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

Geometry modeling and nonlinear reconstruction for X-ray guided breast DOT. Qianqian Fang + , David Boas + , Greg Boverman*, Quan Zhang + , Tina Kauffman + + Massachusetts General Hospital *Northeastern University. NTROI. Outline. Instrument overview Binary function based mesh generator

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

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  1. Geometry modeling and nonlinear reconstruction for X-ray guided breast DOT Qianqian Fang+, David Boas+, Greg Boverman*, Quan Zhang+, Tina Kauffman+ +Massachusetts General Hospital *Northeastern University NTROI SPIE Photonic West 2006

  2. Outline • Instrument overview • Binary function based mesh generator • Iterative block solver • Reconstruction results • Summary SPIE Photonic West 2006

  3. Why combine X-ray with DOT? • Mammography is low-cost and routinely used across the country • X-ray only provide morphological info. • DOT can provide functional measurement but is low-resolution. Combined X-ray/DOT imaging can help doctor’s assessment by overlaying functional image on top of structural image. SPIE Photonic West 2006

  4. System Picture TOBI: tomographical optical breast imaging system Tomosynthesis: 3D Mammography SPIE Photonic West 2006

  5. Binary function based mesh generation Why another mesh generator? • Conventional method for mesh generation from medical images • segmentation and boundary extraction • surface smoothing • advance-front method • mesh optimization • What the new generator can offer? • anyone can understand and implement • no need for boundary extraction and smoothing • high quality elements, no need for mesh optimization • in some cases, may faster than conventional method SPIE Photonic West 2006

  6. Prologue: Distance function based mesh generation • Persson & Strang (MIT), 2004 SIAM Review: Simple mesh generation based on signed-distance functions • Pros: • simple • high quality • Cons: • difficult for compli-cated geometries • slow http://www-math.mit.edu/~persson/mesh/ SPIE Photonic West 2006

  7. Binary Functions • A binary function: inside or out side 1 -2 2 int binary_shape(double x, double y) { return (x>2. && x<-2. && y>2. && y<-2. && x*x+y*y>1.); } For medical images, only memory access is needed, no arithmetic ! SPIE Photonic West 2006

  8. Step 1-3 of 5 • Step 1: initial mesh • Truncating isotropic mesh by binary function • Step 2: boundary layer • Use Laplacian operator to find out the boundary layer • Reduce computationalcomplexity fromO(N) to O(N1/2) for2D, O(N) to O(N2/3)for 3D • Step 3: moving mesh • treat mesh as truss system, solve for physical equilibrium SPIE Photonic West 2006

  9. Step 4 of 5: Boundary Correction • if nodes move outside the geometry • bi-sect search betweenPi and Pi+1 • line search over a circle,find out the closest boundary point Step 5 of 5: Re-Triangulation • Moving mesh will change mesh topology, without timely updating neighbor list, this may cause method to diverge. • Delaunay based triangulation (for example: QHull) can be used (only apply to the nodes within boundary layer) SPIE Photonic West 2006

  10. More complicated cases • Mesh generation in 3D or in Rnspace • Step 1: using uniform grid as initial mesh • Step 4: the second line search in boundary correction is performed on a (hyper-)sphere surface • For medical images: segmentations • Anisotropic elements • Non-uniform mesh density • Quadtree or Octree SPIE Photonic West 2006

  11. Examples SPIE Photonic West 2006

  12. Iterative Block Solver for FEM forward modeling • Solving FEM forward equation: large scale, sparse, complex (or real) entries • Direct methods: SuperLU, UMFPACK, WSMP … • Iterative methods: CG, BiCG, GMRES, QMR … • QMR multi-RHS solver(Boyes&Seidl,1996): A[x1,x2,x3,..,xN]=[b1,b2,b3,…,bN] N: block size SPIE Photonic West 2006

  13. Solver Performance • mesh: 42122 nodes and 230745 elements • RF diffusion equation optimal block size SPIE Photonic West 2006

  14. Results: Simulations • TOMO Slices Forward&Recon meshes Slice#35 Slice#50 SPIE Photonic West 2006

  15. Simulation – Cont’d True absorption Recovered absorption w/o geometry modeling A tumor SPIE Photonic West 2006

  16. Summary • Take home messages: • TOBI: RF+CW+MUX, can co-register with 2D or 3D mammography • Mesh generator is simple and easy to implement, good for mesh generation from medical images • Block solver is efficient in solving forward problems • Problems: mesh generator not entirely stable; some elements close to boundary are not perfect; 3D triangulation produces empty elements. SPIE Photonic West 2006

  17. Acknowledgements • Funding Agencies • NIH • NTROI • Avon Breast Cancer Center • Daniel Kopans • Richard Moore • Dianne Georgian-Smith • Jennifer Curry • Dianne Scourletis • Donna Burgess • Jayne Cormier • Lockheed Palo Alto Research Laboratories • William Boyse • Photon Migration Lab at MGH • Maria Franceschini • Stefan Carp • Juliette Selb • Elizabeth Hillman • Sol Diamond • Phill Jones • Danny Joseph • Ted Hupper • Anand Kumar • GW Krauss • George Themelis • ... SPIE Photonic West 2006

  18. Questions? SPIE Photonic West 2006

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