Biomedical Engineering Reference
In-Depth Information
defined according to a paradigmatic scheme [
12
] derived from ex vivo findings
[
13
,
14
] and it included all MV characteristic types of chordae tendinae: marginal
(first order), basal (second order) and structural anterior chordae.
As above mentioned, in order to reproduce a reliable pre-operative model, struc-
tural FE models were integrated with intra-operative measurements and surgical
findings, thus confirming location and extent of the prolapsing region and the exact
number and type of ruptured chordae.
All tissues were assumed as homogeneous (1.1g/cm
3
, non-linear and elastic:
their behaviour was modeled on the basis of the hyperelasticity theoretical framework
in order to account for their large deformations.
MV leaflet response was described through the hyperelastic constitutive model
originally proposed by May-Newman et al. [
15
], which is based on the strain energy
function
)
that depends on two invariant measures of finite strain I
1
and I
4
:
c
0
e
c
1
(
I
1
−
3
)
1
c
2
(
√
I
4
−
2
4
+
1
)
(
I
1
,
I
4
)
=
−
(1)
F
T
is the right
Cauchy-Green tensor,
a
0
is the unit vector which defines the preferential direction
of the fibers in the material in the undeformed shape and
2
, in which
C
where I
1
=
tr
(
C
)
and I
4
=
a
0
•
C
•
a
0
=
λ
=
F
•
is the stretch of the fibers
in the
a
0
direction.
F
is the deformation gradient tensor, defined as
F
λ
X
, i.e.
the derivative of the current position with respect to the undeformed position. The
constitutive relation was implemented in a user-defined subroutine (VUMAT), as
detailed in [
12
]: the constitutive parameters identified by May-Newman were used
[
15
] and, accordingly, the direction parallel to the annular profile was defined as the
preferential direction of the collagen fibers. Moreover, regionally varying thicknesses
were assigned to the anterior and posterior MV leaflets as reported in [
16
].
Native chordae tendineae response was assumed isotropic and described through
hyperelastic models available in ABAQUS/Explicit and fitting data from literature
[
17
] through a 2nd order polynomial function for marginal and structural chordae
and a 5th order Ogden function for basal chordae.
For each patient-specific model, two simulations were preliminary implemented:
the former represented the pre-operative model (
Pre-Op
model) reproducing the clin-
ical scenario of MV lesions and prolapse dysfunction as derived from CMR images;
the latter, replicating the same geometry of
Pre-Op
model, represented the physi-
ological MV model (
Phy
model) without prolapse, i.e. with a complete and intact
chordal apparatus with respect to pre-operative conditions. For both models, the sys-
tolic MV biomechanics was simulated, applying a physiological pressure load to the
leaflets, for a total simulation time of one second. As above mentioned, boundary
conditions were extracted from CMR-segmentation and reproduced annular kine-
matics and PMs displacement. A general contact algorithm, available in ABAQUS
with a scale-penalty method and a friction coefficient of 0.05 was used to model MV
leaflets coaptation [
18
].
=
∂
x
/
∂
