Global Existence of Classical Solutions to a Cancer Invasion Model

Khadijeh Baghaei, Mohammad Bagher Ghaemi, Mahmoud Hesaaraki

Department of Mathematics, Iran University of Science and Technology, Tehran, Iran.
Department of mathematics, Sharif University of Technology, Tehran, Iran.

DOI: 10.4236/am.2012.34059                

Abstract

This paper deals with a chemotaxis-haptotaxis model of cancer invasion of tissue. The model consists of three reaction- diffusion- taxis partial differential equations describing interactions between cancer cells, matrix degrading enzymes, and the host tissue. The equation for cell density includes two bounded nonlinear density-dependent chemotactic and haptotactic sensitivity functions. In the absence of logistic damping, we prove the global existence of a unique classical solution to this model by some delicate a priori estimate techniques

Keywords

Cancer Invasion Model; Chemotaxis; Haptotaxis; Global Existence

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	M. A. J. Chaplain and G. Lolas, “Mathematical Modeling of Cancer Invasion of Tissue: Dynamic Heterogeneity,” Network and Heterogeneous Media, Vol. 1, No. 3, 2006, pp. 399-439. doi:10.3934/nhm.2006.1.399
[2]	N. Bellomo, N. K. Li and P. K. Maini, “On the Foundation of Cancer Modeling: Selected Topics Speculations, and Perspectives,” Mathematical Model and Methods in Applied Sciences, Vol. 18, No. 4, 2008, pp. 593-646. doi:10.1142/S0218202508002796
[3]	R. A. Gatenby and E. T. Gawlinski, “A Reaction Diffusion Model of Cancer Invasion,” Cancer Research, Vol. 56, No. 24, 1996, pp. 5745-5753.
[4]	A. J. Perumpanani and H. M. Byrne, “Extra Cellular Matrix Concentration Exerts Selection Pressure on Invasive Cells,” European Journal of Cancer, Vol. 35, No. 8, 1999, pp. 1274-1280. doi:10.1016/S0959-8049(99)00125-2
[5]	A. Gerisch and M. A. J. Chaplain, “Mathematical Modeling of Cancer Cell Invasion of Tissue: Local and NonLocal Models and the Effect Adhesion,” Journal of Theoretical Biology, Vol. 250, No. 4, 2008, pp. 684-704. doi:10.1016/j.jtbi.2007.10.026
[6]	Z. Szymańsska, C. Morales-Rodrigo, M. Lachowiz and M. Chaplain, “Mathematical Modelling of Cancer Invasion of Tissue: The Role and Effect of Non-Local Interactions,” Mathematical Model and Methods in Applied Sciences, Vol. 19, No. 2, 2009, pp. 257-281.
[7]	Z. Szymańsska, J. Urbański and A. Marciniak-Czochra, “Mathematical Modeling of the Influence Heat Shock Proteins on Cancer Invasion of Tissue,” Journal of Mathematical Biology, Vol. 58, No. 4-5, 2009, pp. 819-844. doi:10.1007/s00285-008-0220-0
[8]	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. 16941713. doi:10.1137/060655122
[9]	M. A. J. Chaplain and A. R. A. Anderson, “Mathematical Modeling of Tissue Invasion,” Chapman & Hall/CRT, London/Boca Raton, 2003, pp. 267-297.
[10]	C. Walker, “Global Existence for an Age and Spatially Structured Haptotaxis Model with Nonlinear Age-Boundary Conditions,” European Journal of Applied Mathematics, Vol. 19, No. 2, 2008, pp. 113-147. doi:10.1137/060655122
[11]	A. Marciniak-Czochra and M. Ptashnyk, “Boundedness of Solutions of a Haptotaxis Modelsmodel,” Mathematical Model and Methods in Applied Sciences, Vol. 20, No. 3, 2010, pp. 449-476. doi:10.1142/S0218202510004301
[12]	G. Litcanu and C. Morales-Rodrigo,” Global Solutions and Asymptotic Behavior for a Parabolic Degenerate Coupled System Arising from Biology,” Nonlinear Analysis, Vol. 72, No. 1, 2010, pp .77-98.
[13]	G. Litcanu and C. Morales-Rodrigo, “Asymptotic Behavior of Global Solutions to a Model of Cell Invasion,” Mathematical Model and Methods in Applied Sciences, Vol. 20, No. 9, 2009, pp. 1721-1758.
[14]	M. Lachowiz, “Towards Microscopic and Non-Local Models of Tumor Invasion of Tissue,” In: N. Bellomo, M. Chaplain and E. De Angelis, Eds., Selected Topics in Cancer Modeling, Birkh?user, Boston, 2008, pp. 49-63. doi:10.1007/978-0-8176-4713-1_3
[15]	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
[16]	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
[17]	Y. Tao and M. Wang, “A Combined Chemotaxis-Haptotaxis System: The Role of Logistic Source,” IAM Journal on Mathematical Analysis, Vol. 41, No. 4, 2009, pp. 15331558. doi:10.1137/090751542
[18]	Y. Tao and C. Cui, “A Density-Dependent ChemotaxisHaptotaxis System Modeling Cancer Invasion,” Journal of Mathematical Analysis and Applications, Vol. 367, No. 2, 2010, pp. 612-624. doi:10.1016/j.jmaa.2010.02.015
[19]	B. E. Ainseba, M. Bendahman and A. Noussair, “A Reaction-Diffusion System Modeling Predator-Prey with PreTaxis,” Nonlinear Analysis: Real Word Application, Vol. 9, No. 5, 2008, pp. 2086-2105. doi:10.1016/j.nonrwa.2007.06.017
[20]	O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Uarl’ceva, “Linear and Quasi-Linear Equations of Parabolic Type,” American Mathematical Society, Providence, 1968.
[21]	D. Henry. “Geometric Theory of Semilinear Parabolic Equations,” In: Lecture Notes Mathematics, Vol. 840, Springer, Berlin, 1981.
[22]	D. Horstmann and M. Winkler, “Bound Endless vs. BlowUp in a Chemotaxis System,” Journal of Differential Equations, Vol. 215, No. 1, 2005, pp. 52-107. doi:10.1016/j.jde.2004.10.022