Biomedical Engineering Reference
In-Depth Information
the membrane one (12.12) are coupled at the FS interface
Γ
w
by the continuity of the
normal component of the normal stress and of the velocity
−
μ
(
∇
n
·
∂η
∂
u
T
u
+
∇
)
·
n
+
p
=
s
w
,
t
n
=
u
on
Γ
w
.
(12.13)
The grid velocity
w
is then computed as the harmonic extension of
∂η
∂
.
Now, we assume that the displacement of the vessel can be measured by a set of
time resolved images and the sequence of steps
segmentation
+
registration
,aswe
have done in the previous section. After an appropriate cubic spline interpolation (see
Sect. 12.3), we have the time dependent displacement field
t
n
in
Ω
m
η
(
t
,
x
)
defined on
Γ
w
,
m
is assimilated with the numerical model
as indicated by the feedback loop in the Introduction. The FW problem is given by
the system of Eqs. (12
that represents the
Data
. Displacement
η
.
7
1
−
2
, 12.12, 12.13), the unknown being
v
=[
u
,
p
,
η
]
. The post
processing function selects the displacement, i.e.
f
(
v
)=
η
. The CV is represented
by the Young's modulus
E
. The functional
J
reads
T
m
2
d
x
dt
(
η
−
η
J
=
)
+
Regularization
,
0
Γ
w
where
T
is the heart beat duration. Again, the regularizing term enhances the math-
ematical and numerical properties of the problem. A possible form is
T
E
E
re f
2
d
x
dt
α
−
,
0
Γ
w
where
is the usual parameter weighting the effect of the regularizing term on the
minimization process and
E
re f
is a reference value of the Young's modulus available
for instance from the literature. If we assume
a priori
that the CV is positive, we can
also consider the term
α
log
E
E
re f
2
α
max
.
x
∈
Γ
w
,
t
>
0
In both cases the regularizing term penalizes the distance between the control vari-
able
E
and the reference value for the Young's modulus
E
re f
.
The solution of this minimization problem is not trivial in many respects. Here-
after we present a first possible approach, under some simplifying assumptions. Even
though in the more general case, the Young's modulus can be function of time and
space, in the sequel we assume
•
E
constant in time in the interval
, significant changes of the compliance in
an artery being expected over a longer time scale;
[
0
,
T
]
•
E
piecewise constant in space, as we distinguish basically healthy and pathologi-
cal tissues featuring different values of compliance, each value being reasonably
constant in each subregion.
