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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, (1993), 9.
|
[2] |
H. Amann and J. López-Gómez, A priori bounds and multiple solutions for superlinear indefinite elliptic problems,, J. Differential Equations, 146 (1998), 336.
|
[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.
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. Google Scholar |
[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. Google Scholar |
[6] |
T. Cieślak and C. Morales-Rodrigo, Long-time behaviour of an angiogenesis model with flux at the tumor boundary,, Preprint, (). Google Scholar |
[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.
|
[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.
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.
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.
|
[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.
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.
|
[13] |
D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", Lecture Notes Math., (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.
doi: 10.1007/s002850000037. |
[15] |
D. Manoussaki, A mechanochemical model of angiogenesis and vasculogenesis,, ESAIM Math. Modelling Num. Anal., 37 (2003), 581.
|
[16] |
N. V. Mantzaris, S. Webb and H. G. Othmer, Mathematical modeling of tumor induced angiogenesis,, J. Math. Biol., 49 (2004), 111.
|
[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.
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.
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.
|
show all references
References:
[1] |
H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems,, in, (1993), 9.
|
[2] |
H. Amann and J. López-Gómez, A priori bounds and multiple solutions for superlinear indefinite elliptic problems,, J. Differential Equations, 146 (1998), 336.
|
[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.
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. Google Scholar |
[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. Google Scholar |
[6] |
T. Cieślak and C. Morales-Rodrigo, Long-time behaviour of an angiogenesis model with flux at the tumor boundary,, Preprint, (). Google Scholar |
[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.
|
[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.
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.
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.
|
[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.
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.
|
[13] |
D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", Lecture Notes Math., (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.
doi: 10.1007/s002850000037. |
[15] |
D. Manoussaki, A mechanochemical model of angiogenesis and vasculogenesis,, ESAIM Math. Modelling Num. Anal., 37 (2003), 581.
|
[16] |
N. V. Mantzaris, S. Webb and H. G. Othmer, Mathematical modeling of tumor induced angiogenesis,, J. Math. Biol., 49 (2004), 111.
|
[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.
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.
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.
|
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