Slow and fast invasion waves in a model of acid-mediated tumour growth

Mathematical Biosciences

Volumn 220, Issue 1, 2009, Pages 45-56

  Fasano, Antonio ^a   Herrero, Miguel A ^b   Rodrigo, Marianito R ^c  

^a UNIVERSITY OF FLORENCE   (Italy)

^b UNIVERSIDAD COMPLUTENSE DE MADRID   (Spain)

^c INSTITUTO TECNOLÓGICO AUTÓNOMO DE MÉXICO   (Mexico)

Author keywords

35B55; 35K57; 92B05; 92C15; Asymptotic methods; Mathematical biology; Reaction diffusion systems; Tumour growth

Indexed keywords

35B55; 35K57; 92B05; 92C15; ASYMPTOTIC METHODS; MATHEMATICAL BIOLOGY; REACTION-DIFFUSION SYSTEMS; TUMOUR GROWTH;

ACIDS; ASYMPTOTIC ANALYSIS; DIFFUSION IN LIQUIDS;

TUMORS;

BIOLOGICAL INVASION; CANCER; GROWTH MODELING; NUMERICAL METHOD; TUMOR;

ARTICLE; BIOLOGICAL MODEL; CANCER INVASION; DIFFUSION; DYNAMICS; MATHEMATICAL MODEL; TUMOR GROWTH; VELOCITY;

ALGORITHMS; ANIMALS; CARCINOMA, SQUAMOUS CELL; CELL MOVEMENT; DIFFUSION; DISEASE PROGRESSION; HUMANS; HYDROGEN-ION CONCENTRATION; MODELS, BIOLOGICAL; NEOPLASM INVASIVENESS; NEOPLASMS; PROTONS; TIME FACTORS;

EID: 67349225635     PISSN: 00255564     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.mbs.2009.04.001     Document Type: Article

References (32)

1. Ambrosi D., and Mollica F. Mechanical models in tumour growth. In: Preziosi L. (Ed). Cancer Modelling and Simulation (2003), Chapman and Hall 121
2. Anderson A.R.A., Chaplain M.A.J., Newman E.L., Steele R.J.C., and Thompson A.M. Mathematical modelling of tumour invasion and metastasis. J. Theor. Med. 2 (2000) 129
3. Bertuzzi A., Fasano A., Gandolfi A., and Sinisgalli C. ATP production and necrosis formation in a tumour spheroid model. Math. Modell. Nat. Phenom. 2 (2007) 30
4. Bellomo N., Bertotti M.L., and Motta S. Cancer immune system competition: modelling and bifurcation problems. In: Preziosi L. (Ed). Cancer Modelling and Simulation (2003), Chapman and Hall 299
5. Bellomo N., Li N.K., and Maini P.K. On the foundations of cancer modelling: selected topics, speculations, and perspectives. Math. Mod. Meth. Appl. Sci. 18 4 (2008) 1
6. Bender C.M., and Orszag S.A. Advanced Mathematical Methods for Scientists and Engineers (1978), McGraw-Hill, New York
7. Bianchini L., and Fasano A. A model combining acid-mediated tumour invasion and nutrient dynamics. Nonlinear Anal. Real World Appl. 10 (2009) 1955
8. Byrne H.M. Modelling avascular tumour growth. In: Preziosi L. (Ed). Cancer Modelling and Simulation (2003), Chapman and Hall 75
9. Byrne H.M., Chaplain M.A.J., Pettet G.J., and McElwain D.L.S. A mathematical model of trophoblast invasion. J. Theor. Med. 1 (1999) 275
10. Chaplain M.A.J., and Anderson A.R.A. Mathematical modelling of tissue invasion. In: Preziosi L. (Ed). Cancer Modelling and Simulation (2003), Chapman and Hall 269
11. Chaplain M.A.J., and Lolas G. Mathematical modelling of cancer cell invasion of tissue: the role of the urokinase plasminogen activation system. Math. Mod. Meth. Appl. Sci. 15 (2005) 1685
12. Fisher R.A. The wave of advance of advantageous genes. Ann. Eugenics 7 (1937) 353
13. Gatenby R.A. Models of tumour-host interaction as competing populations: Implications for tumour biology and treatment. J. Theor. Biol. 4 21 (1995) 447
14. Gatenby R.A., and Gawlinski E.T. A reaction-diffusion model for cancer invasion. Cancer Res. 56 (1996) 5745
15. Gatenby R.A., and Gawlinski E.T. The glycolitic phenotype in carcinogenesis and tumour invasion: insights through mathematical modelling. Cancer Res. 63 (2003) 3847
16. Gerisch A., and Chaplain M.A.J. Mathematical modelling of cancer cell invasion of tissue: Local and nonlocal models and the effect of adhesion. J. Theor. Biol. 250 (2008) 684
17. Greenspan H.P. Models for the growth of a solid tumour by diffusion. Stud. Appl. Math. 51 (1972) 317
18. Keener J.P. Waves in excitable media. SIAM J. Appl. Math. 39 (1980) 528
19. Keener J.P. A geometrical theory for spiral waves in excitable media. SIAM J. Appl. Math. 46 (1986) 1039
20. A.A. Kolmogorov, I.G. Petrovsky, N.S. Piskunov, Study of the diffusion equation with growth of the quantity of matter and its application to a biology problem, vol. 17, 1937, Bul. Moskovskovo Gos. Univ., Academic Press., p. 1. (in Russian). The Dynamics of Curved Fronts, (P. Pelcé (Ed.), Trans), 1988 (Original work published 1937).
21. Marchant B.P., Norbury J., and Perumpani A.J. Travelling shock waves arising in a model of malignant invasion. SIAM J. Appl. Math. 60 2 (2000) 463
22. Marchant B.P., Norbury J., and Sherratt J.A. Travelling wave solutions to a haptotaxis-dominated model of malignant invasion. Nonlinearity 14 6 (2001) 1653
23. Mikhailov A.S. Foundations of Synergetics I (1994), Springer-Verlag, Heidelberg
24. Perumpani A.J., Sherratt J.A., and Norbury J. Biological inferences from a mathematical model for malignant invasion. Invasion Metastasis 16 4-5 (1996) 209
25. Perumpani A.J., Sherratt J.A., Norbury J., and Byrne H.M. A two-parameter family of travelling waves with a singular barrier arising from the modelling of extracellular matrix-mediated cellular invasion. Physica D 126 (1999) 145
26. Sherratt J.A., and Marchant B.P. Non-sharp travelling wave fronts in the Fisher equation with degenerate nonlinear diffusion. Appl. Math. Lett. 9 5 (1996) 33
27. Smallbone K., Gavaghan D.J., Gatenby R.A., and Maini P.K. The role of acidity in solid tumour growth and invasion. Br. J. Radiol. 76 (2005) S11
28. + compound. Biophys. J. 15 (2007) 641
29. +-ion mobility in the rabbit ventricular myocyte. J. Physiol. 541 1 (2002) 139
30. Venkatasasubramian R., Henson M.A., and Forbes N.S. Incorporating energy metabolism into a growth model of multicellular tumour spheroids. J. Theor. Biol. 242 (2006) 440
31. Winfree A.T. The Geometry of Biological Time (1980), Springer-Verlag, Heidelberg
32. Winfree A.T. When Time Breaks Down (1987), Princeton University Press, Princeton