**AUTHORS:**Akisato Kubo, Yuto Miyata

**Download as PDF**

**ABSTRACT:**
In the present paper we investigate local and non-local type of mathematical models of tumour invasion
with proliferation in order to understand the mechanism of the non-local tumour invasion. We consider the nonlocal
tumour invasion model proposed by Gerisch and Chaplain and an approximation model by expanding the
non-local term into Taylor series, which is closely related to Chaplain and Lolas model describing the local tumour
invasion. We prove the global existence in time and asymptotic profile of the solution to the initial boundary value
problem for the approximation model in one spacial dimension, by applying known mathematical results of the
local tumour invasion model. Finally we show by computer simulations of the approximation model, which are
verified by our mathematical analysis, the time dependent change of the non-local tumour invasion process and
observe the relationship between the value of Taylor coefficients and the tumour cell density or so.

**KEYWORDS:**
Non-local model, mathematical analysis, tumour invasion, Taylor expansion, computational simulation.

**REFERENCES:**

[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, 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 modelling of cancer cell invasion of tissue: The role of the urokinase plasminogen activation system, Math. Modell. Methods. Appl. Sci., 15, 2005, pp.1685-1734.

[5] 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.

[6] 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.

[7] 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.

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

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

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

[11] 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.

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

[13] 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.

[14] 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.

[15] 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.

[16] Kubo, A., Miyata, Y., Mathematical analysis of glioblastoma invasion models from in vitro experiment, Wseas Transaction on Mathematics, 16, 2017, pp.62-68.

[17] Kubo, A., Miyata, Y., Kobayashi, H. and Hayashi, N., Mathematical models and simulations of glioblastoma invasion, International Journal of Mathematical Models and Methods in Applied Sciences, 11, 2017, pp.107-116.

[18] 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.

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

[20] 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.

[21] 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.

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

[23] 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.

[24] 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.

[25] 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.

[26] 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.