63. [CrossRef] [PubMed] [Google Scholar]
- J. Pérez-Beteta, A. Martínez-González, D. Molina, M. Amo-Salas, B. Luque, E. Arregui, M. Calvo, J.M. Borrás, C. López, M. Claramonte and J.A. Barcia, Glioblastoma: Does the pretreatment geometry matter? A postcontrast T1 MRI-based study. Eur. Radiol. 27 (2017) 163–169. [Google Scholar]
- J. Pérez-Beteta, D. Molina-García, J.A. Ortiz-Alhambra, A. Fernández-Romero, B. Luque, E. Arregui, M. Calvo, J.M. Borrás, B. Meléndez, A. Rodriguez de Lope and R. Moreno de la Presa, Tumor surface regularity at MR imaging predicts survival and response to surgery in patients with glioblastoma. Radiology 288 (2018) 218–225. [CrossRef] [PubMed] [Google Scholar]
- J. Pérez-Beteta, D. Molina-García, A. Martínez-González, A. Henares-Molina, M. Amo-Salas, B. Luque, E. Arregui, M. Calvo, J.M. Borrás, J. Martino and C. Velásquez, Morphological MRI-based features provide pretreatment survival prediction in glioblastoma. Eur. Radiol. 29 (2019) 1968–1977. [CrossRef] [PubMed] [Google Scholar]
- J. Pérez-Beteta, J. Belmonte-Beitia and V.M. Pérez-García, Tumor width on T1-weighted MRI images of glioblastoma as a prognostic biomarker: A mathematical model. Math. Model. Nat. Phenom. 15 (2020) 19. [CrossRef] [EDP Sciences] [Google Scholar]
- B. Perthame, PDE models for chemotactic movements: Parabolic, hyperbolic and kinetic. Appl. Math. 49 (2004) 539–564. [CrossRef] [MathSciNet] [Google Scholar]
- M. Protopapa, A. Zygogianni, G.S. Stamatakos, C. Antypas, C. Armpilia, N.K. Uzunoglu and V. Kouloulias, Clinical implications of in silico mathematical modeling for glioblastoma: A critical review. J. Neuro-Oncol. 136 (2018) 1–11. [CrossRef] [PubMed] [Google Scholar]
- B.D. Sleeman and H.A. Levine, A system of reaction diffusion equations arising in the theory of reinforced random walks. SIAM J. Appl. Math. 57 (1997) 683–730. [CrossRef] [MathSciNet] [Google Scholar]
- A. Stevens, The derivation of chemotaxis equations as limit dynamics of moderately interacting stochastic many-particle systems. SIAM J. Appl. Math. 61 (2000) 183–212. [CrossRef] [MathSciNet] [Google Scholar]
- A. Stevens and H.G. Othmer, Aggregation, blowup, and collapse: The ABC’s of taxis in reinforced random walks. SIAM J. Appl. Math. 57 (1997) 1044–1081. [CrossRef] [MathSciNet] [Google Scholar]
- Y. Tao and C. Cui, A density-dependent chemotaxis-haptotaxis system modeling cancer invasion. J. Math. Anal. App. 367 (2010) 612–624. [CrossRef] [Google Scholar]
- Y. Tao and M. Wang, A combined chemotaxis-haptotaxis system: The role of logistic source. SIAM J. Appl. Math. 41 (2009) 1533–1558. [CrossRef] [Google Scholar]
- Y. Tao and M. Winkler, A chemotaxis-haptotaxis model: The roles of nonlinear diffusion and logistic source. SIAM J. Appl. Math. 43 (2012) 685–704. [Google Scholar]
- J. Unkelbach, B.H. Menze, E. Konukoglu, F. Dittmann, M. Le, N. Ayache and H.A. Shih, Radiotherapy planning for glioblastoma based on a tumor growth model: Improving target volume delineation. Phys. Med. Biol. 59 (2014) 747–770. [CrossRef] [PubMed] [Google Scholar]
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.