Akisato Kubo, Yuto Miyata



Mathematical Analysis of Non-Local Tumour Invasion Model and Simulations

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In this paper we investigate a mathematical model of non-local tumour invasion with proliferation proposed by Grisch and Chaplain in 2008. For the better understanding of the model we consider an approximation model expanding the non-local term into Taylor series. We prove the global existence in time and asymptotic profile of the solution to the initial boundary value problem for the approximation model in 1 spacial dimension. Applying known mathematical results of usual(local) tumour invasion models, which are corresponding to the same type problem of Chaplain and Lolas model, we discuss the existence and property of solutions to the problem. Finally we show the time dependent change of the non-local tumour invasion process by computational simulations of the approximation model.


Non-local model, mathematical analysis, tumour invasion, Taylor expansion, simulation


[1] Anderson, A.R.A. and Chaplain, M.A.J., Continuous and discrete mathematical models of tumour-induced angiogenesis, Bull. Math. Biol., Vol.60, Issue.5, 1998, pp.857-899.

[2] Anderson, A.R.A. and Chaplain, M.A.J., Mathematical modelling of tissue invasion, in Cancer Modelling and Simulation, ed. Luigi Preziosi, Chapman Hall/CRC, 2003, pp.269-297.

[3] Chaplain, M.A.J., Lachowicz, M., Szymanska, Z. and Wrzosek, D., Mathematical modeling of cancer invasion: the importance of cell-cell adhesion and cell-matrix adhesion, Math. Models Methods Appl. Sci., Vol.21, No.4, 2011, pp.719- 743.

[4] Chaplain, M.A.J. and Lolas, G., Mathematical modeling of cancer invasion of tissue: Dynamic heterogeneity, Networks and Heterogeneous Media, Vol.1, No.3, 2006, pp.399-439.

[5] Domschke, P., Trucu, D., Gerisch, A. and Chaplain, M.A.J., Mathematical modelling of cancer invasion: Implications of cell adhesion variability for tumour infiltrative growth patterns, Journal of Theoretical Biology, 361, 2014, pp.41-60.

[6] Gerisch, A. and Chaplain, M.A.J., Mathematical modeling of cancer cell invasion of tissue: local and non-local models and the effect of adhesion, J. Theor. Biol., 250, 2008, pp.684-704.

[7] Kubo, A., A mathematical approach to Othmer- Stevens model, Wseas Transactions on Biology and Biomedicine, Issue. 1(2), 2005, pp.45-50.

[8] Kubo, A., Qualitative Characterization of Mathematical Models of Tumour Induced Angiogenesis, Wseas Trans. on Biology and Biomedicine, Issue. 7(3), 2006, pp.546-552.

[9] Kubo, A., Mathematical analysis of some models of tumour Growth and simulations, WSEAS Transactions on Biology and Biomedicine, 2010, pp.31-40.

[10] Kubo, A., Mathematical Analysis of a model of Tumour Invasion and Simulations, International Journal of Mathematical Models and Methods in Applied Science, Vol.4, Issue.3, 2010, pp.187- 194.

[11] Kubo, A., Nonlinear evolution equations associated with mathematical models, Discrete and Continous Dynamical Systems, supplement 2011, 2011, pp.881-890.

[12] Kubo, A. and Hoshino, H., Nonlinear evolution equation with strong dissipation and proliferation, Current Trends in Analysis and its Applications, Birkhauser, Springer, 2015, pp.233-241.

[13] Kubo, A. and Hoshino, H and Kimura K., Global existence and asymptotic behaviour of solutions for nonlinear evolution equations related to tumour invasion, Dynamical Systems, Differential Equations and Applications, AIMS Proceedings, pp.733-744, 2015.

[14] Kubo, A., Kimura K., Mathematical Analysis of Tumour Invasion with Proliferation Model and Simulations, WSEAS Transaction on Biology and Biomedicine, Vol.1, 2014, pp.165-173.

[15] Kubo, A., Miyata, Y., Kobayashi, H., Hoshino, H. and Hayashi, N., Nonlinear evolution equations and its application to a tumour invasion model, Advances in Pure Mathematics, 6(12), 2016, pp.878-893.

[16] Kubo, A., Saito, N., Suzuki, T. and Hoshino, H., Mathematical models of tumour angiogenesis and simulations, Theory of Bio-Mathematics and Its Application., in RIMS Kokyuroku, Vol.1499, 2006, pp.135-146.

[17] Kubo, A. and Suzuki, T., Asymptotic behavior of the solution to a parabolic ODE system modeling tumour growth, Differential and Integral Equations., Vol.17, No.7-8, 2004, pp.721-736.

[18] Kubo, A., Suzuki, T. and Hoshino, H., Asymptotic behavior of the solution to a parabolic ODE system, Mathematical Sciences and Applications, Vol. 22, 2005, pp.121-135.

[19] Kubo, A. and Suzuki, T., Mathematical models of tumour angiogenesis, Journal of Computational and Applied Mathematics, Vol. 204, Issue. 1, 2007, pp.48-55.

[20] Levine, H.A. and Sleeman, B.D., A system of reaction and diffusion equations arising in the theory of reinforced random walks, SIAM J. Appl. math., Vol.57, No.3, 1997, pp.683-730.

[21] Othmer, H.G. and Stevens, A., Aggregation, blowup, and collapse: The ABCs of taxis in reinforced random walks, SIAM J. Appl. Math., Vol.57, No.4, 1997, pp.1044-1081.

[22] Sleeman, B.D. and Levine, H.A., Partial differential equations of chemotaxis and angiogenesis, Math. Mech. Appl. Sci., Vol.24, Issue.6, 2001, pp.405-426.

[23] Yang, Y., Chen, H. and Liu, W., On existence and non-existence of global solutions to a system of reaction-diffusion equations modeling chemotaxis, SIAM J. Math. Anal., Vol.33, No.4, 1997, pp.763-785.

Cite this paper

Akisato Kubo, Yuto Miyata. (2018) Mathematical Analysis of Non-Local Tumour Invasion Model and Simulations. International Journal of Mathematical and Computational Methods, 3, 9-15


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