2005, 2(3): 511-525. doi: 10.3934/mbe.2005.2.511

Critical-Point Analysis For Three-Variable Cancer Angiogenesis Models


Institute of Applied Mathematics, Informatics and Mechanics, Warsaw University, Banacha 2, 02-097 Warsaw, Poland


Institute for Medical BioMathematics, 10 Hate'ena St., POB 282, Bene Ataroth, Israel, Israel

Received  January 2005 Revised  July 2005 Published  August 2005

We perform critical-point analysis for three-variable systems that represent essential processes of the growth of the angiogenic tumor, namely, tumor growth, vascularization, and generation of angiogenic factor (protein) as a function of effective vessel density. Two models that describe tumor growth depending on vascular mass and regulation of new vessel formation through a key angiogenic factor are explored. The first model is formulated in terms of ODEs, while the second assumes delays in this regulation, thus leading to a system of DDEs. In both models, the only nontrivial critical point is always unstable, while one of the trivial critical points is always stable. The models predict unlimited growth, if the initial condition is close enough to the nontrivial critical point, and this growth may be characterized by oscillations in tumor and vascular mass. We suggest that angiogenesis per se does not suffice for explaining the observed stabilization of vascular tumor size.
Citation: Urszula Foryś, Yuri Kheifetz, Yuri Kogan. Critical-Point Analysis For Three-Variable Cancer Angiogenesis Models. Mathematical Biosciences & Engineering, 2005, 2 (3) : 511-525. doi: 10.3934/mbe.2005.2.511

V. Lanza, D. Ambrosi, L. Preziosi. Exogenous control of vascular network formation in vitro: a mathematical model. Networks & Heterogeneous Media, 2006, 1 (4) : 621-637. doi: 10.3934/nhm.2006.1.621


Zejia Wang, Haihua Zhou, Huijuan Song. The impact of time delay and angiogenesis in a tumor model. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021219


Yaodan Huang, Zhengce Zhang, Bei Hu. Bifurcation from stability to instability for a free boundary tumor model with angiogenesis. Discrete & Continuous Dynamical Systems, 2019, 39 (5) : 2473-2510. doi: 10.3934/dcds.2019105


Telma Silva, Adélia Sequeira, Rafael F. Santos, Jorge Tiago. Existence, uniqueness, stability and asymptotic behavior of solutions for a mathematical model of atherosclerosis. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 343-362. doi: 10.3934/dcdss.2016.9.343


Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367-376. doi: 10.3934/proc.2009.2009.367


Marek Bodnar, Monika Joanna Piotrowska, Urszula Foryś, Ewa Nizińska. Model of tumour angiogenesis -- analysis of stability with respect to delays. Mathematical Biosciences & Engineering, 2013, 10 (1) : 19-35. doi: 10.3934/mbe.2013.10.19


Keng Deng, Yixiang Wu. Asymptotic behavior for a reaction-diffusion population model with delay. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 385-395. doi: 10.3934/dcdsb.2015.20.385


Xiaoming Zheng, Gou Young Koh, Trachette Jackson. A continuous model of angiogenesis: Initiation, extension, and maturation of new blood vessels modulated by vascular endothelial growth factor, angiopoietins, platelet-derived growth factor-B, and pericytes. Discrete & Continuous Dynamical Systems - B, 2013, 18 (4) : 1109-1154. doi: 10.3934/dcdsb.2013.18.1109


Zhipeng Qiu, Jun Yu, Yun Zou. The asymptotic behavior of a chemostat model. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 721-727. doi: 10.3934/dcdsb.2004.4.721


Zvia Agur, L. Arakelyan, P. Daugulis, Y. Ginosar. Hopf point analysis for angiogenesis models. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 29-38. doi: 10.3934/dcdsb.2004.4.29


Zhao-Xing Yang, Guo-Bao Zhang, Ge Tian, Zhaosheng Feng. Stability of non-monotone non-critical traveling waves in discrete reaction-diffusion equations with time delay. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 581-603. doi: 10.3934/dcdss.2017029


Jacopo De Simoi. Stability and instability results in a model of Fermi acceleration. Discrete & Continuous Dynamical Systems, 2009, 25 (3) : 719-750. doi: 10.3934/dcds.2009.25.719


Urszula Ledzewicz, Helmut Maurer, Heinz Schättler. Optimal and suboptimal protocols for a mathematical model for tumor anti-angiogenesis in combination with chemotherapy. Mathematical Biosciences & Engineering, 2011, 8 (2) : 307-323. doi: 10.3934/mbe.2011.8.307


Hunseok Kang. Asymptotic behavior of a discrete turing model. Discrete & Continuous Dynamical Systems, 2010, 27 (1) : 265-284. doi: 10.3934/dcds.2010.27.265


Yuncherl Choi, Taeyoung Ha, Jongmin Han, Sewoong Kim, Doo Seok Lee. Turing instability and dynamic phase transition for the Brusselator model with multiple critical eigenvalues. Discrete & Continuous Dynamical Systems, 2021, 41 (9) : 4255-4281. doi: 10.3934/dcds.2021035


Zhi-An Wang. Wavefront of an angiogenesis model. Discrete & Continuous Dynamical Systems - B, 2012, 17 (8) : 2849-2860. doi: 10.3934/dcdsb.2012.17.2849


Mostafa Adimy, Fabien Crauste. Modeling and asymptotic stability of a growth factor-dependent stem cell dynamics model with distributed delay. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 19-38. doi: 10.3934/dcdsb.2007.8.19


Meng Liu, Chuanzhi Bai, Yi Jin. Population dynamical behavior of a two-predator one-prey stochastic model with time delay. Discrete & Continuous Dynamical Systems, 2017, 37 (5) : 2513-2538. doi: 10.3934/dcds.2017108


Chaoying Li, Xiaojing Xu, Zhuan Ye. On long-time asymptotic behavior for solutions to 2D temperature-dependent tropical climate model. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021163


Yu-Hsien Chang, Guo-Chin Jau. The behavior of the solution for a mathematical model for analysis of the cell cycle. Communications on Pure & Applied Analysis, 2006, 5 (4) : 779-792. doi: 10.3934/cpaa.2006.5.779

2018 Impact Factor: 1.313


  • PDF downloads (21)
  • HTML views (0)
  • Cited by (21)

Other articles
by authors

[Back to Top]