Biomedical Engineering Reference
In-Depth Information
bulk behavior of the material.
K
is the bulk modulus of the material,
F
is the
deformation gradient tensor and
J
=
det(
F
).
I
1
and
I
2
are the first and second
deviatoric invariants of the right Cauchy deformation tensor [30]. The scalar
˜
λ
is
the deviatoric stretch ratio along the local fiber direction,
a
, which was oriented
circumferentially for these analyses to correspond with the collagen and smooth
muscle fiber orientations in the arterial wall and plaque cap.
A neo-Hookean form was used to represent the ground substance matrix:
F
1
(
I
1
)
=
µ
(
I
1
−
3)
(12.34)
where
µ
is the shear modulus of the ground substance. The stress-stretch be-
havior for the fiber direction was represented as exponential, with no resistance
to compressive load:
λ
∂
F
2
˜
˜
˜
λ
W
λ
=
∂λ
=
0
,
λ<
1;
(12.35)
λ
W
λ
=
λ
∂
F
2
∂λ
=
C
3
exp(
C
4
(
˜
λ
−
1))
−
1
,
˜
˜
λ
≥
1
where material coefficients
C
3
and
C
4
scale the fiber stress and control its rate
of rise with increasing stretch, respectively. The full Cauchy stress tensor is
defined as.
T
=
2(
W
1
)
B
+
λ
W
λ
a
⊗
a
+
p
1
(12.36)
W
1
,
W
2
and
W
λ
are strain energy derivatives with respect to
I
1
,
I
2
, and
λ
[26],
and
B
is the left deformation tensor. A detailed description of the finite element
implementation of this constitutive model can be found in Weiss
et al
. [19].
The material parameters for the arterial wall were determined by a nonlinear
least squares fit to circumferential stress/strain values presented in the work
of Cox et al. [58] for the canine coronary artery wall using the constitutive
relation described above. The media region of the arterial wall was assigned
material properties based on the curve fit obtained from the Cox et al. data
[57]. The material constants for the media were
µ
=
3
.
57 kPa,
C
3
=
4
.
99 kPa,
and
C
4
=
5
.
49. The bulk modulus was defined as 200.00 kPa. The lesion areas
were assigned identical material properties as were used for the media since
the stress-strain behavior of the arterial wall falls well within the wide range of
values published for the material properties of atherosclerotic lesions [67].
The warping analyses results indicate (Figs. 12.9D and 12.10D) that the pres-
ence of a fully developed lipid core increases the circumferential stretch of the
