Physica D 126 (1999) 145–159 A two parameter family of travelling waves with a singular barrier arising from the modelling of extracellular matrix mediated cellular invasion Abbey J. Perumpanani a,b , Jonathan A. Sherratt c,∗ , John Norbury d , Helen M. Byrne e a Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK b Department of Surgery, Harvard Medical School, Bldg 1400 W, One Kendall Square, Cambridge, MA 02139, USA c Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS, UK d Mathematical Institute, University of Oxford, 24-29 St.Giles’, Oxford OX3 9DU, UK e Department of Theoretical Mechanics, University of Nottingham, University Park, Nottingham NG7 2RD, UK Received 25 May 1998; received in revised form 18 August 1998; accepted 1 October 1998 Communicated by F.H. Busse Abstract Invasive cells variously show changes in adhesion, protease production and motility. In this paper the authors develop and analyse a model for malignant invasion, brought about by a combination of proteolysis and haptotaxis. A common feature of these two mechanisms is that they can be produced by contact with the extracellular matrix through the mediation of a class of surface receptors called integrins. An unusual feature of the model is the absence of cell diffusion. By seeking travelling wave solutions the model is reduced to a system of ordinary differential equations which can be studied using phase plane analysis. The authors demonstrate the presence of a singular barrier in the phase plane and a “hole” in this singular barrier which admits a phase trajectory. The model admits a family of travelling waves which depend on two parameters, i.e. the tissue concentration of connective tissue and the rate of decay of the initial spatial profile of the invading cells. The slowest member of this family corresponds to the phase trajectory which goes through the “hole” in the singular barrier. Using a power series method the authors derive an expression relating the minimum wavespeed to the tissue concentration of the extracellular matrix which is arbitrary. The model is applicable in a wide variety of biological settings which combine haptotaxis with proteolysis. By considering various functional forms the authors show that the key mathematical features of the particular model studied in the early parts of the paper are exhibited by a wider class of models which characterise the behaviour of invading cells. c 1999 Elsevier Science B.V. All rights reserved. Keywords: Cancer; Invasion; Melanoma; Metastasis; Travelling waves 1. Introduction Malignant tumours spread into neighbouring tissue following the acquisition of an invasive phenotype which can be separated into three main characteristics: adhesion, local proteolysis and migration. Adhesion is of two types: ∗ Corresponding author. Tel.: +44-0131-449-5111; e-mail: jas@ma.hw.ac.uk 0167-2789/99/$ – see front matter c 1999 Elsevier Science B.V. All rights reserved. PII:S0167-2789(98)00272-3