On the use of the Bingham statistical distribution in microsphere-based constitutive models for arterial tissue

Abstract

Constitutive models for arterial tissue have been an active research field during the last years. The main micro-constituents of blood vessels are different types of cells and the extra-cellular matrix formed by an isotropic high water content ground substance and a network composed of elastin and collagen fibres. Usually the arterial tissue has been modelled as a hyperelastic material within the framework of continuum mechanics, whereas inclusion of structural tensors into constitutive laws is the most widely used technique to introduce the anisotropy induced by the fibres. Though the different existing fibre bundles present a clear preferential direction, the dispersion inherent to biological tissue advices using of constitutive models including representative structural information associated to the spatial probabilistic distribution of the fibres. Lately, microsphere-based models have demonstrated to be a powerful tool to incorporate this information. The fibre dispersion is incorporated by means of an Orientation Density Function (ODF) that weights the contribution of each fibre in each direction of the micro-sphere. In previous works the rotationally symmetric von Mises ODF was successfully applied to the modelling of blood vessels. In this study the inclusion of the Bingham ODF into microsphere-based model is analysed. This ODF exhibits some advantages with respect to the von Mises one, like a greater versatility and a comparable response to uniaxial and equibiaxial tension tests.

Mathematical modeling of collagen turnover in biological tissue

“…We set the material parameters of the passive behavior of the tissue for the wormlike chain model (Garikipati et al, 2006;Kuhl et al, 2005;Alastrue et al, 2010) to L = 1.532, r 0 = 1.072, k = 1.381 × 10 −23 (J/K), N = 1.38 × 10 21 , θ = 310K and A = 1.107, as adapted from Alastrue et al (2010). The concentration parameters for the Bingham distribution are set to κ 1,2,3 = 0.0, 56.81, 58.70.…”

“…Collagen bears the major part of the load transmitted through the tissue, and substantial research has been devoted to understanding its hierarchical mi-crostructure at and across the different scales, see (Fratzl, 2008) for a review, as in bones (Fratzl et al, 2004), bundles of collagen (Alastrue et al, 2010), collagen fibrils (van der Rijt et al, 2006;Shen et al, 2008), microfibrils (Gautieri et al, 2011) and tropocollagen molecules (Buehler & Wong, 2007), where the latter two used molecular dynamics analysis. There is ample evidence that collagen turnover changes in response to hypertension.…”

“…The worm-like chain model was initially used to model DNA (Bustamante et al, 2003), and was later used for different types of biological tissue. Buehler & Wong (2007) have used this approach to simulate the behavior of tropocollagen molecules and Kuhl et al (2005); Garikipati et al (2006); Alastrue et al (2010) have adopted it to simulate collagen fibers. Despite the mismatch between the different length scales, all these models have successfully characterized the behavior of the underlying hierarchical structure using the following free energy function of an individual worm-like chain,…”

“…The normalizing term A U 2 = 4π is the total area of the unit sphere. The variableρ has been introduced to account for the variability of the collagen fiber distribution, as proposed by (Gasser et al, 2006;Alastrue et al, 2010) in the context of biological tissues. A statistical distribution function characterizes the probability of finding a collagen fibril in a given direction as has been experimentally reported for arterial tissue (Schriefl et al, 2012;Gasser et al, 2012).…”

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

“…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. Alastrué et al [3] first used a p-periodic von Mises orientation distribution function, and then a Bingham orientation distribution function [4] in applying the microsphere model to blood vessels. Waffenschmidt et al [36] used the microsphere approach, with evolving orientation density functions, to model bone remodeling.…”