2. 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 2 MIPS Stanford University Molecular Imaging Program at Stanford School of Medicine Department of Radiology

3. Outline Positron Emission Tomography Data space, reconstruction Compton Scatter, Randoms, Coincidence Pairing, Collimation 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 AMKFoudray Bayesian Inference and Maximum Entropy 2007 07/11/07

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

5. PET: Events “True” “Scatter” “Random” )cos1(1 2 0 0 0    cm E E E sc Energy of the Compton Scattered photon Two decays occur within time window  Multiples: three or more photons detected Randoms: two of the four photons are detected Trues: both photons from a single annihilation event are detected Singles: only one of the annihilation- generated pair of emitted photons are detected AMKFoudray Bayesian Inference and Maximum Entropy 2007 07/11/07

6. Detection Parameters (x 1,y 1,z 1,E 1,t 1 ) (x 2,y 2,z 2,E 2,t 2 ) Need: - good 3D position resolution in the detector (<1mm) - filter scatters: good energy resolution (<10% @ 511 keV) - filter randoms: good time resolution (<2ns) Line of Response AMKFoudray Bayesian Inference and Maximum Entropy 2007 07/11/07

7. - Number of detector elements: ~600,000 - 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 - Image space: 0.5mm pixels, 8cm x 8cm x 8cm FOV => 4 million voxels Data Space Considerations => 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 => ~ 10 11 possible LORs AMKFoudray Bayesian Inference and Maximum Entropy 2007 07/11/07

8. Forward Model Incident High Energy Photon Compton, Rayleigh, Photoelectric Interactions Bremsstrahlung, ionization, x-ray energy, time blurring, device charge centroiding, crystal cross-talk, binning, photon production non- linearities, multiplexing Detection System Blurring (x i,y i,z i,E i,t i ) for i = 1:M 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. AMKFoudray Bayesian Inference and Maximum Entropy 2007 07/11/07

9. Multiple Interaction Based Electronic Collimation (MIBEC) Requirements E i > 10 keV ||x i -x COM,y i -y COM,z i -z COM || < 2cm 450 keV <  i E i < 572 keV Each energy above noise floor All interactions in 2cm nbhd of COM Total energy within energy window t i - min(t 1:M ) < 4ns All interactions within time window 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 AMKFoudray Bayesian Inference and Maximum Entropy 2007 07/11/07

10. 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) AMKFoudray Bayesian Inference and Maximum Entropy 2007 07/11/07

11. 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 (x i,y i,z i,E i ), abbreviated X i where (x’ i,y’ i,z’ i,E i ) is the interaction in system- space, and: COMii  '  i = x i, or y i, or z i This COM reference space had a number of advantages: (1)a significant reduction in the size of the data in measurement space, making further manipulation and searches faster (2)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, (3)calculation of posterior probability map is fully parallelizable, it can scale to any number of processors. x ˆ AMKFoudray Bayesian Inference and Maximum Entropy 2007 07/11/07

12. BEAR: Angle Selection ),...,( )()| ( ) |( 1 1 1 N N N XXP PXXP XXP     For a cluster of N events with information (x i,y i,z i,E i ), 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 .        N i ji ji N XXP XXP PXXP 1 1 )|( ),|( )(),...,|( where X j is (X i-1, X i-2, …, X 1 ). When i=1 in the sum, X j is Ø. The decision rule then is simply max {P(  |X 1,...,X N )} AMKFoudray Bayesian Inference and Maximum Entropy 2007 07/11/07

13. 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). Event space was segmented into: 22x42x52x4 bins in x, y, z, and E and angle space ( ,  ) into 36 and 30 bins, respectively. => Evidence and likelihood AMKFoudray Bayesian Inference and Maximum Entropy 2007 07/11/07

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

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

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

17. 6cm 8cm Case Studies  Look at three total activities: 0.1, 1, 5 mCi, which correspond to 1%, 18%, 50% randoms events The volume is uniformly source- and water-filled A tot = A bkgr + A spheres Plane of sphere sources 2 cm from center 3.5 mm2.5 mm 1.5 mm1.25 mm A spheres ~ 0.002* A tot AMKFoudray Bayesian Inference and Maximum Entropy 2007 07/11/07

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

19. Feature Extraction b 1 = max height of Gaussian 2 1 2 1 2 1 /))()(( 11 ),( fdycx ebayxfitmap   a 1 = constant background (c 1, d 1 ) = peak position sqrt(0.5)* f 1 *2.35 = FWHM Using the multidimensional unconstrained nonlinear minimization (Nelder-Mead) fminsearch algorithm in MATLAB AMKFoudray Bayesian Inference and Maximum Entropy 2007 07/11/07

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

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

22. 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. AMKFoudray Bayesian Inference and Maximum Entropy 2007 07/11/07