Critical-Point Analysis For Three-Variable Cancer Angiogenesis Models
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1.
Institute of Applied Mathematics, Informatics and Mechanics, Warsaw University, Banacha 2, 02-097 Warsaw
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2.
Institute for Medical BioMathematics, 10 Hate'ena St., POB 282, Bene Ataroth
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Received:
01 January 2005
Accepted:
29 June 2018
Published:
01 August 2005
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MSC :
34D05, 34K20, 34K60.
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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.
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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Abstract
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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