ARUP; ISBN: 978-0-9562121-5-3 - CMBBE 2012 - Cardiff University

challenge due to its anisotropic multilayer structure. The anisotropy of annulus fibrosus
(AF) is derived from its alternating layers of fibre-reinforced lamellae. These lamellae
surround a gelatinous central core, the nucleus pulposus (NP) [13]. The orientations of
fibres in each lamella vary with respect to the circumferential and radial positions of the
lamella [14, 15]. To describe such variations, FE models of intervertebral discs
developed in the past assigned the fibre directions based on the shape of the elements
used for the model. The most commonly used element was the hexahedral element, as
its structured vertices act as convenient geometric reference points to describe fibredirections
using either reinforcement bars (discrete elements) or a continuum
characterization. The use of tetrahedral elements was generally avoided despite their
versatility to describe complex geometries [16]. This was because of i) difficulties to
describe fibre-orientations due to element irregularity and ii) slower convergence
compared to hexahedral elements, although the latter is debatable depending on the
initial element shape. The aim of this study was to develop an algorithm to calculate and
assign the fibre-orientations automatically using only the external and internal geometry
of AF as the geometric reference. The novelty of the developed algorithm is the fact that
the computation of fibre-orientations is performed independent to the element shapes.
An idealized semi-elliptical disc model was used to compare the mechanical behaviour
provided by using tetrahedral and hexahedral elements. Compression, flexion, torsion
and lateral bending response of the disc were investigated and compared.
3. METHODS
A representative IVD was designed, based on the work of Little et al. [17], with a major
axis of 50 mm and a minor axis of 37 mm. A height of 8.5 mm was defined. The
volume of the nucleus pulposus was set to 45% of the entire disc geometry [18]. Rigid
body endplates were added at the top and bottom. The model was then meshed and
element sets were created for the AF and the NP. Node sets were defined for the
superior and inferior endplates and for the internal and external surfaces of the AF. The
IVD had the major axis aligned with the x-axis and the minor axis aligned with y-axis of
a xyz system of reference, Figure 1.a.
3.1 Algorithm to compute fibre orientation
The task to determine the fibre orientation at a material point Pi is similar to defining a
local system of reference. Therefore, three mutually orthogonal unit vectors have to be
selected, {n1, n2, n3}, where n1 is a vector tangent to the curve on the transversal crosssection,
n2 is a vector tangent to the bulge and vertically oriented and n3 is determined
by the cross product of n1 and n2. Once this reference has been locally defined, the
rotation of the system around n3 will provide the local fibre orientation, fibre. The
pseudo-code is shown in Box 1. The algorithm was implemented in Matlab ® (The
MathWorks Inc., Natick, Massachusetts).
3.2 Material model and numerical analysis
A large deformation approach was used to describe the mechanical response of the
tissue. Through hyperelastic continuum theory, the stress-strain relationships were
derived assuming given the strain energy function for each tissue. A Neo-Hookean
model was used for the NP, assuming a homogeneous, incompressible material, Eq.(1).
A modified Yeoh's model has been used for the ground matrix of the AF, assuming a