Biomedical Engineering Reference
In-Depth Information
Fig. 6.8
On the
left
,the
colored
vector field consists in the available measures whereas the
black
one corresponds to the noise-free data. In the
center
, the magnitude of the assimilated vector field
is reported. On the
right
,the
colored
vector field corresponds to the assimilated velocity, the
black
one to the noisy data and the
colored field
in the background corresponds to the magnitude of the
velocity
A
k
comes from the discretization of .
u
k
r/
u
k
C
1
C
w
.
u
k
C
1
r/
u
k
(
w
D
0
for
Picard method, 1 for Newton);
Y
k
is the discretization of .
u
k
r/
u
k
.Here
u
k
is
defined as #
u
k
1
C.1#/
u
k
,being# 2 Œ0; 1,
w
is a relaxation parameter, chosen
empirically.
Numerical tests.
In Fig.
6.8
we report the numerical results obtained on two
geometries approximating blood vessels. In Fig.
6.8
(left) the computational grid
is an approximation of a carotid artery; the colored vector field consists in the
actual data used in the assimilation, these are generated adding Gaussian noise to
a reference solution; to appreciate the presence of the noise the noise-free data are
also reported in black. In the center, the magnitude of the assimilated velocity is
displayed; a comparison with a reference solution (conducted in [
18
]) shows that
the noise is filtered and that the assimilated solution is close to the reference one.
On the right, a three-dimensional cylindrical domain is reported, this case
is treated with an axisymmetric formulation. On selected internal surfaces the
assimilated field and its magnitude are reported; it is important to note that the noise
affecting the components of the velocity parallel to the flow is significantly filtered.
Next, we consider the problem of estimating the WSS already described in
the introductory example of Sect.
6.1
. An accurate approximation of the WSS is
fundamental in the investigation of cardiovascular pathologies since it is an index
of the possibility of rupture of the vessel wall and formation of stenosis [
25
].
Approximations of the WSS retrieved from indirect measurements are in general
unreliable because of the post-processing numerical errors and the noise affecting
the measures. Including measurements in simulations is a way for improving the
reliability of the computed solutions and, on the other hand, the introduction of
the mathematical (numerical) model results in noise filtering. For the geometry
of Fig.
6.8
(left), we compute the WSS on a selected internal wall. In order to
quantify the accuracy of this solution we compare the assimilated WSS with
the one associated with a reference solution, we introduce the index of accuracy
E
WSS
DkWSS WSS
FE
k
2
=kWSS
FE
k
2
where WSS
FE
is the value retrieved from
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