The element-free Galerkin method based on moving least squares and moving Kriging approximations for solving two-dimensi
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ORIGINAL ARTICLE
The element‑free Galerkin method based on moving least squares and moving Kriging approximations for solving two‑dimensional tumor‑induced angiogenesis model Mehdi Dehghan1 · Niusha Narimani1 Received: 12 October 2018 / Accepted: 14 May 2019 © Springer-Verlag London Ltd., part of Springer Nature 2019
Abstract The numerical simulation of the tumor-induced angiogenesis process is an useful tool for the prediction of this mechanism and drug targeting using anti-angiogenesis strategy. In the current paper, we study numerically on the continuous mathematical model of tumor-induced angiogenesis in two-dimensional spaces. The studied model is a system of nonlinear time-dependent partial differential equations, which describes the interactions between endothelial cell, tumor angiogenesis factor and fibronectin. We first derive the global weak form of the model and discretize the time variable via a semi-implicit backward Euler method. To approximate the spatial variables of the studied model, we use a meshless technique, namely element-free Galerkin. Also, the shape functions of moving least square and moving Kriging approximations are used in this method. The main difference between two meshless methods proposed here is that the shape functions of moving least squares approximation do not satisfy Kroncker’s delta property, while moving Kriging technique satisfies this property. Also, both techniques do not require the generation of a mesh for approximation, but a background mesh is needed to compute the numerical integrations, which are appeared in the derived global weak form. The full-discrete scheme obtained here gives the linear system of algebraic equations that is solved via an iterative method, namely biconjugate gradient stabilized with zero-fill incomplete lower upper (ILU) preconditioner. Some numerical simulations are provided to illustrate the ability of the presented numerical methods, which show the endothelial cell migration in response to the tumor angiogenesis factors during angiogenesis process as well. Keywords Tumor angiogenesis process · Time-dependent nonlinear partial differential equations · Moving least squares approximation · Moving Kriging interpolation · Element-free Galerkin technique · Time discretization based on backward Euler method · Biconjugate gradient stabilized method · Mathematical Biology Mathematics Subject Classification 92Bxx · 35Q92
1 Introduction Angiogenesis is the growth of new blood vessels from the existing vasculature [1]. This formation of new capillaries occurs in both health and disease [2]. Normal angiogenesis * Mehdi Dehghan [email protected]; [email protected] 1
Niusha Narimani [email protected]; [email protected]
Department of Applied Mathematics, Faculty of Mathematics and Computer Sciences, Amirkabir University of Technology, No. 424, Hafez Ave., Tehran 15914, Iran
happens in the healthy tissues such as fetal growth and wound healing [2], while abnormal angiogenesis occurs in several diseases including cancer (see Fig
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