Simultaneous estimation of the attenuation and emission in
TOF-PET
Joint work VUB (M Defrise), KULeuven (J. Nuyts, A. Rezaei)
8-4-2015
1
Herhaling titel van presentatie
Simultaneous estimation of the emission and attenuation maps in TOF-PET
PET-CT
PET data : gg coincidences
CT data: x-ray attenuation
Reconstruction of the organ
density
Reconstruction with correction
for the g attenuation
"m map"
"Emission map":
tracer's
biodistribution
2
Simultaneous estimation of the emission and attenuation maps in TOF-PET
PET-MRI
MR data: H+ density, T1,T2,..
PET data : gg coincidences
segmentation + a priori m assignments
No direct link with attenuation coefficients
?
"m map"
Reconstruction with correction
for the g attenuation
?
"Emission map":
tracer's
biodistribution
e.g.: V. Keereman, Y. Fierens, C. Vanhove, T. Lahoutte, S. Vandenberghe,
2011 MRI-based attenuation correction for micro-SPECT, Mol. Imag.
Hofmann M et al, Eur J Nucl Mol Imag 2009,
3
Simultaneous estimation of the emission and attenuation maps in TOF-PET
PET without additional data ?
PET data : gg coincidences
?
"m map"
"Emission map"
- Welch A, Campbell C, Clackdoyle R, Natterer F, Hudson M, Bromiley A, Mikecz P, Chillcot F, Dodd M,
Hopwood P, Craib S, Gullberg G T and Sharp P 1998 Attenuation correction in PET using consistency information,
IEEE Trans. Nuclear Science
-Nuyts J, Dupont P, Stroobants S, Benninck R, Mortelmans L, Suetens P 1999 Simultaneous maximum a posteriori
reconstruction of attenuation and activity distributions from emission sinograms, IEEE Trans Med Imag: MLAA
- Papers by A. Bronnikov
4
Time-of-flight PET in a nutshell
detector d2,
time t2
Anger 1966.
1980’s: Allemand et al, Ter Pogossian et al,
Mullani et al, Lewellen et al
h(l - c  /2)
o
l
Measure the arrival time difference t = c (t2 – t1 )/2
localize decay along LOR with uncertainty
l = c  /2
e+
detector d1,
time t1
 FWHM = 500 ps
l FWHM = 75 mm
“Each coincident event yields more information than without time-of-flight,
by a factor ≈ patient diameter/lFWHM ”
Snyder et al 1980’s, Tomitani 1981, Watson 2007,
Surti et al 2006/8, Popescu & Lewitt 2006, …
5
Time-of-flight PET: analytical data model
detector d2,
time t2
h(l - c  /2)
o
l
e+
detector d1,
time t1
m(L,t)  exp(  dl m(l))
measured
data for all lines
L and all t
6
L

L
attenuation
image
dl h(t  l) f (l)  a(L) p(L,t)
TOF profile activity
image
attenuation corrected
data
factor
Simultaneous estimation of the emission and attenuation maps in TOF-PET
TOF-PET without additional data ?
TOF-PET data : gg coincidences
"m map"
"Emission map"
M. Defrise, A. Rezaei, J. Nuyts, TOF-PET
data determine the attenuation sinogram
up to a constant, Phys Med Biol. 2012
- Proof based on the consistency conditions for TOF-PET data
(Continuous and Discrete Data Rebinning in TOF-PET,IEEE Trans Med Imag 2008)
- Analytic algorithm to determine the m map
- Not as stable as a CT scan but benefit: m map is matched to emission map
- Iterative approaches: MLAA and MLACF
7
- Analytic solution for the 2D problem
- Some results
- Analytic solution for the 3D problem
- Discrete methods
- Some results
8-4-2015
8
Herhaling titel van presentatie
Analytic solution for the 2D problem
8-4-2015
9
Herhaling titel van presentatie
The non-attenuated 2D TOF PET data
p(,s,t) 

dl
f
(su
 lu) h(t  l)

u  (sin ,cos ) u   (cos,sin  )
tR
f  C02 (R 2 )
for a gaussian TOF profile satisfy the local consistency condition
2
p p p

p
2
Dp  t 
s 
0
s 
t
st
Proof: insert the definition of p and use
d


0   dl
f
(su
 lu) h(t  l)

dl
dh(t) d t 2 / 2 2
th(t)
 e
 2
dt
dt

Some insight of the consistency condition
The most likely annihilation point (MLA) of a TOF-LOR
p(,s,t) 
 dl
f (su  lu) h(t  l)
is the point x* at the maximum of the TOF profile, i.e at l=t
x*  su  tu  scos  t sin ,ssin   t cos 
t
'
In the limit 0 the equation
2
p p p

p
2
Dp  t 
s 
0
s 
t
st
t
m(,s,t)  a(,s) p(,s,t)  a(,s) f (x*)
a(,s)
m(,s,t)

a( ',s') m( ',s',t')

means that all TOF-LORs sharing the same
MLA x* must be equal

(or: the characteristic curves of this PDE in the 3D space (,s,t)
are loci of constant MLA).

The attenuated 2D TOF PET data are
m(,s,t)  a(,s) p(,s,t)
a(,s)  exp((Rm)(,s))  exp(  dl m(su   lu))
Vocabulary:
a is the attenuation factor
m is the attenuation coefficient
The proof of uniqueness and the algorithm exploit:
p=m/a must be consistent
a is independent of the TOF variable t
2
 (m /a) (m /a)  (m /a)

(m /a)
D(m /a)  t

s
2
0
s

t
st
Since a > 0, we have
2
 (m /a) (m /a)  (m /a)

(m /a)
D(m /a)  t

s
2
0
s

t
st
  loga  loga 
2 m  loga
Dm  m  t


 
t s
 s
 t 
For each LOR (,s) for which m(,s,t)>0, applying a least-square fit
w.r.t. the TOF variable t and with only two unknown parameters
yields
 log a
Rm

and
s
s
 log a
Rm



The solution to the least-square fit is
Rm JsH  J Hs


s
Hss H  Hs2
Rm J Hss  JsHs



Hss H  Hs2
with quantities directly calculated from the measured data:
Hss 
Js 
 dt (mt  m)
2
t
2
Hs 
2
dt
(Dm)
(mt

t m)

 dt
J 
m (mt t m) H 
2
 dt (Dm)
 dt
m
2
m
When does this work ? Answer:
For all LORs such that
-m>0
- along the LOR, the activity is not restricted to a point source
Procedure:
- apply above method to estimate (Rm)( ,s)
in the interior of supp(Rf)
- integrate the gradient within supp(Rf) to find Rm up to a global

additive constant, hence a=exp(-Rm) up to a global factor.
(Least-square solution found by solving a Poisson equation,
we use a Landweber iteration)
- estimate the constant (various methods, e.g. add small external object with known
activity or attenuation)
- either directly use the estimated a
- or reconstruct m
Some details on the implementation (slide J. Nuyts)
The noisy TOF sinogram is first smoothed in 3D. That seems
essential, without this the noise propagation is not acceptable. A
moderate smoothing seems enough: 2.5 pixels radially and
angularly. In TOF-direction I smoothed with a Gaussian with a
width of 70% of the TOF-kernel. This addition smoothing in the
TOF-direction was taken into account by increasing  in the
analytical expressoin.
In a real PET system, there will be such smoothing along the TOFdirection anyway by binning into a relatively small number of bins.
In this test program, I stored the TOF in 128 bins, avoiding
problems due to binning.
Straightforward implementation of the analytical inversion then
yields derivative sinograms with extreme values near the edges,
but with only moderate noise propagation inside the object.
To find a region with reliable values, a sinogram was computed by
summing the original noisy sinogram over all TOF-bins
(nonTOFsino). This sinogram was thresholded, the resulting binary
image was strongly eroded to yield a (overly conservative) mask.
Values inside that mask should definitely be subject to moderate
noise only.
For the radial derivative sinogram, the maximum value inside that
mask is computed. All pixels in the derivative sinogram exceeding
this maximum are labeled. Also their horizontal neighbors are
labeled. In the subsequent Landweber-like iterations, those pixels
will be excluded from the computations. Exactly the same thing
was done for the angular derivative sinogram.
Then the Landweber-like iterations are applied and a calculated
sinogram is obtained. As can be seen in the images, it looks
surprisingly good considering the noise in the derivative
sinograms. I assume this is because this iterative procedure
basically integrates in two directions.
The resulting calculated sinogram should be OK except for a
constant. The calculated sinogram values are typically lower
than the true ones, because the iterations were started from a
zero image. Therefore, I expect that a positive constant must
be added.
To estimate a lower limit of that constant, I compute a
temporary sinogram: I take the median using a 7x7 mask to
exclude extreme values, and I use nonTOFsino to set the
background to zero. Then I compute the minimum of that
temporary sinogram. If that minimum is negative (which was
the case for the few simulations I did) , I subtract it (thus
adding a positive offset) from the calculated sinogram.
Then the resulting calculated sinogram is reconstructed with
FBP. It has terrible streaks in the background, but the part
inside the object looks rather good. Because my offset was
conservative, the reconstructed attenuation is still a bit low.
But if this image can be segmented and if the attenuation is
known in a region, the values can be easily corrected (see
below).
For comparison, I computed a noise-free blank scan from the
known activity image (forward projection without
attenuation), and using nonTOFsino as the transmission scan,
an MLTR and FBP reconstruction were calculated. They look
better of course, but considering that using a noise-free blank
scan is an obvious advantage, I think this indicates that the
analytical inversion is rather stable.
In this simulation, the mean TOF-integrated count was 23.7.
For a 3D PET system with 5 or 7 segments, that would
correspond to a TOF-integrated count of about 4...5 per
segment.
true
activity
true attenuation
Slide J. Nuyts. Attenuated emission sinogram,
summed over all 128 TOF-bins.
Maximum = 106, mean = 23.7.
TOF-resolution was 7.5 cm = 24 pixels FWHM.
The simulation was done oversampling the image
pixels with 3x3 samples/pixel, and with 3 subsamples
per detector.
128 x 128 pixels, pixel size = 3.125 mm
To suppress the noise, the TOF-sinogram was smoothed with a Gaussian in 3D, using the following FWHM:
2.5 pixels radially, 2.5 pixels along the angle, and 0.7 x TOF-FWHM = 16.8 pixels in TOF-direction.
In the reconstruction,  was set to the combined effect of the TOF-kernel and the additional smoothing along the TOF-dimension.
Computed radial derivative (left),
true radial derivative (right).
true attenuation
Defrise-recon.
Computed angular derivative (left),
true angular derivative (right).
Defrise-recon zeroed
outside boundary
Computed attenuation sinogram (left),
true attenuation sinogram (right).
MLTR from true activity
image and noisy sinogram
FBP from true activity
image and noisy sinogram
Simple procedure to find the offset.
As mentioned above, the reconstruction from the calculated
sinogram is not quantitatively accurate, because the analytical
inversion is only accurate except for a constant in the sinogram.
Quantification can be restored, if the attenuation coefficient in a
part of the image is known. Here is a possible way to do this, based
on the linearity of FBP.
1.Let R be the original non-quantitative reconstruction.
2.Make a non-TOF sinogram that is unity inside the object. Here,
this was done by thresholding the TOF-integrated sinogram
nonTOFsino, setting all non-zero values to 1.
3.Reconstruct this sinogram, generating a reconstruction image
corresponding to a sinogram offset of 1. We call it U.
4.Segment the original reconstruction R, identifying a region where
the attenuation coefficient is known a priori. We call this known
attenuation value m.
5. Compute for that region the mean value in R, called MR, and
the mean value in U, called MU.
6. Correct the image R as follows: Rnew = R + (m – MR) / MU * U
For the 2D simulation, the segmentation was done by
thresholding R with a value of mtissue / 2. Because the
quantification of R is not that bad, this separates tissue and
bone from the lungs. We also discard the background by
assuming zero attenuation where the sinogram count is zero
(convex hull). Thus, the segmentation contains bone and
tissue. We compute MR as the median value of the selected
pixels in R. We compute MU as the median of the
corresponding pixels in U. The image was then corrected
using the known attenuation of tissue. The result is shown
below.
True image
Reconstruction R
Corrected image Rnew
True image
Reconstruction R
Corrected image Rnew
Result of the correction applied to the
simulation shown above.
Result of the correction applied to a
simulation with three times less counts
(mean of 7 counts/non-TOF pixel),
obtained with the same parameters.
True image
Result of the correction applied to a
noise-free simulation, using the same
parameters. This illustrates the impact of
the smoothing, which is needed to deal
with noisy data.
Reconstruction R
Corrected image Rnew
Analytic solution for the 3D problem
8-4-2015
20
The non-attenuated 3D TOF PET data
p(,s,z,,t) 
1  2
  tan
 dl
f (scos  lsin ,ssin   lcos,z  l ) h(t  l 1  2 )
f  C02 (R 3 )
for a gaussian TOF profile satisfy two consistency conditions
2
2
p p

p

p


p
Dp 

 s 1  2
 s 
0
2 s
2

t
z
1 
1  st
p
t
p
t p
 2  2 p  2  2 p
Kp 




0
2
2
2
2
2

1  z 1  t
1  zt 1  t
t
The algorithm exploit:
D(m/a)=K(m/a)=0
a is independent of t
estimation of the 4D gradient
 Rm Rm Rm Rm 
Rm  
,
,
,

  s z  
PDEs: Defrise et al TMI 2008. Attenuation problem: submitted to IEEE MIC 2012, A. Rezaei et al

A discrete approach
8-4-2015
22
Notations
activity image
j
j  1,...,M
data
y it
i  1,...,N
system matrix
c ijt
expected data
pit   c ijt  j
t  1,...,T
j
sum dat a
y i   y it
t
sum exp. data
pi   pit  c ij  j wit h c ij   c ijt
t
at tenuation image
j
mj
at tenuation fact ors ai  exp( c ij m j )
j
t


Two possible approaches:
I. MLAA: Poisson maximum-likelihood for ,m
(Nuyts et al, Trans Med Imag 1999; for TOF: Rezaei et al IEEE Med Imag Conf 2011)
T
N
L(y, , m)  

t1
i1

  cij ' m j '

 j ' cij ' m j '

j'


c ijt  j  y it loge
c ijt  j 
e



j
j




II. MLACF: Poisson maximum-likelihood for ,a
ai
(Nuyts et al, submitted IEEE MIC 2012)
T
N
L(y, ,a)  

t1
i1





ai  c ijt  j  y it log ai  c ijt  j 

j
 j



+ simple, fast, and works well on simulated data (see data below)
- in 3D there are more unknowns a than m (more LORs than voxels)






L(y, ,a)    ai  c ijt j  y it log ai  c ijt j 
t1 i1 
j
 j



N 
T

  ai pi  y i log(ai )   y it logpit 

i1 
t1
T
N
is easily maximized w.r.t. attenuation coefficients a at fixed
emission image :
a*i 
yi
pi
1
if
yi
1
pi
if
yi
1
pi
we neglect the non-negativity constraint on m and use
always y/p

We are left with the problem of maximizing
L (y, ) 
*
N

i1
T


y i log(pi )   y it logpit   L(y, ,a* )  F(y)


t1
An iterative algorithm based on a surrogate function:


k 1
j

kj

i
c ij y i
pik

i
t
c ijt y it
k
it
p
MLACF
k  0,1,2,...
j  1,...,M
pitk   cijt kj and pik   pitk
j
The MLACF algorithm is monotone
Convergence is not guaranteed
(since L is not concave)

Generalizations for scatter background OK and implemented
(Nuyts et al submitted MIC 2012, Panin et al submitted MIC 2012)
t
Examples of results with simulated 2D data
8-4-2015
27
Estimating the sinogram of the attenuation map from TOF data.
Results with the MLACF method.
Large ellipse: 300 mm x 480 mm. Background 1, lungs 0.2, tumors 2, 3 and 4
Attenuation coef. Background 0.00966/mm, lungs 0.00266/mm, spine 0.01866/mm
Most attenuated LOR: X 0.03.
32 TOF bins, FWHM = 50 mm, TOF bin spacing is 16 mm
256x256 emission sinogram
(non attenuated, non-TOF)
256x256 mu map sinogram
Simulated 256x256 mu map
Noise-free data
MLACF 4 subsets 16 iterations
TOF FBP w/o (top) and with (bottom)
attenuation correction using exact mumap
Hamming window
3.5
MLACF 4 subsets 16 iterations, rescaled to give same ROI value as FBP
Horizontal profile through reconstructions from noise-free data
3
2.5
2
TOF FBP
MLACF 4 sub 16 iter
1.5
1
0.5
0
0
100
200
300
400
500
600
700
800
900
1000
3.5
MLACF 4 subsets 16 iterations, rescaled to
give same ROI value as FBP
Horizontal profile through
reconstructions from noise-free
data. Zoomed view of previous slide
3
2.5
TOF FBP
MLACF 4 sub 16 iter
2
1.5
1
0.5
0
800
820
840
860
880
900
920
940
960
980
1000
Noise-free data
True mu map (transmission/blank)
Grey scale 0 to 0.25
MLACF 4 subsets 16 iterations
Grey scale 0 to 0.25
Noise-free data
1
MLACF uses 4 subsets 16 iterations
0.9
"exact acf"
0.8
"""MLACF 4 sub 16
iter"""
0.7
0.6
Note that the scaling of the estimated acfs
is obtained by requiring the maximum estimated
acf to be 1 at each iteration of the algorithm.
0.5
0.4
0.3
0.2
0.1
0
0
100
200
300
400
500
600
700
800
900
1000
Noisy data
Maximum count is 100, total count is 15 millions
MLACF uses 4 subsets. FBP is the usual TOF FBP with a hamming window
TOF FBP with attenuation corrections assuming
known mumap
MLACF 4 subsets 8 iterations (top),
16 iterations (bottom)
Noisy data
Maximum count is 100, total count is 15 millions
MLACF uses 4 subsets.
True mu map (transmission/blank)
Grey scale 0 to 0.25
MLACF 4 subsets 12 iterations
Grey scale 0 to 0.25
Noisy data
1
Maximum count is 100, total count is 15 millions
MLACF uses 4 subsets 12 iterations
0.9
"exact acf"
0.8
"MLACF 4 sub 12
iter"
0.7
0.6
Note that the scaling of the estimated acfs
is obtained by requiring the maximum estimated
acf to be 1 at each iteration of the algorithm.
0.5
0.4
0.3
0.2
0.1
0
0
100
200
300
400
500
600
700
800
900
1000
Noisy data
Maximum count is 20, total count is 3 millions
MLACF uses 4 subsets. FBP is the usual TOF FBP with a hamming window
TOF FBP and with attenuation correction
assuming exact mumap
MLACF 4 subsets 8 iterations
Noisy data
5
Maximum count is 20, total count is 3 millions
MLACF uses 4 subsets. FBP is the usual TOF FBP
with a hamming window
Profile along the oblique line in previous slide
4
TOF FBP
3
"MLACF 4 sub 8 iter"
2
1
0
0
100
200
300
400
500
600
700
800
900
1000
Estimating the sinogram of the attenuation map from TOF data.
Results with attenuating medium outside the emission
Large ellipse: 300 mm x 480 mm. Background 1, lungs 0.2, tumors 2, 3 and 4
Attenuation coef. Background 0.00966/mm, lungs 0.00266/mm, spine 0.01866/mm
Bed: 0.01/mm
32 TOF bins, FWHM = 50 mm, TOF bin spacing is 16 mm
Simulated 256x256 emission map
Simulated 256x256 mu map
includes an (uncomfortable) bed
Noise-free data (with bed attenuation)
True mu map (transmission/blank)
Grey scale 0 to 0.35
MLACF 4 subsets 16 iterations
Grey scale 0 to 0.35
No rescaling of the mu map !!
Noisy data (with bed attenuation)
max LOR count = 10, total count 1.26 million
TOF FBP and with attenuation correction
assuming exact mumap
Grey scale [0,2]
MLACF 4 subsets 6 iterations
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MLACF 4 subsets 6 iterations
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