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Fundamental Limits ofPositron Emission Tomography
Fundamental Limits ofPositron Emission Tomography
William W. MosesLawrence Berkeley National Laboratory
Department of Functional ImagingSeptember 5, 2002
Outline:• Spatial Resolution
• Efficiency
• Noise
• Other Modalities
(Best Viewed in “Slide Show” Mode)
Ring of PhotonDetectors • 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
• Reconstruct 2-D Image using Computed Tomography• Multiple Detector Rings 3-D Volumetric Image
Positron Emission Tomography (PET)Positron Emission Tomography (PET)
Fundamental Limits of Spatial ResolutionFundamental 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)
• 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)
d/2
Reconstruction Algorithm1.25 (in-plane)1.0 (axial)
Factor
d
Detector Crystal Width
Photon Noncollinearity
180° ± 0.25°
Positron Range
Shape FWHM
multiplicative factor
0.5 mm (18F)
4.5 mm (82Rb)
1.8 mm
0 (individual coupling)2.2 mm (Anger logic)* *empirically determined from published data
Anger Logic
Radial ElongationRadial Elongation
Tangential Projection
Radial Projection
• 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.
Spatial Resolution Away From CenterSpatial Resolution Away From Center
1 cm
Resolution Degrades Significantly...Resolution Degrades Significantly...
Near Tomograph Center 14 cm from Tomograph Center
Point Source Images in 60 cm Ring Diameter Camera
Theoretical Spatial Resolution(all units in mm)
Theoretical Spatial Resolution(all units in mm)
1.25 Xtal( )2
+ Range( )2
+ Decode( )2
+ Acol.( )2
+ Pen.( )2
Xtal=
d2
d = Crystal WidthR = Detector Ring Radiusr = Distance from Center of Tomograph
Resolution =
Acollinearity=0.0044R
Range=0.5
Penetration=12.5
r
r2 +R2
Decode=2.2(or0)
Caveat:Caveat:
Spatial Resolution is DefinedAssuming Infinite Statistics
Resolution Does Not Include Effects from Noise,but Image Quality Does...
Resolution Does Not Include Effects from Noise,but Image Quality Does...
Effect of Noise On Image QualityEffect of Noise On Image Quality
55M Events3 mm fwhm
1M Events3 mm fwhm
1M Events6 mm fwhm
• All Three Images Have Same Camera Resolution• Statistical Noise Reduced Image Resolution
• All Three Images Have Same Camera Resolution• Statistical Noise Reduced Image Resolution
Low Image Noise High SensitivityLow Image Noise High Sensitivity
Sensitivity Measures Efficiency for Detecting SignalSensitivity Measures Efficiency for Detecting Signal
• Place 20 cm diameter phantom in camera.
• Measure True Event Rate.
• Sensitivity = True Event Rate / µCi / cc.
Sensitivity Definition:
Increase Sensitivity by Removing SeptaIncrease Sensitivity by Removing 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
Inter-PlaneSepta
NoSepta
Sensitivity Includes Noise from BackgroundSensitivity Includes Noise from Background
T = Trues R = RandomsS = Scatter
Image Noise Not Determined by Sensitivity Alone!Image Noise Not Determined by Sensitivity Alone!
Even when you subtract the background, statistical noise from the background remains.
Noise Equivalent Count Rate (NECR)Noise Equivalent Count Rate (NECR)
NECR =T 2
T +S + 2RNECR =
T 2
T +S + 2R
• Like a Signal / Noise Ratio(Sensitivity only Includes Signal)
• Includes Noise from Backgrounds
NECR Properties:
Maximize NECR to Minimize Image NoiseMaximize NECR to Minimize Image Noise
NEC PropertiesNEC Properties
• Obeys Counting Statistics
• Equals Signal x Contrast
• Phantom Geometry Must Be Defined– Usually 20 cm diameter, 20 cm tall cylinder
σNEC = NEC
TT
T +S + 2R
NEC Behavior: Ideal Camera(No Dead Time, No Coincidence Processor Limit)
NEC Behavior: Ideal Camera(No Dead Time, No Coincidence Processor Limit)
0
50
100
150
200
250
0.0 2.0 4.0 6.0 8.0 10.0
NEC (cps)
Activity (µCi/cc)
As ρ → ∞, T2
T +S + 2R→
T2
2R→ Constant
NEC Plateaus as Activity (ρ) Increases!NEC Plateaus as Activity (ρ) Increases!
T ρ
R ρ2
NEC Behavior(With Dead Time, No Coincidence Processor Limit)
NEC Behavior(With Dead Time, No Coincidence Processor Limit)
0
50
100
150
200
250
0.0 2.0 4.0 6.0 8.0 10.0
NEC (cps)
Activity (µCi/cc)
No Dead Time
With Dead Time
• Dead Time (t) reduces T and R by same factor.• As ρ increases, NEC eventually decreases (paralyzing dead time).
Tdead ρe–(ρ t)
Rdead ρ2e–(ρ t)
NEC→
Tdead( )2
2Rdead→ e−ρδt
NEC Behavior(With Dead Time and Coincidence Processor Limit)
NEC Behavior(With Dead Time and Coincidence Processor Limit)
• Total throughput becomes constant (T+2R = Max. Rate).• True / Randoms ratio not affected by rate limit. • R constant, T 1/ρ.
0
50
100
150
200
250
0.0 2.0 4.0 6.0 8.0 10.0
NEC (cps)
Activity (µCi/cc)
No Dead Time
With Dead Time
With DT & CP Limit
NEC→
TCP( )2
2RCP→ ρ−2
RCP = Max. Rate
TCP = Max. Rate
Tdead ρe–(ρ t)
Rdead ρ2e–(ρ t)
Tdead
Rdead
More Solid Angle Is Not Always Better...More Solid Angle Is Not Always Better...
• 3-D has Higher NECR at Low Activity• Peak NECR in 2-D > Peak NECR in 3-D (Less Scatter)
• 3-D has Higher NECR at Low Activity• Peak NECR in 2-D > Peak NECR in 3-D (Less Scatter)
20 cm Phantom 20 cm Phantom
Noise From Reconstruction AlgorithmNoise 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
•Signals from Different Voxels are Coupled•Statistical Noise from One Voxel Affects All Voxels
Object DependenceObject Dependence
“Point-Like” Object100,000 Events
“Uniform” Object1,000,000 Events
Point-Like Objects Reconstruct with Less NoisePoint-Like Objects Reconstruct with Less Noise
PET Take-Home MessagesPET 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!!!Think About Signal/Noise, Not Just Signal!!!
How Does PET Compare to Other Modalities?How Does PET Compare to Other Modalities?
• Parallel Hole Collimator
• Pinhole Collimator
• Coded Aperture
• Compton Camera
Parallel Hole Collimator PropertiesParallel Hole Collimator Properties
Collimator Geometry Defines Acceptance Angle Collimator Geometry Defines Acceptance Angle
Typical Values:
w= 2 mmL= 30 mmt= 0.25 mm
L
w
t
= atanwL⎛ ⎝
⎞ ⎠
Gamma Detector
Collimator
Spatial ResolutionSpatial Resolution
•Resolution Proportional to •Resolution Proportional to Distance from Source
•Resolution Proportional to •Resolution Proportional to Distance from Source
L
w
t
d
R
L
w
td
R
Typical Values:
R= 6 mm (@ 5 cm)R= 12 mm (@ 10 cm)
Resolution (R)=2
wL
d+L2
⎛ ⎝ ⎜ ⎞
⎠ ⎟
EfficiencyEfficiency
•Efficiency Proportional to 2
•Efficiency Independent of Distance from Source•Efficiency Proportional to 2
•Efficiency Independent of Distance from Source
Typical Values:
Efficiency = 0.02%
Efficiency∝θ2 ≈
wL
⎛ ⎝ ⎜ ⎞
⎠ ⎟ 2
L
w
t
L
w
t
Pinhole CameraPinhole CameraL
d
w
Resolution=w
Efficiency∝
wd
⎛
⎝ ⎜
⎞
⎠ ⎟ 2
•Compared to Parallel Hole Collimator, Pinhole • Different Resolution / Efficiency Tradeoff• Higher Resolution, but Smaller Field of View
•Compared to Parallel Hole Collimator, Pinhole • Different Resolution / Efficiency Tradeoff• Higher Resolution, but Smaller Field of View
Coded Aperture CameraCoded Aperture Camera
Resolution=w
Efficiency∝ n
wd
⎛
⎝ ⎜
⎞
⎠ ⎟ 2
•Compared to Pinhole Camera, Many (n) Pinholes • Similar Resolution w/ Higher Efficiency
•Compared to Pinhole Camera, Many (n) Pinholes • Similar Resolution w/ Higher Efficiency
L
w
d
Image Overlap with Coded AperturesImage Overlap with Coded Apertures
•Removing the Overlap Increases the Noise•Noise Increase Depends on Object
•Removing the Overlap Increases the Noise•Noise Increase Depends on Object
“Point-Like” Object “Uniform” Object
Intrinsic Resolution / Efficiency DependencyIntrinsic Resolution / Efficiency DependencyL
d
w
Gamma Detector
Generic Collimating Structure
Dependence on:w d L
Par. HoleResol. w d L-1
Effic. w2 – L-
2
Area – – –
Pinhole/CAResol. w – –Effic. w2 d-2 –Area – d L-1
•Very Different Geometrical Dependencies•Pinhole / Aperture Best for Small Area, High Resol.
•Very Different Geometrical Dependencies•Pinhole / Aperture Best for Small Area, High Resol.
Compton CamerasCompton Cameras
Solid State Detector Array
Anger Camera(No Collimator)
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 DifficultNo Collimator, but Reconstruction Difficult
Compton Camera TradeoffsCompton Camera Tradeoffs
• No Intrinsic Resolution / Efficiency Tradeoff(Resolution Limited by Energy Resolution)
• No Collimator Much Higher Efficiency
• Large Imaging Volume
Advantages:
Disadvantages:
• “Value” of Each Gamma is Lower
• Difference in “Value” Depends on Object(“Point-Like” Objects are Better)
• Random Coincidence Background / NEC
Detected Events Can Have Different ValuesDetected Events Can Have Different Values
Value Inversely Proportional to Volume of Objectthat the Gamma Could Have Come From?
Value Inversely Proportional to Volume of Objectthat the Gamma Could Have Come From?
PETThin Line
CollimatorThin Cone
PinholeThin Cone
CodedAperture
Thin ConesCompton
Cone Surface
ConclusionsConclusions
• 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
AcknowledgementsAcknowledgements
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.
Principle of Computed TomographyPrinciple of Computed Tomography
2-Dimensional Object
1-Dimensional Vertical Projection
1-Dimensional Horizontal Projection
By measuring all 1-dimensional projections of a2-dimensional object, you can reconstruct the object
By measuring all 1-dimensional projections of a2-dimensional object, you can reconstruct the object
Computed TomographyComputed Tomography
Planar X-Ray Computed Tomography
Images courtesy of Robert McGee, Ford Motor Company
Separates Objects on Different PlanesSeparates Objects on Different Planes