Mathematical modelling of cancer invasion: The importance of cell-cell adhesion and cell-matrix adhesion

Mathematical Models and Methods in Applied Sciences

Volumn 21, Issue 4, 2011, Pages 719-743

  Chaplain, Mark A J ^a   Lachowicz, Mirosław ^b   Szymańska, Zuzanna ^c   Wrzosek, Dariusz ^b  

^a UNIVERSITY OF DUNDEE   (United Kingdom)

^b UNIVERSITY OF WARSAW   (Poland)

^c UNIVERSITY OF WARSAW   (Poland)

Author keywords

boundedness of solutions; Cancer invasion of tissue; cell cell adhesion; cell matrix adhesion; computational simulations; existence; nonlocal interactions; uniqueness

Indexed keywords

BOUNDEDNESS OF SOLUTION; CANCER INVASION; CELL-CELL ADHESION; CELL-MATRIX ADHESION; COMPUTATIONAL SIMULATION; EXISTENCE; NON-LOCAL INTERACTIONS; UNIQUENESS;

ADHESION; CELL ADHESION; DISEASES; INTEGRODIFFERENTIAL EQUATIONS; MATHEMATICAL MODELS; TISSUE;

CELLS;

EID: 79955497668     PISSN: 02182025     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218202511005192     Document Type: Article

References (46)

1. A. R. A. Anderson, M. A. J. Chaplain, E. L. Newman, R. J. C. Steele and A. M. Thompson, Mathematical modelling of tumour invasion and metastasis, J. Theor. Med. 2 (2000) 129-154.
2. A. R. A. Anderson, A hybrid mathematical model of solid tumour invasion: The importance of cell adhesion, Math. Med. Biol. 22 (2005) 163-186. (Pubitemid 41511340)
3. N. J. Armstrong, K. J. Painter and J. A. Sherratt, A continuum approach to modelling cell-cell adhesion, J. Theor. Biol. 243 (2006) 98-113. (Pubitemid 44465967)
4. N. Bellomo, N. K. 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. (Pubitemid 351730142)
5. D. Bray, Cell Movements: From Molecules to Motility (Garland Publishing, 2000).
6. H. M. Byrne and M. A. J. Chaplain, Modelling the role of cell-cell adhesion in the growth and development of carcinomas, Math. Comput. Model. 24 (1996) 1-17. (Pubitemid 27118822)
7. H. M. Byrne and M. A. J. Chaplain, Free boundary value problems associated with the growth and development of multicellular spheroids, Euro. J. Appl. Math. 8 (1997) 639-658.
8. H. M. Byrne, M. A. J. Chaplain, G. J. Pettet and D. L. S. McElwain, A mathematical model of trophoblast invasion, J. Theor. Med. 1 (1998) 275-286.
9. M. A. J. Chaplain and B. D. Sleeman, Modelling the growth of solid tumours and incorporating a method for their classification using nonlinear elasticity theory, J. Math. Biol. 31 (1991) 431-473.
10. M. A. J. 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. (Pubitemid 41551043)
11. M. A. J. Chaplain and G. Lolas, Mathematical modelling of cancer invasion of tissue: Dynamic heterogeneity, Net. Hetero. Med. 1 (2006) 399-439.
12. J. W. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems (Cambridge Univ. Press, 2000).
13. A. De Pablo and J. L. V-azquez, Travelling waves and finite propagation in a reactiondi ffusion equation, J. Differential Equations 93 (1991) 19-61.
14. F. Filbet, P. Laurençot and B. Perthame, Derivation of hyperbolic models for chemosensitive movement, J. Math. Biol. 50 (2005) 189-207. (Pubitemid 40240749)
15. H. B. Frieboes, X. Zheng, C.-H. Sun, B. Tromberg, R. A. Gatenby and V. Cristini, An integrated computational/experimental model of tumor invasion, Cancer Res. 66 (2006) 1597-1604. (Pubitemid 43259943)
16. P. Friedl and K. Wolf, Tumour-cell invasion and migration: Diversity and escape mechanisms, Nature Rev. Cancer 3 (2003) 362-374. (Pubitemid 37328856)
17. R. A. Gatenby and E. T. Gawlinski, A reaction-diffusion model of cancer invasion, Cancer Res. 56 (1996) 5745-5753. (Pubitemid 26422183)
18. R. A. Gatenby, E. T. Gawlinski, A. F. Gmitro, B. Kaylor and R. J. Gillies, Acid-mediated tumor invasion: A multidisciplinary study, Cancer Res. 66 (2006) 5216-5223. (Pubitemid 43844945)
19. A. Gerisch and M. A. J. Chaplain, Mathematical modelling of cancer cell invasion of tissue: Local and non-local models and the effect of adhesion, J. Theor. Biol. 250 (2008) 684-704.
20. 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.
21. D. Hanahan and R. A. Weinberg, The hallmarks of cancer, Cell 100 (2000) 57-70. (Pubitemid 30046295)
22. D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes Math., Vol. 840 (Springer-Verlag, 1981).
23. M. Lachowicz, Micro and meso scales of description corresponding to a model of tissue invasion by solid tumours, Math. Models Methods Appl. Sci. 15 (2005) 1667-1683. (Pubitemid 41551042)
24. L. A. Liotta and E. C. Kohn, The microenvironment of the tumour-host interface, Nature 411 (2001) 375-379.
25. P. Macklin and J. S. Lowengrub, Nonlinear simulation of the effect of microenvironment on tumor growth, J. Theor. Biol. 254 (2007) 677-704.
26. A. Marciniak-Czochra and M. Ptashnyk, Derivation of a macroscopic receptor-based model using homogenization techniques, SIAM J. Appl. Math. Anal. 40 (2008) 215-237.
27. V. L. Martins, J. J. Vyas, M. Chen, K. Purdie, C. A. Mein, A. P. South, A. Storey, J. A. McGrath and E. A. O'Toole, Increased invasive behaviour in cutaneous squamous cell carcinoma with loss of basement-membrane type VII collagen, J. Cell Sci. 122 (2009) 1788-1799.
28. M. L. Nyström, G. J. Thomas, M. Stone, I. C. Mackenzie, I. R. Hart and J. F. Marshall, Development of a quantitative method to analyse tumour cell invasion in organotypic culture, J. Pathol. 205 (2005) 468-475. (Pubitemid 40361335)
29. J. T. Oden, A. Hawkins and S. Prudhomme, General diffuse-interface theories and an approach to predictive tumor growth modeling, Math. Models Method Appl. Sci. 20 (2010) 477-517.
30. M. E. Orme and M. A. J. Chaplain, Two-dimensional models of tumour angiogenesis and anti-angiogenesis strategies, IMA J. Math. Appl. Med. Biol. 14 (1997) 189-205. (Pubitemid 127463779)
31. A. J. Perumpanani, J. A. Sherratt, J. Norbury and H. M. Byrne, Biological inferences from a mathematical model for malignant invasion, Invasion & Metastasis 16 (1996) 209-221. (Pubitemid 27399787)
32. A. J. Perumpanani, D. L. Simmons, A. J. H. Gearing, K. M. Miller, G. Ward, J. Norbury, M. Schneemann and J. A. Sherratt, Extracellular matrix-mediated chemotaxis can impede cell migration, Proc. Roy. Soc. London, Ser. B 265 (1998) 2347-2352. (Pubitemid 29015439)
33. I. Ramis-Conde, D. Drasdo, M. A. J. Chaplain and A. R. A. Anderson, Modelling the influence of the E-Cadherin-β-Catenin pathway in cancer cell invasion and tissue architecture: A multi-scale approach, Biophys. J. 95 (2008) 155-165.
34. I. Ramis-Conde, M. A. J. Chaplain, A. R. A. Anderson and D. Drasdo, Multi-scale modelling of cancer cell intravasation: The role of cadherins in metastasis, Phys. Biol. 6 016008 (2009).
35. T. Runst and W. Stickel, Sobolev Spaces of Fractional Order, Nemytskij Operators and Nonlinear Partial Differential Equations (Walter de Gruyter, 1996).
36. F. S-anchez-Garduño and P. K. Maini, Existence and uniqueness of a sharp travelling wave in degenerate non-linear diffusion Fisher-KPP equation, J. Math. Biol. 33 (1994) 163-192.
37. F. S-anchez-Garduño and P. K. Maini, Travelling wave phenomena in some degenerate reaction-diffusion equations, J. Differential Equations 117 (1995) 281-319.
38. J. A. Sherratt, S. A. Gourley, N. J. Armstrong and K. J. Painter, Boundedness of solutions of a non-local reaction-diffusion model for adhesion in cell aggregation and cancer invasion, Euro. J. Appl. Math. 20 (2009) 123-144.
39. M. B. Sporn, The war on cancer, Lancet 347 (1996) 1377-1381. (Pubitemid 26147937)
40. Z. Szymańska, J. Urbański and A. Marciniak-Czochra, Mathematical modelling of the influence of heat shock proteins on cancer invasion of tissue, J. Math. Biol. 58 (2009) 819-844.
41. Z. Szymańska, C. Morales Rodrigo, M. Lachowicz and M. A. J. Chaplain, Mathematical modelling of cancer invasion of tissue: The role and effect of nonlocal interactions, Math. Models Methods Appl. Sci. 19 (2009) 157-281.
42. H. Triebel, Interpolation Theory, Function Spaces, Differential Operators (Veb Deutscher, 1978).
43. H. Triebel, Theory of Function Spaces (Birkhäuser, 1983).
44. R. A. Weinberg, The Biology of Cancer (Garland Science, 2007).
45. R. Weiner, B. A. Schmitt and H. Podhaisky, ROWMAP - a ROW-code with Krylov techniques for large stiff ODEs, Appl. Numer. Math. 25 (1997) 303-319. (Pubitemid 127369181)
46. M. H. Zaman, L. M. Trapani, A. L. Sieminski, D. Mackellar, H. Gong, R. D. Kamm, A. Wells, D. A. Lauffenburger and P. Matsudaira, Migration of tumor cells in 3D matrices is governed by matrix stiffness along with cell-matrix adhesion and proteolysis, PNAS 103 (2006) 10,889-10,894. (Pubitemid 44148391)