On the well-posedness of the Cahn–Hilliard–Biot model and its applications to tumor growth

Marvin Fritz

doi:10.3934/dcdss.2024186

Article Contents

2024, Volume 17, Issue 12: 3533-3563. Doi: 10.3934/dcdss.2024186

This issue Previous Article Global regularity of the 3D generalized Navier-Stokes equations with damping term Next Article Preface

On the well-posedness of the Cahn–Hilliard–Biot model and its applications to tumor growth

Marvin Fritz^1, ,

1.
Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Linz, Austria

^*Corresponding author: Marvin Fritz
^*Corresponding author: Marvin Fritz

Received: October 2023

Revised: October 2024

Early access: October 11, 2024

Published online: October 11, 2024

The author is supported by the state of Upper Austria.

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Abstract

We study the Cahn–Hilliard–Biot model with respect to its mathematical well-posedness. The system models flow through deformable porous media in which the solid material has two phases with distinct material properties. The two phases of the porous material evolve according to a generalized Ginzburg–Landau energy functional, with additional influence from both viscoelastic and fluid effects. The flow-deformation coupling in the system is governed by Biot's theory. This results in a three-way coupled system that can be viewed as an extension of the Cahn–Larché equations by adding a fluid flowing through the medium. We distinguish the cases between a spatially dependent and a state-dependent Biot–Willis function. In the latter case, we consider a regularized system. In both cases, we use a Galerkin approximation to discretize the system and derive suitable energy estimates. Moreover, we apply compactness methods to pass to the limit in the discretized system. In the case of Vegard's law and homogeneous elasticity, we show that the weak solution depends continuously on the data and is unique. Lastly, we present some numerical simulations to highlight the features of the system as a tumor growth model.

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Mathematics Subject Classification: Primary: 35A01, 35A02, 35D30, 35Q92, 65M60.

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Figure 1. Evolution of the tumor mass $ t \mapsto \int_\Omega \varphi(t, x) \, \text{d}x $ for the different models

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Figure 2. Evolution of the tumor volume fraction $ \varphi(t, x) $ over time in the domain $ \Omega $

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Figure 3. Comparison of the contour lines $ \varphi(t, x)=0.5 $ (upper) and $ \varphi(t, x)=0.9 $ (lower) at different times for the Cahn—Hilliard (), Cahn—Larché equation () and Cahn—Hilliard—Biot equations ()

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Figure 4. Evolution of the deformation $ u(t, x) $

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Figure 5. Evolution of the Darcy velocity $ q(t, x)=-\kappa( \varphi)\nabla p $ (which is computed in a post-processing step)

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Table 1. Table of material parameters

Symbol	Value	Symbol	Value
$ \kappa_0 $	0.5	$ \kappa_1 $	5
$ M_0 $	0.5	$ M_1 $	1
$ \alpha_0 $	0.5	$ \alpha_1 $	1
$ E_0 $	2.8	$ E_1 $	1.4
$ \nu_0 $	0.4	$ \nu_1 $	0.2

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References

[1]	H. Abels, H. Garcke and J. Haselböck, Existence of weak solutions to a Cahn–Hilliard–Biot system, Nonlinear Analysis: Real World Applications, 81 (2025), 104194. doi: 10.1016/j.nonrwa.2024.104194.
[2]	M. Alnæs, J. Blechta, J. Hake, A. Johansson, B. Kehlet, A. Logg, C. Richardson, J. Ring, M. E. Rognes and G. N. Wells, The FEniCS project version 1.5, Archive of Numerical Software, 3.
[3]	P. Areias, E. Samaniego and T. Rabczuk, A staggered approach for the coupling of Cahn–Hilliard type diffusion and finite strain elasticity, Computational Mechanics, 57 (2016), 339-351. doi: 10.1007/s00466-015-1235-1.
[4]	L. Bociu, G. Guidoboni, R. Sacco and J. T. Webster, Analysis of nonlinear poro-elastic and poro-visco-elastic models, Archive for Rational Mechanics and Analysis, 222 (2016), 1445-1519. doi: 10.1007/s00205-016-1024-9.
[5]	L. Bociu, B. Muha and J. T. Webster, Mathematical effects of linear visco-elasticity in quasi-static biot models, Journal of Mathematical Analysis and Applications, 527 (2023), 127462. doi: 10.1016/j.jmaa.2023.127462.
[6]	F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier–Stokes Equations and Related Models, Springer Science & Business Media, 2013. doi: 10.1007/978-1-4614-5975-0.
[7]	A. Brunk and M. Fritz, Structure-preserving approximation of the Cahn–Hilliard–Biot system, Numerical Methods for Partial Differential Equations, 1 (2024), to appear.
[8]	G. Cheng, J. Tse, R. Jain and L. Munn, Micro-environmental mechanical stress controls tumor spheroid size and morphology by suppressing proliferation and inducing apoptosis in cancer cells, PLoS One, 4 (2009), e4632. doi: 10.1371/journal.pone.0004632.
[9]	P. Colli, G. Gilardi, E. Rocca and J. Sprekels, Asymptotic analyses and error estimates for a Cahn–Hilliard type phase field system modelling tumor growth, Discrete & Continuous Dynamical Systems-S, 10 (2017), 37-54. doi: 10.3934/dcdss.2017002.
[10]	O. Coussy, Poromechanics, John Wiley & Sons, 2004. doi: 10.1002/0470092718.
[11]	M. Fritz, Tumor evolution models of phase-field type with nonlocal effects and angiogenesis, Bulletin of Mathematical Biology, 85 (2023), Paper No. 44, 34 pp. doi: 10.1007/s11538-023-01151-6.
[12]	M. Fritz, P. K. Jha, T. Köppl, J. T. Oden and B. Wohlmuth, Analysis of a new multispecies tumor growth model coupling 3D phase-fields with a 1D vascular network, Nonlinear Analysis: Real World Applications, 61 (2021), 103331. doi: 10.1016/j.nonrwa.2021.103331.
[13]	M. Fritz, C. Kuttler, M. Rajendran, B. Wohlmuth and L. Scarabosio, On a subdiffusive tumour growth model with fractional time derivative, IMA Journal of Applied Mathematics, 86 (2021), 688-729. doi: 10.1093/imamat/hxab009.
[14]	M. Fritz, E. Lima, V. Nikolić, J. T. Oden and B. Wohlmuth, Local and nonlocal phase-field models of tumor growth and invasion due to ECM degradation, Mathematical Models and Methods in Applied Sciences, 29 (2019), 2433-2468. doi: 10.1142/S0218202519500519.
[15]	M. Fritz, E. A. Lima, J. T. Oden and B. Wohlmuth, On the unsteady Darcy–Forchheimer–Brinkman equation in local and nonlocal tumor growth models, Mathematical Models and Methods in Applied Sciences, 29 (2019), 1691-1731. doi: 10.1142/S0218202519500325.
[16]	M. Fritz, M. L. Rajendran and B. Wohlmuth, Time-fractional Cahn–Hilliard equation: Well-posedness, degeneracy, and numerical solutions, Computers & Mathematics with Applications, 108 (2022), 66-87. doi: 10.1016/j.camwa.2022.01.002.
[17]	H. Garcke, Mechanical effects in the Cahn–Hilliard model: A review on mathematical results, Mathematical Methods and Models in Phase Transitions, 1 (2005), 43-77.
[18]	H. Garcke and K. F. Lam, Global weak solutions and asymptotic limits of a Cahn–Hilliard–Darcy system modelling tumour growth, AIMS Mathematics, 1 (2016), 318-360. doi: 10.3934/Math.2016.3.318.
[19]	H. Garcke and K. F. Lam, Analysis of a Cahn–Hilliard system with non-zero Dirichlet conditions modeling tumor growth with chemotaxis, Discrete and Continuous Dynamical Systems, 37 (2017), 4277-4308. doi: 10.3934/dcds.2017183.
[20]	H. Garcke, K. Lam and A. Signori, On a phase field model of Cahn–Hilliard type for tumour growth with mechanical effects, Nonlinear Analysis: Real World Applications, 57 (2021), 103192. doi: 10.1016/j.nonrwa.2020.103192.
[21]	H. Garcke and D. Trautwein, Approximation and existence of a viscoelastic phase-field model for tumour growth in two and three dimensions, Discrete and Continuous Dynamical Systems-S, 17 (2024), 221-284. doi: 10.3934/dcdss.2023181.
[22]	M. Grasselli and A. Poiatti, A phase-field system arising from multiscale modeling of thrombus biomechanics in blood vessels: Local well-posedness in dimension two, Discrete and Continuous Dynamical Systems-S, 16 (2023), 2364-2425. doi: 10.3934/dcdss.2023105.
[23]	G. Helmlinger, P. A. Netti, H. C. Lichtenbeld, R. J. Melder and R. K. Jain, Solid stress inhibits the growth of multicellular tumor spheroids, Nature Biotechnology, 15 (1997), 778-783. doi: 10.1038/nbt0897-778.
[24]	E. Lima, J. T. Oden, D. Hormuth, T. Yankeelov and R. Almeida, Selection, calibration, and validation of models of tumor growth, Mathematical Models and Methods in Applied Sciences, 26 (2016), 2341-2368. doi: 10.1142/S021820251650055X.
[25]	E. Lima, J. T. Oden, B. Wohlmuth, A. Shahmoradi, D. Hormuth II, T. Yankeelov, L. Scarabosio and T. Horger, Selection and validation of predictive models of radiation effects on tumor growth based on noninvasive imaging data, Computer Methods in Applied Mechanics and Engineering, 327 (2017), 277-305. doi: 10.1016/j.cma.2017.08.009.
[26]	M. Milosevic, S. Lunt, E. Leung, J. Skliarenko, P. Shaw, A. Fyles and R. Hill, Interstitial permeability and elasticity in human cervix cancer, Microvascular Research, 75 (2008), 381-390. doi: 10.1016/j.mvr.2007.11.003.
[27]	A. Miranville, The Cahn–Hilliard Equation: Recent Advances and Applications, SIAM, 2019. doi: 10.1137/1.9781611975925.
[28]	C. Riethmüller, E. Storvik, J. W. Both and F. A. Radu, Well-posedness analysis of the Cahn–Hilliard–Biot model, arXiv preprint, arXiv: 2310.18231.
[29]	T. Roubíček, Nonlinear Partial Differential Equations with Applications, Springer Science & Business Media, 2013.
[30]	R. E. Showalter, Diffusion in poro-elastic media, Journal of Mathematical Analysis and Applications, 251 (2000), 310-340. doi: 10.1006/jmaa.2000.7048.
[31]	E. Storvik, J. W. Both, J. M. Nordbotten and F. A. Radu, A Cahn–Hilliard–Biot system and its generalized gradient flow structure, Applied Mathematics Letters, 126 (2022), 107799. doi: 10.1016/j.aml.2021.107799.
[32]	E. Storvik, C. Riethmüller, J. W. Both and F. A. Radu, Sequential solution strategies for the Cahn–Hilliard–Biot model, arXiv preprint, arXiv: 2401.13358.
[33]	T. Stylianopoulos, J. D. Martin, V. P. Chauhan, S. R. Jain, B. Diop-Frimpong, N. Bardeesy, B. L. Smith, C. R. Ferrone, F. J. Hornicek, Y. Boucher, et al., Causes, consequences, and remedies for growth-induced solid stress in murine and human tumors, Proceedings of the National Academy of Sciences, 109 (2012), 15101-15108. doi: 10.1073/pnas.1213353109.