Fasano, A. and Herrero, Miguel A. and Rodrigo, Marianito R.
(2009)
*Slow and fast invasion waves in a model of acid-mediated tumour growth.*
Mathematical Biosciences, 220
(1).
pp. 45-56.
ISSN 0025-5564

PDF
Restringido a Repository staff only hasta 31 December 2020. 584kB |

Official URL: http://www.sciencedirect.com/science/article/pii/S0025556409000698

## Abstract

This work is concerned with a reaction-diffusion system that has been proposed as a model to describe acid-mediated cancer invasion. More precisely, we consider the properties of travelling waves that can be supported by such a system, and show that a rich variety of wave propagation dynamics, both fast and slow, is compatible with the model. In particular, asymptotic formulae for admissible wave profiles and bounds on their wave speeds are provided.

Item Type: | Article |
---|---|

Uncontrolled Keywords: | Reaction-diffusion systems; Tumour growth; Asymptotic methods; Mathematical biology; Cancer-cell invasion; h+-ion mobility; malignant invasion; ventricular myocyte; excitable media; diffusion; tissue |

Subjects: | Medical sciences > Biology > Biomathematics Medical sciences > Medicine > Oncology Sciences > Mathematics > Operations research |

ID Code: | 16223 |

References: | D. Ambrosi, F. Mollica, Mechanical models in tumour growth, in: L. Preziosi (Ed.), Cancer Modelling and Simulation, Chapman and Hall, 2003, p. 121. A.R.A. Anderson, M.A.J. Chaplain, E.L. Newman, R.J.C. Steele, A.M. Thompson, Mathematical modelling of tumour invasion and metastasis, J. Theor. Med. 2 (2000) 129. A. Bertuzzi, A. Fasano, A. Gandolfi, C. Sinisgalli, ATP production and necrosis formation in a tumour spheroid model, Math. Modell. Nat. Phenom. 2 (2007) 30. N. Bellomo, M.L. Bertotti, S. Motta, Cancer immune system competition: modelling and bifurcation problems, in: L. Preziosi (Ed.), Cancer Modelling and Simulation, Chapman and Hall, 2003, p. 299. N. Bellomo, N.K. Li, P.K. Maini, On the foundations of cancer modelling: selected topics, speculations, and perspectives, Math. Mod. Meth. Appl. Sci. 18 (4) (2008) 1. C.M. Bender, S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York, 1978. L. Bianchini, A. Fasano, A model combining acid-mediated tumour invasion and nutrient dynamics, Nonlinear Anal. Real World Appl. 10 (2009) 1955. H.M. Byrne, Modelling avascular tumour growth, in: L. Preziosi (Ed.), Cancer Modelling and Simulation, Chapman and Hall, 2003, p. 75. H.M. Byrne, M.A.J. Chaplain, G.J. Pettet, D.L.S. McElwain, A mathematical model of trophoblast invasion, J. Theor. Med. 1 (1999) 275. M.A.J. Chaplain, A.R.A. Anderson, Mathematical modelling of tissue invasion, in: L. Preziosi (Ed.), Cancer Modelling and Simulation, Chapman and Hall, 2003, p. 269. M.A.J. Chaplain, 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. R.A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics 7 (1937) 353. R.A. Gatenby, Models of tumour–host interaction as competing populations: Implications for tumour biology and treatment, J. Theor. Biol. 4 (21) (1995) 447. R.A. Gatenby, E.T. Gawlinski, A reaction–diffusion model for cancer invasion, Cancer Res. 56 (1996) 5745. R.A. Gatenby, E.T. Gawlinski, The glycolitic phenotype in carcinogenesis and tumour invasion: insights through mathematical modelling, Cancer Res. 63 (2003) 3847. A. Gerisch, M.A.J. Chaplain, Mathematical modelling of cancer cell invasion of tissue: Local and nonlocal models and the effect of adhesion, J. Theor. Biol. 250 (2008) 684. H.P. Greenspan, Models for the growth of a solid tumour by diffusion, Stud. Appl. Math. 51 (1972) 317. J.P. Keener, Waves in excitable media, SIAM J. Appl. Math. 39 (1980) 528. J.P. Keener, A geometrical theory for spiral waves in excitable media, SIAM J. Appl. Math. 46 (1986) 1039. 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). B.P. Marchant, J. Norbury, A.J. Perumpani, Travelling shock waves arising in a model of malignant invasion, SIAM J. Appl. Math. 60 (2) (2000) 463. B.P. Marchant, J. Norbury, J.A. Sherratt, Travelling wave solutions to a haptotaxis-dominated model of malignant invasion, Nonlinearity 14 (6) (2001) 1653. A.S. Mikhailov, Foundations of Synergetics I, Springer-Verlag, Heidelberg, 1994. A.J. Perumpani, J.A. Sherratt, J. Norbury, Biological inferences from a mathematical model for malignant invasion, Invasion Metastasis 16 (4–5) (1996) 209. A.J. Perumpani, J.A. Sherratt, J. Norbury, H.M. Byrne, 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. J.A. Sherratt, B.P. Marchant, Non-sharp travelling wave fronts in the Fisher equation with degenerate nonlinear diffusion, Appl. Math. Lett. 9 (5) (1996) 33. K. Smallbone, D.J. Gavaghan, R.A. Gatenby, P.K. Maini, The role of acidity in solid tumour growth and invasion, Br. J. Radiol. 76 (2005) S11. P. Swietach, K.W. Spitzer, R.D. Vaughan-Jones, pH-dependence of extrinsic and intrinsic Hþ-ion mobility in the rat ventricular myocyte, investigated using flash photolysis of a caged-Hþ compound, Biophys. J. 15 (2007) 641. R.D. Vaughan-Jones, B.E. Peercy, J.P. Keener, K.W. Spitzer, Intrinsic Hþ-ion mobility in the rabbit ventricular myocyte, J. Physiol. 541 (1) (2002) 139. R. Venkatasasubramian, M.A. Henson, N.S. Forbes, Incorporating energy metabolism into a growth model of multicellular tumour spheroids, J. Theor. Biol. 242 (2006) 440. A.T. Winfree, The Geometry of Biological Time, Springer-Verlag, Heidelberg, 1980. A.T. Winfree, When Time Breaks Down, Princeton University Press, Princeton, 1987. |

Deposited On: | 10 Sep 2012 11:05 |

Last Modified: | 07 Feb 2014 09:25 |

Repository Staff Only: item control page