Open Access
Issue
ESAIM: M2AN
Volume 56, Number 6, November-December 2022
Page(s) 2141 - 2180
DOI https://doi.org/10.1051/m2an/2022064
Published online 01 December 2022
  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. [CrossRef] [MathSciNet] [Google Scholar]
  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. [CrossRef] [MathSciNet] [Google Scholar]
  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. [CrossRef] [Google Scholar]
  4. K. Swanson, Mathematical modeling of the growth and control of tumors. Ph.D. thesis. University of Washington (1999). [Google Scholar]
  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. [CrossRef] [MathSciNet] [Google Scholar]
  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. [CrossRef] [Google Scholar]
  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. [CrossRef] [MathSciNet] [Google Scholar]
  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. [CrossRef] [MathSciNet] [Google Scholar]
  9. J. Sherratt and M. Chaplain, A new mathematical model for avascular tumor growth. J. Math. Biol. 43 (2019) 291–312. [Google Scholar]
  10. Y. Jiang, J. Pjesivac-Grbovic, C. Cantrell and J. Freyer, A multiscale model for avascular tumor growth. Biophys. J. 89 (2005) 3884–3894. [CrossRef] [Google Scholar]
  11. T. Roose, S.J. Chapman and P.K. Maini, Mathematical models of avascular tumor growth. SIAM Rev. 49 (2007) 179–208. [CrossRef] [MathSciNet] [Google Scholar]
  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. [CrossRef] [PubMed] [Google Scholar]
  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. [CrossRef] [PubMed] [Google Scholar]
  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. [Google Scholar]
  15. S. Frigeri, M. Grasselli and E. Rocca, On a diffuse interface model of tumor growth. Eur. J. Appl. Math. 26 (2015) 215–243. [Google Scholar]
  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. [CrossRef] [MathSciNet] [Google Scholar]
  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. [CrossRef] [MathSciNet] [Google Scholar]
  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. [CrossRef] [Google Scholar]
  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. [CrossRef] [Google Scholar]
  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. [CrossRef] [MathSciNet] [Google Scholar]
  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. [CrossRef] [MathSciNet] [Google Scholar]
  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. [CrossRef] [MathSciNet] [Google Scholar]
  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. [Google Scholar]
  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. [CrossRef] [Google Scholar]
  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. [CrossRef] [PubMed] [Google Scholar]
  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. [CrossRef] [Google Scholar]
  27. D. Ambrosi and F. Mollica, On the mechanics of a growing tumor. Int. J. Eng. Sci. 40 (2002) 1297–1316. [CrossRef] [Google Scholar]
  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. [CrossRef] [Google Scholar]
  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. [CrossRef] [Google Scholar]
  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. [CrossRef] [Google Scholar]
  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. [Google Scholar]
  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. [CrossRef] [PubMed] [Google Scholar]
  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. [CrossRef] [PubMed] [Google Scholar]
  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. [Google Scholar]
  35. M. Sarntinoranont, F. Rooney and M. Ferrari, Interstitial stress and fluid pressure within a growing tumor. Ann. Biomed. Eng. 31 (2003) 327–335. [CrossRef] [PubMed] [Google Scholar]
  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. [CrossRef] [PubMed] [Google Scholar]
  37. M. Conti and A. Giorgini, Well-posedness for the Brinkman–Cahn–Hilliard system with unmatched viscosities. J. Differ. Equ. 268 (2020) 6350–6384. [CrossRef] [Google Scholar]
  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. [CrossRef] [MathSciNet] [Google Scholar]
  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. [CrossRef] [Google Scholar]
  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. [CrossRef] [MathSciNet] [Google Scholar]
  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. [CrossRef] [MathSciNet] [Google Scholar]
  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. [CrossRef] [Google Scholar]
  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. [CrossRef] [MathSciNet] [Google Scholar]
  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. [CrossRef] [Google Scholar]
  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. [CrossRef] [Google Scholar]
  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. [CrossRef] [MathSciNet] [Google Scholar]
  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. [CrossRef] [MathSciNet] [Google Scholar]
  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. [CrossRef] [MathSciNet] [Google Scholar]
  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. [CrossRef] [MathSciNet] [Google Scholar]
  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. [CrossRef] [Google Scholar]
  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. [CrossRef] [Google Scholar]
  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. [CrossRef] [Google Scholar]
  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. [Google Scholar]
  54. B. Cockburn, G.E. Karniadakis and C.-W. Shu, The Development of Discontinuous Galerkin methods. Springer, Berlin Heidelberg (2000). [CrossRef] [Google Scholar]
  55. B. Rivière, Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation. SIAM, Philadelphia (2008). [CrossRef] [Google Scholar]
  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. [Google Scholar]
  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. [CrossRef] [MathSciNet] [Google Scholar]
  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. [Google Scholar]
  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. [CrossRef] [MathSciNet] [Google Scholar]
  60. T. Ohta and K. Kawasaki, Equilibrium morphology of block copolymer melts. Macromolecules 9 (1986) 2621–2632. [CrossRef] [Google Scholar]
  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. [CrossRef] [MathSciNet] [Google Scholar]
  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. [MathSciNet] [Google Scholar]
  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. [CrossRef] [MathSciNet] [Google Scholar]
  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. [CrossRef] [Google Scholar]
  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. [CrossRef] [Google Scholar]

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