A fully-decoupled discontinuous Galerkin approximation of the Cahn-Hilliard-Brinkman-Ohta-Kawasaki tumor growth model. (English) Zbl 1503.65255
Summary: In this article, we consider the Cahn-Hilliard-Brinkman-Ohta-Kawasaki tumor growth system, which couples the Brinkman flow equations in the porous medium and the Cahn-Hilliard type equation with the nonlocal Ohta-Kawasaki term. We first construct a fully-decoupled discontinuous Galerkin method based on a decoupled, stabilized energy factorization approach and implicit-explicit Euler method in the time discretization, and strictly prove its unconditional energy stability. The optimal error estimate for the tumor interstitial fluid pressure is further obtained. Numerical results are also carried out to demonstrate the effectiveness of the proposed numerical scheme and verify the theoretical results. Finally, we apply the scheme to simulate the evolution of brain tumors based on patient-specific magnetic resonance imaging, and the obtained computational results show that the proposed numerical model and scheme can provide realistic calculations and predictions, thus providing an in-depth understanding of the mechanism of brain tumor growth.
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 76S05 | Flows in porous media; filtration; seepage |
| 76Z05 | Physiological flows |
| 92C35 | Physiological flow |
| 92C17 | Cell movement (chemotaxis, etc.) |
| 92C37 | Cell biology |
| 92C50 | Medical applications (general) |
| 35Q92 | PDEs in connection with biology, chemistry and other natural sciences |
| 35Q35 | PDEs in connection with fluid mechanics |
Keywords:
Cahn-Hilliard-Brinkman; fully-decoupled; discontinuous Galerkin method; error estimates; brain tumor growthReferences:
| [1] | J. Belmonte-Beitia, G.F. Calvo and V.M. Pérez-Garca, Effective particle methods for Fisher-Kolmogorov equations: theory and applications to brain tumor dynamics. Commun. Nonlinear Sci. Numer. Simul. 19 (2014) 3267-3283. · Zbl 1510.92090 · doi:10.1016/j.cnsns.2014.02.004 |
| [2] | H. Hatzikirou, A. Deutsch, C. Schaller, M. Simon and K. Swanson, Mathematical modelling of glioblastoma tumour development: a review. Math. Models Methods Appl. Sci. 15 (2005) 1779-1794. · Zbl 1077.92032 |
| [3] | V.M. Pérez-Garca, O. León-Triana, M. Rosa and A. Pérez-Martnez, CAR T cells for T-cell leukemias: insights from mathematical models. Commun. Nonlinear Sci. Numer. Simul. 96 (2021) 105684. · Zbl 1459.92045 · doi:10.1016/j.cnsns.2020.105684 |
| [4] | K. Swanson, Mathematical modeling of the growth and control of tumors. Ph.D. thesis. University of Washington (1999). |
| [5] | D. Bresch, T. Colin, E. Grenier, B. Ribba and O. Saut, Computational modeling of solid tumor growth: the avascular stage. SIAM J. Sci. Comput. 32 (2010) 2321-2344. · Zbl 1214.92039 |
| [6] | B. Ribba, O. Saut, T. Colin, D. Bresch, E. Grenier and J.P. Boissel, A multiscale mathematical model of avascular tumor growth to investigate the therapeutic benefit of anti-invasive agents. J. Theor. Biol. 243 (2006) 532-541. · Zbl 1447.92105 |
| [7] | A. Collin, T. Kritter, C. Poignard and O. Saut, Joint state-parameter estimation for tumor growth model. SIAM J. Appl. Math. 81 (2021) 355-377. · Zbl 1485.92033 |
| [8] | T. Michel, J. Fehrenbach, V. Lobjois, J. Laurent, A. Gomes, T. Colin and C. Poignard, Mathematical modeling of the proliferation gradient in multicellular tumor spheroids. J. Theoret. Biol. 458 (2018) 133-147. · Zbl 1406.92059 |
| [9] | J. Sherratt and M. Chaplain, A new mathematical model for avascular tumor growth. J. Math. Biol. 43 (2019) 291-312. · Zbl 0990.92021 |
| [10] | Y. Jiang, J. Pjesivac-Grbovic, C. Cantrell and J. Freyer, A multiscale model for avascular tumor growth. Biophys. J. 89 (2005) 3884-3894. · doi:10.1529/biophysj.105.060640 |
| [11] | T. Roose, S.J. Chapman and P.K. Maini, Mathematical models of avascular tumor growth. SIAM Rev. 49 (2007) 179-208. · Zbl 1117.93011 |
| [12] | S. Sanga, J. Sinek, H. Frieboes, M. Ferrari, J. Fruehauf and V. Cristini, Mathematical modeling of cancer progression and response to chemotherapy. Expert Rev. Anticancer Ther. 6 (2006) 1361-1376. · doi:10.1586/14737140.6.10.1361 |
| [13] | J. Sinek, H. Frieboes, X. Zheng and V. Cristini, Two-dimensional chemotherapy simulations demonstrate fundamental transport and tumor response limitations involving nanoparticles. Biomed. Microdevices 6 (2004) 297-309. · doi:10.1023/B:BMMD.0000048562.29657.64 |
| [14] | M. Dai, E. Feireisl, E. Rocca, G. Schimperna and M. Schonbek, Analysis of a diffuse interface model for multispecies tumor growth. Nonlinearity 30 (2007) 1639-1658. · Zbl 1367.35185 |
| [15] | S. Frigeri, M. Grasselli and E. Rocca, On a diffuse interface model of tumor growth. Eur. J. Appl. Math. 26 (2015) 215-243. · Zbl 1375.92031 |
| [16] | S. Frigeri, K.F. Lam, E. Rocca and G. Schimperna, On a multi-species Cahn-Hilliard-Darcy tumor growth model with singular potentials. Commun. Math. Sci. 16 (2018) 821-856. · Zbl 1404.35456 · doi:10.4310/CMS.2018.v16.n3.a11 |
| [17] | 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. · Zbl 1336.92038 |
| [18] | M. Ebenbeck and H. Garcke, Analysis of a Cahn-Hilliard-Brinkman model for tumour growth with chemotaxis. J. Differ. Equ. 266 (2019) 5998-6036. · Zbl 1410.35058 · doi:10.1016/j.jde.2018.10.045 |
| [19] | J. Jiang, H. Wu and S. Zheng, Well-posedness and long-time behavior of a non-autonomous Cahn-Hilliard-Darcy system with mass source modeling tumor growth. J. Differ. Equ. 259 (2015) 3032-3077. · Zbl 1330.35039 · doi:10.1016/j.jde.2015.04.009 |
| [20] | 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. · Zbl 1380.92029 |
| [21] | J.T. Oden, A. Hawkins and S. Prudhomme, General diffuse-interface theories and an approach to predictive tumor growth modeling. Math. Models Methods Appl. Sci. 20 (2010) 477-517. · Zbl 1186.92024 |
| [22] | E. Rocca and R. Scala, A rigorous sharp interface limit of a diffuse interface model related to tumor growth. J. Nonlinear Sci. 27 (2017) 847-872. · Zbl 1370.92076 |
| [23] | D. Hilhorst, J. Kampmann, T.N. Nguyen and V.D.Z.K. George, Formal asymptotic limit of a diffuse-interface tumor-growth model, Math. Models Methods Appl. Sci. 25 (2015) 1550026. · Zbl 1317.35123 |
| [24] | Z. Xu, X. Yang and H. Zhang, Error analysis of a decoupled, linear stabilization scheme for the Cahn-Hilliard model of two-phase incompressible flows. J. Sci. Comput. 83 (2020) 57. · Zbl 1440.65104 |
| [25] | L. Tang, A.L. van de Ven, D. Guo, V. Andasari, V. Cristini, K.C. Li and X. Zhou, Computational modeling of 3D tumor growth and angiogenesis for chemotherapy evaluation. PLoS One 9 (2014) e83962. |
| [26] | V. Mohammadi and M. Dehghan, Simulation of the phase field Cahn-Hilliard and tumor growth models via a numerical scheme: element-free Galerkin method. Comput. Methods Appl. Mech. Eng. 345 (2019) 919-950. · Zbl 1440.74428 · doi:10.1016/j.cma.2018.11.019 |
| [27] | D. Ambrosi and F. Mollica, On the mechanics of a growing tumor. Int. J. Eng. Sci. 40 (2002) 1297-1316. · Zbl 1211.74161 · doi:10.1016/S0020-7225(02)00014-9 |
| [28] | L. Liu and M. Schlesinger, Interstitial hydraulic conductivity and interstitial fluid pressure for avascular or poorly vascularized tumors. J. Theor. Biol. 380 (2015) 1-8. · Zbl 1343.92110 |
| [29] | Y. Zheng, Y.X. Jiang and Y.P. Cao, Effects of interstitial fluid pressure on shear wave elastography of solid tumors. Extreme Mech. Lett. 47 (2021) 101366. · doi:10.1016/j.eml.2021.101366 |
| [30] | L. Baxter and R. Jain, Transport of fluid and macro molecules in tumors 1. Role of interstitial pressure and convection. Microvasc. Res. 12 (1989) 77-104. |
| [31] | P.A. Netti, D.A. Berk, M.A. Swartz, A.J. Grodzinsky and R.K. Jain, Role of extracellular matrix assembly in interstitial transport in solid tumors. Cancer Res. 60 (2000) 2497-2503. |
| [32] | S.J. Lunt, T.M. Kalliomaki, A. Brown, V.X. Yang, M. Milosevic and R.P. Hill, Interstitial fluid pressure, vascularity and metastasis in ectopic, orthotopic and spontaneous tumours. BMC Cancer 8 (2008) 1-14. · doi:10.1186/1471-2407-8-1 |
| [33] | L.J. Liu, S.L. Brown, J.R. Ewing and M. Schlesinger, Phenomenological model of interstitial fluid pressure in a solid tumor. Phys. Rev. E 84 (2011) 021919. |
| [34] | M. Milosevic, A. Fyles, D. Hedley, M. Pintilie, W. Levin, L. Manchul and R. Hill, Interstitial fluid pressure predicts survival in patients with cervix cancer independent of clinical prognostic factors and tumor oxygen measurements. Cancer Res. 61 (2001) 6400-6405. |
| [35] | M. Sarntinoranont, F. Rooney and M. Ferrari, Interstitial stress and fluid pressure within a growing tumor. Ann. Biomed. Eng. 31 (2003) 327-335. · doi:10.1114/1.1554923 |
| [36] | S. Evje and J.O. Waldeland, How tumor cells can make use of interstitial fluid flow in a strategy for metastasis. Cell. Mol. Bioeng. 12 (2019) 227-254. · doi:10.1007/s12195-019-00569-0 |
| [37] | M. Conti and A. Giorgini, Well-posedness for the Brinkman-Cahn-Hilliard system with unmatched viscosities. J. Differ. Equ. 268 (2020) 6350-6384. · Zbl 1434.35087 · doi:10.1016/j.jde.2019.11.049 |
| [38] | F. Della Porta and M. Grasselli, On the nonlocal Cahn-Hilliard-Brinkman and Cahn-Hilliard-Hele-Shaw systems. Commun. Pure Appl. Anal. 15 (2016) 299-317. · Zbl 1334.35226 · doi:10.3934/cpaa.2016.15.299 |
| [39] | J. Shen and X. Yang, Decoupled energy stable schemes for phase-field models of two-phase complex fluids. SIAM J. Sci. Comput. 36 (2014) B122-B145. · Zbl 1288.76057 |
| [40] | J. Shen and X. Yang, Decoupled energy stable schemes for phase-field models of two-phase incompressible flows. SIAM J. Numer. Anal. 53 (2015) 279-296. · Zbl 1327.65178 |
| [41] | Y. Chen and J. Shen, Efficient, adaptive energy stable schemes for the incompressible Cahn-Hilliard Navier-Stokes phase-field models. J. Comput. Phys. 308 (2016) 40-56. · Zbl 1352.65229 · doi:10.1016/j.jcp.2015.12.006 |
| [42] | S. Minjeaud, An unconditionally stable uncoupled scheme for a triphasic Cahn-Hilliard/Navier-Stokes model. Numer. Methods Partial Differ. Equ. 29 (2013) 584-618. · Zbl 1364.76091 |
| [43] | J. Zhao, H. Li, Q. Wang and X. Yang, Decoupled energy stable schemes for a phase field model of three-phase incompressible viscous fluid flow. J. Sci. Comput. 70 (2017) 1367-1389. · Zbl 1397.76098 |
| [44] | X. Yang, A new efficient fully-decoupled and second-order time-accurate scheme for Cahn-Hilliard phase-field model of three-phase incompressible flow. Comput. Methods Appl. Mech. Eng. 376 (2021) 113589. · Zbl 1506.76193 · doi:10.1016/j.cma.2020.113589 |
| [45] | X. Yang, On a novel fully-decoupled, linear and second-order accurate numerical scheme for the Cahn-Hilliard-Darcy system of two-phase Hele-Shaw flow. Comput. Phys. Commun. 263 (2021) 107868. · Zbl 1539.65146 · doi:10.1016/j.cpc.2021.107868 |
| [46] | C. Collins, J. Shen and S.M. Wise, An efficient, energy stable scheme for the Cahn-Hilliard-Brinkman system. Commun. Comput. Phys. 13 (2013) 929-957. · Zbl 1373.76161 · doi:10.4208/cicp.171211.130412a |
| [47] | A. Diegel, X. Feng and S.M. Wise, Analysis of a mixed finite element method for a Cahn-Hilliard-Darcy-Stokes system. SIAM J. Numer. Anal. 53 (2015) 127-152. · Zbl 1330.76065 |
| [48] | X. Feng and S.M. Wise, Analysis of a Darcy-Cahn-Hilliard diffuse interface model for the Hele-Shaw flow and its fully discrete finite element approximation. SIAM J. Numer. Anal. 50 (2012) 1320-1343. · Zbl 1426.76258 |
| [49] | Y. Liu, W.B. Chen, C. Wang and S.M. Wise, Error analysis of a mixed finite element method for a Cahn-Hilliard-Hele-Shaw system. Numer. Math. 135 (2017) 679-709. · Zbl 1516.65091 |
| [50] | C. Chen and X. Yang, A Second-order time accurate and fully-decoupled numerical scheme of the Darcy-Newtonian-Nematic model for two-phase complex fluids confined in the Hele-Shaw cell. J. Comput. Phys. 456 (2022) 111026. · Zbl 07518104 · doi:10.1016/j.jcp.2022.111026 |
| [51] | X. Yang, A novel decoupled second-order time marching scheme for the two-phase incompressible Navier-Stokes/Darcy coupled nonlocal Allen-Cahn model. Comput. Methods Appl. Mech. Eng. 377 (2021) 113597. · Zbl 1506.76144 · doi:10.1016/j.cma.2020.113597 |
| [52] | Y. Gao, X. He, L. Mei and X. Yang, Decoupled, linear, and energy stable finite element method for the Cahn-Hilliard-Navier-Stokes-Darcy phase field model. SIAM. J. Sci. Comput. 40 (2018) B110-B137. · Zbl 1426.76261 |
| [53] | D.N. Arnold, F. Brezzi, B. Cockburn and L.D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39 (2001) 1749-1779. · Zbl 1008.65080 |
| [54] | B. Cockburn, G.E. Karniadakis and C.-W. Shu, The Development of Discontinuous Galerkin methods. Springer, Berlin Heidelberg (2000). · Zbl 0989.76045 |
| [55] | B. Rivière, Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation. SIAM, Philadelphia (2008). · Zbl 1153.65112 · doi:10.1137/1.9780898717440 |
| [56] | G.N. Wells, E. Kuhl and K. Garikipati, A discontinuous Galerkin method for the Cahn-Hilliard equation. J. Comput. Phys. 218 (2006) 860-877. · Zbl 1106.65086 |
| [57] | D. Kay, V. Styles and E. Süli, Discontinuous Galerkin finite element approximation of the Cahn-Hilliard equation with convection. SIAM J. Numer. Anal. 47 (2009) 2660-2685. · Zbl 1197.65136 |
| [58] | A.C. Aristotelous, O. Karakashian and S.M. Wise, A mixed discontinuous Galerkin, convex splitting scheme for a modified Cahn-Hilliard equation and an efficient nonlinear multigrid solver. Discrete Cont. Dyn. Syst. B 18 (2013) 2211-2238. · Zbl 1278.65149 |
| [59] | X. Feng and Y. Li, Analysis of symmetric interior penalty discontinuous Galerkin methods for the Allen-Cahn equation and the mean curvature flow. IMA J. Numer. Anal. 35 (2015) 1622-1651. · Zbl 1328.65205 · doi:10.1093/imanum/dru058 |
| [60] | T. Ohta and K. Kawasaki, Equilibrium morphology of block copolymer melts. Macromolecules 9 (1986) 2621-2632. |
| [61] | X. Wang, Q. Zhai and R. Zhang, The weak Galerkin method for solving the incompressible Brinkman flow. J. Comput. Appl. Math. 9 (2016) 13-24. · Zbl 1338.76069 · doi:10.1016/j.cam.2016.04.031 |
| [62] | X. Yang and J. Zhao, On linear and unconditionally energy stable algorithms for variable mobility Cahn-Hilliard type equation with logarithmic Flory-Huggins potential. Commun. Comput. Phys. 25 (2019) 703-728. · Zbl 1473.65218 |
| [63] | X. Wang, J. Kou and J. Cai, Stabilized energy factorization approach for Allen-Cahn equation with logarithmic Flory-Huggins potential. J. Sci. Comput. 82 (2020) 1-23. · Zbl 1434.65143 |
| [64] | X. Feng and O.A. Karakashian, Fully discrete dynamic mesh discontinuous Galerkin methods for the Cahn-Hilliard equation of phase transition. Math. Comput. 76 (2007) 1093-1117. · Zbl 1117.65130 |
| [65] | J. Xu, G. Viloanova and H. Gomez, Phase-field model of vascular tumor growth: three-dimensional geometry of the vascular network and integration with imaging data. Comput. Methods Appl. Mech. Eng. 359 (2020) 112648. · Zbl 1441.74133 · doi:10.1016/j.cma.2019.112648 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
