Biomedical Engineering Reference
In-Depth Information
vivo
is still a challenging problem demanding appropriate mathematical tools. In this
section we suggest a DA procedure. The starting point is that the parameters of inter-
est are non trivial functions of measurable quantities. For instance, the compliance
of a tissue affects in a non trivial way its displacement, the latter being retrievable
from images. In some cases (as in elastography) we can prescribe the forces induc-
ing a measured displacement and formulate an inverse problem in the form: given
the force and the consequent displacement, find the stiffness (or more precisely the
Young's modulus, in the case of a linear elastic material) that fits at best the experi-
mental stress-strain data. In other cases, practical reasons prevent the knowledge of
some of the ingredients of this inverse problem. For instance, the natural periodic
motion of a vessel is the result of the interaction with the blood (and the other tis-
sues), in turn forced by the heart action. The forces exerted on the vascular wall by
the blood are not explicitly known but can be computed by solving the complete FSI
problem, as a function of (available) velocity/pressure values on the boundary of the
region of interest. The basic idea of DA approach is then to use numerical simula-
tions for bridging available data to the ingredients needed for solving the inverse
parameter-estimation problem.
In the case of the vascular stiffness, usable data are the images of the vessel dis-
placement (as in Sect. 12.3) and velocity and pressure (Sect. 12.2) on the boundary.
Numerical simulations allow to compute the forces on the wall and eventually to
solve an inverse problem. We present here a first step in this direction. However, it
is worth stressing that this DA approach has potentially a more general use than for
the evaluation of the vascular compliance (that can be currently achieved in several
ways), for different (and more numerous) sets of parameters.
12.4.1 Mathematical and numerical formulation
We formulate the problem of estimating the compliance of a linearly elastic mem-
brane filled by an incompressible fluid as follows. Let
be the volume of interest
of the fluid, where we assume the incompressible Navier-Stokes equations (12.7) to
hold. The membrane
Ω
, i.e. a 2D surface for a 3D fluid, which
is assumed to obey the equation for an elastic membrane
Γ
w
is a portion of
∂Ω
2
ρ
w
∂
η
+
E
θη
=
s
w
,
(12.12)
∂
t
2
where
ρ
w
is the density
of the solid,
s
w
is the stress exerted by the fluid and by surrounding tissues (the latter
will be neglected in the following), and
η
is the membrane displacement assumed to be normal to
Γ
w
,
θ
is a function of the mean and Gaussian
curvatures
ρ
1
and
ρ
2
of the membrane and of the Poisson's ratio
ν
, in particular
h
s
θ
:
=
2
(
4
ρ
1
−
2
(
1
−
ν
)
ρ
2
)
.
1
−
ν
The parameter
accounts for the transversal membrane effects (see [82]). Young's
modulus
E
is the parameter we want to estimate. The fluid subproblem (12.7) and
θ
