Hostname: page-component-745bb68f8f-5r2nc Total loading time: 0 Render date: 2025-02-06T13:15:26.717Z 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: SergioPietro.Frigeri@wias-berlin.de
MAURIZIO GRASSELLI
Affiliation:
Dipartimento di Matematica, Politecnico di Milano, Milano I-20133, Italy email: maurizio.grasselli@polimi.it
ELISABETTA ROCCA
Affiliation:
Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, D-10117 Berlin, Germany email: Elisabetta.Rocca@wias-berlin.de Dipartimento di Matematica “F. Enriques”, Università degli Studi di Milano, Milano I-20133, Italy email: elisabetta.rocca@unimi.it
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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 

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