Scientific Research

An Academic Publisher

**Numerical Solution of Model of Cancer Invasion with Tissue** ()

Chemotaxis-haptotaxis model of cancer invasion with tissue remodeling is one of the important PDE’s systems in medicine, mathematics and biomathematics. In this paper we find the solution of chemotaxis-haptotaxis model of cancer invasion using the new homotopy perturbation method (NHPM). Then by comparing some estimated numerical result with simulation laboratory result, it shows that NHPM is an efficient and exact way for solving cancer PDE’s system.

Share and Cite:

*Applied Mathematics*, Vol. 4 No. 7, 2013, pp. 1050-1058. doi: 10.4236/am.2013.47143.

Conflicts of Interest

The authors declare no conflicts of interest.

[1] | N. Bellomo, N. K. Li and P. K. Maini, “On the Foundations of Cancer Modelling: Selected Topics, Speculations, and Perspectives,” Mathematical Models and Methods in Applied Sciences, Vol. 18, No. 4, 2008, pp. 593-646. doi:10.1142/S0218202508002796 |

[2] | M. A. J. Chaplain and A. R. A. Anderson, “Mathematical modelling of Tissue Invasion, Cancer Modelling and Simulation,” Chapman and Hall/CRT, 2003, pp. 267-297. |

[3] |
M. A. J. Chaplain and G. Lolas, “Mathematical Modelling of Cancer Invasion of Tissue: Dynamic Heterogeneity,” Networks and Heterogeneous Media, Vol. 1, 2006, pp. 399-439. doi:10.3934/nhm.2006.1.399 |

[4] | Y. Tao, “Global Existence of Classical Solutions to a Combined Chemotaxis Haptotaxis Model with Logistic Source,” Journal of Mathematical Analysis and Applications, Vol. 354, No. 1, 2009, pp. 60-69. doi:10.1016/j.jmaa.2008.12.039 |

[5] | C. Walker and G. F. Webb, “Global Existence of Classical Solutions for a Haptotaxis Model,” SIAM Journal on Mathematical Analysis, Vol. 38, No. 5, 2007, pp. 1694-1713. doi:10.1137/060655122 |

[6] |
Y. Tao, “Global Existence for a Haptotaxis Model of Cancer Invasion with Tissue Remodeling,” Nonlinear Analysis: Real World Applications, Vol. 12, No. 1, 2011, pp. 418-435. doi:10.1016/j.nonrwa.2010.06.027 |

[7] | Y. Tao and M. Wang, “Global Solution for a Chemotactic-Haptotactic Model of Cancer Invasion,” Nonlinearity, Vol. 21, No. 10, 2008, pp. 2221-2238. doi:10.1088/0951-7715/21/10/002 |

[8] | Y. Tao and M. Wang, “A Combined Chemotaxis-Haptotaxis System: The Role of Logistic Source,” SIAM Journal on Mathematical Analysis, Vol. 41, No. 4, 2009, pp. 1533-1558. doi:10.1137/090751542 |

[9] | Y. Cherruault and G. Adomian, “Decomposition Methods: A New Proof of Convergence,” Mathematical and Computer Modelling, Vol. 18, No. 12, 1993, pp. 103-106. doi:10.1016/0895-7177(93)90233-O |

[10] | A. M. Wazwaz, “The Decomposition Method Applied to Systems of Partial Differential Equations and to the Reaction Diffusion Brusselator Model,” Applied Mathematics and Computation, Vol. 110, No. 2-3, 2000, pp. 251-264. doi:10.1016/S0096-3003(99)00131-9 |

[11] |
J. H. He, “Variational Iteration Method a Kind of NonLinear Analytical Technique: Some Examples,” International Journal of Non-Linear Mechanics, Vol. 34, No. 4, 1999, pp. 699-708. doi:10.1016/S0020-7462(98)00048-1 |

[12] | S. Momani and S. Abuasad, “Application of He Variational Iteration Method to Helmholtz Equation,” Chaos, Solitons and Fractals, Vol. 27, No. 5, 2005, pp. 1119-1123. doi:10.1016/j.chaos.2005.04.113 |

[13] |
C. K. Chen, “Solving Partial Differential Equations by Two Dimensional Differential Transform,” Applied Mathematics and Computation, Vol. 106, No. 2-3, 1999, pp. 171-179. doi:10.1016/S0096-3003(98)10115-7 |

[14] | F. Ayaz, “On the Two-Dimensional Differential Transform Method,” Applied Mathematics and Computation, Vol. 143, No. 2-3, 2003, pp. 361-374. doi:10.1016/S0096-3003(02)00368-5 |

[15] | J. H. He, “Homotopy Perturbation Technique,” Computer Methods in Applied Mechanics and Engineering, Vol. 178, No. 3-4, 1999, pp. 257-262. doi:10.1016/S0045-7825(99)00018-3 |

[16] | J. Biazar and H. Ghazvini, “Convergence of the Homotopy Perturbation Method for Partial Differential Equations,” Nonlinear Analysis: Real World Applications, Vol. 10, No. 5, 2009, pp. 2633-2640. doi:10.1016/j.nonrwa.2008.07.002 |

[17] | J. Biazar and M. Eslami, “A New Homotopy Perturbation Method for Solving Systems of Partial Differential Equations,” Computers and Mathematics with Applications, Vol. 62, No. 1, 2011, pp. 225-234. doi:10.1016/j.camwa.2011.04.070 |

[18] | K. Baghaei, M. B. Ghaemi and M. Hesaaraki, “Global Existence of Classical Solutions to a Cancer Invasion Model,” Applied Mathematics, Vol. 3, No. 4, 2012, pp. 382-388. doi:10.4236/am.2012.34059 |

[19] | Y. Tao and G. Zhu, “Global Solution to a Model of Tumor Invasion,” Applied Mathematical Sciences, Vol. 1, No. 48, 2007, pp. 2385-2398. |

[20] | P. A. Andreasen, L. Kjoller, L. Christensen and M. J. Duffy, “The Urokinase-Type Plasminogen Activator System in Cancer Metastasis: A Review,” International Journal of Cancer, Vol. 72, No. 1, 1997, pp. 1-22. doi:10.1002/(SICI)1097-0215(19970703)72:1<1::AID-IJC1>3.0.CO;2-Z |

[21] | P. A. Andreasen, R. Egelund and H. H. Petersen, “The Plasminogen Activation System in Tumor Growth, Invasion, and Metastasis,” Cellular and Molecular Life Sciences, Vol. 57, No. 1, 2000, pp. 25-40. doi:10.1007/s000180050497 |

[22] | F. Blasi, J.-D. Vassalli and K. Dan, “Urokinase-Type Plasminogen Activator: Proenzyme, Receptor, and Inhibitors,” Journal of Cell Biology, Vol. 104, No. 4, 1987, pp. 801-804. doi:10.1083/jcb.104.4.801 |

[23] | J. M. Lackie and P. C. Wilkinson, Eds., “Biology of the Chemotactic Response,” Cambridge University Press, Cambridge, 1981. |

[24] | S. B. Carter, “Haptotaxis and the Mechanism of Cell Motility,” Nature, Vol. 213, No. 5073, 1967, pp. 256-260. doi:10.1038/213256a0 |

[25] | A. S. G. Curtis, “The Measurement of Cell Adhesiveness by an Absolute Method,” Journal of Embryology and Experimental Morphology, Vol. 22, 1969, pp. 305-325. |

[26] | A. R. A. Anderson, M. A. J. Chaplain, E. L. Newman, R. J. C. Steele and A. M. Thompson, “Mathematical Modelling of Tumour Invasion and Metastasis,” Journal of Theoretical Medicine, Vol. 2, No. 2, 2000, pp. 129-154. doi:10.1080/10273660008833042 |

[27] |
H. M. Byrne, M. A. J. Chaplain, G. J. Pettet and D. L. S. McElwain, “A Mathematical Model of Trophoblast Invasion,” Journal of Theoretical Medicine, Vol. 1, No. 4, 1998, pp. 275-286. doi:10.1080/10273669908833026 |

Copyright © 2020 by authors and Scientific Research Publishing Inc.

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.