World Scientific
  • Search
  •   
Skip main navigation

Cookies Notification

We use cookies on this site to enhance your user experience. By continuing to browse the site, you consent to the use of our cookies. Learn More
×
Our website is made possible by displaying certain online content using javascript.
In order to view the full content, please disable your ad blocker or whitelist our website www.worldscientific.com.

System Upgrade on Tue, Oct 25th, 2022 at 2am (EDT)

Existing users will be able to log into the site and access content. However, E-commerce and registration of new users may not be available for up to 12 hours.
For online purchase, please visit us again. Contact us at [email protected] for any enquiries.

Local and nonlocal phase-field models of tumor growth and invasion due to ECM degradation

    https://doi.org/10.1142/S0218202519500519Cited by:19 (Source: Crossref)

    We present and analyze new multi-species phase-field mathematical models of tumor growth and ECM invasion. The local and nonlocal mathematical models describe the evolution of volume fractions of tumor cells, viable cells (proliferative and hypoxic cells), necrotic cells, and the evolution of matrix-degenerative enzyme (MDE) and extracellular matrix (ECM), together with chemotaxis, haptotaxis, apoptosis, nutrient distribution, and cell-to-matrix adhesion. We provide a rigorous proof of the existence of solutions of the coupled system with gradient-based and adhesion-based haptotaxis effects. In addition, we discuss finite element discretizations of the model, and we present the results of numerical experiments designed to show the relative importance and roles of various effects, including cell mobility, proliferation, necrosis, hypoxia, and nutrient concentration on the generation of MDEs and the degradation of the ECM.

    Communicated by N. Bellomo

    AMSC: 35K35, 35A01, 35D30, 35Q92, 65M60

    References

    • 1. A. R. Anderson, M. A. Chaplain, E. L. Newman, R. J. Steele and A. M. Thompson, Mathematical modelling of tumour invasion and metastasis, Comput. Math. Methods Med. 2 (2000) 129–154. Google Scholar
    • 2. R. P. Araujo and D. S. McElwain, A history of the study of solid tumour growth: The contribution of mathematical modelling, Bull. Math. Biol. 66 (2014) 1039–1091. Crossref, ISIGoogle Scholar
    • 3. N. J. Armstrong, K. J. Painter and J. A. Sherratt, A continuum approach to modelling cell–cell adhesion, J. Theoret. Biol. 243 (2006) 98–113. Crossref, ISIGoogle Scholar
    • 4. N. Bellomo, N. Li and P. K. Maini, On the foundations of cancer modelling: Selected topics, speculations, and perspectives, Math. Models Methods Appl. Sci. 18 (2008) 593–646. Link, ISIGoogle Scholar
    • 5. F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier–Stokes Equations and Related Models (Springer-Verlag, 2012). Google Scholar
    • 6. H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations (Springer-Verlag, 2010). CrossrefGoogle Scholar
    • 7. M. A. Chaplain, M. Lachowicz, Z. Szymańska and D. Wrzosek, Mathematical modelling of cancer invasion: The importance of cell-cell adhesion and cell-matrix adhesion, Math. Models Methods Appl. Sci. 21 (2011) 719–743. Link, ISIGoogle Scholar
    • 8. M. A. Chaplain and G. Lolas, Mathematical modelling of cancer cell invasion of tissue: The role of the urokinase plasminogen activation system, Math. Models Methods Appl. Sci. 15 (2005) 1685–1734. Link, ISIGoogle Scholar
    • 9. P. Colli, G. Gilardi and D. Hilhorst, On a Cahn–Hilliard type phase field system related to tumor growth, Discrete Contin. Dyn. Syst. – A 35 (2015) 2423–2442. Crossref, ISIGoogle Scholar
    • 10. M. Dai, E. Feireisl, E. Rocca, G. Schimperna and M. E. Schonbek, Analysis of a diffuse interface model of multispecies tumor growth, Nonlinearity 30 (2017) 1639–1658. Crossref, ISIGoogle Scholar
    • 11. T. S. Deisboeck and G. S. Stamatakos, Multiscale Cancer Modeling (CRC Press, 2010). CrossrefGoogle Scholar
    • 12. F. Della Porta, A. Giorgini and M. Grasselli, The nonlocal Cahn–Hilliard–Hele–Shaw system with logarithmic potential, Nonlinearity 31 (2018) 4851–4881. Crossref, ISIGoogle Scholar
    • 13. F. Della Porta and M. Grasselli, On the nonlocal Cahn–Hilliard–Brinkman and Cahn–Hilliard–Hele–Shaw systems, Commun. Math. Sci. 13 (2015) 1541–1567. Crossref, ISIGoogle Scholar
    • 14. Q. Du, M. Gunzburger, R. B. Lehoucq and K. Zhou, A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws, Math. Models Methods Appl. Sci. 23 (2013) 493–540. Link, ISIGoogle Scholar
    • 15. M. Ebenbeck and H. Garcke, Analysis of a Cahn–Hilliard–Brinkman model for tumour growth with chemotaxis, J. Differential Equations 266 (2019) 5998–6036. Crossref, ISIGoogle Scholar
    • 16. 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. Crossref, ISIGoogle Scholar
    • 17. C. Engwer, C. Stinner and C. Surulescu, On a structured multiscale model for acid-mediated tumor invasion: The effects of adhesion and proliferation, Math. Models Methods Appl. Sci. 27 (2017) 1355–1390. Link, ISIGoogle Scholar
    • 18. L. C. Evans, Partial Differential Equations (Amer. Math. Soc., 2010). CrossrefGoogle Scholar
    • 19. S. Frigeri, M. Grasselli and E. Rocca, A diffuse interface model for two-phase incompressible flows with nonlocal interactions and non-constant mobility, Nonlinearity 28 (2015) 1257–1293. Crossref, ISIGoogle Scholar
    • 20. S. Frigeri, K. F. Lam and E. Rocca, On a diffuse interface model for tumour growth with nonlocal interactions and degenerate mobilities, in Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs (Springer, 2017), pp. 217–254. CrossrefGoogle Scholar
    • 21. 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, Math. Models Methods Appl. Sci. 29 (2019) 1691–1731. Link, ISIGoogle Scholar
    • 22. 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. Crossref, ISIGoogle Scholar
    • 23. H. Garcke and K. F. Lam, Well-posedness of a Cahn–Hilliard system modelling tumour growth with chemotaxis and active ort, European J. Appl. Math. 28 (2017) 284–316. Crossref, ISIGoogle Scholar
    • 24. H. Garcke and K. F. Lam, On a Cahn–Hilliard–Darcy system for tumour growth with solution dependent source terms, in Trends in Applications of Mathematics to Mechanics (Springer, 2018), pp. 243–264. CrossrefGoogle Scholar
    • 25. 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. Link, ISIGoogle Scholar
    • 26. 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. Link, ISIGoogle Scholar
    • 27. R. A. Gatenby, Models of tumor-host interaction as competing populations: Implications for tumor biology and treatment, J. Theoret. Biol. 176 (1995) 447–455. Crossref, ISIGoogle Scholar
    • 28. A. Gerisch, On the approximation and efficient evaluation of integral terms in pde models of cell adhesion, IMA J. Numer. Anal. 30 (2010) 173–194. Crossref, ISIGoogle Scholar
    • 29. A. Gerisch and M. Chaplain, Mathematical modelling of cancer cell invasion of tissue: Local and nonlocal models and the effect of adhesion, J. Theoret. Biol. 250 (2008) 684–704. Crossref, ISIGoogle Scholar
    • 30. A. Hawkins-Daarud, K. G. van der Zee and T. J. Oden, Numerical simulation of a thermodynamically consistent four-species tumor growth model, Int. J. Numer. Methods Biomed. Eng. 28 (2021) 3–24. Crossref, ISIGoogle Scholar
    • 31. T. Hillen, K. J. Painter and M. Winkler, Convergence of a cancer invasion model to a logistic chemotaxis model, Math. Models Methods Appl. Sci. 23 (2013) 165–198. Link, ISIGoogle Scholar
    • 32. 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. Differential Equations 259 (2015) 3032–3077. Crossref, ISIGoogle Scholar
    • 33. B. S. Kirk, J. W. Peterson, R. H. Stogner and G. F. Carey, libMesh: A C++ library for parallel adaptive mesh refinement/coarsening simulations, Eng. Comput. 22 (2006) 237–254. Crossref, ISIGoogle Scholar
    • 34. V. Lakshmikantham and M. Rama Mohana Rao, Theory of Integro-Differential Equations (Gordon and Breach Sci. Publ., 1995). Google Scholar
    • 35. K. F. Lam and H. Wu, Thermodynamically consistent Navier–Stokes–Cahn–Hilliard models with mass transfer and chemotaxis, European J. Appl. Math. 29 (2018) 595–644. Crossref, ISIGoogle Scholar
    • 36. G. Leoni and M. Morini, Necessary and sufficient conditions for the chain rule in Wloc1,1(n; d) and BVloc(n; d), J. Eur. Math. Soc. 9 (2007) 219–252. Crossref, ISIGoogle Scholar
    • 37. E. H. Lieb and M. Loss, Analysis (Amer. Math. Soc., 2001). CrossrefGoogle Scholar
    • 38. E. A. Lima, R. C. Almeida and J. T. Oden, Analysis and numerical solution of stochastic phase-field models of tumor growth, Numer. Methods Partial Differential Equations 31 (2015) 552–574. Crossref, ISIGoogle Scholar
    • 39. E. A. Lima, J. T. Oden and R. C. Almeida, A hybrid ten-species phase-field model of tumor growth, Math. Models Methods Appl. Sci. 24 (2014) 2569–2599. Link, ISIGoogle Scholar
    • 40. E. Lima, J. 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, Comput. Methods Appl. Mech. Engrg. 327 (2017) 277–305. Crossref, ISIGoogle Scholar
    • 41. J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications I (Springer-Verlag, 2012). Google Scholar
    • 42. D. H. Madsen and T. H. Bugge, The source of matrix-degrading enzymes in human cancer: Problems of research reproducibility and possible solutions, J. Cell. Biol. 209 (2015) 195–198. Crossref, ISIGoogle Scholar
    • 43. B. Marchant, J. Norbury and J. Sherratt, Travelling wave solutions to a haptotaxis-dominated model of malignant invasion, Nonlinearity 14 (2001) 1653–1671. Crossref, ISIGoogle Scholar
    • 44. T. Mengesha and D. Spector, Localization of nonlocal gradients in various topologies, Calc. Var. Partial Differential Equations 52 (2015) 253–279. Crossref, ISIGoogle Scholar
    • 45. F. Murat and C. Trombetti, A chain rule formula for the composition of a vector-valued function by a piecewise smooth function, Boll. Unione Mat. Ital. 6 (2003) 581–595. Google Scholar
    • 46. N. Nargis and R. Aldredge, Effects of matrix metalloproteinase on tumour growth and morphology via haptotaxis, J. Bioengineer and Biomedical Sci. 6 (2016). https://doi.org/10.4172/2155-9538.1000207 Google Scholar
    • 47. J. T. Oden et al., Toward predictive multiscale modeling of vascular tumor growth, Arch. Comput. Methods Eng. 23 (2016) 735–779. Crossref, ISIGoogle Scholar
    • 48. L. Peng, D. Trucu, P. Lin, A. Thompson and M. A. Chaplain, A multiscale mathematical model of tumour invasive growth, Bull. Math. Biol. 79 (2017) 389–429. Crossref, ISIGoogle Scholar
    • 49. A. Perumpanani, B. Marchant and J. Norbury, Traveling shock waves arising in a model of malignant invasion, SIAM J. Appl. Math. 60 (2000) 463–476. Crossref, ISIGoogle Scholar
    • 50. B. Perumpani, A. Sherratt, J. Norbury and H. Byrne, Biological inferences from a mathematical model for malignant invasion, Invasion Metastasis 16 (1996) 209–221. Google Scholar
    • 51. J. C. Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors (Cambridge Univ. Press, 2001). CrossrefGoogle Scholar
    • 52. T. Roubíček, Nonlinear Partial Differential Equations with Applications (Birkhäuser, 2013). CrossrefGoogle Scholar
    • 53. J. Simon, Compact sets in the space Lp(0,T;B), Ann. Math. Pura Appl. 146 (1986) 65–96. Crossref, ISIGoogle Scholar
    • 54. C. Stinner, C. Surulescu and M. Winkler, Global weak solutions in a PDE-ODE system modeling multiscale cancer cell invasion, SIAM J. Math. Anal. 46 (2014) 1969–2007. Crossref, ISIGoogle Scholar
    • 55. W. Strauss, On continuity of functions with values in various Banach spaces, Pacific J. Math. 19 (1966) 543–551. Crossref, ISIGoogle Scholar
    • 56. Y. Tao and M. Winkler, A chemotaxis-haptotaxis model: The roles of nonlinear diffusion and logistic source, SIAM Math. Anal. 43 (2011) 685–704. Crossref, ISIGoogle Scholar
    • 57. C. Walker and G. F. Webb, Global existence of classical solutions for a haptotaxis model, SIAM J. Math. Anal. 38 (2007) 1694–1713. Crossref, ISIGoogle Scholar
    • 58. W. Walter, Ordinary Differential Equations (Springer-Verlag 1998). CrossrefGoogle Scholar
    • 59. S. M. Wise, J. S. Lowengrub, H. B. Frieboes and V. Cristini, Three-dimensional multispecies nonlinear tumor growth I: Model and numerical method, J. Theoret. Biol. 253 (2008) 524–543. Crossref, ISIGoogle Scholar
    • 60. W. P. Ziemer, Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation (Springer-Verlag, 1989). CrossrefGoogle Scholar
    Remember to check out the Most Cited Articles!

    View our Mathematical Modelling books
    Featuring authors Frederic Y M Wan, Gregory Baker and more!