Open Access
Volume 56, Number 6, November-December 2022
Page(s) 2141 - 2180
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]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.

Recommended for you