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Medical Physics

Nuclear Medicine Tomographic Imaging Techniques

Dr Roger FultonDepartment of PET & Nuclear Medicine

Royal Prince Alfred HospitalSydney

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 are2-D projections of a 3-D sourcedistribution. Internal structures areobscured by overlying and uderlyingstructures. Different projectionviews can assist, but the personinterpreting must mentally assimilatethe 3-D distribution. Deep structuresmay be obscured in all projectionviews.

The alternative is tomographicimaging.

Planar Images

Ant Post

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Tomographic Image

A 2-D representation of structures lying within aselected plane or depth in a 3-D object.

In nuclear medicine tomography the tomographicimage depicts the source distribution, i.e. thedistribution of radiotracer in a “slice” of tissue.

Medical tomographic systems such as x-raycomputed tomography (CT), positron emissiontomography (PET), and single photon emissioncomputed tomography (SPECT) reconstructimages of selected planes within the object fromprojection data obtained at many angles aroundthe object.

The reconstruction of slices from the projectiondata is performed by a mathematical algorithm.

Reconstruction is possible provided enoughprojections are acquired over an adequate rangeof projection angles.

Reconstruction from Projections – “ComputedTomography”

1917 – Initially conceived by Radon,who was concerned withapplications in gravitational theory.

1956 – Radioastronomers Bracewelland Riddle were the first to useRadon’s theory. They used it toreconstruct microwave emissionsfrom the sun.

1973 – Hounsfield was the first touse Radon’s theory in medicalimaging – first x-ray CT scanner(“CAT” Scanner).

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Backprojection

Linear Superposition of Back Projections(LSBP).

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image True image LSBP

where r is the distance from the source.

Blurring factor

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Taking Fourier transform of both sides,

. x image) (LSBP image) (True

frequency, spatial is where,11

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1 x image) (Trueimage) (LSBP FFF

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Fourier Transform

n x image) (LSBP image) (True FF =

Correcting for 1/r Blurring

Taking the inverse Fourier transform of both sides,

gƒ= image LSBP image TrueWhere is the function in the spatial domain whose Fouriertransform is equal to n in the frequency domain.

Ramp filter

CorrectionFilters†

Trueimage = LSBP ƒ g

†

g

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Ramp Filter

Since the correction filter has asimple ramp shape in frequencyspace, it is often called the “ramp”filter.

It can be seen that the 1/r blurring iscompensated by giving increasingweight to higher spatial frequenciesin the Fourier transform of the LSBPimage that are needed to representfine detail. Intuitively, this makessense since blurring suppresseshigh spatial frequency information.Hence blurring ought to be reducedby selectively amplifying the higherfrequencies.

Maximum spatialfrequency present inimage.

Filtered Backprojection (FBP)

Apply filter in frequency space by taking the Fourier transforms of the scanprofiles (projections) and multiplying each frequency component by a factorproportional to the spatial frequency, orConvolve with the scan profiles in the spatial domain prior tobackprojection

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Filtered Backprojection

In practice, instead of the pureramp filter, a modified ramp filteris used that has a somewhatrounded shape in the frequencydomain. This (i) avoids artifactscaused by sharp spatial frequencycut off, and (ii) avoids excessiveenhancement of high frequencynoise in the image.

Filtered backprojection is the mostcommonly used ECTreconstruction method. Otherreconstruction methods will bediscussed later.

Sampling

The projection is not continuous butconsists of discrete point by point samples.The Sampling Theorem tells us that inorder to recover spatial frequencies in asignal up to the maximum frequency, nmax,the linear sampling distance, d, must be

)2(

1

maxn£d

i.e. the highest spatial frequencycomponent to be recovered from the datamust be sampled at least twice per cycle.

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Angular Sampling

The angular sampling intervalshould provide sampling around theperiphery at approximately the sameintervals as the linear samplingdistance, i.e.

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FORMATION OF PROJECTIONS

Single Photon Emission ComputedTomography

(SPECT)

Rotating gamma camera

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Triple Head SPECT Scanner

Heads rotate 120 degrees toobtain 360 degree angularsampling. Rotation is usually“step-and-shoot” althoughacquisition during continuousrotation is also possible on somesystems. Angular incrementtypically 3 degrees. Imaging timeat each angular position istypically 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 intissue, losing energy by Coulombinteractions with electrons, beforecombining with an electron to formpositronium.

Positronium has a very short half-life (10-7

s) and annihilates almost immediately.

Mass is converted to energy in the form of2 photons (511 keV each) according to E= mc2

Photons are emitted in opposite directionsto conserve momentum. Any residualmomentum of positron results in deviationfrom 180°

Positron Decay

Coincidence Detection

Two detectors oriented at 180 degrees to each other are used to detectannhilation photons. A coincidence circuit detects events that occur within anarrow time interval, typically 12 ns.

Coincidence?

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PET Spatial Resolution

The resolution ultimately obtainable by detection of annihilation photons islimited by two factors.1.The non-collinearity of the annihilation photons. The deviation from 180deg is due to the fact that the positron-electron pair is not completely at restwhen annihilation occurs. The angular spread of this deviation has aGaussian distribution with a FWHM of 0.3 degrees. This translates into aresolution effect of about 2.8mm (FWHM) for detectors 100cm apart.2. The range of the positrons prior to annihilation. This contributes aresolution 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 III1975

ECAT II1977

NeuroECAT1978

ECAT 9311985

ECAT EXACT HR+1995

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PET

8 x 8 crystal elements

8 transaxial slices

Detectors in coincidenceBlock detector

Full ring PET scanner

PET "Block" Detector

Scintillatorarray

PMTs

Histogram

X = A + B - C - DA + B + C + D

Y = A - B + C - DA + 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 unscatteredcoincidences contribute useful data for image formation

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PETSinogram

Lines of Response (LORs)

PET Sinogram

Position

Projectionangle

Sinogram Reconstructed slice

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Whole Body PET

Data acquired as bed moves in discreteincrements under computer control

Emission & transmission acquisitionsinterleaved or simultaneous

AxialFOV

overlap

Bed translation

Normal scan

18FDG PET

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3D Acquisition

2D3D

Removing the interplane septa and accepting coincidences between any pair ofdetector rings provides a big increase in count rate sensitivity (factor of 3 to 5). Thissepta-less acquisition is referred to as “3D” PET. (Note: Reconstruction yields a 3Dreconstruction 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.

2DProjection is a 1-D collectionof parallel ray sumsoriginating from a 2D slice

3DProjection is a 2-Dcollection of parallel raysums originating from a3D object

2D PET vs 3D PET

Rays confined to a singleplane. Reconstruct slice byslice.

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 canbe expressed as a sum of sine and cosine waves propagating in variousdirections across the plane. The amplitudes of these waves are denoted bythe Fourier coefficients F(u,v). Fourier reconstruction relies on the fact thatthe amplitudes of the waves propagating at angle q are equal to the Fouriercoefficients of the projection P(k,q) at the same angle, i.e.

( ) . and tan where,),(),( 221 vukuvkPvuF +±=== -qq

CentralSliceTheorem

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Fourier Reconstruction (continued)

By taking projections at severalangles,and then taking their Fouriertransforms, values of F(u,v) can bedetermined along radial lines as shownat right. The dots represent the locationof estimates of the object’s Fouriertransform.

If sufficient projections are available,the original object can be reconstructedby taking the inverse Fourier transform.

In practice it is difficult to calculate theinverse Fourier transform from radiallysampled data, and the transform isusually first interpolated onto arectangular grid. The interpolation mustbe very accurate, especially at largeradii where sampling is relativelysparse, and suitable interpolationalgorithms are very expensivecomputationally.

Interpolate to rectangular grid

Inverse Fourier Transform

Reconstruction

Iterative Methods

Iterative reconstruction methods attempt to progressively improve theestimate of f(x,y) rather than estimate it by direct inversion of the imagetransform. Such methods allow more realistic modelling of the detectionprocess than the simple line integral we had in filtered backprojection,

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The ability to incorporate accurate models for noise, and other effectssuch as attenuation and scatter, can be a significant advantage. Iterativemethods also have the ability to incorporate a priori information about theimage in the reconstruction process. Such information can be used to helpensure an acceptable reconstruction. For example, since f(x,y) representstracer concentration, it is reasonable to require that successive imageestimates 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 volumeelements (voxels).

2. A model of the detection process that specifies the relationship betweenthe observed projection data and the image.

3. A noise model to define the type of distribution of data elements about theirexpectation value.

4. An objective function that provides a measure of the goodness-of-fit of theimage to the measured data and applies any a priori constraints.

5. An iterative algorithm that can successively improve the image estimate byconverging 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 numberof projection bins Nb at each angle and the number of projection angles Nq. Werepresent the m projection values as p={pj, j=1,2,..,m} where pj is the number ofphotons recorded in the jth projection bin. We represent the n voxel values inthe image as f = {fi, i=1,n}.

The required model can then be written p = Af, where A is the mxn transitionmatrix that gives the probability of a photon emitted from voxel i being detectedin projection bin j. This probability can take into account factors affectingprojection measurements such as projection geometry, detector effciency andlocation, resolution, attenuation, scatter and so on. Hence iterativereconstruction techniques can accurately model the detection process and thuscorrect 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 andtransmission 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 of2 or more projections.

Apply ML-EM to one subsetat a time.

Equivalent results to EM butfaster. Acceleration factor isapproximately equal tonumber of subsets.

PET Summary

Cyclotron

Radioisotope Production

Chemistry

Radiotracer Synthesis

Physiological processTissue uptake

Injectioninto blood

Radioactive DecayPositron emission

PET CameraAcquire projections

ComputerTomographic Reconstruction

18FDG PET Brain Scan

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18FDG PET Brain Scan

coronal

sagittal

transaxial

18FDG – Regional Metabolic Activity

listening seeing

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18FDG – Regional Metabolic Activity

thinking moving

6-[18F]Fluoro-DOPA in Parkinson’s Disease

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microPET 18FDG Rat Study

Further reading:

Sorenson JA, Phelps ME, “Physics in Nuclear Medicine”, Grune andStratton, 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 andTreatment”, Churchill Livingstone, Edinburgh, 1998.

Let’s Play PEThttp://www.crump.ucla.edu/software/lpp/lpphome.html

Uniservehttp://science.uniserve.edu.au/school/curric/stage6/phys/fromqtoq.html#radioactivity