Geodesic
Free boundary problems arising in tumor models
Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 15 (2004) no. 3-4, pp. 161-168.

Voir la notice de l'article provenant de la source Biblioteca Digitale Italiana di Matematica

We consider several simple models of tumor growth, described by systems of PDEs, and describe results on existence of solutions and on their asymptotic behavior. The boundary of the tumor region is a free boundary. In §1 the model assumes three types of cells, proliferating, quiescent and necrotic, and the corresponding PDE system consists of elliptic, parabolic and hyperbolic equations. The model in §2 assumes that the tumor has only proliferating cells. Finally in §3 we consider a model for treatment of tumor, described by a system of elliptic and hyperbolic equations. 
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Friedman, Avner. Free boundary problems arising in tumor models. Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 15 (2004) no. 3-4, pp. 161-168. http://geodesic.mathdoc.fr/item/RLIN_2004_9_15_3-4_a0/

[1] B.V. Bazaliy - A. Friedman, A free boundary problem for an elliptic-parabolic system: Application to a model of tumor growth. Communications in PDE, 28, 2003, 517-560. | DOI | MR | Zbl

[2] B.V. Bazaliy - A. Friedman, Global existence and stability for an elliptic-parabolic free boundary problem; An application to a model of tumor growth. Indiana University Math. J., 52, 2003, 1265-1304. | DOI | MR | Zbl

[3] H.M. Byrne - M.A.J. Chaplain, Growth of nonnecrotic tumours in the presence and absence of inhibitors. Mathematical Biosciences, 181, 1995, 130-151. | Zbl

[4] X. Chen - S. Cui - A. Friedman, A hyperbolic free boundary problem modeling tumor growth: Asymptotic behavior. To appear. | Zbl

[5] X. Chen - A. Friedman, A free boundary problem for elliptic-hyperbolic system: An application to tumor growth. SIAM J. Math. Analysis, 35, 4, 2003, 974-976. | DOI | MR | Zbl

[6] S. Cui - A. Friedman, Analysis of a mathematical model of the effect of inhibitors on the growth of tumors. Math. Biosci., 164, 2000, 103-137. | DOI | MR | Zbl

[7] S. Cui - A. Friedman, A free boundary problem for a singular system of differential equations: An application to a model of tumor growth. Trans. AMS, 355, 2003, 3537-3590. | DOI | MR | Zbl

[8] S. Cui - A. Friedman, A hyperbolic free boundary problem modeling tumor growth. Interfaces and Free Boundaries, 5, 2003, 159-181. | DOI | MR | Zbl

[9] M. Fontelos - A. Friedman, Symmetry-breaking bifurcations of free boundary problems in three dimensions. Asymptotic Analysis, 35, 2003, 187-206. | MR | Zbl

[10] A. Friedman - F. Reitich, Analysis of a mathematical model for the growth of tumors. J. Math. Biol., 38, 1999, 262-284. | DOI | MR | Zbl

[11] A. Friedman - F. Reitich, Symmetry-breaking bifurcation of analytic solutions to free boundary problems: An application to a model of tumor growth. Trans. Amer. Math. Soc., 353, 2000, 1587-1634. | DOI | MR | Zbl

[12] A. Friedman - F. Reitich, On the existence of spatially patterned dormant malignancies in a model for the growth of non-necrotic vascular tumor. Math. Models and Methods in Appl. Sciences, 77, 2001, 1-25. | DOI | MR | Zbl

[13] A. Friedman - Y. Tao, Analysis of a model of a virus that replicates selectively in tumor cells. J. Math. Biology, 47, 2003, 391-423. | DOI | MR | Zbl

[14] H. Greenspan, On the growth and stability of cell cultures and solid tumors. J. Theor. Biol., 56, 1976, 229-242. | MR

[15] G. Pettet - C.P. Please - M.J. Tandall - D. Mcelwain, The migration of cells in multicell tumor spheroids. Bull. Math. Biol., 63, 2001, 231-257.

[16] J.T. Wu - H.M. Byrne - D.H. Kirn - L.M. Wein, Modeling and analysis of a virus that replicate selectively in tumor cells. Bull. Math. Biology, 63, 2001, 731-768.