Modelling the growth of solid tumours and incorporating a method for their classification using nonlinear elasticity theory

Medically, tumours are classified into two important classes — benign and malignant. Generally speaking, the two classes display different behaviour with regard to their rate and manner of growth and subsequent possible spread. In this paper, we formulate a new approach to tumour growth using results and techniques from nonlinear elasticity theory. A mathematical model is given for the growth of a solid tumour using membrane and thick-shell theory. A central feature of the model is the characterisation of the material composition of the model through the use of a strain-energy function, thus permitting a mathematical description of the degree of differentiation of the tumour explicitly in the model. Conditions are given in terms of the strain-energy function for the processes of invasion and metastasis occurring in a tumour, being interpreted as the bifurcation modes of the spherical shell which the tumour is essentially modelled as. Our results are compared with actual experimental results and with the general behaviour shown by benign and malignant tumours. Finally, we use these results in conjunction with aspects of surface morphogenesis of tumours (in particular, the Gaussian and mean curvatures of the surface of a solid tumour) in an attempt to produce a mathematical formulation and description of the important medical processes of staging and grading cancers. We hope that this approach may form the basis of a practical application.