Biomedical Engineering Reference
In-Depth Information
Table 2.
NURBS Type and Parameter Assignment of Each
of the Four Models Used to Calculate Strain
NURBS Type
Parameter Assignment
Model 1.1
Cartesian
Cylindrical
Model 1.2
Cartesian
Prolate
Model 2.1
Cylindrical
Cylindrical
Model 2.2
Prolate
Prolate
Left-ventricular geometry prescribes closed, periodic B-splines in the
v
LV
parametric direction and open, clamped B-splines in the
u
LV
and
w
LV
directions.
Selecting the appropriate B-spline parameters (order, number of control points,
knot vector, etc.), we first calculate the observation matrix. Given a sample point
from one of the contours, we calculate its
φ
and
η
values in the prolate spheroidal
coordinate system.
Establishing a parameterization such that 0
≤
v
≤
1 and
0
≤
≤
1, the surface parameter values (
v, w
) corresponding to
φ
and
η
are
simply
v
w
φ
2
π
η
η
max
, where
η
max
is the maximal
η
extent of the
model. The cylindrical parameter assignment is performed similarly except the
w
parameter value is given by
w
=
=
and
w
=
z
z
max
, where
z
max
is the maximal extent of the
model in the
z
direction (Figure 3).
For the Cartesian-based models, we modify the parameterization in the
v
direction slightly by temporarily moving the origin to the center of the sample
points of each short-axis endocardial contour for the specific purpose of assigning
parameteric values to the contour samples. This adjustment is not required, but
we found that such a shift provides a more evenly sampled parameterization in
the
v
direction, especially for nonconcentric short-axis left-ventricular contours.
Due to the discontinuity at
φ
=
−
π
, the shifting of the origin for each short-axis
contour is not possible for the models with a non-Cartesian description.
4.2.3. Right-ventricular NURBS model
Since the morphology of the right ventricle precludes a simple assignment
of parametric values to sample contour points based on base surfaces, we loft a
set of open short-axis epicardial and endocardial B-spline curves to compose the
corresponding surfaces spanned by the parametric directions (
v
RV
,w
RV
) shown
in Figure 5. Linear placement of control points between the epicardial and endo-
cardial surface control points provides the third parametric direction. The right-
ventricular model B-spline composition is similar to the left ventricle, except that
open, clamped B-splines are employed in the
v
RV
parametric direction.
