Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-27T08:03:50.577Z Has data issue: false hasContentIssue false
  • English
  • Français

Analysis of a two-phase model describing the growth of solid tumors

Published online by Cambridge University Press:  19 September 2012

JOACHIM ESCHER  and
ANCA-VOICHITA MATIOC
JOACHIM ESCHER
Affiliation:
Institute for Applied Mathematics, Leibniz University Hanover, Welfengarten 1, 30167 Hanover, Germany emails: escher@ifam.uni-hannover.de, matioca@ifam.uni-hannover.de
ANCA-VOICHITA MATIOC
Affiliation:
Institute for Applied Mathematics, Leibniz University Hanover, Welfengarten 1, 30167 Hanover, Germany emails: escher@ifam.uni-hannover.de, matioca@ifam.uni-hannover.de
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we consider a two-phase model describing the growth of avascular solid tumors when taking into account the effects of cell-to-cell adhesion and taxis due to nutrient. The tumor is surrounded by healthy tissue which is the source of nutrient for tumor cells. In a three-dimensional context, we prove that the mathematical formulation corresponds to a well-posed problem, and find radially symmetric steady-state solutions of the problem. They appear in the regime where the rate of cell apoptosis to cell proliferation is less than the far field nutrient concentration. Furthermore, we study the stability properties of those radially symmetric equilibria and find, depending on the biophysical parameters involved in the problem, both stable and unstable regimes for tumor growth.

Keywords

Radially symmetric stationary solutionClassical solutionStabilityTumor growthTaxis
Type
Papers
Information
European Journal of Applied Mathematics , Volume 24 , Issue 1 , February 2013 , pp. 25 - 48
Copyright
Copyright © Cambridge University Press 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Byrne, H. M. & Chaplain, M. A. (1995) Growth of nonnecrotic tumors in the presence and absence of inhibitors. Math. Biosci. 130, 151–181.Google Scholar
[2]Byrne, H. M. & Preziosi, L. (2003) Modelling solid tumour growth using the theory of mixtures. Math. Med. Biol. 20, 341–366.Google Scholar
[3]Cristini, V., Li, X., Lowengrub, J. & Wise, S. M. (2009) Nonlinear simulation of solid tumor growth using a mixture model: Invasion and branching, J. Math. Biol. 58, 723–763.Google Scholar
[4]Cristini, V., Lowengrub, J. & Nie, Q. (2003) Nonlinear simulation of tumor growth. J. Math. Biol. 46, 191–224.Google Scholar
[5]Cui, S. B. (2005) Analysis of a free boundary problem modeling tumor growth. Acta Math. Sin. Engl. Ser. 21 (5), 1071–1082.CrossRefGoogle Scholar
[6]Cui, S. B. & Escher, J. (2007) Bifurcation analysis of an elliptic free boundary problem modelling the growth of avascular tumors. SIAM J. Math. Anal. 39 1, 210–235.CrossRefGoogle Scholar
[7]Cui, S. B. & Escher, J. (2008) Asymptotic behaviour of solutions of a multidimensional moving boundary problem modeling tumor growth. Comm. Part. Diff. Equ. 33 4, 636–655.Google Scholar
[8]Cui, S. B. & Escher, J. (2009) Well-posedness and stability of a multi-dimensional tumor growth model. Arch. Ration. Mech. Anal. 191, 173–193.Google Scholar
[9]Cui, S. B., Escher, J. & Zhou, F. (2008) Bifurcation for a free boundary problem with surface tension modelling the growth of multi-layer tumors. J. Math. Anal. Appl. 337 1, 443–457.Google Scholar
[10]Cui, S. B. & Friedman, A. (2001) Analysis of a mathematical model of the growth of necrotic tumors. J. Math. Anal. Appl. 255, 636–677.Google Scholar
[11]Escher, J. & Matioc, A.-V. (2010) Radially symmetric growth of nonnecrotic tumors. NoDEA Nonlinear Differ. Equ. Appl. 17 1, 1–20.Google Scholar
[12]Escher, J. & Matioc, A.-V. (2011) Well-posedness and stability analysis for a moving boundary problem modelling the growth of nonnecrotic tumors. Discrete Continuous Dyn. Syst. B 15 3, 573–596.Google Scholar
[13]Escher, J., Matioc, A.-V. & Matioc, B.-V. (2011a) Analysis of a Mathematical Model Describing Necrotic Tumor Growth, Lecture Notes in Applied and Computational Mechanics, Vol. 57, Springer-Verlag, New York, 237–250.Google Scholar
[14]Escher, J. & Matioc, B.-V. (2011b) On the parabolicity of the Muskat problem: Well-posedness, fingering, and stability results. Z. Anal. Anwend. 30 2, 193–218.CrossRefGoogle Scholar
[15]Escher, J., & Matioc, A.-V. & Matioc, B.-V. (2012) A generalised Rayleigh-Taylor condition for the Muskat problem. Nonlinearity 25 1, 73–92.Google Scholar
[16]Escher, J. & Seiler, J. (2008) Bounded H ∞ calculus for pseudo-differential operators and applications to the Dirichlet-Neumann operator. Trans. Amer. Math. Soc. 360 (8), 3945–3973.Google Scholar
[17]Escher, J. & Simonett, G. (1997a) Classical solutions for Hele-Shaw models with surface tension. Adv. Diff. Equ. 2 4, 619–642.Google Scholar
[18]Escher, J. & Simonett, G. (1997b) Classical solutions of multidimensional Hele-Shaw models. SIAM J. Math. Anal. 28 5, 1028–1047.Google Scholar
[19]Friedman, A. & Reitich, F. (1999) Analysis of a mathematical model for the growth of tumors. J. Math. Biol. 38, 262–284.Google Scholar
[20]Friedman, A. & Reitich, F. (2000) Symmetry-breaking bifurcation of analytic solutions to free boundary problems. Trans. Amer. Math. Soc. 353, 1587–1634.Google Scholar
[21]Friedman, A. & Reitich, F. (2001) On the existence of spatially patterned dormant malignancies in a model for the growth of non-necrotic vascular tumors. Math. Models Methods Appl. Sci. 9 4, 601–625.CrossRefGoogle Scholar
[22]Friedman, A. & Tao, Y. (2003) Nonlinear stability of the Muskat problem with capillary pressure at the free boundary. Nonlinear Anal. 53, 45–80.Google Scholar
[23]Greenspan, H. P. (1972) Models for the growth of a solid tumor by diffusion. Stud. Appl. Math. LI (4), 317–340.Google Scholar
[24]Greenspan, H. P. (1976) On the growth and stability of cell cultures and solid tumors. J. Theor. Biol. 56, 229–242.Google Scholar
[25]Ladyzhenskaya, O. & Ural'tseva, N. N. (1968) Linear and Quasilinear Elliptic Equations, Mathematics in Science and Engineering, Vol. 46, Academic Press, Massachusetts.Google Scholar
[26]Lunardi, A. (1995) Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, Basel, Switzerland.Google Scholar