1. Fundamental Limits of Positron Emission Tomography
William W. Moses
Lawrence Berkeley National Laboratory
Department of Functional Imaging
September 5, 2002
Outline:
Spatial Resolution
Efficiency
Noise
Other Modalities
(Best Viewed in “Slide Show” Mode)

2. Positron Emission Tomography (PET)
Ring of Photon
Detectors
Patient injected with positron (+ ) emitting radiopharmaceutical.
+ annihilates with e– from tissue, forming back-to-back 511 keV photon pair.
511 keV photon pairs detected via time coincidence.
Positron lies on line defined by detector pair (i.e., chord).
Reconstruct 2-D Image using Computed Tomography
Multiple Detector Rings  3-D Volumetric Image

3. Fundamental Limits of Spatial Resolution
Dominant Factor is Crystal Width
Limit for 80 cm Ring w/ Block Detectors is 3.6 mm
Ultimate Limit is 0.6 mm (Positron Range)

4. Radial Elongation
Penetration of 511 keV photons into crystal ring blurs measured position.
Blurring worsens as detector’s attenuation length increases.
Effect variously known as Radial Elongation, Parallax Error, or Radial Astigmatism.
Can be removed (in theory) by measuring depth of interaction.
Tangential Projection
Radial Projection

5. Spatial Resolution Away From Center
Point Source Images in 60 cm Ring Diameter Camera
1 cm
Near Tomograph Center
14 cm from Tomograph Center
Resolution Degrades Significantly...

6. Theoretical Spatial Resolution (all units in mm)
d = Crystal Width
R = Detector Ring Radius
r = Distance from Center of Tomograph

7. Spatial Resolution is Defined Assuming Infinite Statistics
Caveat:
Spatial Resolution is Defined Assuming Infinite Statistics
Resolution Does Not Include Effects from Noise, but Image Quality Does...

8. Effect of Noise On Image Quality
55M Events
3 mm fwhm
1M Events
3 mm fwhm
1M Events
6 mm fwhm
All Three Images Have Same Camera Resolution
Statistical Noise  Reduced Image Resolution

9. Low Image Noise  High Sensitivity
Sensitivity Definition:
Place 20 cm diameter phantom in camera.
Measure True Event Rate.
Sensitivity = True Event Rate / µCi / cc.
Sensitivity Measures Efficiency for Detecting Signal

10. Increase Sensitivity by Removing Septa
Inter-Plane
Septa No
Septa
2-D (w/ Septa)
Septa Reduce Scatter
 Smaller Solid Angle for Trues
3-D (w/o Septa)
No Scatter Suppression
 Larger Solid Angle for Trues

11. Sensitivity Includes Noise from Background
T = Trues
S = Scatter
R = Randoms
Even when you subtract the background,
statistical noise from the background remains.
Image Noise Not Determined by Sensitivity Alone!

12. Noise Equivalent Count Rate (NECR)
NECR Properties:
Like a Signal / Noise Ratio (Sensitivity only Includes Signal)
Includes Noise from Backgrounds
Maximize NECR to Minimize Image Noise

13. NEC Properties T Obeys Counting Statistics Equals Signal x Contrast
Phantom Geometry Must Be Defined
Usually 20 cm diameter, 20 cm tall cylinder T

14. NEC Behavior: Ideal Camera (No Dead Time, No Coincidence Processor Limit)
NEC Plateaus as Activity () Increases!

15. NEC Behavior (With Dead Time, No Coincidence Processor Limit)
Tdead  e–( t)
Rdead  2e–( t)
Dead Time (t) reduces T and R by same factor.
As  increases, NEC eventually decreases (paralyzing dead time).

16. NEC Behavior (With Dead Time and Coincidence Processor Limit)
RCP = Max. Rate
TCP = Max. Rate
Tdead  e–( t)
Rdead  2e–( t)
Tdead
Rdead
Total throughput becomes constant (T+2R = Max. Rate).
True / Randoms ratio not affected by rate limit.
R  constant, T  1/.

17. More Solid Angle Is Not Always Better...
20 cm Phantom
3-D has Higher NECR at Low Activity
Peak NECR in 2-D > Peak NECR in 3-D (Less Scatter)

18. Noise From Reconstruction Algorithm
Basic measurement of chord (crystal-crystal coincidence) represents the integral of the activity along that line.
Measurements from other chords needed to constrain activity to its source voxel.
Activity in other voxels complicates the image reconstruction.
Signals from Different Voxels are Coupled
Statistical Noise from One Voxel Affects All Voxels

19. Point-Like Objects Reconstruct with Less Noise
Object Dependence
“Point-Like” Object
100,000 Events
“Uniform” Object
1,000,000 Events
Point-Like Objects Reconstruct with Less Noise

20. PET Take-Home Messages
Spatial Resolution Dominated by Crystal Size
Other effects can be important at high resolution
No Intrinsic Resolution / Efficiency Tradeoff, but Effective Tradeoffs Because of Noise
Noise from Counting Statistics:
Depends on Camera Efficiency / Geometry
Higher Efficiency Doesn’t Always Imply Lower Noise!
Noise from Reconstruction Algorithm:
Depends on Object Geometry
Think About Signal/Noise, Not Just Signal!!!

21. How Does PET Compare to Other Modalities?
Parallel Hole Collimator
Pinhole Collimator
Coded Aperture
Compton Camera

22. Parallel Hole Collimator Properties
Gamma Detector
Typical Values:
w= 2 mm
L= 30 mm
t= 0.25 mm
Collimator
Collimator Geometry Defines Acceptance Angle 

23. Spatial Resolution Resolution Proportional to 
Typical Values:
R= 6 mm 5 cm)
R= 12 mm 10 cm)
Resolution Proportional to 
Resolution Proportional to Distance from Source

24. Efficiency Efficiency Proportional to 2
Typical Values:
Efficiency = 0.02%
Efficiency Proportional to 2
Efficiency Independent of Distance from Source

25. Pinhole Camera Compared to Parallel Hole Collimator, Pinhole 
Different Resolution / Efficiency Tradeoff
Higher Resolution, but Smaller Field of View

26. Coded Aperture Camera Compared to Pinhole Camera, Many (n) Pinholes 
Similar Resolution w/ Higher Efficiency

27. Image Overlap with Coded Apertures
“Point-Like” Object
“Uniform” Object
Removing the Overlap Increases the Noise
Noise Increase Depends on Object

28. Intrinsic Resolution / Efficiency Dependency
Dependence on:
w d L
Par. Hole
Resol. w d L-1
Effic. w2 – L-2
Area – – –
Pinhole/CA
Resol. w – –
Effic. w2 d-2 –
Area – d L-1
Gamma Detector
Generic Collimating Structure
Very Different Geometrical Dependencies
Pinhole / Aperture Best for Small Area, High Resol.

29. No Collimator, but Reconstruction Difficult
Compton Cameras
How They Work:
• Measure first interaction with good Energy resolution.
• Measure first and second interaction with moderate Position resolution.
• Compton kinematics determines scatter angle.
• Source constrained to lie on the surface of a cone.
No Collimator, but Reconstruction Difficult

30. Compton Camera Tradeoffs
Advantages:
No Intrinsic Resolution / Efficiency Tradeoff
(Resolution Limited by Energy Resolution)
No Collimator  Much Higher Efficiency
Large Imaging Volume
Disadvantages:
“Value” of Each Gamma is Lower
Difference in “Value” Depends on Object
(“Point-Like” Objects are Better)
Random Coincidence Background / NEC

31. Detected Events Can Have Different Values
Compton
Cone Surface
PET
Thin Line
Collimator
Thin Cone
Pinhole
Thin Cone
Coded Aperture
Thin Cones
Value Inversely Proportional to Volume of Object that the Gamma Could Have Come From?

32. Conclusions Different Modalities Have Different Imaging Tradeoffs
Resolution, Efficiency, Noise, Imaging Volume
Consider Noise As Well As Signal
Counting Statistics
Background Events
Reconstruction Algorithms
Complex Sources
Some Gammas Are Worth More Than Others
Volume that Detected Gamma Could Have Come From
Value Can Be Estimated Using Simulation
Use Reasonable Source Geometry & Number of Events
Include All Background Sources

33. Acknowledgements U.S. Department of Energy
Office of Environmental and Biological Research
Laboratory Technology Research Division
National Institutes of Health
National Cancer Institute
University of California Office of the President
Breast Cancer Research Program
U.S. Army
Breast Cancer Directive
Commercial Partners
Capintec, Inc.
Digrad, Inc.

34. Principle of Computed Tomography
2-Dimensional Object
1-Dimensional Horizontal Projection
1-Dimensional Vertical Projection
By measuring all 1-dimensional projections of a
2-dimensional object, you can reconstruct the object

35. Separates Objects on Different Planes
Computed Tomography
Planar X-Ray
Computed Tomography
Separates Objects on Different Planes
Images courtesy of Robert McGee, Ford Motor Company