Biomedical Engineering Reference
In-Depth Information
E e is the final value of the elastic modulus and E is the transient modulus dur-
ing the relaxation between the initial and final values of the elastic modulus, while
h e , h 1 , h 2 and a e are strain-dependent material properties. A power law equation
E(ρ)
n is used to fit a stress relaxation curve and the other material proper-
ties are fitted using the rest of the relaxation curves. To fit four different curves, a
total of ten parameters are used and the relaxation function is obtained from curve-
fitting the lowest relaxation profile. The model has been applied to ligament and can
describe its strain and time dependent response (Provenzano et al., 2002 ; Duenwald
et al., 2010 ). However, the relaxation function in the model is not unique. In fact,
for four different relaxation curves, four different relaxation functions can be fit and,
therefore, four separate sets of parameters would describe this behavior, indicating
the method is essentially a curve-fitting based methodology.
The 1D models discussed above describe the uniaxial response of ligaments and
tendons. They must be extended to 3D formulations to be used within a finite ele-
ment framework or even to consider a different mode of deformation analytically,
such as the anterior tibial translation deformation that results in ACL tears. Ex-
tending these models to 3D is likely to require the involvement of more parame-
ters for describing the tissue responses in other orientations because, for instance,
anisotropy is not addressed in these models in their present form. QLV viscoelastic
models and modified QLV models have been implemented into 3D finite element
modeling (Pioletti et al., 1998 ). These continuum models are able to describe the
tissue behavior in 3D settings with curve-fitted parameters. However, because the
QLV and modified QLV models are based on the separable relaxation function, they
cannot fully predict the time and strain dependent responses of soft tissue.
To address this desire for a nonlinear viscoelastic model that can capture the
3D response of ligament and tendon with a reduced number of parameters, and
moreover can potentially be predictive, we have taken a micromechanics approach
that describes the deformation of aligned structural proteins such as collagen and
elastin. Since these are macromolecules we describe their elastic response in terms
of hyperelastic formulations that have been shown to be predictive of the nonlinear
elasticity of biopolymers and inorganic elastomers alike (MacKintosh et al., 1995 ;
von Lockette and Arruda, 1999a , 1999b , 2001 , 2002 ; Bischoff et al., 2000 , 2001 ,
2002a , 2002b , 2002c , 2004 ; Boyce and Arruda, 2000 , 2001 ; Palmer and Boyce,
2008 ).
=
25.2 Constitutive Modeling of Mechanical Response
Ligaments and tendons are largely comprised of aligned viscoelastic collagen fibers
and nonlinear elastic elastin networks. To capture their structure and nonlinear vis-
coelastic behavior, we have developed a micromechanical model incorporating col-
lagen and elastin to describe the nonlinear viscoelastic response of ACLs and our
tissue engineered grafts.
Search WWH ::




Custom Search