An overview of iterative reconstruction
applied to PET (and SPECT)?
Professor Brian F Hutton
Institute of Nuclear Medicine
University College London
brian.hutton@uclh.nhs.uk
m
Outline
• Understanding iterative reconstruction
(ML-EM + OS-EM)
• the flexibility in system modelling
• modelling resolution
• time-of-flight
m
Single Photon Emission Computed
Tomography (SPECT)
• relatively low resolution; long acquisition time (movement)
• noisy images due to random nature of radioactive decay
• tracer remains in body for ~24hrs: radiation dose ~ standard x-ray
• function rather than anatomy
Coincidence Detection: Positron
Emission Tomography (PET)
coincidence
window
time (ns)
detector 1
detector 2
• valid coincidence event if two gammas detected within short time (8-12ns)
Coincidence Lines of Response (LoR)
sinogram
distance
angle
fanbeam
parallel
• data acquired direct to sinogram:
set of projections versus angle
PET / SPECT
Reconstruction
1 angle
2 angles
4 angles
• conventional filtered back projection
• iterative reconstruction
16 angles
128 angles
Understanding iterative reconstruction
Objective
Find the activity distribution whose
estimated projections match the
measurements.
Modelling the system (system matrix)
What is the probability that a photon
emitted from location X will be detected at
detector location Y.
- detector geometry, collimators
- attenuation
- scatter, randoms
detector
(measurement)
m
X
object
m
Y2
Y
estimated
projection
Y1
X
BP
patient
update
(x ratio)
original
projections
ML-EM
reconstruction
NO
original
CHANGE
estimate
FP
estimated
projections
current
estimate
14000
EM reconstruction
12000
Chi-squared
10000
8000
6000
4000
2000
0
0
50
100
150
200
iterations
comparison with projections
300
mean square error
250
200
150
100
50
0
0
50
100
150
200
iterations
comparison with actual object
ML-EM algorithm
 measured _ projections 
new _ estimate  old _ estimate  BP 



FP
old
_
estimate


new
j
 j


1
i
aij
i
new estimate
yi
aij
yˆi
forward projection
old estimate
system matrix
back projection
System matrix
sinogram
0
0
0
0
0
0
0
1
0
0
0
0
pixeli
0
0
0
1
distance
0
0
angle
0
0
0
0
1
voxelj
0
0
0
0
0
0
0
System matrix: with attenuation
0
0
0
0
0
0
0
0.2
0
0
0
0
0
0
0
0.5
0
0
0
0
m
0
0
0.9
0
0
0
0
0
0
0
OS-EM
ML-EM
4 iterations
OS-EM
1 iteration
Update 1
Update 2
Update 3
Update 4
ML-EM: each update involves BP and FP for all projection angles
OSEM: each update only uses a subset of projection angles
EM iterations = OS-EM iterations x no of subsets
6000
chi-squared
5000
4000
3000
2000
em
os2
os4
1000
0
0
50
100
150
iterations
mean square error
250
200
150
100
em
os2
os4
50
0
0
50
100
iterations
150
Image courtesy of Bettinardi et al, Milan
a) ML-EM: noise is proportional to activity
Poisson
FBP
Uniform
ML-EM
FBP
b) ML-EM: noise assumes a Poisson model
ML-EM
Problems with pre-correction
• acquired data assumed to be Poisson
• processing of projections likely to destroy assumption
e.g. scatter correction, randoms correction in PET
• instead incorporate all corrections inside model
Historical Subtract measured randoms and scatter; increases noise
 measured _ projections  scatter 
new _ estimate  old _ estimate  BP 

FPold _ estimate


Instead Add measured randoms and scatter in forward model
 measured _ projections 
new _ estimate  old _ estimate  BP 



FP
old
_
estimate

scatter


Non-uniform convergence
True image
20 iterations
100 iterations
Courtesy Johan Nuyts, KU Leuven, Belgium
iteration
10.70mm
120
11.70mm
11.70mm
12.10mm
normalised uptake
100
13.50mm
13.80mm
80
14.50mm
15.90mm
16.27mm
60
16.30mm
16.6mm
17.70mm
40
18.50mm
20.10mm
20
20.39mm
21.10mm
22.80mm
0
0
J Nucl Med, 2005; 46:469P (abs)
50
100
150
200
250
23.44mm
27.35mm
no of iterations
30.73mm
Convergence rate for 20 lesions (UCL)
true image
8 iter
100 iter
FBP
sinogram
with
noise
smoothed
Image courtesy of J Nuyts, Leuven
Modelling resolution
• potentially improves resolution
• requires many iterations
• slow to compute
• stabilises solution
• better noise properties
detector
(projection)
m
object
w/o resn model
Courtesy: Panin et al
IEEE Trans Med Imaging
2006; 25:907-921
with resn model
System matrix: including resolution model
0
0
0
0
0
0
0.1
0.2
0.1
0
0
0
0
0
0.2
0.5
0
0
0
0
m
0
0.2
0.9
0.2
0
0
0.2
0
0
0
Modelling system resolution
(UltraSPECT, Astonish, Flash, Evolution)
FBP
WBR
10 min scan
5 min scan
FBP
WBR
D-SPECT: reconstruction
includes resolution model
PET resolution
depth of
interaction
detector
fan
depth of interaction results in
asymmetric point spread function
positron range
colinearity
FWHMtotal2 = FWHMdet2 + FWHMrange2 + FWHM1802
Modelling resolution
Simple model:
• assumes no loss of resolution
Account for resolution:
m
• exactly accounts for resolution
• involves higher uncertainty
Contrast v noise:
contrast/recovery
• noise increases with iteration no
• contrast reaches max value
With resolution model:
noise
• need more iterations to reach max
• noise less for same contrast
• better model; better quality
Clinical studies
Reconstruction on
256 pixels x 256 pixels,
28 subsets, 5 iterations
FWHM=4 mm
OSEM
OSEM + smooth
PSF-OSEM
FWHM=5 mm
Courtesy Rapisardi, Bettinardi, Milan
Clinical studies:
14subsets
2 iterations
3D-OP-OSEM
3D-OSEM with PSF
Townsend, Phys Med Biol 2008; 53: R1-R39
Time-of-flight
coincidence
window
t1
m
d Dd
8ns
time (ns)
t1
t2
• both gammas travel with speed of light (c)
• difference in time of detection is (t2-t1)
• emission origin is at distance d from centre
detector 1
t2
detector 2
(t2-t1)
where d = (t2-t1).c/2
• but uncertainty in determining time (dt)
• therefore also uncertainty in determining d (Dd)
dt
600ps
Dd
9cm
Time-of-flight
Normal
back projection:
Using TOF:
• no
knowledge
of position
some
knowledge
of position
• blurred
result
much less
blurring
Adapted from Mike Casey, Siemens white paper
m
Improving signal-to-noise: time-of-flight PET
Detector B
t2
Patient outline
(diameter D)
d1
SNRTOF  √(D/Dd) · SNRnon-TOF
Dd is uncertainty in position due
to limited timing resolution dt;
D is diameter of object (patient)
d
e+ e
t1
Detector A
Dd
dt (ps)
Dd (cm) SNR*
100
1.5
5.2
300
4.5
3.0
500
7.5
2.3
600
9.0
2.1
* SNR gain for 40 cm phantom
= SNRTOF / SNRnon-TOF
TOF converges faster
and achieves better contrast for given noise
TOF
#iter = 1
noTOF
2
5
10
20
35-cm diameter phantom; 5 minute scan time
10, 13, 17, 22-mm hot spheres (6:1 contrast); 28, 37-mm cold spheres
Philips Gemini TF
TOF benefit is more significant as timing resolution improves
TOF:
400ps
TOF:
650ps
NonTOF
1.4M
2.8M
35-cm diameter phantom
5.6M
8.5M
12.7M
16.9M
La-PET proto-type: LaBr
HD·PET
ultraHD·PET
0.57
BMI: 30
2D: FORE+OSEM
3D: HD
3D: ultraHD
0.24
HD·PET images show improved spatial resolution
when compared with 2D reconstruction. The
ultraHD·PET images show incremental improvement
in signal-to-noise such as better liver uniformity and
lower background in cold areas.
Time-of-flight
gain
2.2
Gain
SNR Gain
2006
1990
2.4
2.0
1.8
1.6
Body
1.4
1.2
Body Mass Index (BMI)
1.0
15
20
25
30
35
40
BMI
BMI 30
<10%
10%–14%
15%–19%
20%–24%
25%–29%
≥30%
HD·PET
ultraHD·PET
45
Summary
• Iterative reconstruction is increasingly used in clinical practice
• ML-EM iteration = OS-EM iterations x no of subsets
• Need to be aware of limitations
- bias with low counts
- convergence varies across object
- need to preserve Poisson statistics
• Resolution models potentially improve contrast AND noise
- needs extra iterations
• Time-of-flight information further improves signal to noise
- needs less iterations!
- gain dependent on patient size, application
Acknowledgements
Thanks to Joel Karp, Dave Townsend, Johan Nuyts
for use of material for slides.
OS-EM bias: non-negativity constraint
• Striatal Phantom with 10:1 and 5:1
striatal-to-background uptake ratio
• Background count concentration in 10:1
study half that of 5:1 study
10:1
5:1
• Convergent striatal
count concentration
• Apparent peaking of
measured uptake ratio in
10:1 study
• Non-convergent background
count concentration at low
count level in 10:1 study
Data courtesy of J Dickson, UCL
Meaningful evaluation
• evaluation is difficult!
• wide range of algorithms and parameters
• comparing only two sets of images meaningless!
• conventional performance measures inappropriate
(e.g. resolution, sensitivity)
• measurement is object dependent
• performance is task dependent: ROC analysis!
FBP
OS-EM
contrast/recovery
Comparing performance:
noise
Contrast versus noise
• myocardium to ventricle contrast recovery
• COV from 10 independent noise realisations
• values vary with iteration number / filter parameters
no rr
Data courtesy K Kacperski, UCL
rr+filter