A microsphere-based remodelling formulation for anisotropic biological tissues

Abstract

Biological tissues possess the ability to adapt according to the respective local loading conditions, which results in growth and remodelling phenomena. The main goal of this work is the development of a new remodelling approach that, on the one hand, reflects the alignment of fibrous soft biological tissue with respect to representative loading directions. On the other hand, the continuum approach proposed is based on a sound micro-mechanically motivated formulation. To be specific, use of a worm-like chain model is made to describe the behaviour of long-chain molecules as present in, for instance, collageneous tissues. The extension of such a one-dimensional constitutive equation to the three-dimensional macroscopic level is performed by means of a microsphere formulation. Inherent with the algorithmic treatment of this type of modelling approach, a finite number of unit vectors is considered for the numerical integration over the domain of the unit sphere. As a key aspect of this contribution, remodelling is incorporated by setting up evolution equations for the referential orientations of these integration directions. Accordingly, the unit vectors considered now allow interpretation as internal variables, which characterize the material's anisotropic properties. Several numerical studies underline the applicability of the model that, moreover, nicely fits into iterative finite element formulations so that general boundary value problems can be solved.

Anisotropic microsphere-based approach to damage in soft fibered tissue

“…Menzel and Waffenschmidt (2009) reported a work related to remodeling processes, while Göktepe and Miehe (2005) and Miehe and Göktepe (2005) developed other inelastic models on the microsphere framework for isotropic materials. Moreover, the incorporation of the von Mises ODF could allow, in a more realistic way, the development of remodeling models such as those reported by Kuhl et al (2005) or Menzel et al (2008), with the preferential orientation direction evolving during the simulation.…”

A theoretical model of the endothelial cell morphology due to different waveforms

“…[47] studied this issue by using a von Mises distribution and the evolution of the associated structural tensor. Recently [46,54] presented a work for remodeling within a microsphere approach, where from an initial isotropic state, the reorientation of each of the integration directions leads to an anisotropic behavior. This latter approach follows an approach similar to that described in [28] for one simple fiber, and is similar to the one used here for the evolution of the fibrils.…”

“…Previous authors [1] have used exponential-type models, such as that proposed by [29]. Recently [46] presented a microsphere-based approach for remodeling, where the fibrils behavior was modeled by the Worm-like Chain model (WLC).…”

“…Several remodeling models have been proposed in recent years. Some of them analyze the reorientation of unidimensional fibers driven by different stimuli such as [45] or in a mcirosphere 3D approach [46,62] or [35]. In [28], a complete consistent linearization of the equations in an implicit finite element framework was performed.…”

The effect of general statistical fiber misalignment on predicted damage initiation in composites

“…Since then, the microsphere model has been used extensively in the analysis of biological materials. Menzel and Waffenschmidt [25] used the model, with an evolving orientation distribution function, to simulate remodeling in soft tissues. Murtada et al [30] used the microsphere approach to analyze smooth muscles, wherein a specialized distribution of the muscle contractile fiber orientation, as a function of stretch, was employed.…”