1. Supplementary Material Emission Computed Tomography
Thanks to those that post interesting material on the internet. This supplement is a collection from several.

2. Emission Tomography f(x,y,z) f(x,y,z) Reconstruction projection Slices

3. SPECT Single Photon Emission Computed Tomography
only one gamma photon is detected per decay
Rotating scintillator camera
collimator
NaI(Tl) crystal

4. PET Positron Emission Tomography
What do we want to detect in PET?
2 photons of 511 keV in coincidence, coming in a straight line from the same annihilation
detector e- e+ 
unstable nucleus
emits positron
positron annihilates with electron
two 511 photons are emitted simultaneously in opposite directions
TRUE coincidence

5. Types of Coincidence
True coincidence is the simultaneous detection of the two emissions resulting from a single decay event.
Scatter coincidence is when one or both photons from a single event are scattered and both are detected.
Random coincidence is the simultaneous detection of emission from more than one decay event.
Coincidences: True Scatter Random

6. PET radiation detection
PET scanner
Typical configuration:
whole-body (patient port ~60 cm; axial FOV~15 cm)
scintillator crystals coupled to photomultiplier tubes (PMTs)
cylindrical geometry
~24-32 rings of detector crystals
hundreds of crystals/ring
several millions of Lines Of Response (LORs) (only a few are shown)
Other configurations for special-purpose applications:
- brain imaging
animal PET
mammography, other
PET CT
True
True

7. PET/CT
CT PET CT+PET
General Electric Medical Systems

8. PET data acquisition Organization of data
True counts in LORs are accumulated
In some cases, groups of nearby LORs are grouped into one average LOR (“mashing”)
LORs are organized into projections
etc…

9. PET data acquisition 2D and 3D acquisition modes
In the 3D mode there are no septa: photons from a larger number of incident angles are accepted, increasing the sensitivity.
Note that despite the name, the 2D mode provides three-dimensional reconstructed images (a collection of transaxial, sagittal and transaxial slices), just like the 3D mode!
2D mode (= with septa)
3D mode (= no septa)
septa

10. not detected (septa block photons)
PET data acquisition
2D mode vs. 3D mode
2D mode (= with septa)
3D mode (= no septa)
True
not detected (septa block photons)
True
detected

11. PET data acquisition Organization of data
In 3D, there are extra LORs relative to 2D
oblique planes
...
same planes as 2D +
3D mode

12. PET evolution: spatial resolution
Human brain
Animal PET ~1998
Monkey brain
Image credits: CTI PET Systems
Image credits: Crump Institute, UCLA

13. “Part of the goal is to bring order to this alphabet soup.”
J. Fessler, 2002

14. PET image reconstruction
P(2 ,r)
Radon Transform 1 2
P(1 ,r)
f(x,y) r
Projection
Object

15. PET image reconstruction
Sinogram r 
Projection angle
Projection bin
Object

16. PET image reconstruction
Object
Sinogram  r

17. PET image reconstruction
Object
Sinogram  r

18. PET image reconstruction
Object
Sinogram  r

19. PET image reconstruction
Object
Sinogram  r

20. PET image reconstruction
Object
Sinogram  r

21. PET: 180º (2 opposite photons)
Sinogram
Other representations can be used instead of the sinogram (linogram, planogram)
PET: 180º (2 opposite photons)
SPECT: 360º (1 photon)

22. PET image reconstruction
2D Reconstruction
2D Reconstruction
Each parallel slice is reconstructed independently (a 2D sinogram originates a 2D slice)
Slices are stacked to form a 3D volume f(x,y,z)
2D reconstruction
Plane 5
Slice 5
etc
2D reconstruction
Plane 4
Slice 4
2D reconstruction
Plane 3
Slice 3
2D reconstruction
Plane 2
Slice 2
2D reconstruction
Plane 1
Slice 1

23. PET image reconstruction
2D Reconstruction
Projection and Backprojection
Projection
Backprojection

24. PET image reconstruction
2D Reconstruction
Object
4 projections
16 projections
128 projections
Backprojection
Filtered Backprojection

25. The Importance of counts
4.8mm
6.4mm
12.7 mm
7.9 mm
11.1mm
9.5mm
counts
1 million counts
2 million counts

26. Noise In PET Images
Noise in PET images is dominated by the counting statistics of the coincidence events detected.
Noise can be reduced at the cost of image resolution by using an apodizing window on ramp filter in image reconstruction (FBP algorithm).
counts
Unapodized ramp filter
Hanning window, 4mm
Hanning window, 8mm

27. PET image reconstruction
Data corrections are necessary
the measured projections are not the same as the projections assumed during image reconstruction
integral of the activity along the line or tube of response
Scattered coincidences component
Attenuation
Random coincidences component
Detector efficiency effects
True coincidences component
projection
assumed
projection
measured
Raylman et al.: “An estimated 25% of women receiving mammograms have radiographically dense breasts, which make lesion detection difficult or impossible because the tumors are virtually indistinguishable from dense normal breast tissue. This number is certain to rise in light of the recent recommendation by the National Cancer Institute that women over the age of 40 and at average risk for breast cancer start regular mammographic screening; considering dense breasts are most common in women under the age of 50.”
Object (uniform cylinder)

28. Analytical methods Advantages Disadvantages Fast Simple
Predictable, linear behaviour
Disadvantages
Not very flexible
Image formation process is not modelled  image properties are sub-optimal (noise, resolution)

29. Iterative methods Advantages Disadvantages
Can accurately model the image formation process (use with non-standard geometries, e.g. not all angles measured, gaps)
Allow use of constraints and a priori information (non-negativity, boundaries)
Corrections can be included in the reconstruction process (attenuation, scatter, etc)
Disadvantages
Slow
Less predictive behaviour (noise? convergence?)

30. PET Image reconstruction
Iterative methods
Iteration 1
projection
Estimated projection
image space
projection space
Current estimate
Measured projection
Compare
(e.g. – or / )
Error projection
Update
backprojection
Error image

31. PET Image reconstruction
Iterative methods
Iteration 2
image space
projection space
projection
Estimated projection
Estimated projection
Current estimate
Measured projection
Compare
(e.g. - or / )
Update
backprojection
Error image
Error projection

32. PET Image reconstruction
Iterative methods
Iteration N
image space
projection space
projection
Estimated projection
Estimated projection
Current estimate
Measured projection
Compare
(e.g. - or / )
Update
backprojection
Error image
Error projection

33. Algorithm comparison 600 000 counts (including scatter) original 3DRP
Hanning
3D OSEM + filt.
6 subsets 5 iter. Gauss .5cm
FORE + OSEM
6 subsets 2 iter.
3D OSEM
Image credits: Kris Thielemans
MRC CU, London (now IRSL –

34. Reconstruction of a slice from projections example = myocardial perfusion, left ventricle, long axis
This image should be used to explain the principle of filtered back projection. Note that the image contains an animation.
courtesy of Dr. K. Kouris

35. conventional iterative algebraic methods
Iterative reconstruction methods
conventional iterative algebraic methods
algebraic reconstruction technique (ART) simultaneous iterative reconstruction technique (SIRT) iterative least-squares technique (ILST)
iterative statistical reconstruction methods (with and without using a priori information)
gradient and conjugate gradient (CG) algorithms maximum likelihood expectation maximization (MLEM) ordered-subsets expectation maximization (OSEM) maximum a posteriori (MAP) algorithms

36. algorithm (a recipe)
(1) make the first arbitrary estimate of the slice (homogeneous image),
(2) project the estimated slice into projections analogous to those measured by the camera (important: in this step, physical corrections can be introduced - for attenuation, scatter, and depth-dependent collimator resolution),
(3) compare the projections of the estimate with measured projections (subtract or divide the corresponding projections in order to obtain correction factors - in the form of differences or quotients),
(4) stop or continue: if the correction factors are approaching zero, if they do not change in subsequent iterations, or if the maximum number of iterations was achieved, then finish; otherwise
(5) apply corrections to the estimate (add the differences to individual pixels or multiply pixel values by correction quotients) - thus make the new estimate of the slice,
(6) go to step (2).

37. Iterative reconstruction - multiplicative corrections

38. Iterative reconstruction - differences between individual iterations

39. Iterative reconstruction - multiplicative corrections

40. Filtered back-projection
very fast
direct inversion of the projection formula
corrections for scatter, non-uniform attenuation and other physical factors are difficult
it needs a lot of filtering - trade-off between blurring and noise
quantitative imaging difficult

41. Iterative reconstruction
amplification of noise
long calculation time
discreteness of data included in the model
it is easy to model and handle projection noise, especially when the counts are low
it is easy to model the imaging physics such as geometry, non-uniform attenuation, scatter, etc.
quantitative imaging possible

42. References:
Groch MW, Erwin WD. SPECT in the year 2000: basic principles J Nucl Med Techol 2000; 28: ,
Groch MW, Erwin WD. Single-photon emission computed tomography in the year 2001: instrumentation and quality control J Nucl Med Technol 2001; 20:9-15,
Bruyant PP. Analytic and iterative reconstruction algorithms in SPECT. J Nucl Med 2002; 43: ,
Zeng GL. Image reconstruction - a tutorial Computerized Med Imaging and Graphics 2001; 25(2):97-103,
Vandenberghe S et al. Iterative reconstruction algorithms in nuclear medicine. Computerized Med Imaging and Graphics 2001; 25(2): ,

43. References:
Patterson HE, Hutton BF (eds.). Distance Assisted Training Programme for Nuclear Medicine Technologists. IAEA, Vienna, 2003,
Busemann-Sokole E. IAEA Quality Control Atlas for Scintillation Camera Systems. IAEA, Vienna, 2003, ISBN ,
Steves AM. Review of nuclear medicine technology. Society of Nuclear Medicine Inc., Reston, 1996, ISBN ,
Steves AM. Preparation for examinations in nuclear medicine technology. Society of Nuclear Medicine Inc., Reston, 1997, ISBN ,
Graham LS (ed.). Nuclear medicine self study program II: Instrumentation. Society of Nuclear Medicine Inc., Reston, 1996, ISBN X,
Saha GB. Physics and radiobiology of nuclear medicine. Springer-Verlag, New York, 1993, ISBN