Biomedical Engineering Reference
In-Depth Information
The quadratic corotated FEM was implemented using the
simulation open
framework architecture
(SOFA) toolkit [
12
]. The new algorithm can therefore be
easily compared with the linear corotated formulation implemented in SOFA [
13
].
Furthermore, the framework allows choosing between different time integration
algorithms and linear system solvers. An implicit Euler time integration scheme is
used along with the Pardiso direct sparse solver from the Intel MKL 10.3 for all
simulations presented in this paper.
3 Liver Modeling for Intraoperative Registration
3.1 Viscoelastic Behavior and Model Parameterization
Several groups have developed material laws for the liver based on elasticity theory
(for a comprehensive overview, see Marchesseau et al. [
6
]). The viscoelastic
behavior is usually modeled by quasi-linear viscoelastic (QLV) models, which
use a Prony series in order to describe the time dependence of the hyperelastic
constant. As shown by Suwelack et al., an elaborate viscoelastic model is not
necessary for soft tissue registration [
7
]. Thus, we use Rayleigh damping to increase
the computational efficiency of the method. In this approach, the damping is
determined by the linear combination
C
e
¼ a
M
e
þ b
K
e
(9)
of the elemental stiffness matrix
K
e
and the mass matrix
M
e
.
We use the model parameterization from Suwelack et al. for all simulations in
this paper, i.e., we choose
a ¼
b ¼
:
21, Young's modulus
E
¼
0,
0
2061 Pa and the
n ¼
:
Poisson ration
365 [
7
]. These parameters have been determined in order to
mimic the nonlinear QLV model by Raghunathan et al. [
14
].
At this point it is important to point out that several QLV models with signifi-
cantly different parameterization have been proposed for the liver. In particular,
some groups model the liver as a nearly incompressible organ (Poisson ratio
n>
0
49). We note that according to previous studies, different parameterizations
have only little influence on the registration result [
7
]. Also, it is possible to adapt
the co-rotational model to different parameterization. Furthermore, the quadratic
elements used in the FE formulation significantly reduce volume locking and thus
are very well suited to simulate nearly incompressible materials.
0
:
3.2 Liver Phantom
A liver phantom was built in order to obtain a ground truth for the volumetric
deformations that are predicted by the biomechanical model. For this purpose, a
