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

Positron Emission Tomography (PET)

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)

d/2

Reconstruction Algorithm 1.25 (in-plane)1.0 (axial)

Factor

dDetector 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 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 Center

1 cm

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)

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:

Spatial Resolution is DefinedAssuming Infinite Statistics

Resolution Does Not Include Effects from Noise,but Image Quality Does...

Effect 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

Low Image Noise High Sensitivity

Sensitivity 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 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 Background

T = Trues R = RandomsS = Scatter

Image Noise Not Determined by Sensitivity Alone!

Even when you subtract the background, statistical noise from the background remains.

Noise Equivalent Count Rate (NECR)

NECR =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 Noise

NEC Properties• Obeys Counting Statistics

• Equals Signal x Contrast

• Phantom Geometry Must Be Defined– Usually 20 cm diameter, 20 cm tall cylinder

σNEC = NEC

T TT +S + 2R

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→ Conσtant

NEC Plateaus as Activity (ρ) Increases!

T ρ

R ρ2

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)

• 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...

• 3-D has Higher NECR at Low Activity• Peak NECR in 2-D > Peak NECR in 3-D (Less Scatter)

20 cm Phantom

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

Object Dependence

“Point-Like” Object100,000 Events

“Uniform” Object1,000,000 Events

Point-Like Objects Reconstruct with Less Noise

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!!!

How Does PET Compare to Other Modalities?

• Parallel Hole Collimator• Pinhole Collimator• Coded Aperture• Compton Camera

Parallel Hole Collimator Properties

Collimator Geometry Defines Acceptance Angle

Typical Values:w= 2 mmL= 30 mmt= 0.25 mm

L

wt

= atanwL⎛ ⎝

⎞ ⎠

Gamma Detector

Collimator

Spatial Resolution

•Resolution Proportional to •Resolution Proportional to Distance from Source

L

wt

R

L

wt d

R

Typical Values:R= 6 mm (@ 5 cm)R= 12 mm (@ 10 cm)

Resolution (R)=2w

L d+L2

⎛ ⎝ ⎜ ⎞

⎠ ⎟

Efficiency

•Efficiency Proportional to 2

•Efficiency Independent of Distance from Source

Typical Values:Efficiency = 0.02%

Efficiency∝ θ2 ≈ w

L⎛ ⎝ ⎜ ⎞

⎠ ⎟ 2

L

wt

L

wt

Pinhole CameraL

d

w Resolution=w

Efficiency∝ w

d⎛ ⎝ ⎜

⎞ ⎠ ⎟ 2

•Compared to Parallel Hole Collimator, Pinhole • Different Resolution / Efficiency Tradeoff• Higher Resolution, but Smaller Field of View

Coded Aperture Camera

Resolution=w

Efficiency∝ n w

d⎛ ⎝ ⎜

⎞ ⎠ ⎟ 2

•Compared to Pinhole Camera, Many (n) Pinholes • Similar Resolution w/ Higher Efficiency

L

w

d

Image Overlap with Coded Apertures

•Removing the Overlap Increases the Noise•Noise Increase Depends on Object

“Point-Like” Object “Uniform” Object

Intrinsic Resolution / Efficiency DependencyL

dw

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.

Compton 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 Difficult

Compton 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 Values

Value Inversely Proportional to Volume of Objectthat the Gamma Could Have Come From?

PETThin Line

CollimatorThin Cone

PinholeThin Cone

CodedAperture

Thin ConesCompton

Cone Surface

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

AcknowledgementsU.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 Tomography2-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

Computed TomographyPlanar X-Ray Computed Tomography

Images courtesy of Robert McGee, Ford Motor Company

Separates Objects on Different Planes