Biomedical Engineering Reference
In-Depth Information
values, respectively. The semi-major axes (
a
,
b
, and
c
) are set to halve the distance
between the two control points with extreme
x
,
y
, and
z
values, respectively. The
resulting ellipsoid generally does not pass through the six control points, nor does
it follow the prostate boundary well. To obtain a better initialization, the ellipsoid
is warped using a thin-plate spline transformation, so the six ends of the semi-
major axes of the ellipsoid map into the corresponding control points. Figure 17b
shows the initial mesh for the 3D prostate image shown in Figure 17a.
The deformation of the mesh is described by Eqs. (1) and (2); however, for
the 3D case, the position vector of each vertex has three components, as opposed
to two in the 2D case, i.e.,
p
i
=(
x
i
,y
i
,
z
i
)
T
. Similarly,
velocity, acceleration, and force vectors all have three components as well.
As in the 2D case, image forces in the 3D case drive vertices toward closest
edges and are described by Eq. (3). The values of energy
E
and image forces
f
im
i
are calculated at 3D voxel coordinates (
x
,
y
,
z
). The forces are sampled
using trilinear interpolation to obtain their values,
f
im
i
, at vertex
i
. In order to
prevent bunching of vertices during deformation, only the component of
f
img
i
T
, instead of
p
i
=(
x
i
,y
i
)
that
is locally normal to the surface is applied.
The internal force,
f
in
i
, at each vertex
i
keeps themodel smooth in the presence
of noise and is simulated by treating each edge of themodel as a spring. It is defined
by
M
M
e
ij
e
ij
f
int
i
=
−
,
(21)
j
=1
where
e
ij
=
p
i
−
p
j
is a vector representing the edge connecting vertex
i
with
coordinate
p
i
to an adjacent vertex
j
with coordinate
p
j
, and
M
is the number of
edges connected to vertex
i
. Again, only the normal component of
f
int
i
is applied
at each vertex.
The damping force is again defined by Eq. (5). The values for
w
im
i
,
w
in
i
, and
w
i
were set at 1.0, 0.2, and -0.5, respectively, for each vertex. The initial mesh
of Figure 17b is again shown in Figure 17c after deformation.
We have also developed 3D interactive editing tools, similar to tools used in
the 2D prostate segmentation approach, to allow the user to edit the mesh in 3D.
Specifically, the user can drag vertices to desired locations and optionally clamp
them and re-deform the mesh. To keep the mesh smooth when a vertex is dragged,
vertices within a sphere of a user-defined radius centered at the displaced vertex
are automatically deformed using the thin-plate spline transformation. A larger
radius has the effect of applying this smoothing to more neighboring vertices.
4.3.2. Algorithm evaluation
The algorithm was tested with six 3D TRUS images acquired from patients
who were candidates for prostate brachytherapy. The surface of each prostate was
outlined from the 3D images by a trained technician who is a representative of
