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

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## Abstract

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.

## Keywords

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

### Acknowledgements

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.

## References

- 1.Shivanian E, Jafarabadi A (2018) Capillary formation in tumor angiogenesis through meshless weak and strong local radial point interpolation. Eng Comput 34:603–619Google Scholar
- 2.Folkman J (1971) Tumor angiogenesis: therapeutic implications. N Engl J Med 285(21):1182–1186Google Scholar
- 3.Folkman J (1984) Biology of endothelial cells, vol 27. Springer, Boston, MA, pp 412–428Google Scholar
- 4.Emamjome M, Azarnavid B, Roohani Ghehsareh H (2017) A reproducing kernel Hilbert space pseudospectral method for numerical investigation of a two-dimensional capillary formation model in tumor angiogenesis problem. Neural Comput Appl 9:1–9Google Scholar
- 5.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):801–863zbMATHGoogle Scholar
- 6.Pamuk S, Erdem A (2007) The method of lines for the numerical solution of a mathematical model for capillary formation: the role of endothelial cells in the capillary. Appl Math Comput 186(1):831–835MathSciNetzbMATHGoogle Scholar
- 7.Saadatmandi A, Dehghan M (2008) Numerical solution of a mathematical model for capillary formation in tumor angiogenesis via the Tau method. Commun Numer Methods Eng 24(11):1467–1474MathSciNetzbMATHGoogle Scholar
- 8.Gücüyenen N, Tanoğlu G (2011) Iterative operator splitting method for capillary formation model in tumor angiogenesis problem: analysis and application. Int J Numer Methods Biomed Eng 27(11):1740–1750MathSciNetzbMATHGoogle Scholar
- 9.Pamuk S (2004) Steady-state analysis of a mathematical model for capillary network formation in the absence of tumor source. Math Biosci 189(1):21–38MathSciNetzbMATHGoogle Scholar
- 10.Nayroles B, Touzot G, Villon P (1992) Generalizing the finite element method: diffuse approximation and diffuse elements. Comput Mech 10(5):307–318MathSciNetzbMATHGoogle Scholar
- 11.Belytschko T, Lu YY, Gu L (1994) Element-free Galerkin methods. Int J Numer Methods Eng 37(2):229–256MathSciNetzbMATHGoogle Scholar
- 12.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 300:770–797MathSciNetGoogle Scholar
- 13.Atluri SN, Zhu T (1998) A new meshless local Petrov–Galerkin (MLPG) approach in computational mechanics. Comput Mech 22(2):117–127MathSciNetzbMATHGoogle Scholar
- 14.Shivanian E (2015) Meshless local Petrov–Galerkin ( MLPG) method for three-dimensional nonlinear wave equations via moving least squares approximation. Eng Anal Bound Elem 50:249–257MathSciNetzbMATHGoogle Scholar
- 15.Liu GR, Zhang GY, Dai KY, Wang YY, Zhong ZH, Li GY, Han X (2005) A linearly conforming point interpolation method (LC-PIM) for 2D solid mechanics problems. Int J Comput Methods 2(04):645–665zbMATHGoogle Scholar
- 16.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 Bound Elem 56:129–144MathSciNetzbMATHGoogle Scholar
- 17.Dehghan M, Haghjoo-Saniji M (2017) The local radial point interpolation meshless method for solving Maxwell equations. Eng Comput 33(4):897–918Google Scholar
- 18.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 130:33:1–21Google Scholar
- 19.Shivanian E, Abbasbandy S, Alhuthali MS, Alsulami HH (2015) Local integration of 2-d fractional telegraph equation via moving least squares approximation. Eng Anal Bound Elem 56:98–105MathSciNetzbMATHGoogle Scholar
- 20.Hosseini VR, Shivanian E, Chen W (2016) Local radial point interpolation (mlrpi) method for solving time fractional diffusion-wave equation with damping. J Comput Phys 312:307–332MathSciNetzbMATHGoogle Scholar
- 21.Abbasbandy S, Shivanian E (2016) Numerical simulation based on meshless technique to study the biological population model. Math Sci 10(3):123–130MathSciNetzbMATHGoogle Scholar
- 22.Abbasbandy S, Shivanain E (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
- 23.Shivanian E, Jafarabadi A (2018) The numerical solution for the time-fractional inverse problem of diffusion equation. Eng Anal Bound Elem 91:50–59MathSciNetzbMATHGoogle Scholar
- 24.Lucy LB (1977) A numerical approach to the testing of the fission hypothesis. Astron J 82:1013–1024Google Scholar
- 25.Liu GR, Xu GX (2008) A gradient smoothing method (GSM) for fluid dynamics problems. Int J Numer Methods Fluids 58(10):1101–1133zbMATHGoogle Scholar
- 26.Onate E, Idelsohn S, Zienkiewicz OC, Taylor RL (1996) A finite point method in computational mechanics. Applications to convective transport and fluid flow. Int J Numer Methods Eng 39(22):3839–3866MathSciNetzbMATHGoogle Scholar
- 27.Liszka T, Orkisz J (1980) The finite difference method at arbitrary irregular grids and its application in applied mechanics. Comput Struct 11(1–2):83–95MathSciNetzbMATHGoogle Scholar
- 28.Liu GR, Kee BBT, Chun L (2006) A stabilized least-squares radial point collocation method (LS-RPCM) for adaptive analysis. Comput Methods Appl Mech Eng 195(37):4843–4861MathSciNetzbMATHGoogle Scholar
- 29.Liu GR, Gu YT (2003) A meshfree method: meshfree weak-strong (MWS) form method, for 2-D solids. Comput Mech 33(1):2–14zbMATHGoogle Scholar
- 30.Mukherjee YX, Mukherjee S (1997) The boundary node method for potential problems. Int J Numer Methods Eng 40(5):797–815zbMATHGoogle Scholar
- 31.Shivanian E (2015) A new spectral meshless radial point interpolation (SMRPI) method: a well-behaved alternative to the meshless weak forms. Eng Anal Bound Elem 54:1–12MathSciNetzbMATHGoogle Scholar
- 32.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 25(12):1743–1767MathSciNetzbMATHGoogle Scholar
- 33.Shivanian E, Jafarabadi A (2018) The spectral meshless radial point interpolation method for solving an inverse source problem of the time-fractional diffusion equation. Appl Numer Math 129:1–25MathSciNetzbMATHGoogle Scholar
- 34.Erdem A, Pamuk S (2007) The method of lines for the numerical solution of a mathematical model for capillary formation: the role of tumor angiogenic factor in the extra-cellular matrix. Appl Math Comput 186(1):891–897MathSciNetzbMATHGoogle Scholar
- 35.Shivanian E (2016) Spectral meshless radial point interpolation (SMRPI) method to two-dimensional fractional telegraph equation. Math Methods Appl Sci 39(7):1820–1835MathSciNetzbMATHGoogle Scholar
- 36.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):772–786MathSciNetzbMATHGoogle Scholar
- 37.Dehghan M, Abbaszadeh M, Mohebbi A (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 62:93–111MathSciNetzbMATHGoogle Scholar
- 38.Hardy RL (1971) Multiquadric equations of topography and other irregular surfaces. J Geophys Res 76(8):1905–1915Google Scholar
- 39.Hardy RL (1990) Theory and applications of the multiquadric-biharmonic method 20 years of discovery 1968–1988. Comput Math Appl 19(8):163–208MathSciNetzbMATHGoogle Scholar
- 40.Franke R (1982) Scattered data interpolation: tests of some methods. Math Comput 38(157):181–200MathSciNetzbMATHGoogle Scholar
- 41.Fasshauer GE (2002) Newton iteration with multiquadrics for the solution of nonlinear pdes. Comput Mathe Appl 43(3):423–438MathSciNetzbMATHGoogle Scholar
- 42.Kansa EJ (1990) Multiquadrics—a scattered data approximation scheme with applications to computational fluid-dynamics—ii solutions to parabolic, hyperbolic and elliptic partial differential equations. Comput Math Appl 19(8):147–161MathSciNetzbMATHGoogle Scholar
- 43.Sarra SA, Sturgill D (2009) A random variable shape parameter strategy for radial basis function approximation methods. Eng Anal Bound Elem 33(11):1239–1245MathSciNetzbMATHGoogle Scholar
- 44.Dehghan M, Abbaszadeh M, Mohebbi A (2014) The numerical solution of nonlinear high dimensional generalized Benjamin–Bona-Mahony–Burgers equation via the meshless method of radial basis functions. Comput Math Appl 68(3):212–237MathSciNetzbMATHGoogle Scholar