01/21/2006 SPIE Photonic West 2006 Geometry modeling and nonlinear reconstruction for X-ray guided breast DOT Qianqian Fang +, David Boas +, Greg Boverman*,

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Geometry modeling and Geometry modeling and nonlinear reconstruction for nonlinear reconstruction for X-ray guided breast DOTX-ray guided breast DOT

Qianqian FangQianqian Fang++, David Boas, David Boas++, Greg Boverma, Greg Boverman*, Quan Zhangn*, Quan Zhang++, Tina Kauffman, Tina Kauffman++

++Massachusetts General HospitalMassachusetts General Hospital*Northeastern University*Northeastern University

NTROINTROI

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OutlineOutline

Instrument overviewInstrument overview Binary function based mesh generatorBinary function based mesh generator Iterative block solverIterative block solver Reconstruction resultsReconstruction results SummarySummary

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Why combine X-ray with DOT?Why combine X-ray with DOT?

Mammography is low-cost and routinely Mammography is low-cost and routinely used across the countryused across the country

X-ray only provide morphological info.X-ray only provide morphological info. DOT can provide functional measurement DOT can provide functional measurement

but is low-resolution.but is low-resolution.

Combined X-ray/DOT imaging can help Combined X-ray/DOT imaging can help doctor’s assessment by overlaying doctor’s assessment by overlaying functional image on top of structural image.functional image on top of structural image.

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System PictureSystem Picture

FDM CW unit

RF unit

TDM CW unit

Detector plate

Source plate

Quick Connects

Tomosynthesis: 3D Mammography

TOBI: tomographical optical breast imaging system

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Binary function based mesh Binary function based mesh generationgeneration

Why another mesh generator?Why another mesh generator? Conventional method for mesh generation from medical Conventional method for mesh generation from medical

imagesimages segmentation and boundary extractionsegmentation and boundary extraction surface smoothingsurface smoothing advance-front methodadvance-front method mesh optimizationmesh optimization

What the new generator can offer?What the new generator can offer? anyone can understand and implementanyone can understand and implement no need for boundary extraction and smoothingno need for boundary extraction and smoothing high quality elements, no need for mesh optimizationhigh quality elements, no need for mesh optimization in some cases, may faster than conventional methodin some cases, may faster than conventional method

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Prologue: Distance function based Prologue: Distance function based mesh generationmesh generation

Persson & Strang (MIT), 2004 SIAM Review: SiPersson & Strang (MIT), 2004 SIAM Review: Simple mesh generation based on signed-distance mple mesh generation based on signed-distance functionsfunctions

Pros:Pros: simplesimple high qualityhigh quality

Cons:Cons: difficult for compli-difficult for compli-

cated geometriescated geometries slowslow

http://www-math.mit.edu/~persson/mesh/

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Binary FunctionsBinary Functions

A binary function: inside or out sideA binary function: inside or out side

1( )

0

rf r

r

Î Wìïï= íï Ï Wïî

int binary_shape(double x, double y){ return (x>2. && x<-2. && y>2. && y<-2. && x*x+y*y>1.);}

2-2

1

For medical images, only memory access is needed,no arithmetic !

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Step 1-3 of 5Step 1-3 of 5

Step 1: initial meshStep 1: initial mesh Truncating isotropic mesh by biTruncating isotropic mesh by bi

nary functionnary function

Step 2: boundary layerStep 2: boundary layer Use Laplacian operator to find oUse Laplacian operator to find o

ut the boundary layerut the boundary layer Reduce computationalReduce computational

complexity fromcomplexity fromO(N)O(N) to to O(NO(N1/21/2)) for for2D, 2D, O(N)O(N) to to O(NO(N2/32/3))for 3Dfor 3D

Step 3: moving meshStep 3: moving mesh treat mesh as truss system, solvtreat mesh as truss system, solv

e for physical equilibriume for physical equilibriumi

i

F F= år r

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Step 4 of 5: Boundary CorrectionStep 4 of 5: Boundary Correction if nodes move outside the geometryif nodes move outside the geometry

1.1. bi-sect search betweenbi-sect search betweenPPii and P and Pi+1i+1

2.2. line search over a circle,line search over a circle,find out the closest find out the closest boundary pointboundary point

Moving mesh will change mesh topology, without timely Moving mesh will change mesh topology, without timely updating neighbor list, this may cause method to diverge.updating neighbor list, this may cause method to diverge.

Delaunay based triangulation (for example: QHull) can bDelaunay based triangulation (for example: QHull) can be used (only apply to the nodes within boundary layer)e used (only apply to the nodes within boundary layer)

Step 5 of 5: Re-TriangulationStep 5 of 5: Re-Triangulation

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More complicated casesMore complicated cases

Mesh generation in 3D or in RMesh generation in 3D or in Rnn spacespace

Step 1: using uniform grid as initial meshStep 1: using uniform grid as initial mesh Step 4: the second line search in boundary correction Step 4: the second line search in boundary correction

is performed on a (hyper-)sphere surfaceis performed on a (hyper-)sphere surface

For medical images: For medical images: segmentationssegmentations

Anisotropic elementsAnisotropic elements Non-uniform mesh densityNon-uniform mesh density Quadtree or Octree Quadtree or Octree

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ExamplesExamples

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Iterative Block Solver for FEM Iterative Block Solver for FEM forward modelingforward modeling

Solving FEM forward equation: large scale, Solving FEM forward equation: large scale, sparse, complex (or real) entriessparse, complex (or real) entries

Direct methods: SuperLU, UMFPACK, WSMDirect methods: SuperLU, UMFPACK, WSMP …P …

Iterative methods: CG, BiCG, GMRES, QMIterative methods: CG, BiCG, GMRES, QMR …R …

QMR multi-RHS solver(Boyes&Seidl,1996): QMR multi-RHS solver(Boyes&Seidl,1996): A[x1,x2,x3,..,x A[x1,x2,x3,..,xNN]=[b1,b2,b3,…,b]=[b1,b2,b3,…,bNN] ] N: block size N: block size

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Solver PerformanceSolver Performance

mesh: 42122 nodes and 230745 elementsmesh: 42122 nodes and 230745 elements RF diffusion equationRF diffusion equation

optimal block size

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Results: SimulationsResults: Simulations

TOMO Slices Forward&Recon meshesTOMO Slices Forward&Recon meshesSlice#35

Slice#50

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Simulation – Cont’dSimulation – Cont’d

True absorption Recovered absorption w/o geometry modelingTrue absorption Recovered absorption w/o geometry modeling

A tumor

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SummarySummary

Take home messages:Take home messages: TOBI: RF+CW+MUX, can co-register with 2D or 3D TOBI: RF+CW+MUX, can co-register with 2D or 3D

mammographymammography Mesh generator is simple and easy to implement, Mesh generator is simple and easy to implement,

good for mesh generation from medical imagesgood for mesh generation from medical images Block solver is efficient in solving forward problemsBlock solver is efficient in solving forward problems

Problems: mesh generator not entirely stable; Problems: mesh generator not entirely stable; some elements close to boundary are not some elements close to boundary are not perfect; 3D triangulation produces empty perfect; 3D triangulation produces empty elements.elements.

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AcknowledgementsAcknowledgements Funding AgenciesFunding Agencies

NIHNIH NTROINTROI

Avon Breast Cancer CenterAvon Breast Cancer Center Daniel KopansDaniel Kopans Richard MooreRichard Moore Dianne Georgian-SmithDianne Georgian-Smith Jennifer CurryJennifer Curry Dianne ScourletisDianne Scourletis Donna BurgessDonna Burgess Jayne CormierJayne Cormier

Lockheed Palo Alto ResearLockheed Palo Alto Research Laboratoriesch Laboratories William BoyseWilliam Boyse

Photon Migration Lab at Photon Migration Lab at MGHMGH Maria Franceschini Maria Franceschini Stefan Carp Stefan Carp Juliette Selb Juliette Selb Elizabeth Hillman Elizabeth Hillman Sol Diamond Sol Diamond Phill Jones Phill Jones Danny Joseph Danny Joseph Ted Hupper Ted Hupper Anand Kumar Anand Kumar GW KraussGW Krauss George ThemelisGeorge Themelis ......

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Questions?Questions?