Biomedical Engineering Reference
In-Depth Information
the selection of these parameters, we construct the reference configuration (the
left- and right-ventricular volumetric NURBS models) using contour data from
the end-diastolic images. We denote this initial model as
S
L
(
u, v, w, t
=0).
These parameters remain fixed for the nonrigid registration for each frame of
data. For each time point
t>
0 we construct a separate NURBS model from the
contour data of that time point using the same parameters selected for constructing
S
L
(
u, v, w, t
=0). We denote this model as
S
E
(
u, v, w, t
=
i
). This construction
corresponds to Steps 0 and 1 in Table 1.
The basic principle is the same for formulation of all models: two paramet-
rically identical surfaces are constructed that correspond to the epicardial and en-
docardial surfaces in the (
v
LV
,w
LV
) or (
v
RV
,w
RV
) parametric directions shown
in Figure 5. The identical parameterization for both the endocardial and epicar-
dial surfaces is required to create a volumetric NURBS model with a third set of
control points creating the third parametric direction (
u
LV
or
u
RV
). The maximal
apical extent of the left (right) ventricular model is limited by the most apical
short-axis left (right) ventricular endocardial contour. The maximal basal extent
of both ventricular models is similarly limited by the most basal short-axis image.
A biventricular NURBS model constructed from a canine heart is shown in Fig-
ure 6. In addition to the limits imposed by the image locations, inclusion of the
most apical portions of the ventricles is problematic in that it requires a degenerate
condition of the tensor product volume at that location. Some possible remedies
for future research are discussed in the Conclusions section.
4.2.1. B-spline parameterization of the LV and RV
Consider the biventricular parametric coordinates of Figure 5, examining the
LV and the RV separately. The
u
LV
and
w
LV
directions require open, aperiodic
parameterizations, whereas the
v
LV
direction requires periodic parameterization.
There are several ways to achieve this. (1) We can use Cartesian NURBS with
the constraints that for the
v
LV
direction periodic B-spline basis functions are
used. This would be similar to taking the Cartesian B-spline model described in
[13, 16, 26], attaching two opposite faces of the cubic model together, and ensuring
continuity of the resulting topological torus at the connection interface through
use of periodic B-spline basis functions in the
v
LV
direction. Note that in this
scenario the B-spline control points are represented in the Cartesian coordinates.
(2) We can use non-Cartesian (e.g., cylindrical and prolate spheroidal) B-splines.
For these splines, the control points are represented in the cylindrical or prolate
spheroidal coordinates, while aperiodic basis functions are used in the
u
LV
and
w
LV
directions, and periodic basis functions are used in the
v
LV
direction. In
the case of the RV, all three parametric directions (
u
RV
,v
RV
,w
RV
) require use
of open, aperiodic splines. Thus far, in our work, we have only used Cartesian
B-Splines for modeling the RV.
