A 2D mathematical model for tumor angiogenesis: The roles of certain cells in the extra cellular matrix.

We present a 2D mathematical model of tumor angiogenesis which is an extension of the 1D model originally presented in Levine et al. (2000) [1]. Our model is connected to that 1D model by some transmission and boundary conditions which carry certain cells, the endothelials, pericytes and macrophages from the vessel wall into the extra cellular matrix. In our extended model we also include a mechanism for the action of anti-angiogenic factors such as angiostatin. We present numerical simulations in which we obtain a very good "qualitative agreement" with the time of the onset of vascularization of tumors and with the fact that the capillary tip growth accelerates as it approaches the "tumor".

Steady-state analysis of a mathematical model for capillary network formation in the absence of tumor source.

This paper extends the work done in [S. Pamuk, Ph.D. Thesis, Iowa State University, 2000; Bull. Math. Biol. 63 (5) (2001) 801] in that we investigate the condition that is needed for the degradation of basement membrane in a mathematical model for capillary network formation. To do this, the steady-state behavior of tumor angiogenesis factor is studied under restricted assumptions, and the tumor angiogenesis factor threshold that activates the transport equations in the capillary is estimated using this steady state. Therefore, once the concentration of the tumor angiogenesis factor in the inner vessel wall reaches this threshold value, endothelial cells begin to move into the extracellular matrix for the start of angiogenesis. Furthermore, we do believe that the result we obtain in this paper provides an underlying insight into mechanisms of cell migration which are crucial for tumor angiogenesis.

Solutions of a linearized mathematical model for capillary formation in tumor angiogenesis: an initial data perturbation approximation.

We present a mathematical model for capillary formation in tumor angiogenesis and solve it by linearizing it using an initial data perturbation method. This method is highly effective to obtain solutions of nonlinear coupled differential equations. We also provide a specific example resulting, that even a few terms of the obtained series solutions are enough to have an idea for the endothelial cell movement in a capillary. MATLAB-generated figures are provided, and the stability criteria are determined for the steady-state solution of the cell equation.

A mathematical model for capillary formation and development in tumor angiogenesis: a review.

BACKGROUND: Angiogenesis is a morphogenic process whereby new blood vessels are induced to grow out of a preexisting vasculature. Endothelial cells (EC) form the lining of all blood vessels. Following tumor angiogenic growth factors, EC in neighboring normal capillaries are activated to secrete proteases. These then degrade the basal lamina and permit the EC to migrate into the extracellular matrix. METHODS: We use mechanisms to produce protease, inhibitors, and fibronectin. RESULTS: This article reviews a mathematical model originally presented by Levine et al. [Bull Math Biol 2001;63:801-863] and some results of Pamuk [Math Models Methods Appl Sci 2003;13/1:19-33; Math Biosci 2004;189/1:21-38]. CONCLUSIONS: We obtained a very good computational agreement with the rabbit cornea experiments of Folkman [Sci Am 1976;234:58-64]. We also introduce angiostatin to the model for therapeutic case as studied by Ulukaya et al. [Chemotherapy 2004;50/1:43-50].