Hyperelastic modelling of arterial layers with distributed collagen fibre orientations

Abstract

Constitutive relations are fundamental to the solution of problems in continuum mechanics, and are required in the study of, for example, mechanically dominated clinical interventions involving soft biological tissues. Structural continuum constitutive models of arterial layers integrate information about the tissue morphology and therefore allow investigation of the interrelation between structure and function in response to mechanical loading. Collagen fibres are key ingredients in the structure of arteries. In the media (the middle layer of the artery wall) they are arranged in two helically distributed families with a small pitch and very little dispersion in their orientation (i.e. they are aligned quite close to the circumferential direction). By contrast, in the adventitial and intimal layers, the orientation of the collagen fibres is dispersed, as shown by polarized light microscopy of stained arterial tissue. As a result, continuum models that do not account for the dispersion are not able to capture accurately the stress-strain response of these layers. The purpose of this paper, therefore, is to develop a structural continuum framework that is able to represent the dispersion of the collagen fibre orientation. This then allows the development of a new hyperelastic free-energy function that is particularly suited for representing the anisotropic elastic properties of adventitial and intimal layers of arterial walls, and is a generalization of the fibre-reinforced structural model introduced by Holzapfel & Gasser (Holzapfel & Gasser 2001 Comput. Meth. Appl. Mech. Eng. 190, 4379-4403) and Holzapfel et al. (Holzapfel et al. 2000 J. Elast. 61, 1-48). The model incorporates an additional scalar structure parameter that characterizes the dispersed collagen orientation. An efficient finite element implementation of the model is then presented and numerical examples show that the dispersion of the orientation of collagen fibres in the adventitia of human iliac arteries has a significant effect on their mechanical response.

Figure 1

Histomechanical idealization of a healthy elastic artery with non-atherosclerotic intimal thickening. It is composed of three layers: intima (I), media (M), adventitia (A). I is the innermost layer consisting of a single layer of endothelial cells, a thin basal membrane and a subendothelial layer. The subendothelial layer is comprised mainly of thinly dispersed smooth muscle cells and bundles of collagen fibrils. M is composed of smooth muscle cells, a network of elastic and collagen fibrils and elastic laminae which separate M into a number of transversely isotropic fibre-reinforced units. A is the outermost layer surrounded by loose connective tissue. The primary constituents of A are thick bundles of collagen fibrils arranged in helical structures.

Figure 2

Characterization of an arbitrary unit direction vector M by means of Eulerian angles Θ∈[0,π] and Φ∈[0,2π] in a three-dimensional Cartesian coordinate system {e₁, e₂, e₃}.

Figure 3

Relation between the dispersion parameter κ and the concentration parameter b of the (transversely isotropic) von Mises distribution.

Figure 4

Three-dimensional graphical representation of the orientation of the collagen fibres based on the transversely isotropic density function (4.3).

Figure 5

Two-dimensional graphical representation of the (transversely isotropic) von Mises distribution of the collagen fibres.

Figure 6

Thin-wall approximation of the inflation of the adventitial layer with two embedded families of fibres. The mean orientations and the dispersion of the collagen fibres are characterized by γ and κ, respectively.

Figure 7

Influence of collagen fibre mean alignment γ and dispersion κ on the mechanical response of a thin-walled tube. Solid curves: isotropic tube (κ=1/3). Dashed and dotted curves: anisotropic response corresponding to κ=0.226 and 0, respectively.

Figure 8

Definition of circumferential and axial specimens for the tensile tests.

Figure 9

Finite element computation of a uniaxial tension test on an iliac adventitial strip in the circumferential and axial directions. The Cauchy stress in the direction of the applied load is plotted for a 1.0 N tensile load, and no dispersion of the collagen fibres is taken into account (κ=0).

Figure 10

Finite element prediction of the current (mean) collagen orientations using the parameter c_a=a₁·a₂. Results are shown for circumferential and axial specimens at 1.0 N tensile load: (a) no dispersion of the collagen fibres (κ=0); (b) dispersion of the collagen fibres within each family (κ=0.226).

Figure 11

Computed tensile load/displacement (T/u) response of the circumferential and axial specimens. Dashed and dotted curves are with (κ=0.266) and without (κ=0) dispersion of the collagen fibres.

Figure 12

Finite element computation of the Cauchy stress in the direction of the applied load in an iliac adventitial strip in the circumferential and axial directions. Results are shown for 1.0 N tensile load and dispersion of the collagen fibres is included (κ=0.226).