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Medical Physics
Nuclear Medicine
Tomographic Imaging Techniques
Dr Roger Fulton
Department of PET & Nuclear Medicine
Royal Prince Alfred Hospital
Sydney
Email:
rfulton@mail.usyd.edu.au
Lectures:
http://www.physics.usyd.edu.au/ugrad/sphys/medphys.html
Nuclear Medicine Tomography -
“Emission Tomography”
Planar nuclear medicine images are
2-D projections of a 3-D source
distribution. Internal structures are
obscured by overlying and uderlying
structures. Different projection
views can assist, but the person
interpreting must mentally assimilate
the 3-D distribution. Deep structures
may be obscured in all projection
views.
The alternative is tomographic
imaging.
Planar Images
Ant Post
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Tomographic Image
A 2-D representation of structures lying within a
selected plane or depth in a 3-D object.
In nuclear medicine tomography the tomographic
image depicts the source distribution, i.e. the
distribution of radiotracer in a “slice” of tissue.
Medical tomographic systems such as x-ray
computed tomography (CT), positron emission
tomography (PET), and single photon emission
computed tomography (SPECT) reconstruct
images of selected planes within the object from
projection data obtained at many angles around
the object.
The reconstruction of slices from the projection
data is performed by a mathematical algorithm.
Reconstruction is possible provided enough
projections are acquired over an adequate range
of projection angles.
Reconstruction from Projections – “Computed
Tomography”
1917 – Initially conceived by Radon,
who was concerned with
applications in gravitational theory.
1956 – Radioastronomers Bracewell
and Riddle were the first to use
Radon’s theory. They used it to
reconstruct microwave emissions
from the sun.
1973 – Hounsfield was the first to
use Radon’s theory in medical
imaging – first x-ray CT scanner
(“CAT” Scanner).
Ú
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= '),(),'( dyyxfxp q
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Backprojection
Linear Superposition of Back Projections
(LSBP).
˜
¯
ˆ
Á
Ë
Ê
ƒ=
r
1
imageTrueimageLSBP
where r is the distance from the source.
Blurring factor
˜
¯
ˆ
Á
Ë
Ê
ƒ=
r
1
imageTrueimageLSBP
Taking Fourier transform of both sides,
.ximage)(LSBPimage)(True
frequency,spatialiswhere,
11
Since
1
image)(LSBP
image)(True
n
n
n
FF
F
F
FF
=
=˜
¯
ˆ
Á
Ë
Ê
˜
¯
ˆ
Á
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=
r
r
(Convolution in the spatial domain Multiplication in the frequency domain).≡
˜
¯
ˆ
Á
Ë
Ê
=
r
1
ximage)(Trueimage)(LSBP FFF
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Fourier Transform
nximage)(LSBPimage)(True FF =
Correcting for 1/r Blurring
Taking the inverse Fourier transform of both sides,
gƒ= imageLSBPimageTrue
Where is the function in the spatial domain whose Fourier
transform is equal to n in the frequency domain.
Ramp filter
Correction
Filters
†
Trueimage = LSBP ƒ g
†
g
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Ramp Filter
Since the correction filter has a
simple ramp shape in frequency
space, it is often called the “ramp”
filter.
It can be seen that the 1/r blurring is
compensated by giving increasing
weight to higher spatial frequencies
in the Fourier transform of the LSBP
image that are needed to represent
fine detail. Intuitively, this makes
sense since blurring suppresses
high spatial frequency information.
Hence blurring ought to be reduced
by selectively amplifying the higher
frequencies.
Maximum spatial
frequency present in
image.
Filtered Backprojection (FBP)
Apply filter in frequency space by taking the Fourier transforms of the scan
profiles (projections) and multiplying each frequency component by a factor
proportional to the spatial frequency, or
Convolve with the scan profiles in the spatial domain prior to
backprojection
†
g
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Filtered Backprojection
In practice, instead of the pure
ramp filter, a modified ramp filter
is used that has a somewhat
rounded shape in the frequency
domain. This (i) avoids artifacts
caused by sharp spatial frequency
cut off, and (ii) avoids excessive
enhancement of high frequency
noise in the image.
Filtered backprojection is the most
commonly used ECT
reconstruction method. Other
reconstruction methods will be
discussed later.
Sampling
The projection is not continuous but
consists of discrete point by point samples.
The Sampling Theorem tells us that in
order to recover spatial frequencies in a
signal up to the maximum frequency, nmax,
the linear sampling distance, d, must be
)2(
1
maxn
£d
i.e. the highest spatial frequency
component to be recovered from the data
must be sampled at least twice per cycle.
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Angular Sampling
The angular sampling interval
should provide sampling around the
periphery at approximately the same
intervals as the linear sampling
distance, i.e.
.projectioneach
acrossdistancesamplingnearlitheisand
viewoffieldtheofdiametertheiswhere
,
2
ewsangular viofNumber
d
D
d
Dp
=
FORMATION OF PROJECTIONS
Single Photon Emission Computed
Tomography
(SPECT)
Rotating gamma camera
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Triple Head SPECT Scanner
Heads rotate 120 degrees to
obtain 360 degree angular
sampling. Rotation is usually
“step-and-shoot” although
acquisition during continuous
rotation is also possible on some
systems. Angular increment
typically 3 degrees. Imaging time
at each angular position is
typically 20 – 40 s.
SPECT
Sampled volume
FOV
N x N projection images -> N transaxial slices
N typically 64, 128.
Axis of
rotation
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FORMATION OF PROJECTIONS
Positron Emission Tomography
(PET)
b+
g
g
The positron travels a short distance in
tissue, losing energy by Coulomb
interactions with electrons, before
combining with an electron to form
positronium.
Positronium has a very short half-life (10-7
s) and annihilates almost immediately.
Mass is converted to energy in the form of
2 photons (511 keV each) according to E
= mc2
Photons are emitted in opposite directions
to conserve momentum. Any residual
momentum of positron results in deviation
from 180°
Positron Decay
Coincidence Detection
Two detectors oriented at 180 degrees to each other are used to detect
annhilation photons. A coincidence circuit detects events that occur within a
narrow time interval, typically 12 ns.
Coincidence?
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PET Spatial Resolution
The resolution ultimately obtainable by detection of annihilation photons is
limited by two factors.
1.The non-collinearity of the annihilation photons. The deviation from 180
deg is due to the fact that the positron-electron pair is not completely at rest
when annihilation occurs. The angular spread of this deviation has a
Gaussian distribution with a FWHM of 0.3 degrees. This translates into a
resolution effect of about 2.8mm (FWHM) for detectors 100cm apart.
2. The range of the positrons prior to annihilation. This contributes a
resolution effect of up to ~3mm (FWHM).
The combination of these 2 effects imposes a resolution limit of 1.5 – 3.0mm
(FWHM) in PET.
The reconstructed spatial resolution of a typical PET scanner is ~ 4.5mm (FWHM).
PET Spatial Resolution
PET III
1975
ECAT II
1977
NeuroECAT
1978
ECAT 931
1985
ECAT EXACT HR+
1995
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PET
8 x 8 crystal elements
8 transaxial slices
Detectors in coincidence
Block detector
Full ring PET scanner
PET "Block" Detector
Scintillator
array
PMTs
Histogram
X =
A + B - C - D
A + B + C + D
Y =
A - B + C - D
A + B + C + D
B D
A
X
Y
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PET Scanner Construction
PET vs SPECT Acquisition
PET has electronic collimation.
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2-D PET
For N crystal rings we have (2N-1) transaxial reconstruction planes.
N direct planes (N-1) cross planes
Coincidence Event Types
True
Coincidence
Random/Accidental
Coincidence
Multiple
Coincidences
Scattered
Coincidence
¸ ˚ ˚ ˚
Not all coincidence events are desirable. Only true unscattered
coincidences contribute useful data for image formation
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PET
Sinogram
Lines of Response (LORs)
PET Sinogram
Position
Projection
angle
Sinogram Reconstructed slice
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Whole Body PET
Data acquired as bed moves in discrete
increments under computer control
Emission & transmission acquisitions
interleaved or simultaneous
Axial
FOV
overlap
Bed translation
Normal scan
18FDG PET
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3D Acquisition
2D
3D
Removing the interplane septa and accepting coincidences between any pair of
detector rings provides a big increase in count rate sensitivity (factor of 3 to 5). This
septa-less acquisition is referred to as “3D” PET. (Note: Reconstruction yields a 3D
reconstruction volume in both 2D and 3D PET). Benefits are greater sensitivity,
leading to shorter acquisition times or ability to use smaller doses of radiotracer.
Disadvantages: much larger data sets, reconstruction algorithm more complicated,
longer reconstruction times.
2D
Projection is a 1-D collection
of parallel ray sums
originating from a 2D slice
3D
Projection is a 2-D
collection of parallel ray
sums originating from a
3D object
2D PET vs 3D PET
Rays confined to a single
plane. Reconstruct slice by
slice.
Rays intersect multiple reconstruction planes.
Require “fully 3D” reconstruction algorithm.
Ref: Bendriem and Townsend, 1998
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Reconstruction Methods
• Filtered Backprojection (FBP)
• Iterative Methods – Simple Iterative Reconstruction Technique (SIRT)
• Algebraic Methods - Algebraic Reconstruction Technique (ART)
• Iterative Least Squares Methods (ILST)
• Direct Analytical Methods e.g. Fourier Reconstruction
• Iterative Statistical Methods e.g. Maximum Likelihood Expectation
Maximisation (ML-EM)
Fourier Reconstruction
The 2D function f(x,y) representing the unknown radiotracer distribution can
be expressed as a sum of sine and cosine waves propagating in various
directions across the plane. The amplitudes of these waves are denoted by
the Fourier coefficients F(u,v). Fourier reconstruction relies on the fact that
the amplitudes of the waves propagating at angle q are equal to the Fourier
coefficients of the projection P(k,q) at the same angle, i.e.
( ) .andtanwhere,),(),( 221
vukuvkPvuF +±=== -
qq
Central
Slice
Theorem
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Fourier Reconstruction (continued)
By taking projections at several
angles,and then taking their Fourier
transforms, values of F(u,v) can be
determined along radial lines as shown
at right. The dots represent the location
of estimates of the object’s Fourier
transform.
If sufficient projections are available,
the original object can be reconstructed
by taking the inverse Fourier transform.
In practice it is difficult to calculate the
inverse Fourier transform from radially
sampled data, and the transform is
usually first interpolated onto a
rectangular grid. The interpolation must
be very accurate, especially at large
radii where sampling is relatively
sparse, and suitable interpolation
algorithms are very expensive
computationally.
Interpolate to rectangular grid
Inverse Fourier Transform
Reconstruction
Iterative Methods
Iterative reconstruction methods attempt to progressively improve the
estimate of f(x,y) rather than estimate it by direct inversion of the image
transform. Such methods allow more realistic modelling of the detection
process than the simple line integral we had in filtered backprojection,
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= '),(),'( dyyxfxp q
The ability to incorporate accurate models for noise, and other effects
such as attenuation and scatter, can be a significant advantage. Iterative
methods also have the ability to incorporate a priori information about the
image in the reconstruction process. Such information can be used to help
ensure an acceptable reconstruction. For example, since f(x,y) represents
tracer concentration, it is reasonable to require that successive image
estimates are non-negative in every pixel.
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Iterative Algorithm Requirements
1. A representation of the object to be reconstructed as a finite set of volume
elements (voxels).
2. A model of the detection process that specifies the relationship between
the observed projection data and the image.
3. A noise model to define the type of distribution of data elements about their
expectation value.
4. An objective function that provides a measure of the goodness-of-fit of the
image to the measured data and applies any a priori constraints.
5. An iterative algorithm that can successively improve the image estimate by
converging to a solution that maximizes the objective function.
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The Model of the the Detection Process
Let m be the number of projection values, given by the product of the number
of projection bins Nb at each angle and the number of projection angles Nq. We
represent the m projection values as p={pj, j=1,2,..,m} where pj is the number of
photons recorded in the jth projection bin. We represent the n voxel values in
the image as f = {fi, i=1,n}.
The required model can then be written p = Af, where A is the mxn transition
matrix that gives the probability of a photon emitted from voxel i being detected
in projection bin j. This probability can take into account factors affecting
projection measurements such as projection geometry, detector effciency and
location, resolution, attenuation, scatter and so on. Hence iterative
reconstruction techniques can accurately model the detection process and thus
correct for physical effects involved in the detection process.
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Maximum Likelihood Expecation Maximisation (ML-EM)
EM AlgorithmObjective Function
Seeks the most likely solution consistent with the measured projections, i.e.
finds the image f that is most likely to have produced the projections p.
Lang K and Carson R, “EM reconstruction agorithms for emission and
transmission tomography”, J Comput Assist Tomogr, 8(2):306-316, 1984.
ML-EM Algorithm
Accounts for noise much better than FBP but computationally demanding.
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Ordered Subsets EM
Divide the projections into
“subsets” each consisting of
2 or more projections.
Apply ML-EM to one subset
at a time.
Equivalent results to EM but
faster. Acceleration factor is
approximately equal to
number of subsets.
PET Summary
Cyclotron
Radioisotope Production
Chemistry
Radiotracer Synthesis
Physiological process
Tissue uptake
Injection
into blood
Radioactive Decay
Positron emission
PET Camera
Acquire projections
Computer
Tomographic Reconstruction
18FDG PET Brain Scan
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microPET 18FDG Rat Study
Further reading:
Sorenson JA, Phelps ME, “Physics in Nuclear Medicine”, Grune and
Stratton, Orlando, 1987.
Bendriem B, Townsend DW (ed.), “The Theory and Practice of 3D PET”,
Kluwer Academic Publishers, Dordrecht, 1998.
Henkin RE, “Nuclear Medicine”, Mosby, St Louis, 1996.
Sarper RM, “Nuclear Medicine Instrumentation”, CC Thomas, Springfield,
1984.
Murray IPC, Ell P (ed.), “Nuclear Medicine in Clinical Diagnosis and
Treatment”, Churchill Livingstone, Edinburgh, 1998.
Let’s Play PET
http://www.crump.ucla.edu/software/lpp/lpphome.html
Uniserve
http://science.uniserve.edu.au/school/curric/stage6/phys/fromqtoq.html#radi
oactivity
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