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November  2012, 32(11): 3871-3894. doi: 10.3934/dcds.2012.32.3871

Anti-angiogenic therapy based on the binding to receptors

Manuel Delgado 1, , Cristian Morales-Rodrigo 2, and  Antonio Suárez 3,

1. 

Departamento de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, Calle Tar a s/n, 41012-Seville, Spain
2. 

Departamento de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, Calle Tar a s/n, 41012-Seville, Spain
3. 

Dpto. de Ecuaciones Diferenciales y Análisis Numérico, Fac. de Matemáticas, Univ. de Sevilla, C/. Tarfia s/n, 41012 - Sevilla

Received  May 2011 Revised  September 2011 Published  June 2012

This paper deals with a nonlinear system of partial differential equations modeling the effect of an anti-angiogenic therapy based on an agent that binds to specific receptors of the endothelial cells. We study the time-dependent problem as well as the stationary problem associated to it.
Keywords: Chemotaxis, anti-angiogenic therapy, bifurcation..
Mathematics Subject Classification: Primary: 35K45, 35K57, 92C1.
Citation: Manuel Delgado, Cristian Morales-Rodrigo, Antonio Suárez. Anti-angiogenic therapy based on the binding to receptors. Discrete & Continuous Dynamical Systems, 2012, 32 (11) : 3871-3894. doi: 10.3934/dcds.2012.32.3871
References:
[1]

H. Amann, Nonlinear elliptic equations with nonlinear boundary conditions, in "New Developments in Differential Equations" (Proc. 2$^nd$ Scheveningen Conf., Scheveningen, 1975) (ed. W. Eckhaus), North-Holland Math. Studies, 21, North-Holland, Amsterdam, (1976), 43-63.  Google Scholar

[2]

H. Amann, Maximum principles and principal eigenvalues, in "Ten Mathematical Essays on Approximation in Analysis and Topology" (eds. J. Ferrera, J. López-Gómez and F. R. Ruíz del Portal), Elsevier B. V., Amsterdam, (2005), 1-60.  Google Scholar

[3]

H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in "Function Spaces, Differential Operators and Nonlinear Analysis" (Friedrichroda, 1992) (eds. H. J. Schmeisser and H. Triebel), Teubner-Texte Math., 133, Teubner, Stuttgart, (1993), 9-126.  Google Scholar

[4]

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.  Google Scholar

[5]

H. Berestycki, L. Nirenberg and S. R. S. Varadhan, The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math., 47 (1994), 47-92. doi: 10.1002/cpa.3160470105.  Google Scholar

[6]

S. Cano-Casanova and J. López-Gómez, Properties of the principal eigenvalues of a general class of non-classical mixed boundary value problems, J. Differential Equations, 178 (2002), 123-211. doi: 10.1006/jdeq.2000.4003.  Google Scholar

[7]

R. S. Cantrell and C. Cosner, "Spatial Ecology Via Reaction-Diffusion Equations," Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, Ltd., Chichester, 2003.  Google Scholar

[8]

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. doi: 10.1016/0895-7177(96)00019-2.  Google Scholar

[9]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340.  Google Scholar

[10]

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. doi: 10.1016/j.na.2009.06.057.  Google Scholar

[11]

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.  Google Scholar

[12]

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. doi: 10.1016/j.jde.2007.12.007.  Google Scholar

[13]

J. Dyson, E. Sánchez, R. Villella-Bressan and G. Webb, An age and spatially structured model of tumor invasion with haptotaxis, Discrete Contin. Dyn. Syst. Ser B, 8 (2007), 45-60.  Google Scholar

[14]

H. Enderling, A. R. A. Anderson, M. A. J. Chaplain, A. J. Munro and J. S. Vaidya, Mathematical modelling of radiotherapy strategies for early breast cancer, J. Theor. Biol., 241 (2006), 158-171. doi: 10.1016/j.jtbi.2005.11.015.  Google Scholar

[15]

H. Enderling, M. A. J. Chaplain, A. R. A. Anderson and J. S. Vaidya, A mathematical model of breast cancer development, local treatment and recurrence, J. Theor. Biol., 246 (2007), 245-259. doi: 10.1016/j.jtbi.2006.12.010.  Google Scholar

[16]

M. A. Fontelos, A. Friedman and B. Hu, Mathematical analysis of a model for the initiation of angiogenesis, SIAM J. Math. Anal., 33 (2002), 1330-1355. doi: 10.1137/S0036141001385046.  Google Scholar

[17]

J. García-Melián, J. D. Rossi and J. Sabina de Lis, Existence and uniqueness of positive solutions to elliptic problems with sublinear mixed boundary conditions, Comm. Comtemporary Math. 11 (2009), 585-613.  Google Scholar

[18]

D. Henry, "Geometric Theory Of Semilinear Parabolic Equations," Lecture Notes Math., 840, Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[19]

T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3.  Google Scholar

[20]

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.  Google Scholar

[21]

J. López-Gómez, Nonlinear eigenvalues and global bifurcation: Application to the search of positive solutions for general Lotka-Volterra reaction-diffusion systems with two species, Diff. Int. Eqns., 7 (1994), 1427-1452.  Google Scholar

[22]

J. López-Gómez, Classifying smooth supersolutions for a general class of elliptic boundary value problems, Adv. Diff. Eqns., 8 (2003), 1025-1042.  Google Scholar

[23]

N. V. Mantzaris, S. Webb and H. G. Othmer, Mathematical modeling of tumor-induced angiogenesis, J. Math. Bio., 49 (2004), 111-187. doi: 10.1007/s00285-003-0262-2.  Google Scholar

[24]

C. Walker and G. Webb, Global existence of classical solutions for a haptotaxis model,, SIAM J. Math. Anal., 38 (): 1694.   Google Scholar

[25]

M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905. doi: 10.1016/j.jde.2010.02.008.  Google Scholar

show all references

References:
[1]

H. Amann, Nonlinear elliptic equations with nonlinear boundary conditions, in "New Developments in Differential Equations" (Proc. 2$^nd$ Scheveningen Conf., Scheveningen, 1975) (ed. W. Eckhaus), North-Holland Math. Studies, 21, North-Holland, Amsterdam, (1976), 43-63.  Google Scholar

[2]

H. Amann, Maximum principles and principal eigenvalues, in "Ten Mathematical Essays on Approximation in Analysis and Topology" (eds. J. Ferrera, J. López-Gómez and F. R. Ruíz del Portal), Elsevier B. V., Amsterdam, (2005), 1-60.  Google Scholar

[3]

H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in "Function Spaces, Differential Operators and Nonlinear Analysis" (Friedrichroda, 1992) (eds. H. J. Schmeisser and H. Triebel), Teubner-Texte Math., 133, Teubner, Stuttgart, (1993), 9-126.  Google Scholar

[4]

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.  Google Scholar

[5]

H. Berestycki, L. Nirenberg and S. R. S. Varadhan, The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math., 47 (1994), 47-92. doi: 10.1002/cpa.3160470105.  Google Scholar

[6]

S. Cano-Casanova and J. López-Gómez, Properties of the principal eigenvalues of a general class of non-classical mixed boundary value problems, J. Differential Equations, 178 (2002), 123-211. doi: 10.1006/jdeq.2000.4003.  Google Scholar

[7]

R. S. Cantrell and C. Cosner, "Spatial Ecology Via Reaction-Diffusion Equations," Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, Ltd., Chichester, 2003.  Google Scholar

[8]

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. doi: 10.1016/0895-7177(96)00019-2.  Google Scholar

[9]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340.  Google Scholar

[10]

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. doi: 10.1016/j.na.2009.06.057.  Google Scholar

[11]

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.  Google Scholar

[12]

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. doi: 10.1016/j.jde.2007.12.007.  Google Scholar

[13]

J. Dyson, E. Sánchez, R. Villella-Bressan and G. Webb, An age and spatially structured model of tumor invasion with haptotaxis, Discrete Contin. Dyn. Syst. Ser B, 8 (2007), 45-60.  Google Scholar

[14]

H. Enderling, A. R. A. Anderson, M. A. J. Chaplain, A. J. Munro and J. S. Vaidya, Mathematical modelling of radiotherapy strategies for early breast cancer, J. Theor. Biol., 241 (2006), 158-171. doi: 10.1016/j.jtbi.2005.11.015.  Google Scholar

[15]

H. Enderling, M. A. J. Chaplain, A. R. A. Anderson and J. S. Vaidya, A mathematical model of breast cancer development, local treatment and recurrence, J. Theor. Biol., 246 (2007), 245-259. doi: 10.1016/j.jtbi.2006.12.010.  Google Scholar

[16]

M. A. Fontelos, A. Friedman and B. Hu, Mathematical analysis of a model for the initiation of angiogenesis, SIAM J. Math. Anal., 33 (2002), 1330-1355. doi: 10.1137/S0036141001385046.  Google Scholar

[17]

J. García-Melián, J. D. Rossi and J. Sabina de Lis, Existence and uniqueness of positive solutions to elliptic problems with sublinear mixed boundary conditions, Comm. Comtemporary Math. 11 (2009), 585-613.  Google Scholar

[18]

D. Henry, "Geometric Theory Of Semilinear Parabolic Equations," Lecture Notes Math., 840, Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[19]

T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3.  Google Scholar

[20]

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.  Google Scholar

[21]

J. López-Gómez, Nonlinear eigenvalues and global bifurcation: Application to the search of positive solutions for general Lotka-Volterra reaction-diffusion systems with two species, Diff. Int. Eqns., 7 (1994), 1427-1452.  Google Scholar

[22]

J. López-Gómez, Classifying smooth supersolutions for a general class of elliptic boundary value problems, Adv. Diff. Eqns., 8 (2003), 1025-1042.  Google Scholar

[23]

N. V. Mantzaris, S. Webb and H. G. Othmer, Mathematical modeling of tumor-induced angiogenesis, J. Math. Bio., 49 (2004), 111-187. doi: 10.1007/s00285-003-0262-2.  Google Scholar

[24]

C. Walker and G. Webb, Global existence of classical solutions for a haptotaxis model,, SIAM J. Math. Anal., 38 (): 1694.   Google Scholar

[25]

M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905. doi: 10.1016/j.jde.2010.02.008.  Google Scholar

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