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

  • Elyas ShivanianEmail author
  • Ahmad Jafarabadi
Original Article


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.


Capillary formation Tumor angiogenic factor Spectral meshless radial point interpolation (SMRPI) method Radial basis function Finite difference scheme 



The authors are grateful to the reviewers for carefully reading this paper and for their comments and suggestions which have improved the paper. The second author dedicates this article to Mohammad Reza Shajarian, internationally and critically acclaimed Iranian classical singer, composer, and Ostad (master) of Persian traditional music.


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© Springer-Verlag London Ltd., part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Applied MathematicsImam Khomeini International UniversityQazvinIran

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