BMI I FS05 – Class 4 “Nuclear Imaging: Math” Slide 1

Biomedical Imaging IBiomedical Imaging I

Class 5 – Radionuclide Imaging (PET, SPECT), Part 3: Attenuation

and Scatter Corrections

10/12/05

BMI I FS05 – Class 4 “Nuclear Imaging: Math” Slide 2

Recommended ReadingRecommended Reading

K. Miles, P. Dawson, and M. Blomley (Eds.), Functional Computed Tomography (Isis Medical Media, Oxford, 1997).

R. J. English, SPECT: Single Photon Emission Computed Tomography: A Primer (Society of Nuclear Medicine, Reston, VA, 1995).

M. Reivich and A. Alavi (Eds.), Positron Emission Tomography (A. R. Liss, NY, 1985).

BMI I FS05 – Class 4 “Nuclear Imaging: Math” Slide 3

Characteristics of SPECT & PET imagesCharacteristics of SPECT & PET images

Low spatial resolution

Dose limitations → long acquisition time

Imperfect tissue selectivity

Scatter effects

Low SNR

Dose limitations → poor counting statistics

Attenuation

High CNR

Essentially no signal except from the administered radionuclide

Post-processing operations improve SNR at the expense of spatial resolution

Spatial low pass (“long pass”) filtering

BMI I FS05 – Class 4 “Nuclear Imaging: Math” Slide 4

0 100 200 300 400 500 600 700 800 900 10000

0.2

0.4

0.6

0.8

1

Interplay of spatial resolution and CNRInterplay of spatial resolution and CNR

Ideal case:

– No “bleeding” of signal from one location into the adjacent ones

– Difference between signal arising from adjacent positions determines the contrast, and hence the CNR

Reality:

– Lateral spreading of image information results in lowering of high signal levels and raising of low ones

– The apparent signal difference between adjacent sites is lower than in the ideal case, and so the CNR is lower

BMI I FS05 – Class 4 “Nuclear Imaging: Math” Slide 5

Attenuation CorrectionAttenuation Correction

BMI I FS05 – Class 4 “Nuclear Imaging: Math” Slide 6

Projection OperatorProjection Operator

f(x,y)x

θ

, ,a

ap s c f x y dy

x

s

y

-a

a

BMI I FS05 – Class 4 “Nuclear Imaging: Math” Slide 7

Projection Operator — CT vs. ECTProjection Operator — CT vs. ECT

, ,a

ap s c f x y dx

x–ray CT:

,

0

0

,

, ln ,,

a

ax y dx

a

a

I s cI e

cIp s x y dx

I s

ECT:

,,

,

1) , ,

2) , ,

3) , ,

a

x y

a

a

u v dua

a

a x y

a

p s c x y dx

p s c x y e dx

p s c x y e dx

spatial distribution of attenuation coefficient

spatial distribution of radioactivity

BMI I FS05 – Class 4 “Nuclear Imaging: Math” Slide 8

X-Ray Energy & Contrast X-Ray Energy & Contrast

20 keV 200 keV 2000 keV (= 2MeV)

99mTc γ-ray photons

e--e+ annihilation γ-ray photons

BMI I FS05 – Class 4 “Nuclear Imaging: Math” Slide 9

Other approaches to attenuation correctionOther approaches to attenuation correction

Use co-registered anatomical image (e.g., MRI, x-ray CT) to generate an estimate of the tissue µ at each location, and insert that into Eq. (2) (Slide 7)

Use known-strength γ-emitting standards (e.g., 153Gd (Webb, §2.9.2, p. 79) or 68Ge (§ 2.11.4.1, p. 95)) in conjunction with image data collection, to estimate µ at each tissue location

Iterative image reconstruction algorithms

In “odd-numbered” iterations, treat µ(u,v) as known and fixed, and solve for ρ(x,y)In “even-numbered” iterations, treat ρ(x,y) as known and fixed, and solve for µ(u,v)

Even more elaborate mathematical techniques, e.g.,E. Y. Sidky and X. Pan, “Image reconstruction with a half–detector in single–photon emission computed tomography with nonuniform attenuation,” Optical Engineering 42(9), 2506-2513 (2003).I. Laurette et al., “A three–dimensional ray–driven attenuation, scatter and geometric response correction technique for SPECT in inhomogeneous media,” Physics in Medicine and Biology 45, 3459-3480 (2000).T. Kauppinen et al., “Improvement of brain perfusion SPET using iterative reconstruction with scatter and non–uniform attenuation correction,” European J. Nuclear Medicine 27(9), 1380-1386 (2000).

BMI I FS05 – Class 4 “Nuclear Imaging: Math” Slide 10

Scatter CorrectionScatter Correction

BMI I FS05 – Class 4 “Nuclear Imaging: Math” Slide 11

Origins of scatter artifactsOrigins of scatter artifacts

Accidental coincidences (PET)

Compton scatter events (SPECT and PET)

Can redirect photons within measurement plane, or can redirect photons arising from outside measurement plane into it (assuming ring-shaped detector array)

Spatially and directionally distributed source

Higher photon energy than in x-ray CT means that detector collimation is less effective

More serious problem for area detectors (for whole-body imaging, for example) than for ring-shaped detector arrays

BMI I FS05 – Class 4 “Nuclear Imaging: Math” Slide 12

Inherent DimensionalityInherent Dimensionality

BMI I FS05 – Class 4 “Nuclear Imaging: Math” Slide 13

Scatter correction techniquesScatter correction techniques

Compton scattering (SPECT & PET)

Energy “sub-window” method (Webb, §2.9.2, p. 78)

Extrapolation of scattered “source” strength, from image regions outside the patient’s body, to image regions within it (§2.11.4.2, p. 97)

Accidental coincidences (PET)

True but impractical: if the radius r of the detector ring is varied, the rate of true coincidences is proportional to 1/r, while the accidental coincidence rate is proportional to 1/r2

Accidental coincidence rate for a pair of detector is proportional to the product of the overall count rates for each

By increasing the delay between the timing pulses sent to each detector in a pair, can selectively detect only accidental coincidences