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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

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

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

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

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

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System Components Camera System Three Vicon Cameras Camera Geometry

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

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Experimental Procedure Overview Goal of stereotactic procedure: Goal of stereotactic procedure: align anatomical target with known stereotactic coordinates with proton beam axis with submillimeter accuracy align anatomical target with known stereotactic coordinates with proton beam axis with submillimeter accuracy Experimental procedure: Experimental procedure: align simulated marker with known stereotactic coordinates with laser beam axis 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) 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 determine system alignment error repeatedly (3 independent experiments) for 5 different marker positions

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

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Experimental Procedure Step II- Marker Alignment (Image Processing) 1 cm laser beam from stereotactic cone aligned to phantombase marker 1 cm laser beam from stereotactic cone aligned to phantombase marker digital image shows laser beam spot and marker shadow 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 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) Distance offset between beam-center and marker- center is calculated (typically <0.2 mm)

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

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

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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.: (A T A) -1 A T, 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.

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Results Accuracy of Camera System Method: compare camera- measured distances between markers pairs with DIL-measured values Method: compare camera- measured distances between markers pairs with DIL-measured values Results (15 independent runs) Results (15 independent runs) mean distance error + SD mean distance error + SD caddy: -0.23 + 0.33 mm caddy: -0.23 + 0.33 mm cross: 0.00 + 0.09 mm cross: 0.00 + 0.09 mm

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

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

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Conclusion and Future Directions Currently, Orthogonal Transformation outperforms standard Least-Square based Transformation by more than one order of magnitude 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 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 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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