Biomedical Engineering Reference
In-Depth Information
Fig. 6.17
Mean values (
thick solid lines
) and plus/minus standard deviations (
thin solid lines
)of
the logarithm of estimated Young modulus, for Ǜ
D
4, LJ
D
100 (
left
)andǛ
D
9;LJ
D
20 (
right
).
Adapted from [
6
], with permission of
c
J Wiley & Sons 2012
Numerical Results
Consider the idealized 3D geometry of an abdominal aortic aneurysm showed in
Fig.
6.16
, left. The length of the geometry, the minimum and maximum diameters
are 23 cm, 1.7 cm, and 5 cm, respectively. The Poisson ratio, density, and viscosity
of the structure are 0:46, 1:2g/cm
3
and 10
3
s, respectively. The fluid density
and viscosity are 1 g/cm
3
and 0:035Po, respectively. A Windkessel boundary
condition is used at the outflow (see Fig.
6.16
for details). We assume to have
displacement measures at each grid point of the mesh and that
.k/
.0;
2
I/.In
analogy with the variational approach, comparing the FE discretization of the cost
functional (
6.71
) and the cost functional (
6.72
), we assume the covariance matrix R
k
to be inversely proportional to the M
k
, the mass matrix on †. In particular we take
N
M
k
j†j
D LJ
2
m
T
ref
R
1
k
;
where
m
is the time sampling of the measurements, T
ref
is a reference time, and
LJ is a positive scalar used to weight the importance of the measurements. Also we
assume that
.0/
N.0;ǛI/. Figure
6.17
, shows the reconstructed Young modulus
in the different regions, as a function of Ǜ and LJ. The coefficient LJ represents the
level of confidence attributed to the displacement measures, whereas Ǜ is the a priori
covariance. As expected, the sensitivity with respect to LJ is higher when Ǜ is larger
and the sensitivity with respect to Ǜ is higher in regions with smaller SNR. Together
with the estimated parameters, the Kalman filter provides also their covariances,
which is an important index to evaluate the confidence we should have in the results.
The results are in fact more (less) reliable when the covariances are small (large).
For more results see [
6
].
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