Biomedical Engineering Reference
In-Depth Information
F
T
F
. We denote by
I
1
=
1
2
(I
1
−
tr
C
2
)
,
=
tr
C
,
I
2
=
right Cauchy-Green tensor
C
I
3
=
det
C
, the principal invariants of
C
.
Due to the alignment of cardiac fibers and their organization in sheets, the my-
ocardium exhibits an orthotropic behavior that can be conveniently illustrated in-
troducing the orthogonal unit vector fields
f
0
and
s
0
denoting the orientation of the
fibers and collagen-sheets in the reference configuration. A hyperelastic material
with constitutive response invariant with respect to rotations around
f
0
and
s
0
is de-
scribed by a strain-energy function
(
F
)
that depends on a set of invariant such as
I
1
,
2
,
3
, and also pseudo-invariants defined as follows:
W
C
2
I
4
,f
=
:
f
0
⊗
I
5
,f
=
:
f
0
⊗
I
8
,fs
=
:
sym
(
f
0
⊗
C
f
0
,
f
0
,
C
s
0
),
(14.3)
and analogously
I
4
,s
and
I
5
,s
.
Orthotropic strain-energy functions were suggested in Usyk et al. (
2000
); Costa
et al. (
2001
); Holzapfel and Ogden (
2009
), that in addition are able to represent
the behavior of the laminar sheets in which cardiac myofibers are structured. For
instance, the energy function proposed by Holzapfel and Ogden (
2009
) is given by
2
b
e
b(I
1
−
3
)
1
+
2
b
i
e
b
i
(I
4
,i
−
1
)
2
1
+
2
b
fs
e
b
fs
I
8
,fs
1
,
a
fs
a
a
i
W
=
−
−
−
(
F
)
i
=
f,s
(14.4)
where the eight parameters
a
,
b
,
a
f
,
b
f
,
a
s
,
b
s
,
a
fs
and
b
fs
are experimentally
fitted.
According to Ashikaga et al. (
2008
), the myocardium experiments a change in
myocardial volume of up to 10 %. This is possibly due to blood-filled spaces within
the myocardium which may communicate with the ventricular lumen or from the
coronary arteries from which blood is expelled during systole. However incompress-
ibility of the medium is often assumed as it is mainly constituted by water. In strictly
incompressible models, the pressure field is the Lagrange multiplier enforcing the
constraint, and in slightly compressible models a compressibility modulus penal-
izes the variation in density. For evident reasons, strict incompressibility is more
popular when analytical methods are applied, as one degree of freedom drops out
in homogeneous deformations, while penalization is often preferred in numerical
codes, where no compatibility between spaces of representation of displacement
and pressure fields must be abided.
14.3 Activation and Contraction
Myocardial systolic contraction is usually modeled at the macroscale by incorpo-
rating a possibly anisotropic, additive stress contribution in the force balance (Nash
and Panfilov,
2004
; Smith et al.,
2004
; Göktepe and Kuhl,
2010
; Pathmanathan et
al.,
2010
).
A different approach is to introduce a multiplicative decomposition of the strain.
The active strain method, introduced in the context of biomechanics in Taber and
