A macroscopic approach for stress-driven anisotropic growth in bioengineered soft tissues

Abstract

The simulation of growth processes within soft biological tissues is of utmost importance for many applications in the medical sector. Within this contribution, we propose a new macroscopic approach for modelling stress-driven volumetric growth occurring in soft tissues. Instead of using the standard approach of a-priori defining the structure of the growth tensor, we postulate the existence of a general growth potential. Such a potential describes all eligible homeostatic stress states that can ultimately be reached as a result of the growth process. Making use of well-established methods from visco-plasticity, the evolution of the growth-related right Cauchy–Green tensor is subsequently defined as a time-dependent associative evolution law with respect to the introduced potential. This approach naturally leads to a formulation that is able to cover both, isotropic and anisotropic growth-related changes in geometry. It furthermore allows the model to flexibly adapt to changing boundary and loading conditions. Besides the theoretical development, we also describe the algorithmic implementation and furthermore compare the newly derived model with a standard formulation of isotropic growth.

Mechanical investigations of the peltate leaf of Stephania japonica (Menispermaceae): Experiments and a continuum mechanical material model

“…(see e.g. Reese and Govindjee, 1998;Vladimirov et al, 2008;Lamm et al, 2022). The final form of the update formulas read:…”

A novel anisotropic stress‐driven model for bioengineered tissues accounting for remodeling and reorientation based on homeostatic surfaces

“…a homeostatic stress is tried to reach throughout the whole tissue. Within this contribution, the approach suggested in [4] for modeling growth and remodeling is followed. A so-called 'homeostatic surface' in the principal stress space similar to plasticity is introduced, which describes the preferred or homeostatic stress.…”

“…In the literature, different approaches exist to describe these phenomena, for instance, constraint mixture approaches. In this regard, [4] recently published an approach based on so-called homeostatic surfaces. These surfaces prescribe the preferred state in the principal stress space, such that growth and remodeling is considered in a smeared and phenomenological sense until the current stress state coincides with the surface.…”

A multiphysics modeling approach for in-stent restenosis: Theoretical aspects and finite element implementation

“…More general and elaborate continuum mechanical models for growth and remodeling of soft biological tissues can be derived utilizing the framework for modeling anisotropic inelasticity via structural tensors, introduced in Reese et al [43]. The anisotropic growth formulation developed in Lamm et al [33] is also relevant in this regard wherein the growth is stressdriven.…”

“…Eqs. 33,34,35,36, and 37 are linearized about the states at t n+1 (See Appendix A.2). The computational domain in the reference configuration is spatially approximated via finite elements, i.e.,…”