Biomedical Engineering Reference
In-Depth Information
7.2 Multi-directional Tissues: The Aortic Media Case
The proposed approach can be also employed to describe multi-layered tissues,
comprising layers each having a uni-directional collagen fiber arrangement. This is
the case of the arterial tunica media. In the following, reference will be made to the
aorta.
Following experimental evidences [
31
,
32
,
36
] and well-established modeling
approaches [
47
], aortic media is modelled as a multi-layered thick-walled cylinder
(with internal radius r
i
and thickness h
a
) made up of N identical layers (media
lamellar units MLUs, see Fig.
12
). Such a multi-layered cylinder is assumed to
have a length much greater than r
i
, and to be loaded by a uniform internal pressure
distribution p, undergoing prevailing membranal response with negligible flexural
effects. Let the cylindrical coordinate system
ð
z
;
u
;
q
Þ
be introduced, where q is the
radial coordinate, and u and z the angular and axial coordinates, respectively.
Each MLU comprises an elastin rich elastic lamina (EL) h
e
thick and an
interlamellar substance (IL) with thickness h
IL
.
The EL sub-layer is modelled as a two-phase substance comprising void (or
very soft constituents with negligible stiffness) with volume fraction V
o
, and
elastin. The latter is assumed with a linearly elastic isotropic behavior and
characterized by the Young's modulus E
e
and the Poisson's ratio m
e
. Accordingly,
EL can be reduced by the mixture rule to a homogeneous layer with equivalent
isotropic elastic constants equal to
ð
1
V
o
Þ
E
e
and
ð
1
V
o
Þ
m
e
.
The interlamellar substance, in turn, can be regarded as a multi-layered sub-
structure, made up of concentrically fiber-reinforced layers, comprising elastin,
muscle cells, and crimped collagenous fibers whose main direction is helically
arranged around the vessel axis. In agreement with well-established histological in
vivo measures [
36
] and as previously recalled (see
Sect. 2.2.2
), the wrapping angle
h
F
of collagen fibers can be described as a function of the radial coordinate q
(Fig.
12
).
The kth MLU (k
¼
1...N) is reduced to a homogeneous layer, comprising an
anisotropic (generally with a monoclinic symmetry) non-linearly elastic material
characterized by a tangent equivalent stiffness matrix
k
ð
e
F
Þ
(and compliance
C
matrix
k
¼ð
k
Þ
1
). The latter is obtained by accounting for the kth MLU-based
microstructure through a homogenization step carried out via the standard laminate
theory [
60
,
63
]. To this aim, at the radial position q, the local tangent stiffness
matrix
S
C
is computed referring to a uni-directional collagenous tissue with the fiber
chord direction t
ð
q
Þ
inclined by h
F
ð
q
Þ
with respect to the vessel axis z. Accordingly,
C
C
ð
q
Þ
at each
incremental step. Since
ð
h
e
þ
h
IL
Þ=
r
i
1, any curvature effect is disregarded. It is
worth observing that if q identifies a position within a EL sub-layer, C is described
in agreement with an isotropic response, that is by involving the elastic constants
E
e
and m
e
only.
¼
ð
h
F
;
e
F
Þ
, where h
F
¼
h
F
ð
q
Þ
and e
F
¼
e
F
ð
q
Þ
, and thereby C
¼
C
C
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