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Orthogonal and Least-Squares Coordinate Transforms for Radiosurgery Alignment Verification

This study presents methods for optical alignment verification in radiosurgery using orthogonal and least-squares coordinate transforms. Detailed steps from camera system setup to coordinate transformations and results analysis are outlined.

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Orthogonal and Least-Squares Coordinate Transforms for Radiosurgery Alignment Verification

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  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 • 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 ComponentsCamera System Three Vicon Cameras Camera Geometry

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

  8. Experimental ProcedureOverview • 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 ProcedureStep 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 ProcedureStep 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 ProcedureStep 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 TransformationOrthogonal 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) WhereMA is Rotation for Co-Planarity,MB is rotation for Co-linearity t is Translation vector

  13. 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. Coordinate TransformationLeast-Square Transformation

  14. ResultsAccuracy 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

  15. ResultsSystem 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 • mean system error + SD • orthogonal transform 2.4 + 0.6 mm (1.5 - 3.0) • LS transform 46 + 23 mm (18 - 78)

  16. Least Squares Orthogonal ResultsSystem 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)

  17. Conclusionand 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

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