Orthogonal and Least-Squares Based Coordinate Transforms for Optical Alignment Verification in Radiosurgery

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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
• 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)

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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

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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

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Functional Proton Radiosurgery

• Generation of small functional lesions with
  multiple overlapping proton beams (250 MeV)
• Used to treat functional disorders
• Parkinsons disease (Pallidotomy)
• Tremor (Thalamotomy)
• Trigeminal Neuralgia
• Target definition with MRI

Proton dose distribution for trigeminal neuralgia
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System ComponentsCamera System
Three Vicon Cameras
Camera Geometry
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System ComponentsMarker Systems and
Immobilization
Marker Systems
Marker Caddy Halo
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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

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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

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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 lt0.2 mm)

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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
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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) Where
MA is Rotation for Co-Planarity, MB 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.
  (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.

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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

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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)

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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)

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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