Biomedical Engineering Reference
In-Depth Information
Table 1.
Algorithmic Synopsis for Fitting a 4D Biventricular
NURBS model to
N
+1
Volumetric Short-Axis and
N
+1
Volumetric Long-Axis Tagged Data
0. Initialization: create an NURBS biventricular model for time
t
=0
from short- and
long-axis contour information by performing a least-squares fit to all contour points in all
short- and long-axis slices (each such point is preassigned a
parametric value). From
the resulting NURBS fit, construct the volumetric parameterization
S
L
(
u, v, w, t
=0)
(
v,w
)
by
placing control points linearly in the
u
direction.
for
i
=1
,...,N{
1. For time
t
=
i
create a parameterized NURBS biventricular model from short- and
long-axis contour information by performing a least-squares fit to the contour points (prior
to least-squares, each contour point is assigned a
parametric value). Linearly place
control points in the
u
direction to construct the volumetric parameterization
S
E
(
u, v, w, i
)
(
v,w
)
.
2. Use 1D, 2D, and 3D displacement information to determine corresponding tag
points between
t
=
i
and
t
=0
using the following sources of information: a) Tag plane
normal distances (Figure
??
), b) contour/tag intersections (Figure
??
), and c) intersections
of 3D tag planes. Assign identical parametric values to corresponding points at
t
=0
.
3. Perform NURBS weighted least-squares to determine control points and weights
for the model at
t
=0
(this is the Eulerian fit). The weights reflect confidences in the
correspondences established in step 2. Couple the RV to the LV by assigning corresponding
points on the RV/LV interface at
t
=0
the same parameters as at
t
=
i
. Call the NURBS
parameterization resulting from weighted least-squares at time
t
=0
S
i
E
(
u, v, w,
0)
.
Densely sample
S
i
E
(
u, v, w,
0)
p
j
=
S
i
E
(
u
j
,v
j
,w
j
,
0)
4.
.
For each sample point
determine via conjugate gradient descent the
(
u
j
,v
j
,w
j
)
of
S
L
(
u, v, w,
0)
(constructed
in step 0) that corresponds to the same spatial point
p
j
. Calculate the set of displacements
V
E
(
u
j
,v
j
,w
j
)=
S
E
(
u
j
,v
j
,w
j
,i
)
− S
i
E
(
u
j
,v
j
,w
j
,
0)
.
5.
Perform
the
least-squares
fit
for
the
biventricular
model
at
t
=
i
,
de-
noted
by
S
L
(
u, v, w, i
)
,
from
assigning
the
parameters
(
u
j
,v
j
,w
j
)
to
the
point
S
L
(
u
j
,v
j
,w
j
,
0) +
V
E
(
u
j
,v
j
,w
j
)
for all
j
. This is the Lagrangian fit.
}
6.
Since all models are registered in the common parametric coordinates
(
u, v, w
)
,
temporal smoothing of all models is now possible.
Perform the temporal lofting —
S
L
(
u, v, w, t
)=
l
=0
S
L
(
u, v, w, l
)
N
l,s
(
t
)
.
7.
Lagrangian strain values between time points
t
=0
and
t>
0
can easily be
determined from the displacements
V
L
(
u, v, w, t
)
. Similarly, Eulerian strain values
can be determined from the displacements
V
E
(
u, v, w, t
)
.
From the displacements the
deformation gradient tensor,
, is determined. Subsequently, for any point of interest
within the LV wall or the RV free wall and directio
n
F
n
, the Lagrangian strains in the
can be calculated from
2
n
T
(
F
T
F
−
1
)
n
reference configuration
.
The Eulerian
R
strains in the direction
n
in the deformed configuration
R
t
may be calculated from
2
n
T
(
1
−
(
FF
T
)
−
1
)
n
1
.
