Medically, tumors are classified into two important classes—benign and malignant. Generally speaking, the two classes display different behavior with regard to their rate and manner of growth and subsequent possible spread. In this paper, we formulate a new approach to tumor growth using results and techniques from nonlinear elasticity theory. A mathematical model is given for the growth of a solid tumor using membrane and thick-shell theory. A central feature of the model is the characterization of the material composition of the tumor through the use of a strain energy function, thus permitting a mathematical description of the degree of differentiation of the tumor explicitly in the model. Conditions are given in terms of the strain energy function for the processes of invasion and metastasis occurring in a tumor, being interpreted as the bifurcation modes of the spherical shell, which the tumor is essentially modeled as. Our results are compared with actual medical experimental results and with the general behavior shown by benign and malignant tumors. Finally, we use these results in conjunction with aspects of surface morphogenesis of tumors (in particular, the Gaussian and mean curvatures of the surface of a solid tumor) 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.
Chaplain, M. A. J., & Sleeman, B. D. (1992). A mathematical model for the growth and classification of a solid tumour: a new approach via nonlinear elasticity theory using strain-energy functions. Mathematical Biosciences, 111(2), 169-215. https://doi.org/10.1016/0025-5564(92)90070-D