Travelling wave solutions of a mathematical model for tumor encapsulation.
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The formation of a capsule of dense, fibrous extracellular matrix around a solid tumour is a key prognostic indicator in a wide variety of cancers. However, the cellular mechanisms underlying capsule formation remain unclear. One hypothesised mechanism is the expansive growth hypothesis, which suggests that a capsule may form by rearrangement of existing extracellular matrix without new matrix production. A mathematical model was proposed to study the implications of this hypothesis by Perumpanani, Sherratt and Norbury in 1997. The model consists of conservation equations for tumour cells and extracellular matrix and exhibit travelling wave solutions in which a pulse of extracellular matrix, corresponding to a capsule, moves in parallel with the advancing front of the tumour. This project is based on the work done by Jonathan Sherratt in studying the expansive growth hypothesis. In this work the author presents a detailed study of travelling wave behaviour in the tumour model, the conditions for the existence of travelling waves and their key properties . The analytical work suggests that the traveling waves are stable and are a biologically relevant solution form for the model. Considering an improved model which includes a saturation in the extent of matrix rearrangement per cell, the analytical results show that rate of matrix movement and convection per cell will saturate at higher matrix densities.