Mathematical modelling of capsule formation and multinodularity in benign tumour growth

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Abstract
Tumours that grow locally, but do not invade the surrounding tissue are called benign. Such benign tumours are characterized by the presence of a surrounding band of connective tissue called a capsule. In some cases, the tumours are also broken into a number of discrete nodules. In this paper the authors use a partial differential equation model to study the interactions of a growing tumour with the surrounding tissue. They predict mechanisms for both capsule formation and nodularity. The former has the mathematical form of bifurcation from travelling waves to aggregating waves of connective tissue, resulting in the accretion of connective tissue in a manner corresponding to capsule formation. The cause of multilobularity in tumours is currently not known. Using their model, the authors are able to predict lobulation, when tumour cell motility is retarded by aggregating connective tissue. In the final part of the paper, the authors introduce an enlarged model, and use it to demonstrate both capsule formation and the possible dissolution of the capsule following a mutation resulting in the production of proteases by the cancer cells.