Qualitative Analysis of a Mathematical Model for Capillary Formation in Tumor Angiogenesis

Abstract

Qualitative analysis of a mathematical model for capillary formation is presented under assumptions that enzyme and fibronectin concentrations are in quasi-steady state. The aim of this paper is to prove mathematically that the long-time tendency of endothelial cells will be towards the transition probability density function of enzyme and fibronectin. Endothelial cell steady-state solution is obtained and a numerical simulation is provided to show that there is a close agreement between the steady-state solution obtained analytically and the numerically calculated steady-state of the related initial value problem, which provides strong evidence for the stability of this steady-state.

Stability Analysis of the Steady-State Solution of a Mathematical Model in Tumor Angiogenesis

“…The result we have obtained in Eq. (11) agrees with the result obtained in [6]. Therefore, we now have…”

“…(1) is an equation for Endothelial cell migration in Tumor angiogenesis. We have shown that this simpler model, consisting of a single pde for the endothelial cell density, captures almost all of the features of the original model [4,6]. The analysis performed on the model has permitted us to focus upon the behaviour of the endothelial cells at the capillary.…”

“…As in [6], we consider that there is no angiostatin supplied to the circulatory system for simplicity. Therefore, the active enzyme is the same as the total enzyme, i.e., c a (y,t) ≡ c(y,t).…”

Nonlinear modelling of cancer: bridging the gap between cells and tumours

“…Theoretical nonlinear analyses of various tumour models mentioned above have been performed (e.g. [53, 78, 122, 131–134, 136–142, 155, 182, 199, 202–204, 207, 208, 389, 404, 480, 515, 517, 539, 558, 562, 564, 576, 577]). Recently, mixture models have been developed that are capable of describing the detailed interactions among multiple solid (cell) species and extra-/intra- cellular liquids (see section 2.5).…”

Numerical solution of a mathematical model for capillary formation in tumor angiogenesis via the tau method

“…Over the last few years, so many mathematical models of tumor growth have appeared in the research literature, for example, see [1][2][3][4][5][6]. In this paper, we shall consider the following parabolic equation:…”