IntroductionIn positron emission tomography (PET), each line of response (LOR) has a different sensitivity due to the scanner's geometry and detector properties. In order to achieve artifact-free and good quality images, these non-uniformities must be corrected using normalization factors (NFs). A widely used normalization method is the component-based normalization (CBN), where the number of parameters to estimate is dramatically reduced by modelling the efficiency of each LOR as a product of multiple factors.

In the CBN, the components can be separated into time-invariant, time-variant and acquisition-dependent components. The crystal efficiencies are the main time-variant factor and are updated by a regular normalization scan. However, the effective crystal efficiencies can change between normalization scans for many reasons, including, for example, variations in temperature. Therefore, it would be advantageous to be able to estimate the effective crystal efficiencies directly for each unique emission scan.

Self-Normalization of 3D PET Data by EstimatingScan-Dependent Effective Crystal EfficienciesMartin A. Belzunce and Andrew J. ReaderKing's College London, Division of Imaging Sciences and Biomedical Engineering, St Thomas’ Hospital, London, UK

21,2,121 ...... ddzrbbddri ddpN

crystal interferencegeometric factors

crystal efficiencies

dead-time

axial factors

OverviewThe aim is to estimate scan-dependent effective crystal efficiencies directly from emission data using a component-based normalization (CBN). The proposed method, which includes exploitation of time-invariant normalization factors, enables the reconstruction of good quality images without the need for a separate normalization scan.

Component-Based Normalization

Results with Simulated Data

AQTVTIC NNNN ..NC: complete normalization factorsNTI: time-invariant components (geometric, crystal interference, axial factors)NTV: time-variant components (crystal efficiencies)NAQ: acquisition-dependent components (dead-time)

Normalization Factor in Bin i:

Component Classification:

Scan-Dependent Crystal Efficiencies rsXfNAq C ..

rsXfNAq TITI ..

Complete Forward Model in EM reconstruction:Time-Invariant Forward Model:

cf

TVt

cartifacts NXff .

Reconstruction

q: projected image using the complete forward modelqTI: projected image using the a forward model with only time-invariant factorsfc: image reconstructed with complete normalization factorsfartifact: image reconstructed with time-invariant normalization factorsX: x-ray transform that projects image into a sinogram

TITITV q

b

q

qN Ratio of two

noisy vectors

We can estimate time variant normalization

factors with:

b: emission sinogram

NTV depends only on the crystal efficiencies. If we work in crystal space, the number of parameters to estimate is

reduced by 4 orders of magnitude and a variance reduction is achieved.

Self-Normalization Algorithm1. The time-variant NFs (NTV) are generated from the current crystal efficiencies

estimate xk.

2. The complete NFs are generated with NC = NTV · NTI.

3. The measured emission fully 3D sinogram b is reconstructed into image fk and using NC.

4. fk is projected into a sinogram pk using NC and the complete forward model.

5. C[] is applied to the projected and to the input sinograms.

6. The ratio between C[b] and C[pk] is computed to estimate the current crystal efficiencies.

srAfNxDxDC

bCx

kTI

kkk

)(21

1

Sinogram to Crystal Operator • Fan-sum algorithm to get the crystal

efficiency of one crystal from the NTV sinogram:

• We use both detectors in a sinogram bin to reduce uncertainty.

• The operator in matrix notation:

1.1.][

21

21TT

TVT

TVT

TV DD

NDNDNCx

)(1).())(().())(().()( 112121 ixixjixixjixixjNJJJ

TV

Crystal to Sinogram Operator• Each sinogram bin represents a unique

combination of two crystal elements (span 1 and without polar mashing).

• Efficiency of a sinogram bin:

• Matrix operator:

D1: matrix with as many rows as bins and columns as crystal element. It has a 1 for each row to identify detector 1.D2: likewise for detector 2.

)().()( 21 ixixjNTV

1

...

1

...

1

)()( 21 xDxDNTV

: Hadamard product (element by element)xD1 : Matrix multiplication

kkTV xDxDN 21

kx

0x

OP-OSEM reconstruction

with TITVC NNN

Project reconstructed

imagesrAfNN k

TITV

Transform to crystal-space

Ratio between emission and projected sinograms in crystal

space

Iter 1 Iter 2 Xtals in Norm

• Four sets of crystal efficiencies were simulated. Set 1 had a normal distribution with the same standard deviation as the one found in the Biograph mMR normalization files. Sets 2 and 3 had a normal distribution with, respectively, 5 and 10 times the standard deviation of set 1. The remaining set had a uniform distribution between 0 and 2 (set 4).• For each set of crystal efficiencies, 3D sinograms

were simulated by projecting a uniform cylinder and a brain phantom and multiplying them by the complete normalization factors. Poisson noise was introduced in the sinograms.

ConclusionsA method to estimate successfully scan-dependent crystal efficiencies is presented. The algorithm permits the reconstruction of good quality images without having performed a recent normalization scan, by using only the time-invariant NFs. With real data, it was found that an inaccurate scatter estimate can introduce a bias that needs to be compensated for in order to achieve good results.

Results with Real Data

Set 2Set2Set 4Set4

BrainWeb Phantom

• A striatal phantom (model RS-901T, Radiology Support Devices Inc.) was scanned filled with 18F in the background and 5 times that activity in the caudate and the putamen.

Unit crystal efficiencies

1kx

bC srAfNxDxDC k

TIkk )(21

Estimated crystal

efficiencies

Example of reconstruction

after 1 iteration

Reconstruct with new crystal efficiencies

References[1] A. Salomon et al, ”A Self-Normalization Reconstruction Technique for PET Scans Using the Positron Emission Data,” IEEE TMI, vol. 31, pp 2234-40, 2012.[2] G. Delso et al, ”Performance Measurements of the Siemens mMR Integrated Whole-Body PET/MR Scanner,” Journal of Nuclear Medicine, vol. 52, pp. 1914-1922, December 01, 2011.[3] M. E. Casey et al, ”A component based method for normalization in volume PET,” Proc. 3rd Int. Meeting of Fully 3D Image Reconstruction, 67-71, 1995.[4] R. D. Badawi et al, ”Algorithms for calculating detector efficiency normalization coefficients for true coincidences in 3D PET,” Phys Med Biol, vol 43, 1998.

Self-Normalizatio

n for simulations with set 2

and set 4 of crystal

efficiencies

Image quality

parameters

Set

2S

et

4