Biomedical Engineering Reference
In-Depth Information
On the wall the fluid velocity is prescribed equal to the vessel velocity (Dirichlet
condition), while inflow and outflow boundary data can be retrieved by measures or
designed to reproduce a physiological or pathological behavior.
12.3.1.1 Assimilation of segmented vascular surfaces
Registration is a procedure to align images taken from different devices, from dif-
ferent viewpoints or at different time instants. Many different methodologies exist
depending on the source of images, their dimensionality and the type of movement
to be recovered, particularly whether we have small or large deformations. In partic-
ular, a wide number of different approaches has been detailed for surface registration
(see e.g. [4, 11, 13, 71]).
Here we resort to an algorithm relying upon a minimization procedure [28]. The
registration is performed over 3D surfaces representing the vessel at the different
time frames. More precisely, given M+1 time frames corresponding to M+1 trian-
gulated surfaces, the tracking process consists in M registration steps between each
couple of consecutive time steps. Within each stage the points of one surface, the
source surface
.A
displacement field for the whole surface mesh is computed so that at the end of this
tracking procedure, M displacement fields are available describing the vessel wall
motion at the instants of the image acquisitions.
The map between two consecutive frames is computed by minimizing a func-
tional in the form (12.1). In particular, let us denote with
S
, are mapped to the subsequent one, called the
target surface
T
ϕ
(
·
)
the (unknown) map
from
. Referring to the feedback loop in the Introduction, the forward prob-
lem FW is the actual application of
S
to
T
(
Input
), so that
2
ϕ
to the source surface
S
v
. The control
variable set CV is given by the mathematical representation of the map
=
f
(
v
)=
ϕ
(
S
)
. The
Data
are represented by the target surface
T
. This
can be parametrically described by assuming, e.g., that it belongs to a functional fi-
nite dimensional space spanned by a basis function set
ϕ
(
·
)
a
i
ψ
i
, being
a
i
real coefficients. In this case
a
i
are the CV. However, since the map is supposed to
be strongly space-dependent, in [69] we resorted to a
non-parametric map
implicitly
defined with a collocation approach by the position of the nodes on the source im-
age. This means that the coordinates of the vertices computed by the minimization
process implicitly define the map point-wise. The map is then extended to the entire
source surface by a piecewise linear interpolation of the values at the vertices.
Finally, to complete the picture, we need to specify the definition of the distance
between
ψ
i
so that
ϕ
=
∑
and the regularizing term. Different choices are avail-
able, strictly problem dependent. Let us introduce the distance of the image of a point
on
ϕ
(
S
)
and the data
T
S
to the surface
T
as
δ
(
ϕ
(
x
)
,T
)=
inf
{
ϕ
(
x
)
−
y
:
y
∈T },
x
∈S .
(12.8)
2
The post-processing in this case is trivially the identity application.
