A hyperbolic free boundary problem modeling tumor growth: Asymptotic behavior

Authors:
Xinfu Chen, Shangbin Cui and Avner Friedman

Journal:
Trans. Amer. Math. Soc. **357** (2005), 4771-4804

MSC (2000):
Primary 34B15; Secondary 35C10, 35Q80, 92C15

Published electronically:
July 20, 2005

MathSciNet review:
2165387

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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we study a free boundary problem modeling the growth of radially symmetric tumors with two populations of cells: proliferating cells and quiescent cells. The densities of these cells satisfy a system of nonlinear first order hyperbolic equations in the tumor, and the tumor's surface is a free boundary . The nutrient concentration satisfies a diffusion equation, and satisfies an integro-differential equation. It is known that this problem has a unique stationary solution with . We prove that (i) if , then , and (ii) the stationary solution is linearly asymptotically stable.

**1.**J. Adam,*A simplified mathematical model of tumor growth*, Math. Biosci.**81**(1986), 224-229.**2.**Borys V. Bazaliy and Avner Friedman,*A free boundary problem for an elliptic-parabolic system: application to a model of tumor growth*, Comm. Partial Differential Equations**28**(2003), no. 3-4, 517–560. MR**1976462**, 10.1081/PDE-120020486**3.**Borys Bazaliy and Avner Friedman,*Global existence and asymptotic stability for an elliptic-parabolic free boundary problem: an application to a model of tumor growth*, Indiana Univ. Math. J.**52**(2003), no. 5, 1265–1304. MR**2010327**, 10.1512/iumj.2003.52.2317**4.**N. Bellomo and L. Preziosi,*Modelling and mathematical problems related to tumor evolution and its interaction with the immune system*, Math. Comput. Modelling**32**(2000), no. 3-4, 413–452. MR**1775113**, 10.1016/S0895-7177(00)00143-6**5.**N. Britton and M. Chaplain,*A qualitative analysis of some models of tissue growth*, Math. Biosci.**113**(1993), 77-89.**6.**Helen M. Byrne,*A weakly nonlinear analysis of a model of avascular solid tumour growth*, J. Math. Biol.**39**(1999), no. 1, 59–89. MR**1705626**, 10.1007/s002850050163**7.**H. Byrne and M. Chaplain,*Growth of nonnecrotic tumors in the presence and absence of inhibitors*, Math. Biosci.**131**(1995), 130-151.**8.**H. Byrne and M. Chaplain,*Growth of necrotic tumors in the presence and absence of inhibitors*, Math. Biosci.**135**(1996), 187-216.**9.**H. M. Byrne and M. A. J. Chaplain,*Free boundary value problems associated with the growth and development of multicellular spheroids*, European J. Appl. Math.**8**(1997), no. 6, 639–658. MR**1608619**, 10.1017/S0956792597003264**10.**Shangbin Cui,*Analysis of a mathematical model for the growth of tumors under the action of external inhibitors*, J. Math. Biol.**44**(2002), no. 5, 395–426. MR**1908130**, 10.1007/s002850100130**11.**Shangbin Cui and Avner Friedman,*Analysis of a mathematical model of the effect of inhibitors on the growth of tumors*, Math. Biosci.**164**(2000), no. 2, 103–137. MR**1751267**, 10.1016/S0025-5564(99)00063-2**12.**Shangbin Cui and Avner Friedman,*Analysis of a mathematical model of the growth of necrotic tumors*, J. Math. Anal. Appl.**255**(2001), no. 2, 636–677. MR**1815805**, 10.1006/jmaa.2000.7306**13.**Shangbin Cui and Avner Friedman,*A free boundary problem for a singular system of differential equations: an application to a model of tumor growth*, Trans. Amer. Math. Soc.**355**(2003), no. 9, 3537–3590 (electronic). MR**1990162**, 10.1090/S0002-9947-03-03137-4**14.**Shangbin Cui and Avner Friedman,*A hyperbolic free boundary problem modeling tumor growth*, Interfaces Free Bound.**5**(2003), no. 2, 159–181. MR**1980470**, 10.4171/IFB/76**15.**Avner Friedman and Fernando Reitich,*Analysis of a mathematical model for the growth of tumors*, J. Math. Biol.**38**(1999), no. 3, 262–284. MR**1684873**, 10.1007/s002850050149**16.**Avner Friedman and Fernando Reitich,*Symmetry-breaking bifurcation of analytic solutions to free boundary problems: an application to a model of tumor growth*, Trans. Amer. Math. Soc.**353**(2001), no. 4, 1587–1634 (electronic). MR**1806728**, 10.1090/S0002-9947-00-02715-X**17.**Avner Friedman and Fernando Reitich,*On the existence of spatially patterned dormant malignancies in a model for the growth of non-necrotic vascular tumors*, Math. Models Methods Appl. Sci.**11**(2001), no. 4, 601–625. MR**1832995**, 10.1142/S021820250100101X**18.**H. Greenspan,*Models for the growth of solid tumor by diffusion*, Stud. Appl. Math.**51**(1972), 317-340.**19.**H. P. Greenspan,*On the growth and stability of cell cultures and solid tumors*, J. Theoret. Biol.**56**(1976), no. 1, 229–242. MR**0429164****20.**D. McElwain and G. Pettet,*Cell migration in multicell spheroids*:*swimming against the tide*, Bull. Math. Biol.**55**(1993), 655-674.**21.**G. Pettet, C. P. Please, M. J. Tindall and D. McElwain,*The migration of cells in multicell tumor spheroids*, Bull. Math. Biol.**63**(2001), 231-257.**22.**J. Sherrat and M. Chaplain,*A new mathematical model for avascular tumor growth*, J. Math. Biol.**43**(2001), 291-312.**23.**K. Thompson and H. Byrne,*Modelling the internalisation of labelled cells in tumor spheroids*, Bull. Math. Biol.**61**(1999), 601-623.**24.**J. Ward and J. King,*Mathematical modelling of avascular-tumor growth*II:*Modelling growth saturation*, IMA J. Math. Appl. Med. Biol.**15**(1998), 1-42.

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Additional Information

**Xinfu Chen**

Affiliation:
Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260

Email:
xinfu@pitt.edu

**Shangbin Cui**

Affiliation:
Department of Mathematics, Zhongshan University, Guangzhou, Guangdong 510275, People’s Republic of China

Email:
mcinst@zsu.edu.cn

**Avner Friedman**

Affiliation:
Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus, Ohio 43210-1174

Email:
afriedman@mbi.osu.edu

DOI:
https://doi.org/10.1090/S0002-9947-05-03784-0

Keywords:
Tumor growth,
free boundary problem,
stationary solution,
asymptotic behavior

Received by editor(s):
September 24, 2002

Published electronically:
July 20, 2005

Article copyright:
© Copyright 2005
American Mathematical Society