The influence of external free energy and homeostasis on growth and shape change

Mechanical forces are essential for the proper growth and biomechanical remodeling of several biological tissues. However, the relationship between mechanical forces and changes in tissue volume and shape is only beginning to be revealed in experiments. Theoretical contributions have provided an appropriate framework to interpret growth and remodeling in terms of mechanical loading. We present a volumetric growth model that shares features of these previous models. We focus on three key parameters for predicting tissue growth: (1) the characterization of the homeostatic reference state in which no growth occurs, (2) the mechanical energy of newly created tissue, and (3) the effect of shear forces on growth and inelastic shape change. To facilitate this analysis, we decompose the deformation gradient into a product of the inelastic growth and elastic deformations. We then place the evolution equation for the growth deformation in a thermodynamic context. Homeostasis, incoming free energy, and shear forces each directly affect the evolution of the tissue. We demonstrate this using numerical examples.