No Access

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

Technical University of Munich, Department of Mathematics, Boltzmannstraße 3, 85748 Garching, Germany

Corresponding author.

The University of Texas at Austin, Oden Institute for Computational Engineering and Sciences, 201 East 24th St, Austin, TX 78712-1229, USA

E-mail Address: [email protected]

Search for more papers by this author

Vanja Nikolić

Technical University of Munich, Department of Mathematics, Boltzmannstraße 3, 85748 Garching, Germany

E-mail Address: [email protected]

Search for more papers by this author

J. Tinsley Oden

The University of Texas at Austin, Oden Institute for Computational Engineering and Sciences, 201 East 24th St, Austin, TX 78712-1229, USA

E-mail Address: [email protected]

Search for more papers by this author

, and

Barbara Wohlmuth

Technical University of Munich, Department of Mathematics, Boltzmannstraße 3, 85748 Garching, Germany

E-mail Address: [email protected]

Search for more papers by this author

https://doi.org/10.1142/S0218202519500519Cited by:24 (Source: Crossref)

Abstract

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

Keywords:

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, Web of Science, Google 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, Web of Science, Google 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, Web of Science, Google 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). Crossref, Google 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, Web of Science, Google 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, Web of Science, Google 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, Web of Science, Google 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, Web of Science, Google Scholar
11. T. S. Deisboeck and G. S. Stamatakos, Multiscale Cancer Modeling (CRC Press, 2010). Crossref, Google 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, Web of Science, Google 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, Web of Science, Google 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, Web of Science, Google 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, Web of Science, Google 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, Web of Science, Google 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, Web of Science, Google Scholar
18. L. C. Evans, Partial Differential Equations (Amer. Math. Soc., 2010). Crossref, Google 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, Web of Science, Google 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. Crossref, Google 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, Web of Science, Google 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, Web of Science, Google 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, Web of Science, Google 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. Crossref, Google 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, Web of Science, Google 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, Web of Science, Google 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, Web of Science, Google 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, Web of Science, Google 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, Web of Science, Google 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, Web of Science, Google 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, Web of Science, Google 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, Web of Science, Google 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, Web of Science, Google 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, Web of Science, Google Scholar
36. G. Leoni and M. Morini, Necessary and sufficient conditions for the chain rule in $W_{loc}^{1, 1} (ℝ^{n}; ℝ^{d})$ and ${B V}_{loc} (ℝ^{n}; ℝ^{d})$ , J. Eur. Math. Soc. 9 (2007) 219–252. Crossref, Web of Science, Google Scholar
37. E. H. Lieb and M. Loss, Analysis (Amer. Math. Soc., 2001). Crossref, Google 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, Web of Science, Google 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, Web of Science, Google 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, Web of Science, Google 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, Web of Science, Google 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, Web of Science, Google Scholar
44. T. Mengesha and D. Spector, Localization of nonlocal gradients in various topologies, Calc. Var. Partial Differential Equations 52 (2015) 253–279. Crossref, Web of Science, Google 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, Web of Science, Google 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, Web of Science, Google 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, Web of Science, Google 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). Crossref, Google Scholar
52. T. Roubíček, Nonlinear Partial Differential Equations with Applications (Birkhäuser, 2013). Crossref, Google Scholar
53. J. Simon, Compact sets in the space $L^{p} (0, T; B)$ , Ann. Math. Pura Appl. 146 (1986) 65–96. Crossref, Web of Science, Google 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, Web of Science, Google Scholar
55. W. Strauss, On continuity of functions with values in various Banach spaces, Pacific J. Math. 19 (1966) 543–551. Crossref, Web of Science, Google 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, Web of Science, Google 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, Web of Science, Google Scholar
58. W. Walter, Ordinary Differential Equations (Springer-Verlag 1998). Crossref, Google 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, Web of Science, Google Scholar
60. W. P. Ziemer, Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation (Springer-Verlag, 1989). Crossref, Google Scholar