Capillary Formation In Tumor Angiogenesis Research Articles

Two-dimensional capillary formation model in tumor angiogenesis problem through spectral meshless radial point interpolation

In this paper, the spectral meshless radial point interpolation (SMRPI) technique is applied to a mathematical model for two-dimensional capillary formation model in tumor angiogenesis problem. This is a natural continuation of capillary formation in tumor angiogenesis (Shivanian and Jafarabadi in Eng Comput 34:603–619, 2018), where the capillary (1D problem) has been considered. The mathematical model describes the progression of tumor angiogenic factor in a unit square space domain, namely the extracellular matrix. First, we obtain a time discrete scheme by approximating time derivative via a finite difference formula, and then, we use the SMRPI approach to approximate the spatial derivatives. This approach is based on a combination of meshless methods and spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct shape functions which act as basis functions in the frame of SMRPI. Because of non-availability of the exact solution, we consider two strategies for checking the stability of time difference scheme and for survey the convergence of the fully discrete scheme. The obtained numerical results show that the SMRPI provides high accuracy and efficiency with respect to the other classical methods in the literature.

A Moving Mesh Method for Mathematical Model of Capillary Formation in Tumor Angiogenesis

Using adaptive mesh methods is one of the strategies to improve numerical solutions in time-dependent partial differential equations. Moving mesh method is an adaptive mesh method. In this method, an increase in the number of mesh points is not needed for increasing the accuracy and efficiency of numerical solutions. Highly accurate solutions only can be obtained by concentrating points in areas where more accuracy is needed and decreasing the number of points in smooth areas. In this paper, moving mesh method is applied for the mathematical model of capillary formation in tumor angiogenesis. Numerical results show that this method leads to stable solutions for large values of cell diffusion coefficient in addition to enjoying necessary accuracy and efficiency.

Capillary formation in tumor angiogenesis through meshless weak and strong local radial point interpolation

This article is devoted to showing the efficiency and accuracy of two numerical procedures for the numerical solution of the capillary formation in tumor angiogenesis. A meshless local radial point interpolation (MLRPI) scheme based on Galerkin weak form is analyzed. The reason for choosing MLRPI approach is that it does not require any background integration cells; instead, integrations are implemented over local quadrature domains, which are further simplified for reducing the complication of computation using regular and simple shape. The spectral meshless radial point interpolation (SMRPI) method does not need any integration. The radial point interpolation method is proposed to construct shape functions for MLRPI and basis functions for SMRPI. A weak formulation with a Heaviside step function transforms the set of governing equations into local integral equations on local subdomains in MLRPI, whereas the operational matrices convert easily the governing equations (even high order) into a linear system of equations in SMRPI. The finite difference technique is employed to approximate the time derivatives. A stability analysis technique based on the eigenvalue of the matrices is employed to check the stability condition for the presented meshless methods. Furthermore, the methods are successfully applied to solve the problem with high values of the cell diffusion constant which many of the available methods did not investigate for solving these cases. Because of non-availability of the exact solution, we consider two strategies for checking the stability of time difference scheme and for surveying the convergence of the fully discrete scheme. From numerical results, it is considerable that the present methods provide the similar results. Moreover, less CPU time and less computational complexity are two advantages of the SMRPI method with respect to the MLRPI method.

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.

Numerical analysis of a mathematical model for capillary formation in tumor angiogenesis using a meshfree method based on the radial basis function

Abstract In this paper, an analysis is performed to find the numerical solution of a mathematical model for capillary formation in tumor angiogenesis. Firstly, a time stepping approach is employed for the time derivative, then a meshfree process based on a global collocation method using the radial basis functions (RBFs) is applied for solving the problem. Stability analysis of the method is investigated. Because of non-availability of the exact solutions, efficiency and accuracy of the method is demonstrated, by comparison with existing methods. Also the method is successfully applied for solving the problem with high values of the cell diffusion constant, which many of the available methods are not applicable for solving these cases.

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

AbstractA numerical procedure is developed to obtain the solution of a mathematical model for capillary formation in tumor angiogenesis. The proposed method is based on the shifted Legendre tau technique. Our approach consists of reducing the problem to a set of algebraic equations by expanding the approximate solution as a shifted Legendre polynomial with unknown coefficients. The operational matrices of integral and derivative together with the tau method are then utilized to evaluate the unknown coefficients of shifted Legendre polynomials. Illustrative example is included to demonstrate the validity and applicability of the presented technique. Copyright © 2007 John Wiley & Sons, Ltd.

The method of lines for the numerical solution of a mathematical model for capillary formation: The role of endothelial cells in the capillary

In this paper we present the method of lines to obtain the numerical solution of a mathematical model for capillary formation in tumor angiogenesis. This method is an approach to the numerical solution of partial differential equations that involve a time variable t and space variables x, y, … As the number of lines is increased, the accuracy of the method of lines representation of the original system increases. The method provides very accurate numerical solutions for linear and nonlinear problems in comparison with other existing methods. We also provide computer programs written in Matlab and figures that show the endothelial cell movement.

A mathematical model for the roles of pericytes and macrophages in the initiation of angiogenesis. I. The role of protease inhibitors in preventing angiogenesis

In this paper, a simple mathematical model developed in H.A. Levine, B.D. Sleeman, M. Nilsen-Hamilton [J. Math. Biol., in press] to describe the initiation of capillary formation in tumor angiogenesis is extended to include the roles of pericytes and macrophages in regulating angiogenesis. The model also allows for the presence of anti-angiogenic (angiostatic) factors. The model is based on the observation that angiostatin can prevent the degradation of fibronectin in the basal lamina by inhibiting the catalytic action of active proteolytic enzyme. That is, it is proposed that the inhibitor `deactivates' the protease but that it does not reduce the over all concentration of the protease. It consequently explores the possibility of preventing neovascular capillaries from migrating through the extra-cellular matrix toward the tumor by inhibiting protease action. The model is based on the theory of reinforced random walks coupled with Michaelis–Menten mechanisms which view endothelial cell receptors as the catalysts for transforming both tumor and macrophage derived angiogenic factors into proteolytic enzyme which in turn degrade the basal lamina. A simple catalytic reaction is proposed for the degradation of the basal lamina by the active proteases. A mechanism, in which the angiostatin acts as a protease inhibitor is discussed which has been substantiated experimentally. A second mechanism for the production of protease inhibitor from angiostatin by endothelial cells is proposed to be of Michaelis–Menten type. Mathematically, this mechanism includes the former as a subcase.