Cahn–Hilliard–Brinkman systems for tumour growth

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November 2021, 14(11): 3989-4033. doi: 10.3934/dcdss.2021034

Cahn–Hilliard–Brinkman systems for tumour growth

Matthias Ebenbeck ^1, , Harald Garcke ^1,, and Robert Nürnberg ^2,

1.	Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany
2.	Department of Mathematics, University of Trento, Trento, Italy

^* Corresponding author: Harald Garcke

Received March 2020 Revised January 2021 Published November 2021 Early access March 2021

A phase field model for tumour growth is introduced that is based on a Brinkman law for convective velocity fields. The model couples a convective Cahn–Hilliard equation for the evolution of the tumour to a reaction-diffusion-advection equation for a nutrient and to a Brinkman–Stokes type law for the fluid velocity. The model is derived from basic thermodynamical principles, sharp interface limits are derived by matched asymptotics and an existence theory is presented for the case of a mobility which degenerates in one phase leading to a degenerate parabolic equation of fourth order. Finally numerical results describe qualitative features of the solutions and illustrate instabilities in certain situations.

Keywords: Tumour growth, Cahn–Hilliard equation, phase field model, Brinkman model, existence, singular limit, finite elements.

Mathematics Subject Classification: Primary: 35K35; Secondary: 35K57, 35Q92, 35R35, 35C20, 65M60, 92C42.

Citation: Matthias Ebenbeck, Harald Garcke, Robert Nürnberg. Cahn–Hilliard–Brinkman systems for tumour growth. Discrete & Continuous Dynamical Systems - S, 2021, 14 (11) : 3989-4033. doi: 10.3934/dcdss.2021034

References:

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show all references

References:

[1]	H. Abels, H. Garcke and G. Grün, Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities, Math. Models Methods Appl. Sci., 22 (2012), 1150013. doi: 10.1142/S0218202511500138. Google Scholar
[2]	A. Agosti, P. F. Antonietti, P. Ciarletta, M. Grasselli and M. Verani, A Cahn-Hilliard-type equation with application to tumor growth dynamics, Math. Methods Appl. Sci., 40 (2017), 7598-7626. doi: 10.1002/mma.4548. Google Scholar
[3]	D. Ambrosi and L. Preziosi, On the closure of mass balance models for tumor growth, Math. Models Methods Appl. Sci., 12 (2002), 737-754. doi: 10.1142/S0218202502001878. Google Scholar
[4]	P. R. Amestoy, T. A. Davis and I. S. Duff, Algorithm 837: AMD, an approximate minimum degree ordering algorithm, ACM Trans. Math. Software, 30 (2004), 381-388. doi: 10.1145/1024074.1024081. Google Scholar
[5]	L. Baňas and R. Nürnberg, Finite element approximation of a three dimensional phase field model for void electromigration, J. Sci. Comp., 37 (2008), 202-232. doi: 10.1007/s10915-008-9203-y. Google Scholar
[6]	J. W. Barrett, H. Garcke and R. Nürnberg, Chapter 4 - Parametric finite element approximations of curvature-driven interface evolutions, in Geometric Partial Differential Equations - Part I, Handbook of Numerical Analysis, 21, Elsevier (2020), 275–423. doi: 10.1016/bs.hna.2019.05.002. Google Scholar
[7]	J. W. Barrett, H. Garcke and R. Nürnberg, Stable phase field approximations of anisotropic solidification, IMA J. Numer. Anal., 34 (2014), 1289-1327. doi: 10.1093/imanum/drt044. Google Scholar
[8]	J. W. Barrett, R. Nürnberg and V. Styles, Finite element approximation of a phase field model for void electromigration, SIAM J. Numer. Anal., 42 (2004), 738-772. doi: 10.1137/S0036142902413421. Google Scholar
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[10]	H. 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), 639-658. doi: 10.1017/S0956792597003264. Google Scholar
[11]	P. G. Ciarlet, Mathematical Elasticity. Vol. I. Three-Dimensional Elasticity, North-Holland Publishing Co., Amsterdam, 1988. Google Scholar
[12]	P. Colli, G. Gilardi and D. Hilhorst, On a Cahn–Hilliard type phase field system related to tumor growth, Discrete Contin. Dyn. Syst., 35 (2015), 2423-2442. doi: 10.3934/dcds.2015.35.2423. Google Scholar
[13]	V. Cristini, H. B. Frieboes, R. Gatenby, S. Caserta, M. Ferrari and J. Sinek, Morphologic instability and cancer invasion, Clin. Cancer Res., 11 (2005), 6772-6779. doi: 10.1158/1078-0432.CCR-05-0852. Google Scholar
[14]	V. Cristini, H. B. Frieboes, X. Li, J. S. Lowengrub, P. Macklin, S. Sanga, S. M. Wise and X. Zheng, Nonlinear modeling and simulation of tumor growth, Selected Topics in Cancer Modeling, Birkhäuser Boston, (2008), 113–181. Google Scholar
[15]	V. Cristini, X. Li, J. S. Lowengrub and S. M. Wise, Nonlinear simulations of solid tumor growth using a mixture model: Invasion and branching, J. Math. Biol., 58 (2009), 723-763. doi: 10.1007/s00285-008-0215-x. Google Scholar
[16]	V. Cristini and J. Lowengrub, Multiscale Modeling of Cancer: An Integrated Experimental and Mathematical Modeling Approach, Cambridge University Press, Cambridge, 2010. doi: 10.1017/CBO9780511781452. Google Scholar
[17]	V. Cristini, J. Lowengrub and Q. Nie, Nonlinear simulation of tumor growth, J. Math. Biol., 46 (2003), 191-224. doi: 10.1007/s00285-002-0174-6. Google Scholar
[18]	T. A. Davis, Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method, ACM Trans. Math. Software, 30 (2004), 196-199. doi: 10.1145/992200.992206. Google Scholar
[19]	T. A. Davis, Algorithm 849: A concise sparse Cholesky factorization package, ACM Trans. Math. Software, 31 (2005), 587-591. doi: 10.1145/1114268.1114277. Google Scholar
[20]	M. Ebenbeck, Cahn–Hilliard–Brinkman Models for Tumour Growth: Modelling, Analysis and Optimal Control, Ph.D thesis, University Regensburg, 2020. Google Scholar
[21]	M. Ebenbeck and H. Garcke, Analysis of a Cahn–Hilliard–Brinkman model for tumour growth with chemotaxis, J. Differential Equations, 266 (2019), 5998-6036. doi: 10.1016/j.jde.2018.10.045. Google Scholar
[22]	M. Ebenbeck and H. Garcke, On a Cahn–Hilliard–Brinkman model for tumor growth and its singular limits, SIAM J. Math. Anal., 51 (2019), 1868-1912. doi: 10.1137/18M1228104. Google Scholar
[23]	C. Eck, H. Garcke and P. Knabner, Mathematical Modeling, Springer, Cham, 2017. doi: 10.1007/978-3-319-55161-6. Google Scholar
[24]	C. M. Elliott and H. Garcke, On the Cahn–Hilliard equation with degenerate mobility, SIAM J. Math. Anal., 27 (1996), 404-423. doi: 10.1137/S0036141094267662. Google Scholar
[25]	J. Eyles, J. R. King and V. Styles, A tractable mathematical model for tissue growth, Interfaces Free Bound., 21 (2019), 463-493. doi: 10.4171/IFB/428. Google Scholar
[26]	S. J. Franks and J. R. King, Interactions between a uniformly proliferating tumour and its surroundings: Stability analysis for variable material properties, Internat. J. Engrg. Sci., 47 (2009), 1182-1192. doi: 10.1016/j.ijengsci.2009.07.004. Google Scholar
[27]	S. J. Franks and J. R. King, Interactions between a uniformly proliferating tumour and its surroundings: Uniform material properties, Math. Med. Biol., 20 (2003), 47-89. doi: 10.1093/imammb/20.1.47. Google Scholar
[28]	H. B. Frieboes, J. S. Lowengrub, S. Wise, X. Zheng, P. Macklin, E. L. Bearer and V. Cristini, Computer simulation of glioma growth and morphology, NeuroImage, 37 (2007), 59-70. doi: 10.1016/j.neuroimage.2007.03.008. Google Scholar
[29]	A. Friedman, A free boundary problem for a coupled system of elliptic, hyperbolic, and Stokes equations modeling tumor growth, Interfaces Free Bound., 8 (2006), 247-261. doi: 10.4171/IFB/142. Google Scholar
[30]	A. Friedman, Free boundary problems associated with multiscale tumor models, Math. Model. Nat. Phenom., 4 (2009), 134-155. doi: 10.1051/mmnp/20094306. Google Scholar
[31]	A. Friedman, Mathematical analysis and challenges arising from models of tumor growth, Math. Models Methods Appl. Sci., 17 (2007), 1751-1772. doi: 10.1142/S0218202507002467. Google Scholar
[32]	A. Friedmann and B. Hu, Bifurcation for a free boundary problem modeling tumor growth by Stokes equation, SIAM J. Math. Anal., 39 (2007), 174-194. doi: 10.1137/060656292. Google Scholar
[33]	S. Frigeri, M. Grasselli and E. Rocca, On a diffuse interface model of tumour growth, European J. Appl. Math., 26 (2015), 215-243. doi: 10.1017/S0956792514000436. Google Scholar
[34]	S. Frigeri, K. F. Lam and E. Rocca, On a diffuse interface model for tumour growth with non-local interactions and degenerate mobilities, in Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs (Springer INdAM ser.), 22 Springer Cham, (2017), 217–254. Google Scholar
[35]	M. Fritz, E. A. B. F. Lima, J. T. Oden and B. Wohlmuth, On the unsteady Darcy–Forchheimer–Brinkman equation in local and nonlocal tumor growth models, Math. Models Methods Appl. Sci., 29 (2019), 1691-1731. doi: 10.1142/S0218202519500325. Google Scholar
[36]	H. Garcke and K. F. Lam, Global weak solutions and asymptotic limits of a Cahn–Hilliard–Darcy system modelling tumour growth, AIMS Math., 1 (2016), 318-360. doi: 10.3934/Math.2016.3.318. Google Scholar
[37]	H. Garcke and K. F. Lam, On a Cahn–Hilliard–Darcy system for tumour growth with solution dependent source terms, Trends in Applications of Mathematics to Mechanics, Springer, Cham 27 (2018), 243–264. Google Scholar
[38]	H. Garcke, K. F. Lam, R. Nürnberg and E. Sitka, A multiphase Cahn–Hilliard–Darcy model for tumour growth with necrosis, Math. Models Methods Appl. Sci., 28 (2018), 525-577. doi: 10.1142/S0218202518500148. Google Scholar
[39]	H. Garcke, K. F. Lam and A. Signori, On a phase field model of Cahn-Hilliard type for tumour growth with mechanical effects, Nonlinear Anal. Real World Appl., 57 (2021), 103192. doi: 10.1016/j.nonrwa.2020.103192. Google Scholar
[40]	H. Garcke, K. F. Lam, E. Sitka and V. Styles, A Cahn–Hilliard–Darcy model for tumour growth with chemotaxis and active transport, Math. Models Methods Appl. Sci., 26 (2016), 1095-1148. doi: 10.1142/S0218202516500263. Google Scholar
[41]	H. Garcke and B. Stinner, Second order phase field asymptotics for multi-component systems, Interfaces Free Bound., 8 (2006), 131-157. doi: 10.4171/IFB/138. Google Scholar
[42]	Y. Giga and A. Novotný, eds, Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, Springer, Cham, 2018. doi: 10.1007/978-3-319-13344-7. Google Scholar
[43]	H. P. Greenspan, On the growth and stability of cell cultures and solid tumors, J. Theoret. Biol., 56 (1976), 229-242. doi: 10.1016/S0022-5193(76)80054-9. Google Scholar
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Figure 1. Typical situation for the formal asymptotic analysis

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Figure 2. The tumour and healthy regions $ \Omega_T $ and $ \Omega_H $

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Figure 3. Schematic sketch of the inner region close to $ \Sigma(0) $

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Figure 4. Initial tumour size for initial data $ r $: A slightly perturbed sphere

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Figure 5. Comparison of Cahn--Hilliard--Darcy and Cahn--Hilliard--Brinkman models: Tumour at time $ t = 12 $ for $ \beta = 0.1 $, left side for the CHD model, right side for the CHB model with $ \eta = 10^{-5} $

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Figure 6. Influence of different mobilities: Tumour at time $ t = 9 $ for $ \eta = 10^{-5} $, $ \beta = 0.1 $ and $ \alpha = \rho_S = 2 $, but with different mobilities, left $ m(\varphi) = \frac{1}{2}(1+\varphi)^2 $, middle $ m(\varphi) = \epsilon $, right $ m(\varphi) = 10^{-3}\epsilon $

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Figure 7. Influence of the adhesion parameter $ \beta $: Evolution of the tumour with $ m(\varphi) = \tfrac{1}{2}(1+\varphi)^2 $ and $ \eta = 0.1 $, above for $ \beta = 0.1 $ at time $ t = 1, 3, 6, 10 $, below for $ \beta = 0.01 $ at time $ t = 1, 1.5, 2, 2.5 $

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Figure 8. Influence of viscosiy I: Tumour and velocity for $ \beta = 0.01 $ at time $ t = 2.5 $, left for $ \eta = 0.1 $, right for $ \eta = 100 $, on top the tumour and below the velocity magnitude

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Figure 9. Influence of viscosity II: Evolution of the tumour at time $ t = 1, 3, 6, 10 $ with $ \beta = 0.1 $, $ \nu = 0 $ and a no-slip boundary condition on the left boundary, on top for $ \eta = 0.1 $ and below for $ \eta = 10 $

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Figure 10. Velocity profiles for different viscosities: The velocity magnitude at time $ t = 10 $ with $ \beta = 0.1 $, $ \nu = 0 $ and a no-slip boundary condition on the left boundary, left for $ \eta = 0.1 $, right for $ \eta = 10 $

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Figure 11. Influence of viscosity contrast: Tumour at time $ t = 10 $ with $ \beta = 0.1 $, $ \nu = 0 $ and a no-slip b. c. on the left boundary, with $ \eta_- = 0.01 $, $ \eta_+ = 1 $; $ \eta_- = 1 $, $ \eta_+ = 0.01 $; $ \eta_- = 0.01 $, $ \eta_+ = 10 $; $ \eta_- = 10 $, $ \eta_+ = 0.01 $

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Figure 12. Influence of initial profile I: Tumour at time $ t = 0, 0.3, 1, 1.6 $ with $ \eta = 100 $ and with the initial profile corresponding to $ r_1 $

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Figure 13. Influence of initial profile II: Tumour at time $ t = 0, 0.7, 1.2, 2.6 $ with $ \eta = 100 $ and with the initial profile corresponding to $ r_2 $

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Figure 14. Influence of initial profile III: Tumour at time $ t = 0, 0.5, 1.2, 2.4 $ with $ \eta = 100 $ and with the initial profile corresponding to $ r_3 $

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Figure 15. Influence of initial profile IV: Tumour at time $ t = 0, 0.3, 1.3, 2.3 $ with $ \eta = 0.01 $ and with the initial profile corresponding to $ r_4 $

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