Biomedical Engineering Reference
In-Depth Information
additional computational cost with respect to the case in which the geometry is as-
sumed to be fixed. We will denote this approach 4D Image Based (4DIB). A similar
technique has been proposed in [63, 68] where the authors apply this image-based
motion approach to intra-cranial aneurysms and coronary arteries respectively, even
if implementing different strategies for some steps of the procedure.
12.3.1 Mathematical and numerical formulation
The workflow of the 4DIB approach consists in the following steps (for more details,
we refer to [69]). We assume to have an image set that represents the vessel of interest
at several time frames
{
t
k
}
within a heart beat.
Segmentation.
Depending on the nature of the source images, their dimensionality
and the complexity of the geometry to be reconstructed, segmentation can be per-
formed on single 2D planes or directly on 3D datasets. Different segmentation
methodologies and different ways to represent the final models are available. For an
introduction, see [24, 62]. In the applications presented here, a level set technique
was used for the 3D segmentation of vessels, specifically the segmentation tool avail-
able within the Vascular Modelling Toolkit (VMTK) software package [3]. At the
end of this step, a triangulated surface is available for each time frame.
Motion tracking.
This consists in solving a
registration problem
(see e.g. [28]),
i.e. finding the alignment of the geometries of two consecutive time frames, so to
have a displacement field that maps the points on the surface of the lumen at a given
time frame to the surface in the subsequent one. This is another example of inverse
problems that can be cast in the form of the feedback loop in the Introduction. For
this reason, we detail this step in the next subsection.
Simulation.
From the sequence of maps describing the motion of the surface points
from one time frame to the subsequent one, the displacement of the boundary of the
moving domain is retrieved at the image acquisition times. Then, this is interpolated
to define the displacement of the boundary at each time instant in the simulation.
The velocity of the boundary is obtained as the time derivative of the displacement.
To ensure also the continuity of the time derivative of the points velocity, a cubic
spline time interpolation is chosen. The displacement and the grid velocity
w
of the
whole domain, computed at each time step of the simulation, are obtained as the
harmonic extension of the boundary fields. Once the domain motion is available,
the incompressible Navier-Stokes equations for a Newtonian fluid in a moving do-
main can be written in the
Arbitrary Lagrangian Eulerian
(ALE) formulation (see,
e.g., [70])
∂
u
T
u
t
−
μ∇
·
(
∇
u
+
∇
)+(
u
−
w
)
·
∇
u
+
∇
p
=
s
,
in
Ω
(
t
)
∂
[
1
mm
]
∇
·
u
=
0
in
Ω
(
t
)
,
(12.7)
u
=
w
on
Γ
w
(
t
)
,
+
Boundary Conditions
on
Γ
in
(
t
)
and
Γ
out
(
t
)
.
