Orthogonal and Least-Orthogonal and Least-Squares Based Coordinate Squares Based Coordinate

Transforms for Optical Transforms for Optical Alignment Verification in Alignment Verification in

RadiosurgeryRadiosurgeryErnesto Gomez PhD, Yasha Karant PhD, Ernesto Gomez PhD, Yasha Karant PhD,

Veysi Malkoc, Veysi Malkoc, Mahesh R. NeupaneMahesh R. Neupane,, Keith E. Schubert PhD, Reinhard W. Keith E. Schubert PhD, Reinhard W.

Schulte, MDSchulte, MD

ACKNOWLEDGEMENTACKNOWLEDGEMENT Henry L. Guenther Foundation Henry L. Guenther Foundation

Instructionally Related Programs (IRP), Instructionally Related Programs (IRP), CSUSB CSUSB

ASI (Associated Student Inc.), CSUSBASI (Associated Student Inc.), CSUSB

Department of Radiation Medicine, Loma Department of Radiation Medicine, Loma Linda University Medical Center (LLUMC)Linda University Medical Center (LLUMC)

Michael Moyers, Ph.D. (LLUMC)Michael Moyers, Ph.D. (LLUMC)

OVERVIEWOVERVIEW IntroductionIntroduction System ComponentsSystem Components 1. Camera System1. Camera System

2. Marker System2. Marker System Experimental ProcedureExperimental Procedure

1. Phantombase Alignment1. Phantombase Alignment2. Alignment Verification (Image Processing)2. Alignment Verification (Image Processing)

3. Marker Image Capture3. Marker Image Capture Coordinate TransformationsCoordinate Transformations 1. Orthogonal Transformation1. Orthogonal Transformation 2. Least Square Transformation2. Least Square Transformation Results and AnalysisResults and Analysis Conclusions and Future directionsConclusions and Future directions Q &AQ &A

INTRODUCTIONINTRODUCTION Radiosurgery is a non-Radiosurgery is a non-

invasive stereotactic invasive stereotactic treatment technique treatment technique applying focused radiation applying focused radiation beamsbeams

It can be done in several It can be done in several ways:ways:

1. Gamma Knife1. Gamma Knife

2. LINAC Radiosurgery2. LINAC Radiosurgery

3. Proton Radiosurgery3. Proton Radiosurgery

Requires sub-millimeter Requires sub-millimeter positioning and beam positioning and beam delivery accuracydelivery accuracy

Functional Proton Functional Proton RadiosurgeryRadiosurgery

Generation of small Generation of small functional lesions with functional lesions with multiple overlapping multiple overlapping proton beams (250 proton beams (250 MeV)MeV)

Used to treat functional Used to treat functional disorders:disorders: Parkinson’s disease Parkinson’s disease

(Pallidotomy)(Pallidotomy) Tremor Tremor

(Thalamotomy)(Thalamotomy) Trigeminal NeuralgiaTrigeminal Neuralgia

Target definition with Target definition with MRIMRI

Proton dose distribution for trigeminal neuralgia

System ComponentsSystem ComponentsCamera SystemCamera System

Three Vicon Cameras

Camera Geometry

System ComponentsSystem ComponentsMarker Systems and ImmobilizationMarker Systems and Immobilization

Marker Caddy & HaloMarker Systems

Marker Cross

Marker Caddy

Stereotactic Halo

Experimental ProcedureExperimental ProcedureOverviewOverview

Goal of stereotactic procedure:Goal of stereotactic procedure: align anatomical target with align anatomical target with

known stereotactic coordinates known stereotactic coordinates with proton beam axis with with proton beam axis with submillimeter accuracysubmillimeter accuracy

Experimental procedure:Experimental procedure: align simulated marker with align simulated marker with

known stereotactic coordinates known stereotactic coordinates with laser beam axiswith laser beam axis

let system determine distance let system determine distance between (invisible) predefined between (invisible) predefined marker and beam axis based on marker and beam axis based on (visible) markers (caddy & (visible) markers (caddy & cone)cone)

determine determine system alignment system alignment errorerror repeatedly (3 repeatedly (3 independent experiments) for 5 independent experiments) for 5 different marker positionsdifferent marker positions

Experimental ProcedureExperimental ProcedureStep I- Phantombase AlignmentStep I- Phantombase Alignment

Platform attached to Platform attached to stereotactic halostereotactic halo

Three ceramic Three ceramic markers attached to markers attached to pins of three different pins of three different lengthslengths

Five hole locations Five hole locations distributed in distributed in stereotactic spacestereotactic space

Provides 15 marker Provides 15 marker positions with positions with known known stereotactic stereotactic coordinatescoordinates

Experimental ProcedureExperimental ProcedureStep II- Marker Alignment (Image Step II- Marker Alignment (Image

Processing)Processing) 1 cm laser beam from 1 cm laser beam from stereotactic cone aligned stereotactic cone aligned to phantombase markerto phantombase marker

digital image shows laser digital image shows laser beam spot and marker beam spot and marker shadowshadow

image processed using image processed using MATLAB 7.0 by using MATLAB 7.0 by using customized circular fit customized circular fit algorithm to beam and algorithm to beam and marker imagemarker image

Distance offset between Distance offset between beam-center and marker-beam-center and marker-center is calculated center is calculated (typically <0.2 mm)(typically <0.2 mm)

Experimental ProcedureExperimental ProcedureStep III- Capture of Cone and Caddy Step III- Capture of Cone and Caddy

MarkersMarkers Capture of all visible Capture of all visible

markers with 3 Vicon markers with 3 Vicon camerascameras

Selection of 6 markers in Selection of 6 markers in each system, forming two each system, forming two large, independent large, independent trianglestriangles

Cross marker triangles

Caddy marker triangles

Coordinate Coordinate TransformationTransformation

Orthogonal TransformationOrthogonal Transformation

Involves 2 coordinate systemsInvolves 2 coordinate systems Local (L) coordinate system Local (L) coordinate system

(Patient Reference System)(Patient Reference System) Global (G) coordinate system Global (G) coordinate system

(Camera Reference System)(Camera Reference System) Two-Step Transformation of 2 Two-Step Transformation of 2

triangles:triangles: RotationRotation

L-plane parallel to G-planeL-plane parallel to G-plane L-triangle collinear with G-L-triangle collinear with G-

triangletriangle TranslationTranslation

Transformation equation usedTransformation 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 Coordinate TransformationTransformation

Least-Square TransformationLeast-Square Transformation Also involves global (G) and local (L) coordinate Also involves global (G) and local (L) coordinate systemssystems

Transformation is represented by a single Transformation is represented by a single homogeneous coordinates with 4D vector & matrix homogeneous coordinates with 4D vector & matrix representation.representation.

General Least-Square transformation matrix:General Least-Square transformation matrix:

AX = BAX = B The regression procedure is used:The regression procedure is used:

X = AX = A++ B B Where AWhere A++ is the pseudo-inverse of A (i.e.: (A is the pseudo-inverse of A (i.e.: (ATTA)A)-1-1AATT, use QR), use QR) X is homogenous 4 x 4 transformation matrixX is homogenous 4 x 4 transformation matrix..

The transformation matrix or its inverse can be applied The transformation matrix or its inverse can be applied to local or global vector to determine the to local or global vector to determine the corresponding vector in the other system.corresponding vector in the other system.

ResultsResultsAccuracy of Camera Accuracy of Camera

SystemSystem Method: compare camera-Method: compare camera-

measured distances between measured distances between markers pairs with DIL-markers pairs with DIL-measured valuesmeasured values

Results (15 independent runs)Results (15 independent runs) mean distance error mean distance error ++ SD SD

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

-1.00

-0.50

0.00

0.50

1.00

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Data runE

rro

r [m

m]

Caddy

Cross

ResultsResultsSystem Error - Initial System Error - Initial

ResultsResults (a) (a) First 12 data runs:First 12 data runs: mean system error mean system error ++ SD SD

orthogonal transformorthogonal transform

2.8 + 2.2 mm (0.5 - 5.5)2.8 + 2.2 mm (0.5 - 5.5) LS transformLS transform

61 + 33 mm (8.9 - 130)61 + 33 mm (8.9 - 130)

(b) 8 data runs, after (b) 8 data runs, after improving calibrationimproving calibration mean system error mean system error ++ SD SD

orthogonal transformorthogonal transform

2.4 + 0.6 mm (1.5 - 3.0)2.4 + 0.6 mm (1.5 - 3.0) LS transformLS transform

46 46 ++ 23 mm (18 - 78) 23 mm (18 - 78)

ResultsResultsSystem Error - Current System Error - Current

ResultsResults (c) Last 15 data runs, (c) Last 15 data runs,

5 target positions, 3 runs 5 target positions, 3 runs per position:per position: mean system error mean system error ++ SD SD

orthogonal transformorthogonal transform

0.6 0.6 ++ 0.3 mm (0.2 - 1.3) 0.3 mm (0.2 - 1.3) LS transformLS transform

25 25 ++ 8 mm (14 - 36) 8 mm (14 - 36)

0.1

1

10

100

1 6 11 16

Data run

Err

or

[mm

]

Least Squares

Orthogonal

ConclusionConclusionand Future Directionsand Future Directions

Currently, Orthogonal Transformation Currently, Orthogonal Transformation outperforms standard Least-Square based outperforms standard Least-Square based Transformation by more than one order of Transformation by more than one order of magnitudemagnitude

Comparative analysis between Orthogonal Comparative analysis between Orthogonal Transformation and more accurate version Transformation and more accurate version of Least-Square based Transformation (e.g. of Least-Square based Transformation (e.g. Constrained Least Square) needs to be doneConstrained Least Square) needs to be done

Various optimization options, e.g., different Various optimization options, e.g., different marker arrangements, will be applied to marker arrangements, will be applied to attain an accuracy of better than 0.5 mmattain an accuracy of better than 0.5 mm