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
2MIPS Stanford UniversityMolecular ImagingProgram at Stanford
School of MedicineDepartment of Radiology
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
AMKFoudrayBayesian Inference and Maximum Entropy 200707/11/07
PET: An Inverse Problem
Detectors
Subject’s Body
Radio-isotope probe
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PET: Events
“True”
“Scatter”
“Random”
)cos1(1 20
0
0
cm
EE
Esc
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
AMKFoudrayBayesian Inference and Maximum Entropy 200707/11/07
Detection Parameters(x1,y1,z1,E1,t1)
(x2,y2,z2,E2,t2)
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
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- 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
=> ~ 1011 possible LORs
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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
(xi,yi,zi,Ei,ti) 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.
AMKFoudrayBayesian Inference and Maximum Entropy 200707/11/07
Multiple Interaction Based Electronic Collimation (MIBEC)
Requirements
Ei > 10 keV ||xi-xCOM,yi-yCOM,zi-zCOM|| < 2cm
450 keV < i Ei < 572 keV
Each energy above noise floor All interactions in 2cm nbhd of COM
Total energy within energy window
ti - 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
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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)
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BEAR: Bayesian Classifier
After filtering the interactions for energy, position and time constraints, a cluster of N interactions is formed (NM), 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:
COMii ' i = xi, or yi, or zi
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.
x̂
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BEAR: Angle Selection
),...,()()|,...,(
),...,|(1
11
N
NN
XXPPXXP
XXP
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 .
N
i ji
jiN
XXP
XXPPXXP
11
)|(
),|()(),...,|(
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)}
AMKFoudrayBayesian Inference and Maximum Entropy 200707/11/07
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
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Testing BEAR
MarginalPSF
Posteriorprobability
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Angle Prediction
Deviation RMS Deviation RMS
The RMS deviation of the 2D PSF in (left) and (right) (, )
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AMKFoudrayBayesian Inference and Maximum Entropy 200707/11/07
SNR vs Activity
5cm D=2.5cm
15cmL=7cm
0.1, 1, 5 mCi correspond to about 1%, 18%, 50% randoms events
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 Atot= Abkgr + Aspheres
Plane of sphere sources 2 cm from center
3.5 mm2.5 mm
1.5 mm 1.25 mm
Aspheres ~ 0.002* Atot
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Reconstructed ImagesU
nfi
lter
edB
EA
R1% 18% 50%
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Feature Extraction
b1 = max height of Gaussian
21
21
21 /))()((
11),( fdycxebayxfitmap
a1 = constant background
(c1 , d1) = peak position
sqrt(0.5)* f1 *2.35 = FWHM
Using the multidimensional unconstrained nonlinear minimization (Nelder-Mead) fminsearch algorithm in MATLAB
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Feature Size
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Feature Contrast
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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.
AMKFoudrayBayesian Inference and Maximum Entropy 200707/11/07
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