American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Genome characterization through dichotomic classes: An analysis of the whole chromosome 1 of A. thaliana
  • MBE Home
  • This Issue
  • Next Article
    A structural model of the VEGF signalling pathway: Emergence of robustness and redundancy properties
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, (1993), 9.   Google Scholar

[2]

H. Amann and J. López-Gómez, A priori bounds and multiple solutions for superlinear indefinite elliptic problems,, J. Differential Equations, 146 (1998), 336.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[13]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", Lecture Notes Math., (1981).   Google Scholar

[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.  Google Scholar

[15]

D. Manoussaki, A mechanochemical model of angiogenesis and vasculogenesis,, ESAIM Math. Modelling Num. Anal., 37 (2003), 581.   Google Scholar

[16]

N. V. Mantzaris, S. Webb and H. G. Othmer, Mathematical modeling of tumor induced angiogenesis,, J. Math. Biol., 49 (2004), 111.   Google Scholar

[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.  Google Scholar

[18]

R. M. Sutherland, Cell and environment interactions in tumor microregions: The multicell spheroid model,, Science, 240 (1988), 177.  doi: 10.1126/science.2451290.  Google Scholar

[19]

M. Winkler, Aggregation vs global diffusive behavior in the higher-dimensional Keller-Segel model,, J. Differential Equations, 248 (2010), 2889.   Google Scholar

show all references

References:
[1]

H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems,, in, (1993), 9.   Google Scholar

[2]

H. Amann and J. López-Gómez, A priori bounds and multiple solutions for superlinear indefinite elliptic problems,, J. Differential Equations, 146 (1998), 336.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[13]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", Lecture Notes Math., (1981).   Google Scholar

[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.  Google Scholar

[15]

D. Manoussaki, A mechanochemical model of angiogenesis and vasculogenesis,, ESAIM Math. Modelling Num. Anal., 37 (2003), 581.   Google Scholar

[16]

N. V. Mantzaris, S. Webb and H. G. Othmer, Mathematical modeling of tumor induced angiogenesis,, J. Math. Biol., 49 (2004), 111.   Google Scholar

[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.  Google Scholar

[18]

R. M. Sutherland, Cell and environment interactions in tumor microregions: The multicell spheroid model,, Science, 240 (1988), 177.  doi: 10.1126/science.2451290.  Google Scholar

[19]

M. Winkler, Aggregation vs global diffusive behavior in the higher-dimensional Keller-Segel model,, J. Differential Equations, 248 (2010), 2889.   Google Scholar

[1]

Manuel Delgado, Cristian Morales-Rodrigo, Antonio Suárez. Anti-angiogenic therapy based on the binding to receptors. Discrete & Continuous Dynamical Systems - A, 2012, 32 (11) : 3871-3894. doi: 10.3934/dcds.2012.32.3871
[2]

Maciej Leszczyński, Urszula Ledzewicz, Heinz Schättler. Optimal control for a mathematical model for anti-angiogenic treatment with Michaelis-Menten pharmacodynamics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2315-2334. doi: 10.3934/dcdsb.2019097
[3]

Filippo Cacace, Valerio Cusimano, Alfredo Germani, Pasquale Palumbo, Federico Papa. Closed-loop control of tumor growth by means of anti-angiogenic administration. Mathematical Biosciences & Engineering, 2018, 15 (4) : 827-839. doi: 10.3934/mbe.2018037
[4]

Nasser Sweilam, Fathalla Rihan, Seham AL-Mekhlafi. A fractional-order delay differential model with optimal control for cancer treatment based on synergy between anti-angiogenic and immune cell therapies. Discrete & Continuous Dynamical Systems - S, 2020, 13 (9) : 2403-2424. doi: 10.3934/dcdss.2020120
[5]

Reihaneh Mostolizadeh, Zahra Afsharnezhad, Anna Marciniak-Czochra. Mathematical model of Chimeric Anti-gene Receptor (CAR) T cell therapy with presence of cytokine. Numerical Algebra, Control & Optimization, 2018, 8 (1) : 63-80. doi: 10.3934/naco.2018004
[6]

John D. Nagy, Dieter Armbruster. Evolution of uncontrolled proliferation and the angiogenic switch in cancer. Mathematical Biosciences & Engineering, 2012, 9 (4) : 843-876. doi: 10.3934/mbe.2012.9.843
[7]

Urszula Ledzewicz, James Munden, Heinz Schättler. Scheduling of angiogenic inhibitors for Gompertzian and logistic tumor growth models. Discrete & Continuous Dynamical Systems - B, 2009, 12 (2) : 415-438. doi: 10.3934/dcdsb.2009.12.415
[8]

Baba Issa Camara, Houda Mokrani, Evans K. Afenya. Mathematical modeling of glioma therapy using oncolytic viruses. Mathematical Biosciences & Engineering, 2013, 10 (3) : 565-578. doi: 10.3934/mbe.2013.10.565
[9]

Danthai Thongphiew, Vira Chankong, Fang-Fang Yin, Q. Jackie Wu. An on-line adaptive radiation therapy system for intensity modulated radiation therapy: An application of multi-objective optimization. Journal of Industrial & Management Optimization, 2008, 4 (3) : 453-475. doi: 10.3934/jimo.2008.4.453
[10]

Jianjun Paul Tian. Finite-time perturbations of dynamical systems and applications to tumor therapy. Discrete & Continuous Dynamical Systems - B, 2009, 12 (2) : 469-479. doi: 10.3934/dcdsb.2009.12.469
[11]

Urszula Ledzewicz, Helen Moore. Optimal control applied to a generalized Michaelis-Menten model of CML therapy. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 331-346. doi: 10.3934/dcdsb.2018022
[12]

Avner Friedman, Xiulan Lai. Free boundary problems associated with cancer treatment by combination therapy. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 6825-6842. doi: 10.3934/dcds.2019233
[13]

Rachid Ouifki, Gareth Witten. A model of HIV-1 infection with HAART therapy and intracellular delays. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 229-240. doi: 10.3934/dcdsb.2007.8.229
[14]

Avner Friedman, Xiulan Lai. Antagonism and negative side-effects in combination therapy for cancer. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2237-2250. doi: 10.3934/dcdsb.2019093
[15]

Ben Sheller, Domenico D'Alessandro. Analysis of a cancer dormancy model and control of immuno-therapy. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1037-1053. doi: 10.3934/mbe.2015.12.1037
[16]

Harsh Vardhan Jain, Avner Friedman. Modeling prostate cancer response to continuous versus intermittent androgen ablation therapy. Discrete & Continuous Dynamical Systems - B, 2013, 18 (4) : 945-967. doi: 10.3934/dcdsb.2013.18.945
[17]

Chunpeng Wang, Jingxue Yin, Bibo Lu. Anti-shifting phenomenon of a convective nonlinear diffusion equation. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 1211-1236. doi: 10.3934/dcdsb.2010.14.1211
[18]

Sergiu Aizicovici, Simeon Reich. Anti-periodic solutions to a class of non-monotone evolution equations. Discrete & Continuous Dynamical Systems - A, 1999, 5 (1) : 35-42. doi: 10.3934/dcds.1999.5.35
[19]

Hengki Tasman, Edy Soewono, Kuntjoro Adji Sidarto, Din Syafruddin, William Oscar Rogers. A model for transmission of partial resistance to anti-malarial drugs. Mathematical Biosciences & Engineering, 2009, 6 (3) : 649-661. doi: 10.3934/mbe.2009.6.649
[20]

Urszula Ledzewicz, Heinz Schättler. On the optimality of singular controls for a class of mathematical models for tumor anti-angiogenesis. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 691-715. doi: 10.3934/dcdsb.2009.11.691

2018 Impact Factor: 1.313

Tools

Metrics

  • PDF downloads (16)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences