American Institute of Mathematical Sciences

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

Critical-Point Analysis For Three-Variable Cancer Angiogenesis Models

Urszula Foryś 1, , Yuri Kheifetz 2, and  Yuri Kogan 2,

1. 

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

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.
Keywords: instability, stability, angiogenesis, critical point, time delay., asymptotic behavior, mathematical model, vascular network.
Mathematics Subject Classification: 34D05, 34K20, 34K6.
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
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