Critical-Point Analysis For Three-Variable Cancer Angiogenesis Models
We perform critical-point analysis for three-variable systems thatrepresent essential processes of the growth of the angiogenic tumor, namely,tumor growth, vascularization, and generation of angiogenic factor (protein) asa function of effective vessel density. Two models that describe tumor growthde...
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AIMS Press
2005-07-01
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| Series: | Mathematical Biosciences and Engineering |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/mbe.2005.2.511 |
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| author | Urszula Foryś Yuri Kheifetz Yuri Kogan |
| author_facet | Urszula Foryś Yuri Kheifetz Yuri Kogan |
| author_sort | Urszula Foryś |
| collection | DOAJ |
| description | We perform critical-point analysis for three-variable systems thatrepresent essential processes of the growth of the angiogenic tumor, namely,tumor growth, vascularization, and generation of angiogenic factor (protein) asa function of effective vessel density. Two models that describe tumor growthdepending on vascular mass and regulation of new vessel formation througha key angiogenic factor are explored. The first model is formulated in termsof ODEs, while the second assumes delays in this regulation, thus leadingto a system of DDEs. In both models, the only nontrivial critical point isalways unstable, while one of the trivial critical points is always stable. Themodels predict unlimited growth, if the initial condition is close enough to thenontrivial critical point, and this growth may be characterized by oscillationsin tumor and vascular mass. We suggest that angiogenesis per se does notsuffice for explaining the observed stabilization of vascular tumor size. |
| format | Article |
| id | doaj-art-92d2d5a2c82241bfab235d1c69673422 |
| institution | Kabale University |
| issn | 1551-0018 |
| language | English |
| publishDate | 2005-07-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Mathematical Biosciences and Engineering |
| spelling | doaj-art-92d2d5a2c82241bfab235d1c696734222025-01-24T01:49:00ZengAIMS PressMathematical Biosciences and Engineering1551-00182005-07-012351152510.3934/mbe.2005.2.511Critical-Point Analysis For Three-Variable Cancer Angiogenesis ModelsUrszula Foryś0Yuri Kheifetz1Yuri Kogan2Institute of Applied Mathematics, Informatics and Mechanics, Warsaw University, Banacha 2, 02-097 WarsawInstitute for Medical BioMathematics, 10 Hate'ena St., POB 282, Bene AtarothInstitute for Medical BioMathematics, 10 Hate'ena St., POB 282, Bene AtarothWe perform critical-point analysis for three-variable systems thatrepresent essential processes of the growth of the angiogenic tumor, namely,tumor growth, vascularization, and generation of angiogenic factor (protein) asa function of effective vessel density. Two models that describe tumor growthdepending on vascular mass and regulation of new vessel formation througha key angiogenic factor are explored. The first model is formulated in termsof ODEs, while the second assumes delays in this regulation, thus leadingto a system of DDEs. In both models, the only nontrivial critical point isalways unstable, while one of the trivial critical points is always stable. Themodels predict unlimited growth, if the initial condition is close enough to thenontrivial critical point, and this growth may be characterized by oscillationsin tumor and vascular mass. We suggest that angiogenesis per se does notsuffice for explaining the observed stabilization of vascular tumor size.https://www.aimspress.com/article/doi/10.3934/mbe.2005.2.511instabilitystabilityangiogenesiscritical pointtime delay.asymptotic behaviormathematical modelvascular network |
| spellingShingle | Urszula Foryś Yuri Kheifetz Yuri Kogan Critical-Point Analysis For Three-Variable Cancer Angiogenesis Models Mathematical Biosciences and Engineering instability stability angiogenesis critical point time delay. asymptotic behavior mathematical model vascular network |
| title | Critical-Point Analysis For Three-Variable Cancer Angiogenesis Models |
| title_full | Critical-Point Analysis For Three-Variable Cancer Angiogenesis Models |
| title_fullStr | Critical-Point Analysis For Three-Variable Cancer Angiogenesis Models |
| title_full_unstemmed | Critical-Point Analysis For Three-Variable Cancer Angiogenesis Models |
| title_short | Critical-Point Analysis For Three-Variable Cancer Angiogenesis Models |
| title_sort | critical point analysis for three variable cancer angiogenesis models |
| topic | instability stability angiogenesis critical point time delay. asymptotic behavior mathematical model vascular network |
| url | https://www.aimspress.com/article/doi/10.3934/mbe.2005.2.511 |
| work_keys_str_mv | AT urszulaforys criticalpointanalysisforthreevariablecancerangiogenesismodels AT yurikheifetz criticalpointanalysisforthreevariablecancerangiogenesismodels AT yurikogan criticalpointanalysisforthreevariablecancerangiogenesismodels |