Biomedical Engineering Reference
In-Depth Information
S
u
z
d
σ
coefficient, defined by
is the fluid mass density. For a parabolic
profile, the friction parameter is defined as
K
r
=
α
=
,and
ρ
A u
2
8
πμ
[
6
], which is the value
generally used in practice. The Coriolis coefficient is set to
α
=
1, corresponding to a
flat profile [
6
], in order to simplify the analysis. The density
ρ
and the fluid dynamic
viscosity
are considered constant. Hence, the 1D model does not account for the
non-Newtonian behavior of blood.
The previous system of two equations for the three unknown variables
A
,
Q
,and
P
needs to be closed. In order to do that, a structural model for the vessel wall
movements, relating pressure and area, must be given. Here, the simplest pressure-
area algebraic relation [
6
,
7
]isused
μ
√
A
−
√
A
0
A
0
P
(
t
,
z
)
−
P
ext
=
β
,
(12)
where
A
0
is the initial area and
is a single parameter that describes the mechanical
and physical properties of the vessel wall
β
√
π
hE
β
=
2
,
(13)
1
−
ξ
where
h
the wall thickness,
E
the vessel wall Young, or elasticity, modulus, and
ξ
is constant along
z
only when
E
,
h
,or
A
0
are
constant, since they may be functions of
z
. In this work, the wall parameters are
assumed constant along
z
, and the external pressure is neglected:
P
ext
=
the vessel wall Poisson ratio.
β
0.
Numerical Discretization of the 1D Model
The 1D model is numerically discretized in time and space by means of a second-
order Taylor-Galerkin scheme [
6
]. It consists in using the Lax-Wendroff scheme to
discretize in time and the finite element method to obtain the space approximation.
This discretization can be considered as a finite element counterpart of the Lax-
Wendroff scheme, which has a very good dispersion error characteristic and can be
easily implemented [
6
].
A uniform mesh is used, meaning that the elements size is constant and equal to
h
.
Moreover, linear (
P
1) finite elements are considered. The Lax-Wendroff scheme is
obtained using a Taylor series of the solution
U
T
truncated to the second
order, resulting in an explicit scheme. Being an explicit time advancing method, the
Lax-Wendroff scheme requires the verification of a condition bounding the time
step [
6
]
=[
QA
]
√
3
3
h
Δ
t
≤
|
)
,
(14)
(
+
|
max
c
u
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