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Bayesian Estimation for Angle Recovery: Event Classification and Reconstruction in Positron Emission Tomography

Bayesian Estimation for Angle Recovery: Event Classification and Reconstruction in Positron Emission Tomography. A.M.K. Foudray, C.S. Levin Department of Radiology and Molecular Imaging Program Stanford University, Stanford, CA 94305 Department of Physics

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Bayesian Estimation for Angle Recovery: Event Classification and Reconstruction in Positron Emission Tomography

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  1. Bayesian Estimation for Angle Recovery: Event Classification and Reconstruction in Positron Emission Tomography • A.M.K. Foudray, C.S. Levin • Department of Radiology and Molecular Imaging Program Stanford University, Stanford, CA 94305 • Department of Physics • University of California San Diego, La Jolla, CA 92092 Stanford University MIPS 2 School of Medicine Department of Radiology Molecular Imaging Program at Stanford

  2. Outline Positron Emission Tomography Compton Scatter, Randoms, Coincidence Pairing, Collimation Data space, reconstruction Multiple Interaction Based Electronic Collimation (MIBEC) Instrumentation Considerations BEAR: A Naïve Bayesian Classifier Prediction Capabilities Reconstruction in Biologically Relevant Noise Regimes Reconstructed Spatial Resolution and Contrast Bayesian Inference and Maximum Entropy 2007 07/11/07 AMKFoudray

  3. PET: An Inverse Problem Detectors Subject’s Body Radio-isotope probe Bayesian Inference and Maximum Entropy 2007 07/11/07 AMKFoudray

  4. E = 0 E sc E + - 1 0 ( 1 cos ) q 2 m c 0 Trues: both photons from a single annihilation event are detected PET: Events “True” “Scatter” Singles: only one of the annihilation-generated pair of emitted photons are detected Two decays occur within time window  “Random” Energy of the Compton Scattered photon Randoms: two of the four photons are detected Multiples: three or more photons detected Bayesian Inference and Maximum Entropy 2007 07/11/07 AMKFoudray

  5. Detection Parameters (x1,y1,z1,E1,t1) Line of Response (x2,y2,z2,E2,t2) Need: - good 3Dposition resolutionin the detector (<1mm) - filter scatters: goodenergy resolution(<10% @ 511 keV) - filter randoms: goodtime resolution(<2ns) Bayesian Inference and Maximum Entropy 2007 07/11/07 AMKFoudray

  6. Data Space Considerations - Number of detector elements: ~600,000 => ~ 1011 possible LORs - Image space: 0.5mm pixels, 8cm x 8cm x 8cm FOV => 4 million voxels - Cannot give biological entity too high of a dose, and have to perform acquisitions over “reasonable” time periods (for it to be useful) – images are usually constructed from a few hundred million counts => Solution to reconstruction problem is ill-posed and is generally treated by expectation maximization algorithms (here, OSEM), but can be treated with Bayesian Estimation schemes Bayesian Inference and Maximum Entropy 2007 07/11/07 AMKFoudray

  7. Forward Model Incident High Energy Photon Interactions Compton, Rayleigh, Photoelectric Bremsstrahlung, ionization, x-ray Complex forward model: many kinds of interactions, many sources of blur, lossy detection schemes (non/inherent multiplexing) A Bayes approach, which has “tunable” strictness about the forward model, is an ideal choice. Detection System Blurring energy, time blurring, device charge centroiding, crystal cross-talk, binning, photon production non-linearities, multiplexing (xi,yi,zi,Ei,ti) for i = 1:M Bayesian Inference and Maximum Entropy 2007 07/11/07 AMKFoudray

  8. Multiple Interaction Based Electronic Collimation (MIBEC) Use these interactions, these bits of insight into the transport of the high energy photon, to give us more information about where it was generated. A B Requirements Each energy above noise floor All interactions in 2cm nbhd of COM ||xi-xCOM,yi-yCOM,zi-zCOM|| < 2cm Ei > 10 keV Total energy within energy window All interactions within time window 450 keV < i Ei < 572 keV ti - min(t 1:M) < 4ns Bayesian Inference and Maximum Entropy 2007 07/11/07 AMKFoudray

  9. LOR assignment What is the size of the blur simply from the forward model? (methods of energy deposition; blurring, non-linearities, discreteness in detection; position assignment method) Bayesian Inference and Maximum Entropy 2007 07/11/07 AMKFoudray

  10. a = a - a ' i i COM • This COM reference space had a number of advantages: • a significant reduction in the size of the data in measurement space, making further manipulation and searches faster • the construction of COM space does not depend on measurement location (always – pointing towards the detection volume), it takes advantage of measurement symmetries, and data can be added to the training set without knowledge and recalculation of prior training data, • calculation of posterior probability map is fully parallelizable, it can scale to any number of processors. ˆ x BEAR: Bayesian Classifier After filtering the interactions for energy, position and time constraints, a cluster of N interactions is formed (NM), each interaction defined by its energy and relative position (xi,yi,zi,Ei), abbreviated Xi where (x’i,y’i,z’i,Ei) is the interaction in system-space, and: i = xi, or yi, or zi Bayesian Inference and Maximum Entropy 2007 07/11/07 AMKFoudray

  11. Q P ( X ,..., X | ) P ( ) Q = Q 1 N P ( | X ,..., X ) 1 N P ( X ,..., X ) 1 N P ( X | X , ) Q N  i j = . P ( | X ,..., X ) P ( ) Q Q 1 N P ( X | X ) = 1 i i j BEAR: Angle Selection For a cluster of N events with information (xi,yi,zi,Ei), or X, we would like to see if we have enough information to give Bayes’ theorem to get any kind of predictive capabilities for the incident photon direction (, ), abbreviated . where Xj is (Xi-1, Xi-2, …, X1). When i=1 in the sum, Xj is Ø. The decision rule then is simply max {P(|X1,...,XN)} Bayesian Inference and Maximum Entropy 2007 07/11/07 AMKFoudray

  12. Training the BEAR Use a point source to sample the  data space, spanning the  range of the LOR. Record all clusters, constrained to the energy, position and time requirements. Then fill PDF matrices (or look-up tables when the matrices are *extremely* sparse). => Evidence and likelihood Event space was segmented into: 22x42x52x4 bins in x, y, z, and E and angle space (, ) into 36 and 30 bins, respectively. Bayesian Inference and Maximum Entropy 2007 07/11/07 AMKFoudray

  13. Testing BEAR Posterior probability Marginal PSF Bayesian Inference and Maximum Entropy 2007 07/11/07 AMKFoudray

  14. Angle Prediction  Deviation RMS  Deviation RMS The RMS deviation of the 2D PSF in  (left) and  (right)  (, ) Bayesian Inference and Maximum Entropy 2007 07/11/07 AMKFoudray

  15. SNR vs Activity 15cm L=7cm 5cm D=2.5cm 0.1, 1, 5 mCi correspond to about 1%, 18%, 50% randoms events Bayesian Inference and Maximum Entropy 2007 07/11/07 AMKFoudray

  16. Case Studies Plane of sphere sources 2 cm from center 8cm 2.5 mm 3.5 mm 6cm 1.5 mm 1.25 mm The volume is uniformly source- and water-filled Atot= Abkgr + Aspheres Aspheres ~ 0.002* Atot  Look at three total activities: 0.1, 1, 5 mCi, which correspond to 1%, 18%, 50% randoms events Bayesian Inference and Maximum Entropy 2007 07/11/07 AMKFoudray

  17. Reconstructed Images 1% 18% 50% Unfiltered BEAR Bayesian Inference and Maximum Entropy 2007 07/11/07 AMKFoudray

  18. 2 2 2 - - + - = + (( x c ) ( y d ) ) / f fitmap ( x , y ) a b e 1 1 1 1 1 Feature Extraction a1 = constant background b1 = max height of Gaussian (c1 , d1) = peak position sqrt(0.5)* f1 *2.35 = FWHM Using the multidimensional unconstrained nonlinear minimization (Nelder-Mead) fminsearch algorithm in MATLAB Bayesian Inference and Maximum Entropy 2007 07/11/07 AMKFoudray

  19. Feature Size Bayesian Inference and Maximum Entropy 2007 07/11/07 AMKFoudray

  20. Feature Contrast Bayesian Inference and Maximum Entropy 2007 07/11/07 AMKFoudray

  21. Summary - Constructed a Bayesian method to utilize novel detection capabilities to create a multiple-interaction based electric collimation algorithm – i.e. determine properties of the photon before interaction (incident angle). - Used this angular information to create a filter for “weeding out” N>1 clusters (and ultimately the coincidence event) that didn’t corroborate the information gained from coincidence pairing. This filter improved the contrast ratio in the reconstructed image by 40% on average. - Future work will include using the histogrammed posterior PDFs for weighted projector functions, reconstructing singles, selecting pairs from multiples, to increase the usage of counts acquired by the detector. - More optimal methods for prior construction, as well as likelihood and evidence look up procedures. Bayesian Inference and Maximum Entropy 2007 07/11/07 AMKFoudray

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