Critical-Point Analysis For Three-Variable Cancer Angiogenesis Models

Abstract

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.

Keywords

• instability
• stability
• angiogenesis
• critical point
• time delay.
• asymptotic behavior
• mathematical model
• vascular network