Free boundary problems arising in tumor models

Avner Friedman

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (2004)

  • Volume: 15, Issue: 3-4, page 161-168
  • ISSN: 1120-6330

Abstract

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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.

How to cite

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Friedman, Avner. "Free boundary problems arising in tumor models." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 15.3-4 (2004): 161-168. <http://eudml.org/doc/252409>.

@article{Friedman2004,
abstract = {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.},
author = {Friedman, Avner},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Free boundary problems; Asymptotic stability; Tumor models; Treatment of cancer},
language = {eng},
month = {12},
number = {3-4},
pages = {161-168},
publisher = {Accademia Nazionale dei Lincei},
title = {Free boundary problems arising in tumor models},
url = {http://eudml.org/doc/252409},
volume = {15},
year = {2004},
}

TY - JOUR
AU - Friedman, Avner
TI - Free boundary problems arising in tumor models
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 2004/12//
PB - Accademia Nazionale dei Lincei
VL - 15
IS - 3-4
SP - 161
EP - 168
AB - 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.
LA - eng
KW - Free boundary problems; Asymptotic stability; Tumor models; Treatment of cancer
UR - http://eudml.org/doc/252409
ER -

References

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  2. BAZALIY, B.V. - FRIEDMAN, A., 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. Zbl1089.35079MR2010327DOI10.1512/iumj.2003.52.2317
  3. BYRNE, H.M. - CHAPLAIN, M.A.J., Growth of nonnecrotic tumours in the presence and absence of inhibitors. Mathematical Biosciences, 181, 1995, 130-151. Zbl0836.92011
  4. CHEN, X. - CUI, S. - FRIEDMAN, A., A hyperbolic free boundary problem modeling tumor growth: Asymptotic behavior. To appear. Zbl1082.35166
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  7. CUI, S. - FRIEDMAN, A., 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. Zbl1036.34018MR1990162DOI10.1090/S0002-9947-03-03137-4
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  9. FONTELOS, M. - FRIEDMAN, A., Symmetry-breaking bifurcations of free boundary problems in three dimensions. Asymptotic Analysis, 35, 2003, 187-206. Zbl1054.35145MR2011787
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  13. FRIEDMAN, A. - TAO, Y., Analysis of a model of a virus that replicates selectively in tumor cells. J. Math. Biology, 47, 2003, 391-423. Zbl1052.92027MR2029005DOI10.1007/s00285-003-0199-5
  14. GREENSPAN, H., On the growth and stability of cell cultures and solid tumors. J. Theor. Biol., 56, 1976, 229-242. MR429164
  15. PETTET, G. - PLEASE, C.P. - TANDALL, M.J. - MCELWAIN, D., The migration of cells in multicell tumor spheroids. Bull. Math. Biol., 63, 2001, 231-257. 
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