Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-13T07:05:42.947Z Has data issue: false hasContentIssue false
  • English
  • Français

On a diffuse interface model of tumour growth

Published online by Cambridge University Press:  20 January 2015

SERGIO FRIGERI ,
MAURIZIO GRASSELLI  and
ELISABETTA ROCCA
SERGIO FRIGERI
Affiliation:
Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, D-10117 Berlin, Germany email: [email protected]
MAURIZIO GRASSELLI
Affiliation:
Dipartimento di Matematica, Politecnico di Milano, Milano I-20133, Italy email: [email protected]
ELISABETTA ROCCA
Affiliation:
Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, D-10117 Berlin, Germany email: [email protected] Dipartimento di Matematica “F. Enriques”, Università degli Studi di Milano, Milano I-20133, Italy email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a diffuse interface model of tumour growth proposed by A. Hawkins-Daruud et al. ((2013) J. Math. Biol.67 1457–1485). This model consists of the Cahn–Hilliard equation for the tumour cell fraction ϕ nonlinearly coupled with a reaction–diffusion equation for ψ, which represents the nutrient-rich extracellular water volume fraction. The coupling is expressed through a suitable proliferation function p(ϕ) multiplied by the differences of the chemical potentials for ϕ and ψ. The system is equipped with no-flux boundary conditions which give the conservation of the total mass, that is, the spatial average of ϕ + ψ. Here, we prove the existence of a weak solution to the associated Cauchy problem, provided that the potential F and p satisfy sufficiently general conditions. Then we show that the weak solution is unique and continuously depends on the initial data, provided that p satisfies slightly stronger growth restrictions. Also, we demonstrate the existence of a strong solution and that any weak solution regularizes in finite time. Finally, we prove the existence of the global attractor in a phase space characterized by an a priori bounded energy.

Keywords

diffuse interfacetumour growthCahn–Hilliard equationsreaction–diffusion equationsweak solutionswell-posednessglobal attractors
Type
Papers
Information
European Journal of Applied Mathematics , Volume 26 , Issue 2 , April 2015 , pp. 215 - 243
Copyright
Copyright © Cambridge University Press 2015 

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]Agmon, S. (2010) Lectures on Elliptic Boundary Value Problems, Revised edition of the 1965 original. AMS Chelsea Publishing, Providence, RI.Google Scholar
[2]Araujo, R. P. & McElwain, D. L. S. (2004) A history of the study of solid tumour growth: The contribution of mathematical modelling. Bull. Math. Biol. 66, 1039–1091.Google Scholar
[3]Barbu, V. (2010) Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer, New York.Google Scholar
[4]Besov, O. V., Il'in, V. P. & Nikol'skiĭ, S. M. (1979) Integral Representations of Functions and Embedding Theorems, Taibleson, M. H. (editor), Scripta Series in Mathematics, Vol. 2, V. H. Winston & Sons, Washington, D.C.; Halsted Press [John Wiley & Sons], New York-Toronto, Ont.-London.Google Scholar
[5]Bosia, S., Conti, M. & Grasselli, M. On the Cahn-Hilliard-Brinkman system. Commun. Math. Sci, to appear.Google Scholar
[6]Brezis, H. (1973) Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Math. Stud. 5, North-Holland, Amsterdam.Google Scholar
[7]Chatelain, C., Balois, T., Ciarletta, P. & Ben Amar, M. (2011) Emergence of microstructural patterns in skin cancer: A phase separation analysis in a binary mixture. New J. Phys. 13, 115013 (21 pp.).Google Scholar
[8]Colli, P., Gilardi, G. & Hilhorst, D. (2015) On a Cahn-Hilliard type phase fields system related to tumor growth. Discrete Contin. Dyn. Syst. Ser. A, 35, 2423–2442, doi:10.3934/dcds.2015.35.2423.Google Scholar
[9]Colli, P., Krejčí, P., Rocca, E. & Sprekels, J. (2007) Nonlinear evolution inclusions arising from phase change models. Czechoslovak Math. J. 57, 1067–1098.Google Scholar
[10]Cristini, V., Li, X., Lowengrub, J. S. & Wise, S. M. (2009) Nonlinear simulations of solid tumor growth using a mixture model: Invasion and branching. J. Math. Biol. 58, 723–763.CrossRefGoogle ScholarPubMed
[11]Cristini, V. & Lowengrub, J. (2010) Multiscale Modeling of Cancer. An Integrated Experimental and Mathematical Modeling Approach, Cambridge University Press, Cambridge.CrossRefGoogle Scholar
[12]Elliott, C. M. & Garcke, H. (1996) On the Cahn-Hilliard equation with degenerate mobility. SIAM J. Math. Anal. 27, 404–423.Google Scholar
[13]Frieboes, H. B., Jin, F., Chuang, Y.-L., Wise, S. M., Lowengrub, J. S. & Cristini, V. (2010) Three-dimensional multispecies nonlinear tumor growth-II: Tumor invasion and angiogenesis. J. Theoret. Biol. 264, 1254–1278.Google Scholar
[14]Friedman, A. (1969) Partial Differential Equations, Holt, Rinehart and Winston, New York.Google Scholar
[15]Frigeri, S., Grasselli, M. & Rocca, E. (2013) A diffuse interface model for two-phase incompressible flows with nonlocal interactions and nonconstant mobility. arXiv 1303.6446(2013), 1–47.Google Scholar
[16]Gagliardo, E. (1959) Ulteriori proprietà di alcune classi di funzioni in più variabili. Ricerche Mat. 8, 24–51.Google Scholar
[17]Giacomin, G. & Lebowitz, J. L. (1997) Phase segregation dynamics in particle systems with long range interactions. I. Macroscopic limits. J. Stat. Phys. 87, 37–61.Google Scholar
[18]Giacomin, G. & Lebowitz, J. L. (1998) Phase segregation dynamics in particle systems with long range interactions. II. Interface motion. SIAM J. Appl. Math. 58, 1707–1729.Google Scholar
[19]Gilbarg, D. & Trudinger, N. (1977) Elliptic Partial Differential Equations of Second Order, Grundlehren der Mathematischen Wissenschaften, Vol. 224, Springer-Verlag, Berlin-New York.CrossRefGoogle Scholar
[20]Hawkins-Daarud, A., Prudhomme, S., van der Zee, K. G. & Oden, J. T. (2013) Bayesian calibration, validation, and uncertainty quantification of diffuse interface models of tumor growth. J. Math. Biol. 67, 1457–1485.Google Scholar
[21]Hawkins-Daarud, A., van der Zee, K. G. & Oden, J. T. (2011) Numerical simulation of a thermodynamically consistent four-species tumor growth model. Int. J. Numer. Methods Biomed. Eng. 28, 3–24.CrossRefGoogle Scholar
[22]Hilhorst, D., Kampmann, J., Nguyen, T. N. & Van der Zee, K. G. Formal asymptotic limit of a diffuse-interface tumor-growth model. Math. Models Methods Appl. Sci., DOI: 10.1142/S0218202515500268.Google Scholar
[23]Lions, J.-L. (1969) Quelques méthodes de résolution des problémes aux limites non linéaires, Dunod, Gauthier-Villars, Paris.Google Scholar
[24]Lowengrub, J., Titi, E. & Zhao, K. (2013) Analysis of a mixture model of tumor growth. European J. Appl. Math. 24, 1–44.Google Scholar
[25]Lowengrub, J. S., Frieboes, H. B., Jin, F., Chuang, Y.-L., Li, X., Macklin, P., Wise, S. M. & Cristini, V. (2010) Nonlinear modelling of cancer: Bridging the gap between cells and tumours. Nonlinearity 23, R1–R91.Google Scholar
[26]Nečas, J. (2012) Direct methods in the theory of elliptic equations, Translated from the 1967 French original by G. Tronel and A. Kufner. Editorial coordination and preface by Š. Nečasová and a contribution by C.G. Simader. Springer Monographs in Mathematics. Springer, Heidelberg, 2012.Google Scholar
[27]Nirenberg, L. (1959) On elliptic partial differential equations. Ann. Sc. Norm. Super. Pisa 13, 115–162.Google Scholar
[28]Oden, J. T., Hawkins, A. & Prudhomme, S. (2010) General diffuse-interface theories and an approach to predictive tumor growth modeling. Math. Models Methods Appl. Sci. 20, 477–517.Google Scholar
[29]Oden, J. T., Prudencio, E. E. & Hawkins-Daarud, A. (2013) Selection and assessment of phenomenological models of tumor growth. Math. Models Methods Appl. Sci. 23, 1309–1338.CrossRefGoogle Scholar
[30]Pata, V. & Zelik, S. (2007) A result on the existence of global attractors for semigroups of closed operators. Commun. Pure Appl. Anal. 6, 481–486.Google Scholar
[31]Temam, R. (1997) Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York.Google Scholar
[32]Wang, X. & Wu, H. (2012) Long-time behavior for the Hele-Shaw-Cahn-Hilliard system. Asymptot. Anal. 78, 217–245.Google Scholar
[33]Wang, X. & Zhang, Z. (2013) Well-posedness of the Hele-Shaw-Cahn-Hilliard system. Ann. Inst. H. Poincaré Anal. Non Linéaire 30, 367–384.Google Scholar
[34]Wise, S. M., Lowengrub, J. S., Frieboes, H. B. & Cristini, V. (2008) Three-dimensional multispecies nonlinear tumor growth-I: Model and numerical method. J. Theoret. Biol. 253, 524–543.Google Scholar
[35]Wu, X., van Zwieten, G. J. & van der Zee, K. G. (2014) Stabilized second-order convex splitting schemes for Cahn-Hilliard models with applications to diffuse-interface tumor-growth models. Int. J. Numer. Methods Biomed. Eng. 30, 180–203.Google Scholar