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Mathematical Biosciences and Engineering

2005, Volume 2, Issue 3: 511-525. doi: 10.3934/mbe.2005.2.511
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Critical-Point Analysis For Three-Variable Cancer Angiogenesis Models

  • 1.
    Institute of Applied Mathematics, Informatics and Mechanics, Warsaw University, Banacha 2, 02-097 Warsaw
  • 2.
    Institute for Medical BioMathematics, 10 Hate'ena St., POB 282, Bene Ataroth
  • Received: 01 January 2005 Accepted: 29 June 2018 Published: 01 August 2005

  • MSC : 34D05, 34K20, 34K60.

  • 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[J]. Mathematical Biosciences and Engineering, 2005, 2(3): 511-525. doi: 10.3934/mbe.2005.2.511

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  • 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.


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  • © 2005 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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