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

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


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


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


- 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


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.


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


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)


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


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)}


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


Testing BEAR

MarginalPSF

Posteriorprobability


Angle Prediction

Deviation RMS Deviation RMS

The RMS deviation of the 2D PSF in (left) and (right) (, )



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


Reconstructed ImagesU

nfi

lter

edB

EA

R1% 18% 50%


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


Feature Size


Feature Contrast


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.


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

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