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2013, 10(1): 185-198. doi: 10.3934/mbe.2013.10.185

A therapy inactivating the tumor angiogenic factors

Cristian Morales-Rodrigo 1,

1. 

Departamento de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, Calle Tar a s/n, 41012-Seville

Received  April 2012 Revised  September 2012 Published  December 2012

This paper is devoted to a nonlinear system of partial differential equations modeling the effect of an anti-angiogenic therapy based on an agent that binds to the tumor angiogenic factors. The main feature of the model under consideration is a nonlinear flux production of tumor angiogenic factors at the boundary of the tumor. It is proved the global existence for the nonlinear system and the effect in the large time behavior of the system for high doses of the therapeutic agent.
Keywords: Chemotaxis, anti-angiogenic therapy..
Mathematics Subject Classification: Primary: 92C17; Secondary: 35K57, 92C5.
Citation: Cristian Morales-Rodrigo. A therapy inactivating the tumor angiogenic factors. Mathematical Biosciences & Engineering, 2013, 10 (1) : 185-198. doi: 10.3934/mbe.2013.10.185
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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