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

muscle fascicle forces from EMG, kinematic or anthropometric measurements [2], as
well as multi-level optimization models [3] were developed to better approximate the
action of lumbar muscles. However, none of these models addressed the constitutive
behaviour of the spine muscles. Indeed, muscle constitutive modelling would add to the
existing simulation approaches by allowing predictive calculations of the forces that
apply on the osteo-ligamentous spine for any simulated deformation, at any strain rate.
Accordingly, such models were already explored for the cervical spine [4]. Yet, no
constitutive models have been reported for lumbar spine to our knowledge, despite the
presence of particularly high mechanical loads in this area. As such, this study proposes
to address this contribution by developing a novel active lumbar spine muscle model
and assess its potential to be calibrated specifically to simulate the action of the lumbar
spine muscles.
2. MATERIALS AND METHODS
A modified quasi-incompressible transversely isotropic hyperelastic constitutive model
was adopted for the passive and active behavior of the lumbar muscles. The model was
based on a modified hyperelastic model originally proposed for cardiac tissues [5] and a
similar approach used previously to accommodate transversely isotropic material
symmetry of soft tissues [6,7]. An uncoupled form of the strain energy was used to
simulate the nearly incompressible behavior of the passive matrix of the muscle
fascicles through the superimposition of the dilatational ( (Eq. 1) and deviatoric
( ̅ (Eq. 2) components of the tissue mechanics. The embedded muscular fibres
accounted for both a passive ( ̅ (Eq. 3) and an active ( ̅ (Eq. 4) fibre
strain energy terms, where ̅ is the uncoupled fibre stretch ratio, is the muscle peak
stress and is a strain-like quantity measured from the rest state of the muscle. The
whole constitutive relation was the additive contribution of the fibre and of the matrix
deviatoric and volumetric stresses (Eq. 6), where J is the Jacobian determinant, K is the
matrix bulk modulus, G is the matrix shear modulus, N is the undeformed fibre
direction and C is the right Cauchy-Green strain tensor.
and
̅
̅
( ̅ ) ∫
̅
( ̅ ) ∫
( ̅ ) { ̅ ̅
(1)
̅ (2)
(
)
(3)
(4) where
, ( ̅ ) { { [ ( ̅ )] } ̅
[ ( ̅ ) ( ̅ )] [
(
) (
̅
) (5)
̅ ]
(
̅ ) (6)
A single unidirectional element (L=40 mm, A=100 mm 2 ) was used to assess the
implemented model by means of user-defined material subroutines. 30% traction and
2