An Eulerian Nonlinear Elastic Model for Compressible and Fluidic Tissue with Radially Symmetric Growth

Abstract

Cell proliferation, apoptosis, and myosin-dependent contraction can generate elastic stress and strain in living tissues, which may be dissipated by internal rearrangement through cell topological transition and cytoskeletal reorganization. Moreover, cells and tissues can change their sizes in response to mechanical cues. The present work demonstrates the role of tissue compressibility and internal rearranging activities on its size and mechanics regulation in the context of differential growth induced by a field of growth-promoting chemical factors. We develop a mathematical model based on finite elasticity and growth theory and the reference map techniques to describe the coupled tissue growth and mechanics in the Eulerian frame. We incorporate the tissue rearrangement by introducing a rearranging rate to the reference map evolution, leading to elastic-energy dissipation when tissue growth and deformation are in radial symmetry. By linearizing the model, we show that the stress follows the Maxwell-type viscoelastic relaxation. The rearrangement rate, which we call tissue fluidity, sets the stress relaxation time, and the ratio between the shear modulus and the fluidity sets the tissue viscosity. By nonlinear simulation of growing tissue spheroids and discs with graded growth rates along the radius, we find that the tissue compressibility and fluidity influence their equilibrium size. By comparing the nonlinear simulations with the linear analytical solutions, we show the size change as a nonlinear effect due to the advection of the tissue density flow, which only occurs when both tissue compressibility and fluidity are small. We apply the model to study tumor spheroid growth and epithelial disc growth when a reaction-diffusion process determines the growth-promoting factor field.