1. Orthogonal and Least-Squares Based Coordinate Transforms for Optical Alignment Verification in Radiosurgery
Ernesto Gomez PhD, Yasha Karant PhD, Veysi Malkoc, Mahesh R. Neupane,
Keith E. Schubert PhD, Reinhard W. Schulte, MD

2. ACKNOWLEDGEMENT Henry L. Guenther Foundation
Instructionally Related Programs (IRP), CSUSB
ASI (Associated Student Inc.), CSUSB
Department of Radiation Medicine, Loma Linda University Medical Center (LLUMC)
Michael Moyers, Ph.D. (LLUMC)

3. OVERVIEW Experimental Procedure Introduction System Components
1. Camera System
2. Marker System
Experimental Procedure
1. Phantombase Alignment
2. Alignment Verification (Image Processing)
3. Marker Image Capture
Coordinate Transformations
1. Orthogonal Transformation
2. Least Square Transformation
Results and Analysis
Conclusions and Future directions
Q &A

4. INTRODUCTION
Radiosurgery is a non-invasive stereotactic treatment technique applying focused radiation beams
It can be done in several ways:
1. Gamma Knife
2. LINAC Radiosurgery
3. Proton Radiosurgery
Requires sub-millimeter positioning and beam delivery accuracy

5. Functional Proton Radiosurgery
Generation of small functional lesions with multiple overlapping proton beams (250 MeV)
Used to treat functional disorders:
Parkinson’s disease (Pallidotomy)
Tremor (Thalamotomy)
Trigeminal Neuralgia
Target definition with MRI
Proton dose distribution for trigeminal neuralgia

6. System Components Camera System
Three Vicon Cameras
Camera Geometry

7. System Components Marker Systems and Immobilization
Marker Cross
Marker Caddy
Stereotactic Halo
Marker Systems
Marker Caddy & Halo

8. Experimental Procedure Overview
Goal of stereotactic procedure:
align anatomical target with known stereotactic coordinates with proton beam axis with submillimeter accuracy
Experimental procedure:
align simulated marker with known stereotactic coordinates with laser beam axis
let system determine distance between (invisible) predefined marker and beam axis based on (visible) markers (caddy & cone)
determine system alignment error repeatedly (3 independent experiments) for 5 different marker positions

9. Experimental Procedure Step I- Phantombase Alignment
Platform attached to stereotactic halo
Three ceramic markers attached to pins of three different lengths
Five hole locations distributed in stereotactic space
Provides 15 marker positions with known stereotactic coordinates

10. Experimental Procedure Step II- Marker Alignment (Image Processing)
1 cm laser beam from stereotactic cone aligned to phantombase marker
digital image shows laser beam spot and marker shadow
image processed using MATLAB 7.0 by using customized circular fit algorithm to beam and marker image
Distance offset between beam-center and marker-center is calculated (typically <0.2 mm)

11. Experimental Procedure Step III- Capture of Cone and Caddy Markers
Capture of all visible markers with 3 Vicon cameras
Selection of 6 markers in each system, forming two large, independent triangles
Caddy marker triangles
Cross marker triangles

12. Coordinate Transformation Orthogonal Transformation
Involves 2 coordinate systems
Local (L) coordinate system (Patient Reference System)
Global (G) coordinate system (Camera Reference System)
Two-Step Transformation of 2 triangles:
Rotation
L-plane parallel to G-plane
L-triangle collinear with G-triangle
Translation
Transformation equation used:
pn(g) = MB . MA . pn(l) + t (n = 1 - 3)
Where MA is Rotation for Co-Planarity, MB is rotation for Co-linearity
t is Translation vector

13. Coordinate Transformation Least-Square Transformation
Also involves global (G) and local (L) coordinate systems
Transformation is represented by a single homogeneous coordinates with 4D vector & matrix representation.
General Least-Square transformation matrix:
AX = B
The regression procedure is used:
X = A+ B
Where A+ is the pseudo-inverse of A (i.e.: (ATA)-1AT, use QR)
X is homogenous 4 x 4 transformation matrix.
The transformation matrix or its inverse can be applied to local or global vector to determine the corresponding vector in the other system.

14. Results Accuracy of Camera System
Method: compare camera-measured distances between markers pairs with DIL-measured values
Results (15 independent runs)
mean distance error + SD
caddy: mm
cross: mm

15. Results System Error - Initial Results
(a) First 12 data runs:
mean system error + SD
orthogonal transform
mm ( )
LS transform
mm ( )
(b) 8 data runs, after improving calibration
mm ( )
mm ( )

16. Results System Error - Current Results
(c) Last 15 data runs,
5 target positions, 3 runs per position:
mean system error + SD
orthogonal transform
mm ( )
LS transform
mm ( )
Least Squares
Orthogonal

17. Conclusion and Future Directions
Currently, Orthogonal Transformation outperforms standard Least-Square based Transformation by more than one order of magnitude
Comparative analysis between Orthogonal Transformation and more accurate version of Least-Square based Transformation (e.g. Constrained Least Square) needs to be done
Various optimization options, e.g., different marker arrangements, will be applied to attain an accuracy of better than 0.5 mm