Continuum elasticity models for tissue growth and mechanotransduction.

Abstract

We here consider modelling tissue growth and mechanotransduction, utilising a continuum approach based on the theory of elasticity. Our models are valid for both unhealthy and healthy tissues and assume a spherical symmetry is present. In principal, there are a number of tissue types which can be modelled using our framework, however, we choose to focus on three main tissue paradigms; epithelial cysts, ovarian follicles and avascular tumour spheroids, which all share common geometric features of a central core, surrounded by a proliferating rim of cells. In all cases, the tissues are embedded in a constraining outer gel. We show that growth within the tissue leads to the build-up of internal stress, and we find potential mechanotransductive mechanisms occurring as a result of this growth. These mechanisms are deemed switching points and we show that they can occur both as a result of the internal stress and of the associated strains. We also find that, for systems with a deformable central core, this core can expand and shrink passively as a result of growth within the surrounding material in parameter regimes that we identify. We also relax the assumption that the inner core is non-growing and consider that both phases of the tissue undergo growth. This leads to a competition of growth and we show that switching still occurs, but is now dependent upon the growth in both regions. This is considered specifically in the case of ovarian follicles, where we further observe that the cuboidalisation of cells can be produced as a consequence of the mechanics within the system. We next consider the addition of the effect of contractility upon the linear model using two techniques of implementation. The necessity of the use of nonlinear elasticity is then tested, from which we show that for most parameters within the realms of soft biological tissues, the linear approximation to the nonlinear model is of a sufficient likeness to warrant the use of the linear scheme. We find that the ratio of the Young's modulus of the tissue and the surrounding medium is key in determining the effectiveness of the linear model. These studies consistently highlight the importance of the mechanical properties of the tissue and surrounding extracellular matrix, specifically stressing the significance of the Young's modulus upon tissue growth dynamics.