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

with a distinctive set of mechanical and physiological properties, given by its
complex anatomical structure [3]. Consequently, an accurate constitutive modeling
regarding the biomechanical parameters of the IVD is a challenge. In addition, it
may also be a very important step to fully understand the causes of DDD, even if
genetics, ageing or metabolic factors also play a significant role [1].
MATERIALS AND METHODS
Fig. 1 The generic anatomical features of an IVD [1]
The paramount tool for this work is a FORTRAN home-developed open source FE
solver. This solver comprises several relevant features of biomechanics, namely
the almost incompressibility of soft-tissues, the most general isotropic and
anisotropic hyperelastic laws, viscoelastic effects, fully implicit time integration
scheme and different types of finite elements [4]. Mixed u/p elements are available
for monophasic simulations, but the most relevant elements for IVD modeling are
the u/p-c ones, namely the linear 4-node tetrahedron (with an extra bubble node) or
the quadratic 10-node tetrahedron and 27-node hexahedron [5]. In case of u/p-c
elements, the continuity of the pressure field is consistent with a multiphasic
formulation. Currently, biphasic formulation is implemented, considering solid and
fluid parts. The innovative formulation here adopted was based on the one adopted
by Huyghe (1986) [6]. The strain energy density potential W was adopted in the
following form [5]:
( ) = W ( C)
+ W ( J ) Q
C , where ( ) 2
Q = − p − p and ( ) ( ) 2
J = + J −1
W H +
1
2k
~
k
WH (1)
2
where W (C)
and WH (J ) are, respectively, the isochoric and volumetric strain
energy densities, with an additional energy term Q which has the merit of coupling
the mixed formulation (displacements and pressure fields). F is the deformation
T
gradient, J = det(F)
, and C = F F is the right Cauchy-Green strain tensor. k is a
penalty parameter playing the role of a bulk modulus, p is the pressure computed from
the (unknown) displacement fields and ~ p is the pressure interpolated from the
(unknown) pressure field. The contribution of the isochoric strain energy density
changes with the adopted hyper-visco-elastic constitutive model. Finally, in case of a
biphasic formulation, the elemental stiffness matrix and the corresponding elemental
system of equations can be written as:
⎡K
⎢
⎣K
UU
PU
K
KUP
⎤ ⎡Δu⎤
⎡ R ⎤ ⎡FU
⎤
⎥ : ⎢ ⎥ = ⎢ ⎥ − ⎢ ⎥ , (2)
− Kαβ
⎦ ⎣Δp⎦
⎣U
+ T1
+ T2
⎦ ⎣ FP ⎦
PP