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

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)

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

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

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

System Components Camera System Three Vicon Cameras Camera Geometry

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

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

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

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)

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

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

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.

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: -0.23 + 0.33 mm cross: 0.00 + 0.09 mm

Results System Error - Initial Results (a) First 12 data runs: mean system error + SD orthogonal transform 2.8 + 2.2 mm (0.5 - 5.5) LS transform 61 + 33 mm (8.9 - 130) (b) 8 data runs, after improving calibration 2.4 + 0.6 mm (1.5 - 3.0) 46 + 23 mm (18 - 78)

Results System Error - Current Results (c) Last 15 data runs, 5 target positions, 3 runs per position: mean system error + SD orthogonal transform 0.6 + 0.3 mm (0.2 - 1.3) LS transform 25 + 8 mm (14 - 36) Least Squares Orthogonal

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