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xx
ij ijk
k
=
=
∑
1
10
1
10
(1.4b)
where j = 1, 2, 3 are the three orthogonal axes components (i.e., x, y, or z), x
ijk
is the
kth measurement for the ith fiducial point, and k is the number of measurements for
each fiducial point. For the calibrations of the image and robot, n = 10 and
x
ij
is the
mean measurement for the jth component of the ith fiducial point.
1.10.1.2 Fiducial Registration Error
The exact positions of N fiducials, which are given by P = [p
j
; j = 1, …, N] in the
coordinate system of the transducer, are known for either the image calibration
phantom or the robot calibration plates. For the calibration of the image, we mea-
sured the positions of N fiducials (the intersection of the nylon strings), which are
given by Q = [q
j
; j =1, …, N], in the 3-D coordinate system of the image. For the cali-
bration of the robot, we measured the positions of N fiducials (the small divots),
which are given by Q = [q
j
; j = 1, …, N], in the coordinate system of the robot by
moving it sothat it would touchthe small divot holes. The FRE was calculated as the
error of the rms distance between the corresponding fiducial positions before and
after the registration process. The FRE is given by:
(
)
FRE =
=
=
∑
qFp
N
jj
j
N
2
1
(1.5)
The rigid-body transformation that registers the exact fiducial positions P with
the measured fiducial positions Q is given by F.
1.10.1.3 Target Registration Error
TRE is defined as the distance between the corresponding points (not the fiducial
points) and is calculated using (1.5) before and after the registration process. We
used four targets in the image calibration phantom to determine the TRE; four other
markers on the plates were used to determine the TRE for robot calibration.
1.10.2 Results of Calibration
Our ability to localize the intersections of the nylon strings in the 3-D TRUS image
was analyzed along the x, y, and z directions and tabulated in Table 1.1 as the aver-
age FLE. As shown in Table 1.1, the FLE for localizing the intersection of the strings
is similar in the x or y directions and larger in the z direction. These values are attrib-
uted to errors of measurement in the z direction, which has the poorest resolution in
3-D TRUS images, corresponding to the elevation (i.e., out-of-plane) direction of the
acquired 2-D images [116].
For the calibration of the robot, the FLE for localizing the divots on the two
orthogonal plates is approximately the same in all three orthogonal directions. This
error, which is caused by the flexibility of the arm and the backlash in the joints of
24 3-D Ultrasound Imaging
the arm, is greater than the localization of the intersection of the strings; therefore,
this error will dominate the calibration errors of the system.
As shown in Table 1.2, the mean FRE and TRE for the calibration of the robot
were calculated to be 0.52 ± 0.18 mm and 0.68 ± 0.29 mm, respectively. Because the
FRE is larger for the calibration of the robot than the FRE for the calibration of the
image, both the mean FRE and TRE are larger for the calibration of the robot than
the FRE and the TRE for the calibration of the image. Because system errors will
result from both calibrationsof theimage andthe robot,the errors inthe calibration
of the robot will dominate the accuracy of integration for the two coordinate
systems.
1.10.3 Accuracy of Needle Placement
The accuracy of the robot’s needle placement was determined by moving the tip of
the needle to nine locations on a 5 cm × 5 cm grid that represented the skin of the
patient (i.e., a 3 × 3 grid of targeting points). The tip of the needle was then located
by using a three-axis stage (Parker Hannifin Co., Irwin, Pennsylvania) with a mea-
suring accuracy of 2
m. The displacement error, ε
d
, between the measured position
and targeted position of the needle tip was calculated using (1.6):
()()()
ε
d iii
xx yy zz= − +− +−
222
(1.6)
where (x, y, z) are the coordinates for the targeted point and (x
i
,y
i
,z
i
) are the coordi-
nates for the ith measured point. The mean error of the needle placement,
ε
d
, and
the standard deviation (STD) from 10 measurements at each position were found to
be 0.15 ± 0.06 mm.
1.10 Evaluation of 3-D TRUS-Guided Brachytherapy System 25
Table 1.1 FLE for Identification of the Fiducials Used for
Both Image and Robot Calibration (in Millimeters)
Image Calibration Robot Calibration
xyz xyz
Mean 0.05 0.06 0.11 0.17 0.21 0.20
Standard deviation 0.01 0.02 0.01 0.07 0.11 0.10
Table 1.2 FRE and TRE for the
Image and Robot Calibration of
the 3-D US-Guided Brachytherapy
System (in Millimeters)
Image
Calibration
Robot
Calibration
FRE TRE FRE TRE
Mean 0.12 0.23 0.52 0.68
STD 0.07 0.11 0.18 0.29
Source: [45].
1.10.4 Accuracy of Needle Angulation
The accuracy of the robot’s needle angulation was determined by attaching a small
plate to the needle holder. The robot was then used to angulate the plate vertically
and laterally (0°,5°,10°,15°). We measured the orientation of the plate using the
three-axis stage and determined the angulation error by comparing the angle of the
measured plate and of the expected plate.
As shown in Table 1.3, the mean angular differences between the measured
plate and the expected plate were less than 0.12° with a mean angle of 0.07°. How-
ever, angulation errors will propagate displacement errors as the distance from the
needle insertion is increased.
A needle was inserted 10 cm from the needle guide and the displacement error
was estimated. Using the mean and maximum angulation errors, we calculated the
mean and maximum displacement errors for a 10-cm insertion to be ±0.13 and
±0.50 mm, respectively.
1.10.5 Accuracy of Needle Targeting
The accuracy of the needle targeting system was evaluated using two tissue-mimick-
ing phantoms constructed from agar [131] and contained in a box. One side of the
box was removed toallow theinsertion of theneedle intothe agar gel (Figure 1.13).
As shown in Figure 1.13, each phantom contained two rows of stainless steel
beads with diameters of 0.8 mm to provide four different bead-targeting configura-
tions: two different needle insertion depths and two different distances from the
ultrasound transducer. The bead configuration formed a 4 cm × 4cm× 4 cm cube
that simulated the approximate size of a large prostate. The targeting experiments
involved: (1) producing a 3-D TRUS image, (2) identifying a bead to be targeted, (3)
choosing a trajectory, (4) positioning the robot to allow insertion of the needle to the
target, and (5) inserting the needle into the target. The accuracy of the needle target-
ing system was calculated by determining the deviations made by the tip of the nee-
dle from the location of the preinsertion bead in the 3-D TRUS image. By averaging
the results from all targeting experiments, we calculated the mean error to be 0.74 ±
0.24 mm. In addition, we plotted the results of the accuracy of the needle targeting
system as a 3-D scatterplot of the needle tips relative to the target (Figure 1.14).
1.10.6 Accuracy of Needle Tracking
Because the parallel trajectory of the needle insertion to the axis of the US transducer
can be verified by observing the insertion of the needle in the set of real-time 2-D
26 3-D Ultrasound Imaging
Table 1.3 Guidance Error Analysis of Needle Angulation by the Robot for
Vertical and Lateral Angulation of the Needle (in Degrees)
Vertical Angulation Lateral Angulation
Targeted angles 5° 10° 15° 5° 10° 15°
Mean angle difference
ε
a
0.02° 0.04° 0.06° 0.07° 0.10° 0.12°
Standard deviation 0.02° 0.06° 0.05° 0.05° 0.03° 0.08°
Maximum angle difference 0.08° 0.22° 0.18° 0.15° 0.15° 0.28°
TRUS images, we compared the results from the oblique needle tracking to the
results obtained from the parallel trajectory tracking. We used the robot to guide
1.10 Evaluation of 3-D TRUS-Guided Brachytherapy System 27
TRSP
TRLP
BRLP
BRSP
Z
X
Y
Figure 1.13 Schematic diagram of the prostate phantom. The four rows of circles represents the
four different bead configurations. The needle entered the phantom from the front, parallel to the
x-axis and was inserted into the cylindrical channel representing the rectum. TRLP = top row, long
penetration; TRSP = top row, short penetration; BRLP = bottom row, long penetration; BRSP = bot-
tom row, short penetration.
−2
−1
−1 −0.5
2
1
1
1
0
0
0.5
0
2
Figure 1.14 Needle targeting accuracy is displayed in a 3-D plot as the 95% confidence ellipsoid.
The origin of the coordinate system represents the target, and the needle tip positions after inser-
tion (relative to the targets) are represented by the squares. The projections of the needle tip posi-
tions and the ellipsoid (on the three orthogonal planes) are also shown. These results are for the
targets near the transducer and for a short penetration as shown in Figure 1.13. (From: [45].
© 2004 American Association of Physicists in Medicine. Reprinted with permission.)
the insertion of a needle at different angles (±5°, ±10°, ±15°) with respect to the tra
-
jectory of parallel insertion. For each oblique trajectory, the needle was automati-
cally tracked and its angle of insertion was automatically determined and compared
with the angle of parallel insertion. This test, which used an agar gel phantom and a
chicken tissue phantom [3, 132], was repeated five times for each angle in order to
determine the mean tracking error as well as the standard deviation.
Table 1.4 shows the evaluation results of the needle-tracking algorithm. Using
the chicken tissue phantom, the average execution time was 0.13 ± 0.01 second, and
the average angulation error was 0.54°±0.16°. Using the agar gel phantoms, the
average execution time was 0.12 ± 0.01 second, and the average angulation error
was 0.58°±0.36°.
The results shown in Table 1.4 demonstrate that the insertion of the needle can
be tracked in near real time, and that the tracking error doesnot significantly depend
on the angle of insertion. Figure 1.14 shows the views provided to the physician dur-
ing needle tracking.
1.10.7 Accuracy of Seed Implantation
The accuracy of seed implantation and the guidance of 3-D TRUS were evaluated
using a brachytherapy needle to implant spherical steel beads with diameters of 0.8
mm. The beads were implanted into several agar gel phantoms at different positions
and different needle trajectories (0°, ±5°, ±10°, ±15°) with respect to the axis of the
TRUS probe. These tests were carried out by: (1) choosing a trajectory and position
in the 3-D TRUS image, (2) positioning the robot at the chosen trajectory, (3) guid-
ing a needle to its target, and (4) implanting a bead. The error of implantation was
determined by: (1) obtaining a 3-D TRUS image after the implantation, (2) identify-
ing the bead in the 3-D TRUS image, and (3) calculating the distance between the
actual positions of the bead to the planned position.
Table 1.5 shows the accuracy of the bead implantation. The mean error was cal-
culated to be 2.59 ± 0.76 mm. The largest error was observed in the y direction (in
the vertical plane), corresponding to the bevel direction of the needle.
1.11 Conclusions
We have described the principles of 3-D ultrasound imaging and its use for
image-guided prostate brachytherapy. We have also demonstrated the technical fea-
sibility of a robot-aided prostate brachytherapy procedure using 3-D TRUS guid-
28 3-D Ultrasound Imaging
Table 1.4 Needle Segmentation Results for Chicken Tissue and Agar
Phantoms
Angle
–15
–10 –5 5 10 15
Test results for
chicken tissue
phantom
Time (sec)
0.13 0.11 0.12 0.12 0.12 0.14
Accuracy (deg)
0.50 0.51 0.43 0.37 0.74 0.74
Test results for
agar phantoms
Time (sec)
0.12 0.12 0.12 0.11 0.12 0.13
Accuracy (deg)
0.30 0.71 0.48 0.68 0.42 0.86
ance. Using robotic assistance, our experimental results indicate that the
brachytherapy needle can be guided accurately and consistently to target points in
the 3-D TRUS image. We expect that the introduction of a robotic system and the
implementation of 3-D TRUS tools for automatic needle detection of oblique inser-
tion, localization of the implanted seeds, and monitoring of prostate changes will
produce an effective intraoperative prostate brachytherapy procedure.
In our robot-assisted and 3-D TRUS-guided system, the robot and 3-D TRUS
image coordinate systems were registered using specially designed calibration phan-
toms to position the hole of the needle guidance in relation to the 3-D TRUS image.
This approach allowed us to remove the constraints of the parallel needle trajectory
that are presentin currentprostate brachytherapy procedures. Rather than having it
fixed onto the US transducer holder, the needle guide was mounted onto a robotic
arm. This mount provided greater flexibility over the needle’s trajectory and
reduced interference with the pubic arch.
In registering the coordinate systems of the robot and the imaging system,
fiducial points in the 3-D TRUS image must be determined. Errors in locating these
fiducial points will propagate and result in targeting errors, which lead to system-
atic errors. By comparing Table 1.2 with Table 1.1, we can see that the FLE for the
robot fiducials is greater than the FLE for the 3-D TRUS fiducials; therefore, errors
in calibrating the robot coordinate system will dominate and deteriorate the overall
accuracy of the system.
By comparing the results obtained in the evaluation of needle-targeting accu-
racy using a rigid rod with the results obtained for bead implantation accuracy with
a brachytherapy needle, we can see that the errors observed in the latter are larger
and dominate the overall performance of the system. Due to the bevel at the needle
tip, the large implantation error was primarily caused by the deflection of the needle
from the planned trajectory. Solving this source of error will greatly improve the
performance of the system [115].
Acknowledgments
The authors acknowledge the help of Kerry Knight in the preparation of this manu-
script. The authors also gratefully acknowledge the financial support of the Cana
-
dian Institutes of Health Research, and the Ontario R&D Challenge Fund. The first
Acknowledgments 29
Table 1.5 Brachytherapy Seed Implantation Error for Different
Angulations of the Needle (in Millimeters)
Angle
–20
–15 –10 –5 0 5 10 15 20
x
i
0.62 0.33 0.34 0.78 0.41 0.54 0.65 0.40 0.51
y
i
2.79 2.01 1.02 2.97 1.58 2.97 3.27 3.03 1.99
z
i
0.89 0.61 0.79 0.75 0.53 0.75 0.98 0.63 0.87
Total error
2.99 2.13 1.33 3.16 1.72 3.11 3.48 3.12 2.23
Mean
STD
2.59 ± 0.76 mm
Note: The error is shown in each of the 3-D orthogonal directions as well as the total error.
Source: [29].
author holds a Canada Research Chair and acknowledges the supportof theCanada
Research Chair program and the Canadian Foundation for Innovation.
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Acknowledgments 33