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A structural model of the VEGF signalling pathway: Emergence of robustness and redundancy properties
A therapy inactivating the tumor angiogenic factors
1. | Departamento de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, Calle Tara s/n, 41012-Seville |
References:
[1] |
H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in "Function Spaces, Differential Operators and Nonlinear Analysis" (editors, H. J. Schmeisser and H. Triebel), Teubner, Stuttgart, Leipzig, (1993), 9-126. |
[2] |
H. Amann and J. López-Gómez, A priori bounds and multiple solutions for superlinear indefinite elliptic problems, J. Differential Equations, 146 (1998), 336-374. |
[3] |
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.
doi: 10.1006/bulm.1998.0042. |
[4] |
M. A. J. Chaplain and A. M. Stuart, A model mechanism for the chemotactic response of endothelial cells to tumour angiogenesis factor, IMA J. Math. Appl. Med. Biol., 10 (1993), 149-168. |
[5] |
M. A. J. Chaplain, Avascular growth, angiogenesis and vascular growth in solid tumours: the mathematical modelling of the stages of tumor development, Math. Comput. Modelling, 23 (1996), 47-87. |
[6] |
T. Cieślak and C. Morales-Rodrigo, Long-time behaviour of an angiogenesis model with flux at the tumor boundary, Preprint, arXiv:1202.4695. |
[7] |
M. Delgado, I. Gayte, C. Morales-Rodrigo and A. Suárez, An angiogenesis model with nonlinear chemotactic response and flux at the tumor boundary, Nonlinear Anal., 72 (2010), 330-347. |
[8] |
M. Delgado, C. Morales-Rodrigo and A. Suárez, Anti-angiogenic therapy based on the binding receptors, Discrete Contin. Dyn. Syst. Ser A, 32 (2012), 3871-3894.
doi: 10.3934/dcds.2012.32.3871. |
[9] |
M. Delgado, C. Morales-Rodrigo, A. Suárez and J. I. Tello, On a parabolic-elliptic chemotactic model with coupled boundary conditions, Nonlinear Analysis RWA, 11 (2010), 3884-3902.
doi: 10.1016/j.nonrwa.2010.02.016. |
[10] |
M. Delgado and A. Suárez, Study of an elliptic system arising from angiogenesis with chemotaxis and flux at the boundary, J. Differential Equations, 244 (2008), 3119-3150. |
[11] |
M. A. Fontelos, A. Friedman and B. Hu, Mathematical analysis of a model for the initiation of angiogenesis, SIAM J. Math. Anal., 33 (2002), 1330-1350.
doi: 10.1137/S0036141001385046. |
[12] |
J. García-Melián, J. D. Rossi and J. Sabina, Existence and uniqueness of positive solutions to elliptic problems with sublinear mixed boundary conditions, Comm. Comtemporary Math., 11 (2009), 585-613. |
[13] |
D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes Math., 840, Springer 1981. |
[14] |
H. A. Levine, B. D. Sleeman and M. Nilsen-Hamilton, Mathematical modeling of the onset of capillary formation initiating angiogenesis, J. Math. Biol., 42 (2001), 195-238.
doi: 10.1007/s002850000037. |
[15] |
D. Manoussaki, A mechanochemical model of angiogenesis and vasculogenesis, ESAIM Math. Modelling Num. Anal., 37 (2003), 581-599. |
[16] |
N. V. Mantzaris, S. Webb and H. G. Othmer, Mathematical modeling of tumor induced angiogenesis, J. Math. Biol., 49 (2004), 111-187. |
[17] |
M. Owen, T. Alarcón, P. K. Maini and H. M. Byrne, Angiogenesis and vascular remodelling in normal and cancerous tissues, J. Math. Biol., 58 (2009), 689-721.
doi: 10.1007/s00285-008-0213-z. |
[18] |
R. M. Sutherland, Cell and environment interactions in tumor microregions: The multicell spheroid model, Science, 240 (1988), 177-184.
doi: 10.1126/science.2451290. |
[19] |
M. Winkler, Aggregation vs global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905. |
show all references
References:
[1] |
H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in "Function Spaces, Differential Operators and Nonlinear Analysis" (editors, H. J. Schmeisser and H. Triebel), Teubner, Stuttgart, Leipzig, (1993), 9-126. |
[2] |
H. Amann and J. López-Gómez, A priori bounds and multiple solutions for superlinear indefinite elliptic problems, J. Differential Equations, 146 (1998), 336-374. |
[3] |
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.
doi: 10.1006/bulm.1998.0042. |
[4] |
M. A. J. Chaplain and A. M. Stuart, A model mechanism for the chemotactic response of endothelial cells to tumour angiogenesis factor, IMA J. Math. Appl. Med. Biol., 10 (1993), 149-168. |
[5] |
M. A. J. Chaplain, Avascular growth, angiogenesis and vascular growth in solid tumours: the mathematical modelling of the stages of tumor development, Math. Comput. Modelling, 23 (1996), 47-87. |
[6] |
T. Cieślak and C. Morales-Rodrigo, Long-time behaviour of an angiogenesis model with flux at the tumor boundary, Preprint, arXiv:1202.4695. |
[7] |
M. Delgado, I. Gayte, C. Morales-Rodrigo and A. Suárez, An angiogenesis model with nonlinear chemotactic response and flux at the tumor boundary, Nonlinear Anal., 72 (2010), 330-347. |
[8] |
M. Delgado, C. Morales-Rodrigo and A. Suárez, Anti-angiogenic therapy based on the binding receptors, Discrete Contin. Dyn. Syst. Ser A, 32 (2012), 3871-3894.
doi: 10.3934/dcds.2012.32.3871. |
[9] |
M. Delgado, C. Morales-Rodrigo, A. Suárez and J. I. Tello, On a parabolic-elliptic chemotactic model with coupled boundary conditions, Nonlinear Analysis RWA, 11 (2010), 3884-3902.
doi: 10.1016/j.nonrwa.2010.02.016. |
[10] |
M. Delgado and A. Suárez, Study of an elliptic system arising from angiogenesis with chemotaxis and flux at the boundary, J. Differential Equations, 244 (2008), 3119-3150. |
[11] |
M. A. Fontelos, A. Friedman and B. Hu, Mathematical analysis of a model for the initiation of angiogenesis, SIAM J. Math. Anal., 33 (2002), 1330-1350.
doi: 10.1137/S0036141001385046. |
[12] |
J. García-Melián, J. D. Rossi and J. Sabina, Existence and uniqueness of positive solutions to elliptic problems with sublinear mixed boundary conditions, Comm. Comtemporary Math., 11 (2009), 585-613. |
[13] |
D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes Math., 840, Springer 1981. |
[14] |
H. A. Levine, B. D. Sleeman and M. Nilsen-Hamilton, Mathematical modeling of the onset of capillary formation initiating angiogenesis, J. Math. Biol., 42 (2001), 195-238.
doi: 10.1007/s002850000037. |
[15] |
D. Manoussaki, A mechanochemical model of angiogenesis and vasculogenesis, ESAIM Math. Modelling Num. Anal., 37 (2003), 581-599. |
[16] |
N. V. Mantzaris, S. Webb and H. G. Othmer, Mathematical modeling of tumor induced angiogenesis, J. Math. Biol., 49 (2004), 111-187. |
[17] |
M. Owen, T. Alarcón, P. K. Maini and H. M. Byrne, Angiogenesis and vascular remodelling in normal and cancerous tissues, J. Math. Biol., 58 (2009), 689-721.
doi: 10.1007/s00285-008-0213-z. |
[18] |
R. M. Sutherland, Cell and environment interactions in tumor microregions: The multicell spheroid model, Science, 240 (1988), 177-184.
doi: 10.1126/science.2451290. |
[19] |
M. Winkler, Aggregation vs global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905. |
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