Cell Adhesion Mechanisms and Elasto-Viscoplastic Mechanics of Tumours

Tumour cells usually live in a environment formed by other host cells, extra-cellular matrix and extra-cellular liquid. Cells duplicate, reorganise and deform while binding each other thanks to adhesion molecules exerting forces of measurable strength. In this paper it is illustrated a macroscopic mechanical model of solid tumour which takes such adhesion mechanisms into account. The extracellular matrix is treated as an elastic compressible material, while, in order to define the relationship between stress and strain for the cellular constituents, the deformation gradient is decomposed in a multiplicative way distinguishing the contribution due to growth, to plastic rearrangement and to elastic deformation. On the basis of experimental results at a cellular level, it is proposed that at a macroscopic level there exists a yield condition separating the elastic and viscoplastic regimes. Previously proposed models are obtained as limit cases, e.g. fluid-like models are obtained in the limit of fast cell reorganisation and negligible yield stress. A numerical test case shows that the model is able to account for several complex interactions: how tumour growth can be influenced by stress, how and where it can generate plastic reorganisation of the cells, how it can lead to capsule formation and compression of the surrounding tissue.