Biomedical Engineering Reference
In-Depth Information
σ
(
,
)
·
=
0 will always
be considered at the main vessel outflow (see Fig.
2
b). Concerning the outflow
section of the side-branch, four types of outflow boundary conditions are explored:
zero velocity,
u
In this work a traction free boundary condition
u
p
n
=
0
, meaning that the side-branch is neglected and modeled as a
no-slip wall [
19
]; zero normal stress,
σ
(
,
)
·
=
0[
4
]; coupling with a 0D model
corresponding to a simple resistance [
13
]; and coupling with a one-dimensional
(1D) model equivalent to the three-dimensional (3D) side-branch [
6
]. Thus, the first
two approaches neglect the effects of the remaining parts of the cardiovascular
system, as opposed to the last two which resort to the Geometrical Multiscale
Approach [
6
] to account for the global circulation on the localized numerical
simulation.
u
p
n
3.4
The 1D Model
The 1D simplified model is formulated assuming that an artery is a cylindrical
compliant tube, with axial symmetry and fixed cylinder axis. The velocity com-
ponents orthogonal to the vessel axis are neglected and the wall displacements are
only accounted for in the radial direction. Moreover, no body forces are considered
and the pressure,
P
, is assumed constant on each axial section, varying only
coaxially. The area of each cross-section
S
is given by
A
(
t
,
z
)
)=
S
d
(
t
,
z
σ
, and the mean
A
−
1
S
u
z
d
velocity is defined as
u
,where
u
z
is the axial velocity. The area,
A
,
the averaged pressure,
P
, and the mean flux,
Q
=
σ
A u
, are the unknown variables to
be determined. The average pressure and flow rate are related to the 3D pressure
and velocity, respectively, while the area is related to the 3D wall displacement.
Thus, the 1D model provides a fluid-structure interaction (FSI) description of blood
flow in arteries, accounting for the wall compliance due to the blood load. For that
reason, the 1D model captures very well the wave propagation nature of blood flow
in arteries.
Integrating the Navier-Stokes equations on a generic cross-section S of the
cylindrical vessel, and after the above mentioned simplifications, explored in [
6
],
the reduced 1D form of the continuity and momentum equations for the flow of
blood in arteries is given, for all
t
=
>
0, by
⎧
⎨
∂
A
t
+
∂
Q
z
=
0
,
∂
∂
z
∈
(
a
,
b
)
,
(11)
Q
2
A
K
r
Q
A
⎩
∂
Q
∂
t
+
α
∂
A
ρ
∂
P
+
z
+
=
,
0
∂
z
∂
where
z
is the axial direction,
L
=
b
−
a
denotes the vessel length,
K
r
is the friction
parameter,
α
is the momentum flux correction coefficient, also known as Coriolis
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