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

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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
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Outline

• Instrument overview
• Binary function based mesh generator
• Iterative block solver
• Reconstruction results
• Summary

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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 doctors
  assessment by overlaying functional image on top
  of structural image.

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System Picture
TOBI tomographical optical breast imaging system
Tomosynthesis 3D Mammography
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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

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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/
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Binary Functions

• A binary function inside or out side

1
-2
2
int binary_shape(double x, double y) return
(xgt2. xlt-2. ygt2. ylt-2.
xxyygt1.)
For medical images, only memory access is
needed, no arithmetic !
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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

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Step 4 of 5 Boundary Correction

• if nodes move outside the geometry
• bi-sect search betweenPi and Pi1
• 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)

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More complicated cases

• Mesh generation in 3D or in Rn space
• 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

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Examples
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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(BoyesSeidl,1996)
  Ax1,x2,x3,..,xNb1,b2,b3,,bN
  N block size

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Solver Performance

• mesh 42122 nodes and 230745 elements
• RF diffusion equation

optimal block size
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Results Simulations

• TOMO Slices ForwardRecon meshes

Slice35
Slice50
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Simulation Contd

• True absorption Recovered absorption
  w/o geometry modeling

A tumor
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Summary

• Take home messages
• TOBI RFCWMUX, 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.

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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
• ...

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• Questions?