A novel stress-induced anisotropic growth model driven by nutrient diffusion: Theory, FEM implementation and applications in bio-mechanical problems

摘要

In this paper, a novel physically-motivated anisotropic model for growth driven by nutrient diffusion is proposed and the mathematical framework is extensively presented. Growth phenomena usually occur in living tissues under different mechanobiological stimuli. Here the growth is driven by the diffusion of a chemical substance which reflects, in fact, the extent of nutrients availability or other growth factors at the cellular level. Due to its simplicity, a commonly used assumption is the isotropy of the growth tensor. In other words, the magnitude of the growth is determined by the nutrient diffusion without incorporating the effect of a preferred direction for cell growth. Since the macroscopic volumetric growth is the resultant of mitosis (binary fission) at cellular scale, it makes sense to confer directionality to the growth tensor. This will render the growth tensor anisotropic and consequently more complex. In this work, the anisotropy of the growth tensor is dictated by the principal directions of the stress tensor in an intuitive and physically motivated fashion. One can imagine that the growth is powered by nutrient diffusion while it is steered by the stress. A fully implicit and monolithic scheme is implemented for this coupled and multiphysics problem in an FEM framework. Several numerical examples are presented to demonstrate the applicability and versatility of the proposed model for reproducing biofilm growth in confined geometries; tumor growth within the brain in the avascular stage; and bone ingrowth in the vicinity of a rough implant surface.