Journal Article

Biphasic behaviour in malignant invasion

Ben P. Marchant, John Norbury and Helen M. Byrne

in Mathematical Medicine and Biology: A Journal of the IMA

Published on behalf of Institute of Mathematics and its Applications

Volume 23, issue 3, pages 173-196
Published in print September 2006 | ISSN: 1477-8599
Published online September 2006 | e-ISSN: 1477-8602 | DOI:
Biphasic behaviour in malignant invasion

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  • Applied Mathematics
  • Biomathematics and Statistics


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Invasion is an important facet of malignant growth that enables tumour cells to colonise adjacent regions of normal tissue. Factors known to influence such invasion include the rate at which the tumour cells produce tissue-degrading molecules, or proteases, and the composition of the surrounding tissue matrix. A common feature of experimental studies is the biphasic dependence of the speed at which the tumour cells invade on properties such as protease production rates and the density of the normal tissue. For example, tumour cells may invade dense tissues at the same speed as they invade less dense tissue, with maximal invasion seen for intermediate tissue densities. In this paper, a theoretical model of malignant invasion is developed. The model consists of two coupled partial differential equations describing the behaviour of the tumour cells and the surrounding normal tissue. Numerical methods show that the model exhibits steady travelling wave solutions that are stable and may be smooth or discontinuous. Attention focuses on the more biologically relevant, discontinuous solutions which are characterised by a jump in the tumour cell concentration. The model also reproduces the biphasic dependence of the tumour cell invasion speed on the density of the surrounding normal tissue. We explain how this arises by seeking constant-form travelling wave solutions and applying non-standard phase plane methods to the resulting system of ordinary differential equations. In the phase plane, the system possesses a singular curve. Discontinuous solutions may be constructed by connecting trajectories that pass through particular points on the singular curve and recross it via a shock. For certain parameter values, there are two points at which trajectories may cross the singular curve and, as a result, two distinct discontinuous solutions may arise.

Keywords: travelling wave; hole in the wall; hyperbolic; shock

Journal Article.  0 words. 

Subjects: Applied Mathematics ; Biomathematics and Statistics

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