Engineering with Computers

, Volume 34, Issue 3, pp 603–619 | Cite as

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

  • Elyas Shivanian
  • Ahmad Jafarabadi
Original Article


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.


Capillary formation Tumor angiogenesis Spectral meshless radial point interpolation (SMRPI) method Meshless local radial point interpolation (MLRPI) 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.


  1. 1.
    Gücüyenen Nurcan, Tanoğlu Gamze (2011) Iterative operator splitting method for capillary formation model in tumor angiogenesis problem: analysis and application. Int J Numer Methods Biomed Eng 27(11):1740–1750MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Levine HA, Pamuk S, Sleeman BD, Nilsen-Hamilton M (2001) Mathematical modeling of capillary formation and development in tumor angiogenesis: penetration into the stroma. Bull Math Biol 63(5):801–863CrossRefMATHGoogle Scholar
  3. 3.
    David D (1990) Reinforced random walks. Probab. Theory Rel Fields 84:203–229MathSciNetCrossRefGoogle Scholar
  4. 4.
    Stevens A, Othmer HG (1997) G Othmer. Aggregation, blowup, and collapse: the abc’s of taxis in reinforced random walks. SIAM J Appl Math 57(4):1044–1081MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Fornberg Bengt, Flyer Natasha (2015) Solving PDEs with radial basis functions. Acta Numerica 24:215–258MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Fornberg B, Natasha F (2015) A primer on radial basis functions with applications to the geosciences. 87. SIAMGoogle Scholar
  7. 7.
    Dehghan M, Abbaszadeh M (2016) Variational multiscale element free Galerkin (VMEFG) and local discontinuous Galerkin (LDG) methods for solving two-dimensional Brusselator reaction-diffusion system with and without cross-diffusion. Comput Methods Appl Mech Eng 300:770–797MathSciNetCrossRefGoogle Scholar
  8. 8.
    Zhuo-Jia Fu, Chen Wen, Ling Leevan (2015) Method of approximate particular solutions for constant-and variable-order fractional diffusion models. Eng Anal Boundary Elem 57:37–46MathSciNetCrossRefGoogle Scholar
  9. 9.
    Zhuo-Jia Fu, Xi Qiang, Ling Leevan, Cao Chang-Yong (2017) Numerical investigation on the effect of tumor on the thermal behavior inside the skin tissue. Int J Heat Mass Transfer 108:1154–1163CrossRefGoogle Scholar
  10. 10.
    Nayroles B, Touzot G, Villon P (1992) Generalizing the finite element method: diffuse approximation and diffuse elements. Computat Mech 10(5):307–318CrossRefMATHGoogle Scholar
  11. 11.
    Shivanian E, Khodabandehlo HR (2015) Application of meshless local radial point interpolation (MLRPI) on a one-dimensional inverse heat conduction problem. Ain Shams Engineering JournalGoogle Scholar
  12. 12.
    Shivanian E (2013) Analysis of meshless local radial point interpolation (MLRPI) on a nonlinear partial integro-differential equation arising in population dynamics. Eng Anal Boundary Elem 37(12):1693–1702MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Dehghan Mehdi, Mirzaei Davoud (2009) Meshless local Petrov–Galerkin (MLPG) method for the unsteady magnetohydrodynamic (MHD) flow through pipe with arbitrary wall conductivity. Appl Numer Math 59(5):1043–1058MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Shivanian E (2015) Meshless local Petrov-Galerkin (MLPG) method for three-dimensional nonlinear wave equations via moving least squares approximation. Eng Anal Boundary Elem 50:249–257MathSciNetCrossRefGoogle Scholar
  15. 15.
    Shirzadi A, Takhtabnoos F (2016) A local meshless method for Cauchy problem of elliptic pdes in annulus domains. Inverse Probl Sci Eng 24(5):729–743MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Dehghan M, Abbaszadeh M (2017) The meshless local collocation method for solving multi-dimensional Cahn-Hilliard, Swift-Hohenberg and phase field crystal equations. Eng Anal Boundary Elem 78:49–64MathSciNetCrossRefGoogle Scholar
  17. 17.
    Dehghan M, Abbaszadeh M, Mohebbi A (2015) A meshless technique based on the local radial basis functions collocation method for solving parabolic-parabolic Patlak-Keller-Segel chemotaxis model. Eng Anal Boundary Elem 56:129–144MathSciNetCrossRefGoogle Scholar
  18. 18.
    Dehghan M, Mohammadi V (2017) Comparison between two meshless methods based on collocation technique for the numerical solution of four-species tumor growth model. Commun Nonlinear Sci Numer Simul 44:204–219MathSciNetCrossRefGoogle Scholar
  19. 19.
    Elyas S, Ahmad J (2016) Inverse Cauchy problem of annulus domains in the framework of spectral meshless radial point interpolation. Eng Comput 1–12Google Scholar
  20. 20.
    Elyas S, Ahmad J (2017) Rayleigh–Stokes problem for a heated generalized second grade fluid with fractional derivatives: a stable scheme based on spectral meshless radial point interpolation. Eng Comput 1–14Google Scholar
  21. 21.
    Chen W, Fu Z-J, Chen C-S (2014) Recent advances in radial basis function collocation methods. SpringerGoogle Scholar
  22. 22.
    Shivanian E, Abbasbandy S, Alhuthali MS, Alsulami HH (2015) Local integration of 2-D fractional telegraph equation via moving least squares approximation. Eng Anal Boundary Elem 56:98–105MathSciNetCrossRefGoogle Scholar
  23. 23.
    Hosseini VR, Shivanian E, Chen W (2015) Local integration of 2-D fractional telegraph equation via local radial point interpolant approximation. Eur Phys J Plus 130(2):1–21Google Scholar
  24. 24.
    Aslefallah M, Shivanian E (2015) Nonlinear fractional integro-differential reaction-diffusion equation via radial basis functions. Eur Phys J Plus 130(3):1–9CrossRefGoogle Scholar
  25. 25.
    Shivanian E (2016) On the convergence analysis, stability, and implementation of meshless local radial point interpolation on a class of three-dimensional wave equations. Int J Numer Methods Eng 105(2):83–110MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Dehghan M, Ghesmati A (2010) Numerical simulation of two-dimensional sine-Gordon solitons via a local weak meshless technique based on the radial point interpolation method (RPIM). Comput Phys Commun 181(4):772–786MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Tadeu A, Chen CS, António J, Simoes Nuno (2011) A boundary meshless method for solving heat transfer problems using the Fourier transform. Adv Appl Math Mech 3(05):572–585MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Abbasbandy S, Roohani H (2014) Ghehsareh, Ishak Hashim, and A Alsaedi. A comparison study of meshfree techniques for solving the two-dimensional linear hyperbolic telegraph equation. Eng Anal Bound Elem 47:10–20MathSciNetCrossRefGoogle Scholar
  29. 29.
    Saadatmandi Abbas, Dehghan Mehdi (2008) Numerical solution of a mathematical model for capillary formation in tumor angiogenesis via the tau method. Commun Numer Methods Eng 24(11):1467–1474MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Abbasbandy S, Roohani H (2012) Ghehsareh, and Ishak Hashim. Numerical analysis of a mathematical model for capillary formation in tumor angiogenesis using a meshfree method based on the radial basis function. Eng Anal Bound Elem 36(12):1811–1818MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Dehghan Mehdi (2006) Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices. Math Comput Simul 71(1):16–30MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Shivanian Elyas, Khodayari Arman (2017) Meshless local radial point interpolation (mlrpi) for generalized telegraph and heat diffusion equation with non-local boundary conditions. J Theor Appl Mech 55(2):571–582CrossRefGoogle Scholar
  33. 33.
    Shivanian E (2014) Analysis of meshless local and spectral meshless radial point interpolation (mlrpi and smrpi) on 3-D nonlinear wave equations. Ocean Eng 89:173–188CrossRefGoogle Scholar
  34. 34.
    Eur Phys J Plus (2014) Meshless local radial point interpolation (mlrpi) on the telegraph equation with purely integral conditions. 129(11):241CrossRefGoogle Scholar
  35. 35.
    Shivanian E, Abbasbandy S (2015) The effects of mhd flow of third grade fluid by means of meshless local radial point interpolation (mlrpi). Int J Ind Math 7(1):1–11Google Scholar
  36. 36.
    Khodabandehlo HR, Shivanian E (2015) Application of meshless local radial point interpolation (mlrpi) on generalized one-dimensional linear telegraph equation. Int J Adv Appl Math Mech 2(3):38–50Google Scholar
  37. 37.
    Shivanian Elyas (2015) A meshless method based on radial basis and spline interpolation for 2-d and 3-d inhomogeneous biharmonic bvps. Zeitschrift für Naturforschung A 70(8):673–682CrossRefGoogle Scholar
  38. 38.
    Shivanian E, Rahimi A, Hosseini M (2016) Meshless local radial point interpolation to three-dimensional wave equation with neumann’s boundary conditions. Int J Comput Math 93(12):2124–2140MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Shivanian E (2016) Local integration of population dynamics via moving least squares approximation. Eng Comput 32(2):331–342CrossRefGoogle Scholar
  40. 40.
    Local radial point interpolation (mlrpi) method for solving time fractional diffusion-wave equation with damping. J Comput Phys 312:307–332MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Shivanian E (2016) Spectral meshless radial point interpolation (smrpi) method to two-dimensional fractional telegraph equation. Math Methods Appl Sci 39(7):1820–1835MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Fatahi H, Saberi-Nadjafi J, Shivanian E (2016) A new spectral meshless radial point interpolation (smrpi) method for the two-dimensional fredholm integral equations on general domains with error analysis. J Comput Appl Math 294:196–209MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Abbasbandy Saeid, Shivanian Elyas (2016) Numerical simulation based on meshless technique to study the biological population model. Math Sci 10(3):123–130MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Shivanian Elyas (2017) Analysis of the time fractional 2-D diffusion-wave equation via moving least square (mls) approximation. Int J Appl Comput Math 3(3):2447–2466MathSciNetCrossRefGoogle Scholar
  45. 45.
    Shivanian Elyas, Jafarabadi Ahmad (2017) Error and stability analysis of numerical solution for the time fractional nonlinear schrödinger equation on scattered data of general-shaped domains. Numer Methods Partial Differ Equ 33(4):1043–1069CrossRefMATHGoogle Scholar
  46. 46.
    Shivanian E, Jafarabadi A (2017) Numerical solution of two-dimensional inverse force function in the wave equation with nonlocal boundary conditions. Inverse Probl Sci Eng 1–25Google Scholar
  47. 47.
    Shivanian Elyas, Ghadiri Majid, Shafiei Navvab (2017) Influence of size effect on flapwise vibration behavior of rotary microbeam and its analysis through spectral meshless radial point interpolation. Appl Phys A 123(5):329CrossRefGoogle Scholar
  48. 48.
    Shivanian Elyas, Jafarabadi Ahmad (2017) An improved spectral meshless radial point interpolation for a class of time-dependent fractional integral equations: 2d fractional evolution equation. J Comput Appl Math 325:18–33MathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    Hedayat F, Elyas S, Hosseini SJ, Ghoncheh (2017) Buckling of doubly clamped nano-actuators in general form through spectral meshless radial point interpolation (smrpi). J Nanoanalysis 4(1):76–84Google Scholar
  50. 50.
    Wendland H (2005) Scattered data approximation. Cambridge University PressGoogle Scholar
  51. 51.
    Satya N (2002) Atluri1 & Shengping Shen. The meshless local Petrov-Galerkin (MLPG) method: a simple & less-costly alternative to the finite element and boundary element methods. Comput Model. Eng Sci 3:11–51Google Scholar
  52. 52.
    Nath Datta B (2010) Numerical linear algebra and applications. SiamGoogle Scholar
  53. 53.
    Dehghan Mehdi, Abbaszadeh Mostafa, Mohebbi Akbar (2016) The use of element free Galerkin method based on moving Kriging and radial point interpolation techniques for solving some types of Turing models. Eng Anal Bound Elem 62:93–111MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of Applied MathematicsImam Khomeini International UniversityQazvinIran

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