Biomedical Engineering Reference
In-Depth Information
Fig. 12.12.
3D simplified aorta, Forward simulation. Geometry at time
t
=
0
ms
,
t
=
75
ms
,
t
=
cm
2
]. The displacement has been scaled
150
ms
and
t
=
300
ms
. Colored with blood pressure [
dyne
/
by a factor 2 for the sake of visualization
Table 12.3.
Estimates (to be multiplied by 10
6
, top) and relative error for different values of
ξ
.
10
6
dyne
cm
2
Exact
E
is 4
·
/
↓
E
0
\
ξ
→
40
20
10
10
6
dyne
/
cm
2
3
.
955
3
.
915
3
.
679
1
.
1%
2
.
1%
8
.
0%
accuracy is maintained also for smaller values of
. Moreover, the BFGS applied
to the membrane case is pretty robust with respect to the noise, both in terms of
accuracy and convergence.
Other 2D cases on non-trivial geometries (like a bifurcation) have been reported
in [83].
We now consider the three dimensional geometry shown in Fig. 12.12, repre-
senting a simplified aorta. This geometry consists of a cylinder of radius 1
ξ
5
cm
and
height 10
cm
and half a torus with curvature radius 4 cm. The physical parameters are
E
.
10
6
dyne/cm
2
,
2
cm
. We prescribe a parabolic velocity profile
at the inlet, with the peak velocity
u
M
(
=
4
·
μ
=
0
.
5,
h
s
=
0
.
10
3
π
t
)
given by
u
M
=
100 max
t
(
sin
(
t
))
cm
/
s
,
and absorbing boundary conditions at the outlet [82]. The time-step used is
Δ
t
=
2
. The pressure field
and the geometry displacement are shown in Fig. 12.12. As before, we estimate the
compliance using the displacement of the forward simulation perturbed by an arti-
ficial noise. The estimation has been performed over the first 150
ms
. In Table 12.3
we report the results related to a single realization of the noise.
In the 3D case, the method seems to be more sensitive to the noise. However even
with a noise with
.
5
ms
. In this case
η
M
is defined as
η
M
(
t
)=
ma
x
|
η
fwd
(
x
,
t
)
|
ξ
=
10, the parameter estimation is accurate enough for most of
clinical applications.
12.4.3 Perspectives
As the preliminary results indicate, the assimilation of data and numerical models is
a worthwhile approach for estimating patient-specific parameters to be used either
