Mathematical modelling of cancer invasion of tissue: The role and effect of nonlocal interactions

Mathematical Models and Methods in Applied Sciences

Volumn 19, Issue 2, 2009, Pages 257-281

  Szymańska, Zuzanna ^a   Rodrigo, Cristian Morales ^a   Lachowicz, Miroslław ^a   Chaplain, Mark A J ^b  

^a UNIVERSITY OF WARSAW   (Poland)

^b UNIVERSITY OF DUNDEE   (United Kingdom)

Author keywords

Cancer invasion of tissue; Computational simulations; Existence, uniqueness, regularity of solutions; Haptotaxis; Nonlocal interactions; Travelling waves

Indexed keywords

CANCER INVASION OF TISSUE; COMPUTATIONAL SIMULATIONS; EXISTENCE, UNIQUENESS, REGULARITY OF SOLUTIONS; HAPTOTAXIS; NONLOCAL INTERACTIONS; TRAVELLING WAVES;

CATALYSTS; CELL PROLIFERATION; COMPUTER SIMULATION; DEGRADATION; ENZYMES; GREEN'S FUNCTION; MATHEMATICAL MODELS; TISSUE; WAVE TRANSMISSION;

PARTIAL DIFFERENTIAL EQUATIONS;

EID: 65249129127     PISSN: 02182025     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218202509003425     Document Type: Article

References (40)

1. R. A. Adams, Sobolev Spaces (Academic Press, 1975).
2. 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.
3. A. R. A. Anderson, A hybrid mathematical model of solid tumour invasion: The importance of cell adhesion, Math. Med. Biol. 22 (2005) 163-186.
4. S. Aznavoorian, M. L. Stracke, H. Krutzsch, E. Schiffmann and L. A. Liotta, Signal transduction for chemotaxis and haptotaxis by matrix molecules in tumor cells, J. CellBiol. 110 (1990) 1427-1438.
5. 3 mediates chemotactic and haptotactic motility in human melanoma cells through different signaling pathways, J. Biol. Chem. 271 (1996) 3247-3254.
6. N. Bellomo and P. Maini, Preface Challenging mathematical problems in cancer modelling, Math. Mod. Meth. Appl. Sci 7 (2007) 1641-1646.
7. N. Bellomo, A. Bellouquid, J. Nieto and J. J. Soler, Multicellular growing systems: Hyperbolic limits towards macroscopic description, Math. Mod. Meth. Appl. Sci. 17 (2007) 1675-1693.
8. A. Bellouquid and M. Delitala, Mathematical methods and tools of kinetic theory towards modelling complex biological systems, Math. Mod. Meth. Appl. Sci. 15 (2005) 1639-1666.
9. D. Bray, Cell Movements: From Molecules to Motility (Garland Publishing, 2000).
10. N. F. Britton, Aggregation and the competitive exclusion principle, J. Theor. Biol. 136 (1989) 57-66.
11. N. F. Britton, Spatial structures and periodic travelling waves in an integro-differential reaction-diffusion population model, SIAM J. Appl. Math. 50 (1990) 1663-1688.
12. A. Q. Cai, A. Lindman and B. D. Huges, Modelling directional guidance and motility regulation in cell migration Bull. Math. Biol. 68 (2006) 25-52.
13. S. B. Carter, Haptotaxis and the mechanism of cell motility, Nature 213 (1967) 256-260.
14. M. A. J. Chaplain and A. R. A. Anderson, Mathematical Modelling of Tissue Invasion, in Cancer Modelling and Simulation, ed. L. Preziosi (Chapman & Hall/CRT, 2003), pp. 269-297.
15. M. A. J. Chaplain and G. Lolas, Mathematical modelling of cancer cell invasion of tissue: The role of the urokinase plasminogen activation system, Math. Mod. Meth.Appl. Sci. 15 (2005) 1685-1734.
16. M. A. J. Chaplain and G. Lolas, Mathematical modelling of cancer invasion of tissue: Dynamic heterogeneity, Net. Hetero. Med. 1 (2006) 399-439.
17. W. D. Comper, Extracellular Matrix, Vol. 2 Molecular Components and Interactions (Hartwood Academic, 1996).
18. L. Corrias, B. Perthame and H. Zaag, Global solutions of some chemotaxis and angiogenesis systems in high space dimensions, Milan J. Math. 72 (2004) 1-28.
19. A. De Pablo and J. L. Vázquez, Travelling waves and finite propagation in a reactiondiffusion equation, J. Diff. Eqns. 93 (1991) 19-61.
20. F. Filbet, P. Laurencot and B. Perthame, Derivation of hyperbolic models forchemosensitive movement, J. Math. Biol. 50 (2005) 189-207.
21. S. A. Gourley, M. A. J. Chaplain and F. A. Davidson, Spatio-temporal pattern formation in a nonlocal reaction-diffusion equation, Dyn. Sys. 16 (2001) 173-192.
22. D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes Math., Vol. 840 (Springer, 1981).
23. M. Lachowicz, Micro and meso scales of description corresponding to a model of tissue invasion by solid tumours, Math. Mod. Meth. Appl. Sci. 15 (2005) 1667-1683.
24. J. B. McCarthy, S. L. Palm and L. T. Furcht, Migration by haptotaxis of a Schwanncell tumor line to the basement membrane glycoprotein laminin, J. Cell Biol. 97 (1983) 772-777.
25. J. B. McCarthy, S. T. Hagen and L. T. Furcht, Human fibronectin contains distinct adhesion- and motility-promoting domains for metastatic melanoma cells, J. Cell Biol. 102(1986) 179-188.
26. C. Morales-Rodrigo, Local existence in a model of tissue invasion by solid tumours, Math. Comp. Model., in press doi: 10.1016/j.mcm.2007.031.
27. L. Olsen, J. A. Sherratt, P. K. Maini and F. Arnold, A mathematical model for the capillary endothelial cell-extracellular matrix interactions in wound-healing angiogenesis, IMA J. Math. Appl. Med. Biol. 14 (1997) 261-281.
28. 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.
29. B. Perthame, Transport Equation in Biology (Birkhäuser, 2007).
30. Ed. L. Preziosi, Cancer Modelling and Simulation (Chapman & Hall/CRT, 2003).
31. I. Rubinstein and L. Rubinstein, Partial Differential Equations in MathematicalPhysics (Cambridge Univ. Press, 1998).
32. F. Sánchez-Garduñeo 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.
33. F. Sánchez-Garduñeo and P. K. Maini, Travelling wave phenomena in some degenerate reaction-diffusion equations, J. Diff. Eqns. 117 (1995) 281-319.
34. J. A. Sherratt and J. D. Murray, Models of epidermal wound healing, Proc. Roy. Soc.London, Ser. B 241 (1990) 29-36.
35. J. A. Sherratt, Chemotaxis and chemokinesis in eukaryotic cells: the Keller-Segel equations as an approximation to a detailed model, Bull. Math. Biol. 56 (1994 129-146.
36. C. L. Stokes and D. Lauffenburger, Analysis of the roles of microvessel endothelial cell random motility and chemotaxis in angiogenesis, J. Theor. Biol. 152 (1991) 377-403.
37. G. Taraboletti, D. D. Roberts and L. A. Liotta, Thrombospondin-induced tumor cell migration: Haptotaxis and chemotaxis are mediated by different molecular domains, J. Cell Biol. 105 (1987) 2409-2415.
38. W. Yu, J. Kim and L. Ossowski, Reduction in surface urokinase receptor forces malignant cells into a protracted state of dormancy, J. Cell Biol. 137 (1997) 767-777.
39. 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) 10889-10894.
40. E. Zauderer, Partial Differential Equations of Applied Mathematics, 2nd edn. (John Wiley & Sons, 1989).