The images in positron emission tomography (PET) represent three dimen- sional dynamic ...... is combined with a proton to form a neutron and a neutrino.
Improving Attenuation Corrections Obtained Using Singles–Mode Transmission Data in Small–Animal PET by Eric Vandervoort B.Sc., University of Guelph, 2000 M.Sc., The University of British Columbia, 2004 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF Doctor of Philosophy in The Faculty of Graduate Studies (Physics)
The University Of British Columbia (Vancouver, Canada) March, 2008 c Eric Vandervoort 2008
Abstract The images in positron emission tomography (PET) represent three dimensional dynamic distributions of biologically interesting molecules labelled with positron emitting radionuclides (radiotracers). Spatial localisation of the radio–tracers is achieved by detecting in coincidence two collinear photons which are emitted when the positron annihilates with an ordinary electron. In order to obtain quantitatively accurate images in PET, it is necessary to correct for the effects of photon attenuation within the subject being imaged. These corrections can be obtained using singles–mode photon transmission scanning. Although suitable for small animal PET, these scans are subject to high amounts of contamination from scattered photons. Currently, no accurate correction exists to account for scatter in these data. The primary purpose of this work was to implement and validate an analytical scatter correction for PET transmission scanning. In order to isolate the effects of scatter, we developed a simulation tool which was validated using experimental transmission data. We then presented an analytical scatter correction for singles–mode transmission data in PET. We compared our scatter correction data with the previously validated simulation data for uniform and non–uniform phantoms and for two different transmission source radionuclides. Our scatter calculation correctly predicted the contribution from scattered photons to the simulated data for all phantoms and both transmission sources. We then applied our scatter correction as part of an iterative reconstruction algorithm for simulated and experimental PET transmission data for uniform and non–uniform phantoms. We also tested our reconstruction and scatter correction procedure using transmission data for several animal studies (mice, rats and primates). For all studies considered, we found that the average reconstructed linear ii
Abstract attenuation coefficients for water or soft–tissue regions of interest agreed with expected values to within 4%. Using a 2.2 GHz processor, the scatter correction required between 6 to 27 minutes of CPU time (without any code optimisation) depending on the phantom size and source used. This extra calculation time does not seem unreasonable considering that, without scatter corrections, errors in the reconstructed attenuation coefficients were between 18 to 45% depending on the phantom size and transmission source used.
iii
Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ii
Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iv
Abstract
List of Tables
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
x
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiii 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1
Basic PET Imaging Principles
. . . . . . . . . . . . . . . . .
1
1.2
Applications of PET . . . . . . . . . . . . . . . . . . . . . . .
2
1.2.1 1.2.2
Oncology . . . . . . . . . . . . . . . . . . . . . . . . . Cardiology . . . . . . . . . . . . . . . . . . . . . . . .
3 5
1.2.3
Neurology
6
1.2.4
Small–Animal PET
1.3
1.4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
Positron Decay . . . . . . . . . . . . . . . . . . . . . . . . . .
10
1.3.1
Positron Range and Photon Acollinearity . . . . . . .
11
1.3.2
The production of PET tracers . . . . . . . . . . . . .
15
Photon Interactions in Matter
. . . . . . . . . . . . . . . . .
15
1.4.1
Photoelectric Absorption . . . . . . . . . . . . . . . .
15
1.4.2
Rayleigh Scattering
. . . . . . . . . . . . . . . . . . .
16
1.4.3
Compton Scattering . . . . . . . . . . . . . . . . . . .
17
1.5
Photon Attenuation . . . . . . . . . . . . . . . . . . . . . . .
18
1.6
PET Instrumentation . . . . . . . . . . . . . . . . . . . . . .
19
1.7
Resolution and the Depth of Interaction . . . . . . . . . . . .
23 iv
Table of Contents 1.8
Data Acquisition in PET . . . . . . . . . . . . . . . . . . . .
26
1.9
Quantitative PET . . . . . . . . . . . . . . . . . . . . . . . .
29
1.9.1
Attenuation Corrections
. . . . . . . . . . . . . . . .
30
1.9.2
Scatter Corrections
. . . . . . . . . . . . . . . . . . .
31
1.9.3
Randoms Corrections . . . . . . . . . . . . . . . . . .
32
1.9.4
Normalisation
. . . . . . . . . . . . . . . . . . . . . .
35
1.9.5
Dead–Time Corrections . . . . . . . . . . . . . . . . .
37
1.9.6
Limits to the Absolute Quantification of PET images
38
1.10 Image Reconstruction Methods . . . . . . . . . . . . . . . . . 1.10.1 Filtered Back Projection Reconstruction
40
. . . . . . .
40
1.10.2 Methods for 3D–PET . . . . . . . . . . . . . . . . . .
42
1.10.3 Iterative Reconstruction Algorithms . . . . . . . . . .
44
1.10.4 Maximum Likelihood-Expectation Maximisation . . .
46
1.10.5 Ordered Subsets -Expectation Maximisation
47
. . . . .
1.10.6 Maximum a Posteriori (MAP) penalised reconstructions
. . . . . . . . . . . . . . . . . . . . . . . . . . .
48
1.11 Purpose of this work . . . . . . . . . . . . . . . . . . . . . . .
51
1.12 Outline of Thesis
. . . . . . . . . . . . . . . . . . . . . . . .
52
. . . . . . . . . . . . . . . . . . . . . . . . .
56
2.1
Attenuation Corrections . . . . . . . . . . . . . . . . . . . . .
56
2.2
Attenuation Corrections using Photon Transmission . . . . .
58
2.2.1
Photon Transmission Post–Processing . . . . . . . . .
61
2.2.2
Reconstruction of Photon Transmission Data . . . . .
63
2.2.3
Segmentation of Attenuation–maps
. . . . . . . . . .
65
2.2.4
Post–injection Photon Transmission . . . . . . . . . .
67
2.3
Attenuation Corrections using X–ray transmission . . . . . .
69
2.4
Segmented MRI–based Attenuation Corrections
. . . . . . .
73
2.5
Scatter–Corrections in PET
. . . . . . . . . . . . . . . . . .
74
2.6
Theoretically–Based Scatter Corrections
2.7
Monte Carlo-Based Scatter Corrections
2.8
2 Literature Review
. . . . . . . . . . .
78
. . . . . . . . . . . .
82
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
v
Table of Contents . . . . . . . . . . . . . . .
85
Methods and Analysis . . . . . . . . . . . . . . . . . . . . . .
86
3.1.1
The microPET R4 Tomograph . . . . . . . . . . . . .
86
3.1.2
The microPET Focus 120 Tomograph . . . . . . . . .
87
3.1.3
Simulations . . . . . . . . . . . . . . . . . . . . . . . .
87
3.1.4
Coincidence–Mode Studies
. . . . . . . . . . . . . . .
89
3.1.5
Singles–Mode Studies . . . . . . . . . . . . . . . . . .
92
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
94
3 Validation of the Simulation Data 3.1
3.2
3.3
3.4
Results 3.2.1
Coincidence–Mode Data
3.2.2
Singles–Mode Data
Discussion
. . . . . . . . . . . . . . . .
94
. . . . . . . . . . . . . . . . . . .
96
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
3.3.1
Coincidence-Mode Data . . . . . . . . . . . . . . . . . 102
3.3.2
Singles-Mode Data
Conclusion
. . . . . . . . . . . . . . . . . . . 105
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4 Development and Validation of the Scatter Correction . . 110 4.1 4.2
The Scatter Correction . . . . . . . . . . . . . . . . . . . . . 110 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.3
Results
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.3.1
Validation of the Efficiency Model . . . . . . . . . . . 118
4.3.2
Influence of the 136.5 keV γ–ray on
4.3.3
Effects of multiple and out of FOV scatter
57 Co
data . . . . 123 . . . . . . 123
4.4
Discussion
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.5
Conclusion
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
5 Development and Validation of the Reconstruction Procedure
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.1
Reconstructions and Scatter Rescaling . . . . . . . . . . . . . 132
5.2
Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
5.3
Experiments
5.4
Results
. . . . . . . . . . . . . . . . . . . . . . . . . . . 135
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.4.1
Simulations . . . . . . . . . . . . . . . . . . . . . . . . 136
5.4.2
Experimental Data
. . . . . . . . . . . . . . . . . . . 145
vi
Table of Contents 5.5
5.6
Discussion
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
5.5.1
Simulations . . . . . . . . . . . . . . . . . . . . . . . . 156
5.5.2
Experimental Data
Conclusion
. . . . . . . . . . . . . . . . . . . 159
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
6 Extension of the Scatter Correction for a Primate Scanner 163 6.1
Experiments
. . . . . . . . . . . . . . . . . . . . . . . . . . . 164
6.2
Reconstructions and Scatter Corrections
6.3
Results
6.4
Discussion
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
6.5
Conclusion
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
. . . . . . . . . . . 165
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
7 Post–Injection Transmission Scanning . . . . . . . . . . . . . 172 7.1
Experiments
7.2
Transmission Data Reconstructions and Correction Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
7.3
Emission Data Reconstructions and Correction Procedures
7.4
Results
7.5
7.6
. . . . . . . . . . . . . . . . . . . . . . . . . . . 172
. 174
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
7.4.1
Reconstructed Attenuation–Maps
7.4.2
Image Quantification
Discussion
. . . . . . . . . . . 176
. . . . . . . . . . . . . . . . . . 176
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
7.5.1
Reconstructed Attenuation–Maps
7.5.2
Image Quantification
. . . . . . . . . . . 181
. . . . . . . . . . . . . . . . . . 183
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
8 Future Work and Conclusions . . . . . . . . . . . . . . . . . . 187 8.1
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
8.2
Comparison of the Proposed Method With Potential Alternatives
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
8.3
Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
8.4
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
vii
List of Tables 1.1
Summary of the properties of some positron–emitting nuclei of interest for PET, based on a similar table in Bailey et al. [1] with updated values from the Evaluated Nuclear Structure Data Files [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2
12
Properties of some of the inorganic scintillator crystals that have been used in PET based on the data provided in the following citations [3, 4, 5]. . . . . . . . . . . . . . . . . . . .
3.1
22
Summary of the source models used for each simulated dataset (emission and transmission) and details concerning the experimental data to which the simulated data were compared. . .
3.2
95
Comparison of the estimated scatter fraction (in percent) for the simulated and experimental transmission data using the 68 Ge
3.3
source. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
Comparison of the estimated scatter fraction (in percent) for the simulated and experimental transmission data using the 57 Co
5.1
sources. . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
Reconstructed attenuation coefficients (in cm −1 ) averaged over all axial slices (excluding edges) of the uniform water cylinder attenuation–maps for the simulated
68 Ge
and
57 Co
transmis-
sion data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
viii
List of Tables 5.2
Reconstructed attenuation coefficients (in cm −1 ) for each of the three ROIs (Teflon, water and air) averaged over all axial slices (excluding edges) of the non–uniform phantoms reconstructed using simulated
68 Ge
and
57 Co
transmission data.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 5.3
Average reconstructed attenuation coefficients (in cm −1 ) in ROIs for the uniform water cylinders and both transmission sources. The data shown here is for the scatter–free simulation data and is compared for different maximum oblique angles used in the single–slice rebinning (SSRB). All the axial sinograms were summed prior to reconstruction to reduce noise.
5.4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
Reconstructed attenuation coefficients (in cm −1 ) averaged over all axial slices (excluding edges) of the uniform water cylinder attenuation–maps for the experimental
68 Ge
and
57 Co
trans-
mission data. . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 5.5
Reconstructed attenuation coefficients (in cm −1 ) for each of the three ROIs (Teflon, water and air) averaged over all axial slices (excluding edges) of the non–uniform phantoms reconstructed using experimental
68 Ge
and 57 Co transmission data.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 7.1
Comparison of total amounts of activity within a circular regions–of–interest (ROIs) for emission images reconstructed using attenuation–maps with different combinations of emission contamination (EC) and scatter corrections (SC). Based on well-counter measurements, we estimate that approximately 0.9 MBq and 6.0 MBq of activity were present within the ROIs for the cylinder and rodent studies, respectively. . . . . 181
ix
List of Figures 1.1
A photograph of the microPET Focus 120 imaging system. Also visible in the photo is the animal bed and a plastic head holder which is used to constrain rats and provide oxygen during the acquisition of PET data. . . . . . . . . . . . . . .
1.2
8
Energy Spectrum (relative number of emitted positrons with a given energy) is shown for the dominant positron decay mode (relative intensity 89%) for
68 Ga.
The spectrum data
is based on theory [6] and is shown courtesy of Dr. Ross Schmidtlein, Sloan–Kettering Institute. Experimental values for the mean (Emean ) and maximum positron energies (Emax β β ) are from Burrows [7]. . . . . . . . . . . . . . . . . . . . . . . . 1.3
11
Simulated data showing the two–dimensional displacement from the origin (upper–plots) and total distance from the point of emission (lower–plots) for the positrons emitted by 18 F
and
gies for
15 O 18 F
in water. The maximum positron kinetic ener-
and
15 O
are 634 keV and 1732 keV, respectively.
The data was generated using Monte–Carlo simulation software [8] and is based on a similar figure in Levin and Hoffman [9]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4
Total mass attenuation coefficients (in units of
cm 2 /g)
13
for wa-
ter as a function of energy between 0.001 MeV and 1.0 MeV. Also included is the individual contribution from the photoelectric effect, coherent and incoherent scattering. Data is from Berger et al. [10]. . . . . . . . . . . . . . . . . . . . . . .
20
x
List of Figures 1.5
Schematic diagram to illustrate a conventional PET detector block based on similar figures in Cherry et al. [3] and Bailey et al. [1].
1.6
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
Schematic diagram, based on a figure appearing in Bailey et al. [1], to illustrate the parallax effect. There is a degradation of the resolution in both the radial direction (a) and axial direction (b) because it is impossible to distinguish between the two annihilation–photon flight paths, shown as the solid and dashed arrows in each figure. PET detector crystals are shown as the grey rectangles. . . . . . . . . . . . . . . . . . .
1.7
25
Schematic diagram to illustrate organisation of conventional PET sinograms. The figures shown in (a) and (c) show data that would be grouped into a particular projection angle in the sinogram data, with and without increased effective radial sampling. The figures shown in (b) and (d) show data that would be grouped into a particular oblique angle in the sinogram data, with and without increased effective axial sam-
1.8
pling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Schematic diagram to illustrate 2D (a) and 3D (b) data ac-
27
quisition in PET. The figures represent the arrangement of the detectors in the axial direction. The detector rings are shown in grey and collimators are shown in black. LORs are shown as the thin lines connecting the detectors. The acquisition shown in (a) allows for both direct and cross planes to contribute data. While figure (b) illustrates fully 3D data acquisition (only selected LORs are shown for clarity). This figure was adapted from a similar illustration in Cherry et al. [3].
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
xi
List of Figures 1.9
Schematic diagram to illustrate the process of photon attenuation as it applies to PET coincidence measurements. A pair ~ passing of annihilation photons are emitted from position B through an arbitrary attenuating medium µ(~r) to be detected ~ and C ~ which are operated in the two detectors at positions A in coincidence. . . . . . . . . . . . . . . . . . . . . . . . . . .
31
1.10 Schematic diagram, based on a figure appearing in Zaidi and Koral [11], to illustrate the effects of scatter within the object being imaged (a) and from within the crystals (b). The paths of annihilation–photons are shown as solid arrows and the LORs to which they are assigned are shown as dashed lines. .
33
1.11 A schematic diagram of the line integral data acquisition model is shown in figure (a).
Figure (b) shows the back
projection of this data. Intersection of back-projected lines through the object correspond to possible source locations. .
42
1.12 A schematic diagram to illustrate the truncated portion of the total activity viewed in an oblique sinogram . . . . . . . . 2.1
43
Comparison of reconstructed images (FBP) for simulated PET data for a 30 mm radius water cylinder with uniform activity concentration. . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
xii
List of Figures 2.2
Comparison of three different transmission geometries which have been employed for cylindrical PET detector designs. The figures (a) (b) and (c) respectively show: a stationary ring of positron emitting activity (the PET scanner is operated in coincidence–mode in this case); rotating rods of positron emitting activity (coincidence–mode); and a rotating source where the detectors are operated in singles–mode, that is, “coincidence”–events are made using the known source position and the recorded photons in the side of the detector ring opposite the source. The grey–scale image in the centre represents a possible attenuation distribution (µ–map) for the thorax. Examples of photon paths are shown as the solid arrows while the directions of source motion (if applicable) are shown as dashed arrows. This illustration is based on a similar figure appearing in Bailey et al. [12]. . . . . . . . . . .
2.3
61
Comparison of linear attenuation coefficient (µ) and mass attenuation coefficient (µ/ρ) for several materials of biological interest in the energy range applicable to PET/CT data. Data shown was compiled by Hubbell and Seltzer [13]. Note that, in the lower figure, the data for lung and soft tissue are very similar and cannot be clearly distinguished. . . . . . . .
2.4
72
Simulated data showing the energy (left–plot) and radial sinogram profile (right–plot) for scattered and unscattered photons in PET emission data. The simulation modelled a 30 mm radius water cylinder, a positron emitting line source centred radially in a scanner based on the microPET R4. Data are shown separately for the total events (scatter+unscattered), unscattered photons and scattered photons. The data were generated using Monte–Carlo simulation software [8].
. . . .
75
xiii
List of Figures 2.5
Schematic diagram to illustrate the single–scatter simulation scatter correction [14] as given by equation (2.5). The correction uses estimates of the distribution of attenuation (µ– map) and activity (λ–map) to compute the contribution from single–scattered photons. For the detector pair labelled A and B, one possible path for scattered photons is indicated by the dashed lines in the diagram. The line–of–response to which the scatter events would be assigned is shown as a solid line.
3.1
79
Graphics showing the objects included in the GATE simulations for the line sources (left image), the scatter phantom (centre image) and the transmission scan using the 30 mm radius cylinder (right image). The simulated scanner model shown here is the microPET R4. . . . . . . . . . . . . . . . .
3.2
90
Comparison of experimental and β + simulation sinograms (top images) for the line source data summed over all sinograms with ring difference zero and ±1. Middle plots show profiles through the simulated and experimental data for two different angular views. Lower plots show the same profile data on a logarithmic scale. . . . . . . . . . . . . . . . . . . .
3.3
Comparison of experimental and
β+
97
simulation reconstruc-
tions (top images) for the line source data. Middle plots show the horizontal and vertical full-widths at half–maximum (FWHM) and lower plots show full–widths at one–tenth maximum (FWTM) for the reconstructed data. . . . . . . . . . . 3.4
98
Comparison of simulated and experimental profiles, summed over all sinograms with ring difference zero and ±1, for the scatter phantom data. Middle plots show profiles through the simulated and experimental data for two different angular views. The bottom plot shows the estimated scatter fractions for each angular view for the experimental and all simulated data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
xiv
List of Figures 3.5
Comparison of experimental and simulated effective ACFs for the R4 (left–hand plots) and Focus 120 (right–hand plots). Data shown is for the
3.6
68
Ge transmission source.
. . . . . . . . . . . . . . . . 100
Comparison of experimental and simulated effective ACFs for the R4 (left–hand plots) and Focus 120 (right–hand plots). Data shown is for the
4.1
57
Co transmission source.
. . . . . . . . . . . . . . . . 101
Schematic diagram illustrating the assumed geometry used to compute the distribution of scattered and unscattered photons given by equations (4.1) to (4.3). One possible path for scattered photons is indicated by the thick solid arrows in each diagram. The unscattered path is shown as a dashed arrow. Figures (a) and (b) show the two source positions, corresponding to i = 1 and i = 2 in the equations, which contribute to each line–of–response defined by the crystal pair shown in the diagram. . . . . . . . . . . . . . . . . . . . . . . 112
4.2
Average radial profiles for the simulated and calculated transmission data (top–row) and single–scatter sinogram data (bottom– row). Data shown is for the uniform water cylinders using the 68
4.3
Ge transmission source.
. . . . . . . . . . . . . . . . . . . . . 118
Average radial profiles for the simulated and calculated transmission data (top–row) and single–scatter sinogram data (bottom–row) for the
4.4
57
Co transmission source. . . . . . . . . . . . . . . . . . . 119
Comparison of the simulated and calculated transmission sinogram data (top-row), transmission profiles (middle-row) and single-scatter sinogram data (bottom-row) for the non-uniform phantom using the
68
Ge transmission source. The profile data in the plots have
been summed over the dashed and dotted lines shown in the sinogram images to reduce noise.
. . . . . . . . . . . . . . . . . . . 120
xv
List of Figures 4.5
Comparison of the simulated and calculated transmission sinogram data (top-row), transmission profiles (middle-row) and single-scatter sinogram data (bottom-row) for the non–uniform phantom using the
57
Co transmission source. The profile data in the plots have
been summed over the dashed and dotted lines shown in the sinogram images to reduce noise. . . . . . . . . . . . . . . . . . . . . 121
4.6
Percent scatter fractions per sinogram for the simulated and calculated single–scatter sinogram data. Data shown is for all three uniform and the non–uniform phantoms and for both transmission sources. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.7
Comparison of the intrinsic efficiency estimated using simulated point source data with two different efficiency models one which uses the energy absorption coefficients µ en (left– hand plot) and one which used the linear attenuation coefficients µ (right–hand plot).
4.8
. . . . . . . . . . . . . . . . . . . 123
Average radial profiles for the experimental, simulated and analytically calculated transmission data (top–row). The simulated and calculated total–scatter sinogram data are shown in the bottom– row. Data shown are for the uniform water cylinders using the 68 Ge transmission source. . . . . . . . . . . . . . . . . . . . . . . . . 126
4.9
Average radial profiles for the experimental, simulated and analytically calculated transmission data (top–row). The simulated and calculated total–scatter sinogram data are shown in the bottom– row. Data shown are for the uniform water cylinders using the 57 Co transmission source. . . . . . . . . . . . . . . . . . . . . . . . . 127
5.1
Slices through the µ–map images with (upper images) and without scatter corrections (middle images) for the uniform water cylinders reconstructed using simulated
68 Ge
transmis-
sion data. Profiles through the reconstructed data are shown in the lower plots.
. . . . . . . . . . . . . . . . . . . . . . . . 138
xvi
List of Figures 5.2
Slices through the µ–map images with (upper images) and without scatter corrections (middle images) for the uniform water cylinders reconstructed using simulated
57 Co
transmis-
sion data. Profiles through the reconstructed data are shown in the lower plots. 5.3
. . . . . . . . . . . . . . . . . . . . . . . . 139
Average value of attenuation coefficients (in cm −1 ) in ROIs for each axial slice of the uniform water cylinders reconstructed using simulated 68 Ge (top–row) and 57 Co (bottom–row) transmission data. . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
5.4
Slices through the µ–map images with and without scatter corrections (upper images) for the non–uniform phantom reconstructed using simulated
68 Ge
data. Profiles through the
reconstructed data are shown in the lower plots. 5.5
. . . . . . . 141
Slices through the µ–map images with and without scatter corrections (upper images) for the non–uniform phantom reconstructed using simulated
57 Co
data. Profiles through the
reconstructed data are shown in the lower plots. 5.6
Average attenuation coefficients (in cm −1 ) for three ROIs (Teflon, water and air) for each axial slice of the non–uniform phantom reconstructed using simulated 57 Co
5.7
. . . . . . . 142
68 Ge
(top–row) and
(bottom–row) transmission data. . . . . . . . . . . . . . 143
Top rows show images for three of the reconstruction methods applied to the analytically calculated scatter–free data. The images shown used blank data with an average of 500 counts per LOR. Lower plots show average attenuation coefficients (in cm−1 ) in ROIs for data with different levels of statistical noise. Data shown is for the 30 mm radius uniform water cylinder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
xvii
List of Figures 5.8
Slices through the µ–map images with (upper images) and without scatter corrections (middle images) for the uniform water cylinders reconstructed using
68 Ge
transmission data.
Profiles through the reconstructed data are shown in the lower plots. Note that these data were reconstructed using experimental transmission data, unlike those shown in figure 5.1 which were reconstructed from simulated data. The half circular structure beneath each cylinder is the animal bed. . . . . 148 5.9
Slices through the µ–map images with (upper images) and without scatter corrections (middle images) for the uniform water cylinders reconstructed using
57 Co
transmission data.
Profiles through the reconstructed data are shown in the lower plots. Note that these data were reconstructed using experimental transmission data, unlike those shown in figure 5.2 which were reconstructed from simulated data. The half circular structure beneath each cylinder is the animal bed. . . . . 149 5.10 Average value of attenuation coefficients (in cm −1 ) in ROIs for each axial slice of the uniform water cylinders reconstructed using experimental 68 Ge (top–row) and 57 Co (bottom–row) transmission data. . . . . . . . . . . . . . . . . . . . . . . . . 150 5.11 Slices through the µ–map images with and without scatter corrections (upper images) for the non–uniform phantom reconstructed using experimental
68 Ge
data. Profiles through
the reconstructed data are shown in the lower plots. . . . . . 151 5.12 Slices through the µ–map images with and without scatter corrections (upper images) for the non–uniform phantom reconstructed using experimental
57 Co
data. Profiles through
the reconstructed data are shown in the lower plots. . . . . . 152 5.13 Average attenuation coefficients (in cm −1 ) for three ROIs (Teflon, water and air) for each axial slice of the non–uniform phantom reconstructed using experimental and
57 Co
68 Ge
(top–row)
(bottom–row) transmission data. . . . . . . . . . . . 153
xviii
List of Figures 5.14 Slices through the µ–map images with (upper images) and without scatter corrections (middle images) for the rat study reconstructed using experimental
57 Co
data. The average
value and standard deviation for all voxels within an ROI (the boundaries of which are indicated by the dashed lines in the images) are shown below each image. The lower plots show µ–value histograms for the reconstructed data with and without scatter corrections (lower left and right–hand plots, respectively). Please refer to section 5.5.2 for a detailed description of each of the objects visible in the image. . . . . . . 154 5.15 Slices through the µ–map images with (upper images) and without scatter corrections (middle images) for the mouse study reconstructed using experimental
57 Co
data. The aver-
age value and standard deviation for all voxels within an ROI (the boundaries of which are indicated by the dashed lines in the images) are shown below each image. The lower plots show µ–value histograms for the reconstructed data with and without scatter corrections (lower left and right–hand plots, respectively). Please refer to section 5.5.2 for a detailed description of each of the objects visible in the image. . . . . . . 155 6.1
Slices through the reconstructed µ–map images with (upper images) and without scatter corrections (middle images) for the water cylinder (50 mm radius) reconstructed using experimental transmission data acquired using a
57 Co
transmission
source. Profiles through the reconstructed data (averaged over 10 axial planes), the results of an axial ROI analysis (for the ROI shown in the transverse images), and the µ–value histograms for both µ–maps are shown in the lower plots. For these data, the cylinder was centred vertically in the scanner and rests above the animal–bed (the half–circular structure visible below the cylinder).
. . . . . . . . . . . . . . . . . . . 167
xix
List of Figures 6.2
Slices through the reconstructed µ–map images with (upper images) and without scatter corrections (middle images) for the shifted uniform water cylinder (50 mm radius) reconstructed using experimental transmission data acquired using a
57 Co
transmission source. Profiles through the recon-
structed data (averaged over 10 axial planes), the results of an axial ROI analysis (for the ROI shown in the transverse images), and the µ–value histograms for both µ–maps are shown in the lower plots. For these data, the cylinder has been shifted by 27 mm in the vertical direction. . . . . . . . . 168 6.3
Slices through the reconstructed µ–map images with (upper images) and without scatter corrections (middle images) for the primate transmission scan acquired using the microPET Focus 220 with a
57 Co
transmission source. Profiles through
the reconstructed data (averaged over 3 axial planes) and the µ–value histogram for both µ–maps are shown in the lower plots. The profile shown passes through the skull, brain, the sinus and the monkey’s snout. Note the improved contrast between tissue and sinus for the With SC µ-maps relative to No SC. 7.1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
Comparison of transverse attenuation–map images (averaged over 3 axial planes) reconstructed using 57 Co
68 Ge
(top–row) and
(centre–row) transmission data. The lower–row shows
µ–value histograms corresponding to the 68 Ge, 57 Co and 57 Co attenuation–maps which have been rescaled for 511 keV photons. Data shown are for the line source attached to a water cylinder and demonstrates the influence of emission contamination (EC) and scatter corrections (SC). . . . . . . . . . . . 177
xx
List of Figures 7.2
Comparison of transverse attenuation–map images (averaged over 3 axial planes) reconstructed using 57 Co
68 Ge
(top–row) and
(centre–row) transmission data. The lower–row shows
µ–value histograms corresponding to the 68 Ge, 57 Co and 57 Co attenuation–maps which have been rescaled for 511 keV photons. Data shown are for the rodent experiment and demonstrates the influence of emission contamination (EC) and scatter corrections (SC). These images correspond to a transverse slice of the rat’s torso and include parts of its lungs and a circular cross–section of the external tumour on the left hand side of the image. . . . . . . . . . . . . . . . . . . . . . . . . . 178 7.3
Comparison of transverse activity–concentration images (averaged over 3 axial planes) reconstructed using attenuation and scatter corrections computed using the attenuation–maps corresponding to the
68 Ge
(top–row) and
57 Co
(centre–row)
transmission data. Profile data through the reconstructed activity images are also shown in the bottom–row plots. Data 7.4
shown are for the line source attached to a water cylinder. . . 179 Comparison of transverse activity–concentration images (averaged over 3 axial planes) reconstructed using attenuation and scatter corrections computed using the attenuation-maps corresponding to the
68 Ge
(top-row) and
57 Co
(centre-row)
transmission data. Profile data through the reconstructed activity are also shown in the bottom-row plots. Data shown are for the rodent study with a simulated external tumour. . . 180
xxi
List of Figures 8.1
Comparison of transverse attenuation images (top–row) and activity concentration images (second–row) for a water cylinder filled with a uniform concentration of
18 F
activity. Data
were averaged over all axial planes to reduce noise and illustrate the systematic effects of scatter in the transmission data. The third and forth rows show profiles through the attenuation and activity images, respectively. Note that the small regions of lower attenuation and activity at the top of the images correspond to an air bubble in the cylinder. . . . . 191 8.2
Transverse attenuation images (top–row) and profiles (bottom– row) for a central slice of the 45 mm radius water cylinder using
68 Ge
transmission data. Data was reconstructed using
the automated reconstruction–based segmentation method of Nuyts et al. [15]. . . . . . . . . . . . . . . . . . . . . . . . . . 193 8.3
Comparison of the experimental and analytically calculated (model) emission contamination sinograms (summed over all axial planes) for the cylinder study using the
68 Ge
transmis-
sion acquisition settings. Profiles are shown in the lower plots, summed between the dashed and dotted lines shown on the sinograms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 8.4
Comparison of µ-value histograms for reconstructed attenuationmaps corrected using either the measured or analytically calculated emission contamination sinograms for the cylinder study. Data shown has also been corrected for scatter and was acquired using the
68 Ge
transmission acquisition settings. 197
xxii
Acknowledgements I would first like to thank my supervisor, Vesna Sossi, for all of her support and help during my PhD. She has provided me thoughtful guidance and advice for the project and has helped set me along the path for my future career. She has encouraged me to pursue my own ideas and has been absolutely supportive at every turn. She is a great role–model and I can’t thank her enough! I would also like to thank my supervisory committee, Anna Celler, David Measday, Alex MacKay, for their time, support and patience. Their guidance for this work has been extremely valuable and I appreciate all of their efforts. Thanks also to my external examiner and my university examiners: Dr. Chris Waltham and Dr. Chris Orvig. I look forward to your input on my thesis. My thanks also go out to all the current and former members of the UBC PET imaging group. In particular, I’d like to thank Marie-Laure Camborde, Siobhan McCormick and Rick Kornelsen for helping me with experiments and helping me understand how the microPET works; Ste´ phan Blinder for all his computing help; Barry Pointon for his valuable discussions and nerdy jokes; and Katie Dinelle and Sarah Lidstone for their assistance around the lab and for being good friends. This work could not have been completed without the financial support provided by NSERC, the UBC graduate fellowship and my girlfriend. I also appreciate the encouragement and helpful suggestions provided by Danny Newport at Siemens Preclinical Solutions. The simulation work was also greatly assisted by S´ebastien Jan and many other members of the GATE users group. I would like to also thank my parents, whose love and support have carried me through all these years of schooling. Special thanks also go out to Ally, my partner and my best friend.
xxiii
Chapter 1
Introduction 1.1
Basic PET Imaging Principles
Positron emission tomography (PET) is a non–invasive diagnostic medical imaging technique which provides information about physiology and biological function in living organisms. In PET, the subject (either a patient or an experimental animal) is administered a compound, called a radio– pharmaceutical or radio–tracer, which consists of a biologically interesting molecule labelled with a positron–emitting radioactive atom. PET images represent three-dimensional (and often dynamically changing) distributions of the radio–labelled molecules in the body. The radio–tracers are ingested, inhaled or administered to the patient intravenously and then selectively metabolised or otherwise taken up by organs or organ systems in the body. The distribution of radio–pharmaceutical compounds in a living subject provides clinicians and medical researchers with a non–invasive way to monitor metabolic pathways and observe cellular processes in vivo as well as a way to investigate an organ’s function and changes in metabolic activity due to disease. The primary function of the hardware associated with a PET imaging system is to detect the radiation emitted in the process of positron annihilation. As the radioactive atoms that label the radio–tracer decay, they emit energetic positrons (the anti–particle of the electron). The positrons travel a short distance through tissue and then annihilate with electrons. The mass of the two particles is converted into photon energy. The most probable type of photon emission that can occur is one in which two γ–rays, referred to as annihilation photons, are emitted back–to–back (i.e. they are emitted with directions that are approximately 180 o with respect to each 1
Chapter 1. Introduction other). Each annihilation photon has an energy equal to the rest mass of an electron (511 keV). The PET scanner consists of an array of γ–ray detectors, usually taking the form of discrete scintillation crystals arranged in fixed cylindrical rings. These detectors are operated in coincidence, meaning that an event is only recorded if two different crystals each detect a photon “simultaneously”, that is, within a small time interval (on the order of several nanoseconds) constrained by the timing resolution of the imaging system. The images in PET are obtained using the fact that annihilation interactions and, therefore, the approximate location of the original radioactive atom, lies somewhere along the line–of–response (LOR), connecting the pair of detectors associated with the coincidence event. The process of estimating the spatial distribution of radioactive atoms from these events is known as tomography or image reconstruction.
1.2
Applications of PET
The radio–tracers used in PET are developed by first identifying a biological process and then synthesising a positron labelled molecule with which the process can be visualised and quantified. Ideally the labelled molecules only take part in a small number of steps in the biological process, since the initial tracer cannot be distinguished from its reaction products using PET imaging. The most important and commonly–used PET tracer is the
18 F–labelled
glucose analog fluoro–deoxy–glucose ([ 18 F]DG) which was
introduced in 1977 [16] and imaged using conventional PET tomography shortly afterwards in 1979 [17]. FDG allows for in vivo imaging of the initial stages of glucose metabolism. Unlike glucose, FDG is not fully metabolised, rather it is taken into the cell and phosphorylated by hexokinase (the first reaction in glycolysis, the initial anaerobic stage of cellular respiration) at which point further metabolic processes cease. The phosphorylated FDG molecules remain trapped in the intracellular space providing a record of glycolysis in cells throughout the body [18]. The process requires that patients be imaged 45 to 60 minutes after the injection for equilibrium to be reached. Currently, FDG is the most commonly used clinical PET tracer 2
Chapter 1. Introduction due to: (a) the general usefulness of the glycolysis metabolic pathway for monitoring disease; (b) the fact that it can be synthesised with an extremely high radiochemical yield [19]; and (c) due to favourable imaging properties of
18 F
(e.g. relatively long half–life and short positron range as shown in
table 1.1). In addition to FDG, more than 500 PET tracers have been developed and consist of various labelled enzyme and transporter substrates, ligands for receptor systems, hormones, antibodies, peptides, drugs (both medical and illicit), and oligonucleotides [20]. If the tracer reaches an equilibrium state (such as in FDG imaging) static images can be obtained which represent the equilibrium distribution of radio–tracer within the subject. Alternatively, a series of PET images can be obtained over time. Dynamic image data can be combined with a specific model which describes the uptake and washout of the tracer molecule within different tissue types or cellular environments. For PET tracers which are administered intravenously, tracer kinetic modelling usually also requires a measure of the concentration of tracer in the plasma (derived from arterial sampling or from the image itself). The parameters of a properly constructed model can be estimated by fitting to the dynamic image data and reflect the physiological or biochemical rate constants which are often of greater interest to clinicians and researchers than static images.
1.2.1
Oncology
The most common clinical application for PET is in oncology. It has long been known [21] that cancerous cells metabolise glucose at a much faster rate than other tissue types. The growth of a tumour is accompanied with a loss of efficient production of adenosine triphosphate (ATP) in the citric acid cycle (the aerobic portion of cellular respiration) in individual cancerous cells. The energy required for the rapid replication of tumour cells is then supplied by ATP generated from glycolysis only. As a result, 19 times more glucose consumption is required to generate the same amount of ATP in cancerous cells as compared to normal tissue [18], leading to extremely high concentrations of FDG in tumours (i.e. the tumours appear as extremely
3
Chapter 1. Introduction “bright” spots in FDG imaging). The accuracy of a diagnostic test, such as the use of imaging to identify a tumour, can be characterised by its sensitivity and specificity. The sensitivity is the proportion of patients correctly identified as being in a diseased state divided by the total number of patients identified as diseased including false positives. Specificity is a complementary ratio representing the proportion of patients correctly identified as not being diseased over the total number of patients identified as not diseased including false negatives. The literature regarding the specificity and sensitivity of PET as a tool for diagnosis, staging (the classification of a tumour based on its extent) and restaging (reclassifying a tumour following treatment) in oncology have recently been reviewed [18, 22] for solitary pulmonary nodules, lung cancer, colorectal cancer, melanoma, lymphoma, breast cancer, and other cancers. In the exhaustive review of Gamghir et al. [22], the average sensitivity and specificity across all oncology applications of FDG PET were estimated at 84% (based on 18,402 patient studies) and 88% (based on 14,264 patient studies), respectively. Only small variations in the sensitivity (84–87%) and specificity (88–93%) were observed. In cases where comparisons were available, PET was also found to provide better diagnostic accuracy than X–ray computed tomography (CT) or magnetic resonance imaging (MRI) [18] and result in treatment changes for 20–40% of the oncology patients [20]. In light of this evidence, the U.S. Food and Drug Administration (FDA) has approved FDG-PET for the imaging of all cancers and the Health Care Finance Administration for Medicare now provides reimbursement for diagnosing, staging, and restaging lung cancer, colorectal cancer, lymphoma, melanoma, head and neck cancer, and esophageal cancer. In Canada the move to widespread use of PET for cancer diagnosis and staging has been slower (especially in Ontario), due to provincial differences in funding levels and the lower availability of scanners and cyclotrons [23]. The development of new tracers which target specific biological properties of cancer cells is an extremely active field of research. One of the most promising new tracers for oncology is
18 F
fluoro-thymidine (FLT) which
can be used to image cell replication and proliferation of tumours in vivo 4
Chapter 1. Introduction [24]. Cancer cells exhibit higher proliferation rates and consequently increased DNA replication and up-regulation of thymidine transport. FLT– PET imaging has a number of advantages over FDG for monitoring the response to therapy, since glucose metabolism may increase as part of the normal inflammatory response in both normal and surviving cancerous tissue following therapy (especially for chemo–therapy). FLT also provides better sensitivity for detecting brain tumours because of the relatively high rate of glucose metabolism for all brain tissue but a lack of significant neuronal cell divisions [25]. Another important development in oncological PET is the emergence of combined PET/CT imaging devices [26]. The combination of molecular and anatomic imaging in one scanner has many advantages including: (a) biological and anatomical imaging which be performed in one examination; (b) simplified co–registration of the PET and CT images, since the two scans are acquired close to the same time and space; and (c) improved localisation of the functional information from PET to the anatomical structures from CT especially for regions which are difficult to image using PET alone, such as the head, neck, mediastinum and post-surgical abdomen [18]. These advantages have led to the widely reported estimate that PET/CT now accounts for 80% of current PET sales in the United States. It is anticipated that PET/CT will dramatically change the planning of radiation therapy and the monitoring of surgical, medical, and radiation treatments [27].
1.2.2
Cardiology
One of the earliest applications of tomographic PET [28], recognised the usefulness of positron emitting nuclei to monitor cardiac metabolism and perfusion. Using an anaesthetised dog, Phelps et al. [28] monitored the equilibrium distribution of in the myocardium using ing
11 CO–labelled
15 O
labelled water in tissue, ammonium uptake
13 N–ammonia
and general distribution of blood us-
hemoglobin. Some of the radio–tracers which have been
used in cardiac PET are: FDG to measure myocardial glucose utilisation; 82 Rb,
H2 15 O and
13 N–ammonia
which provide measurements of blood–flow
5
Chapter 1. Introduction and perfusion of the myocardium; and
18 F–FTHA
to monitor fatty acid
utilisation [29]. One of the primary advantages of PET imaging for the heart is its ability to distinguish between reversible and irreversible damage to myocardial tissue following coronary artery disease. This is achieved by comparing patterns of myocardial blood flow (e.g. using or
82 Rb)
13 N–ammonia
and glucose utilisation (using FDG) in images of the same pa-
tient [30]. Reduced blood flow with good FDG uptake in the same region of the myocardium is indicative of reversibly damaged myocardium, while lower rates for both metabolic processes indicates irreversibly damaged scar tissue. In spite of its proved usefulness, cardiac PET has yet to become commonplace in the clinics due to the higher availability and/or lower cost of other diagnostic imaging techniques which can also provide measurements of myocardial viability such as MRI, single–photon computed tomography (SPECT) and ultrasound [29].
1.2.3
Neurology
PET has been used in neurology to monitor and quantify neuroreceptor binding, cerebral glucose metabolism, and blood flow in degenerative brain disorders [29]. Although Alzheimer’s disease (AD) is the most common form of dementia, it can only be diagnosed with certainty by post–mortem examination of brain tissue. Histologically, AD is characterised by the deposition of amyloid–beta plaques and neurofibrillary tangles, neuronal loss and atrophy [31]. FDG–PET provides early diagnosis of Alzheimer’s, based on characteristic reduced metabolism in the parietal cortex [32]. It has also been shown that PET can differentiate AD from other causes of dementia and from the normal effects of ageing with a sensitivity of 94% and specificity of 73% [33]. The second most common neurodegenerative disorder is Parkinson’s disease (PD), in which a progressive loss of pigmented dopaminergic neurons in the substantia nigra and other brain stem nuclei occurs [31]. This neuronal loss leads to a depletion of the neurotransmitter dopamine in the striatum particularly within the putamen. PD is characterised by motor impairment
6
Chapter 1. Introduction (resting tremor, rigidity, balance problems and slowed or otherwise impaired ability to adjust the body’s position), autonomic nervous dysfunction and other cognitive and psychiatric features. The progression and severity of PD can be visualised using tracers associated with dopamine binding and storage such as (18 F)–DOPA (an analog of the dopamine precursor LDOPA) [34]. (18 F)–DOPA can be used to estimate dopaminergic cell density [35] and its decline as PD disease progresses [36]. One of the disadvantages of ( 18 F)– DOPA is the complexity of the kinetics and metabolism of this radio–tracer [37]. Ligands that bind to the dopamine transporter (DAT) have been used to quantify pre-synaptic dopaminergic degeneration in PD [38] PET has also provided insights into the underlying pathology and physiology of PD [31] using tracers such as
11 C–raclopride
and
11 C–SCH23390.
These tracers are
antagonists for dopamine D2 and D1 receptors, respectively (antagonists occupy specific receptor sites without triggering the normal cellular response associated with the corresponding agonist or endogenous ligand).
1.2.4
Small–Animal PET
The research presented in this thesis is concerned primarily with the application of PET to small animal studies. Mice have become the preferred research animal for use in molecular biology due to their short life span and rapid rate of reproduction. In addition, many human diseases can be approximated by genetically modifying mice due to the wide number of similarities between the human and mouse genomes (fully sequenced in 2002 [39]). In the field of neuroscience, rats have historically been preferred over mice because of the relatively larger size of the rat brain which more easily permits surgical modification [40]. Until recently, the use of PET for the imaging of small animals such as rats and mice has been limited by technological difficulties, primarily, insufficient resolution and sensitivity [41]. Although the use of smaller crystals and a reduced detector diameter are obvious methods to improve spatial resolution and sensitivity, implementing these changes has required new approaches to scanner design and technology [42]. Researchers in small animal PET have, for example, helped to pioneer imaging using new
7
Chapter 1. Introduction types of photon detectors (LSO [43] and multi-wire proportional chambers [44]) and novel scintillation light detectors (avalanche photo–diodes [45] and position–sensitive photo–multiplier tubes [43]). Figure 1.1 shows a photograph of the microPET Focus 120 which was used throughout this thesis. The microPET Focus 120 scanner is a third-generation animal PET scanner dedicated to rodent imaging with spatial resolution characterised by a 1.1 to 1.5 mm full width at half maximum (FWHM) and absolute sensitivities between 7.0 to 4.0% in the centre of the field–of–view [46].
Figure 1.1: A photograph of the microPET Focus 120 imaging system. Also visible in the photo is the animal bed and a plastic head holder which is used to constrain rats and provide oxygen during the acquisition of PET data. There are a number of considerations which are unique to small animal PET which warrant some discussion. During a conventional human PET scan it is necessary for the patient to remain motionless during image acquisition (the duration of which is typically between 20 and 60 minutes). 8
Chapter 1. Introduction Since this is not possible for animals, almost all small animal PET studies require the use of anaesthesia [40]. Some tracers (such as FDG) are irreversibly trapped in tissue after injection and, therefore, it is only necessary to anaesthetise the animal just before it is to be imaged. However, for many tracers (e.g. most of those used for neuroscience) the biological parameters are derived from a dynamic sequence of images which may require some form of pharmacological paralysis or physical constraint to keep the animal still during imaging. The effects of these sources of stress must be characterised and well understood in order to correctly interpret the PET data [47]. Another important consideration for small animal PET is the amount of injected tracer. If the amount of radio-ligand administered is too high, the compound can perturb the biological system under investigation [48]. For some tracers, such as FDG and other probes for endogenous compounds that are naturally present at fairly high concentrations [40], large amounts of radioactivity can be injected into an animal with little effect on the rate of incorporation. The signal–to–noise ratio for FDG–PET imaging can, therefore, be improved simply by increasing the amount of injected activity [41]. However, this is not possible for many receptor studies, where higher tracer concentrations lead to undesired changes in the measured biological parameters (lower specific binding) due to significant receptor occupancy [48]. A distinct advantage of using PET over conventional laboratory techniques, such as auto-radiography or post–mortem dissection, is that the animal can survive the study. This opens up the possibility of serial longitudinal studies in which a particular lab animal is repeatedly imaged, in vivo, to monitor the progression of a disease or evaluate the efficacy of some medical treatment. Since the same animal can be used repeatedly, the variability caused by inter–animal differences is removed [40]. For many studies each animal can also serve as its own control (e.g. the same animal can be imaged before and after it has been lesioned [41]).
9
Chapter 1. Introduction
1.3
Positron Decay
PET provides information about physiology and biological function by detecting the radiation emitted by compounds labelled with positron–emitting, radioactive atoms. The following discussion of the process of positron decay has been summarised from more detailed descriptions presented elsewhere [1, 3]. Radioactive decay by positron–emission is a process in which, essentially, a proton in the nucleus is transformed into a neutron. In addition a positron (β + ), which is the positively charged anti–matter conjugate of the electron, and a neutrino (ν) are ejected from the nucleus. The process can be represented schematically as: A ZX
A → Z−1 Y + β + + ν + Q (+ e)
(1.1)
where Q is energy and X is a neutron deficient atom. Since the daughter nucleus has a charge, Z − 1, which is one less than its parent, one of the orbital electrons must be ejected from the daughter atom to conserve charge. The ejected electron is represented by the (+ e) term included in the above equation. An electron is often ejected from the atom in a process called internal conversion, if, after the decay, the daughter nucleus is left in an excited state with sufficient energy to overcome the binding energy of the orbital electron. Energy conservation requires that the daughter atom Y must have a ground state mass at least two electron masses (2×511 keV) lower than its parent X. Any energy in excess of 1022 keV is shared between the positron and neutrino in the form of kinetic energy. This energy is divided between the two particles in an essentially random manner from one decay to the next. As a result, the distribution of possible energies for each particle has a particular maximum energy, E max β , and an average energy, , with values that are characteristic for a given nucleus. An example Emean β of the distribution of kinetic energies for positrons is shown in figure 1.2 for the dominant positron decay mode for
68 Ga.
This particular spectrum is of
interest to us, since 68 Ga is the positron–emitting daughter nucleus of which is used throughout this thesis.
68 Ge,
Alternatively, a neutron deficient atom can decay in a process known as 10
Chapter 1. Introduction
Relative Number of β Emitted (Arbitrary Units)
Emean
Emax β
+
β
0
500
1000 1500 Energy (KeV)
2000
Figure 1.2: Energy Spectrum (relative number of emitted positrons with a given energy) is shown for the dominant positron decay mode (relative intensity 89%) for 68 Ga. The spectrum data is based on theory [6] and is shown courtesy of Dr. Ross Schmidtlein, Sloan–Kettering Institute. Experimental values for the mean (Emean ) and maximum positron energies (Emax β β ) are from Burrows [7]. electron capture (EC). Here the nucleus captures an orbital electron which is combined with a proton to form a neutron and a neutrino. This process is more common with heavier elements (where orbital electrons are distributed closer to the nucleus) and is frequently in competition with positron–decay. An example of this is competition is observed in the decay of
68 Ga,
where
11% of nuclei decay by EC while 89% of decays are by positron decay.
1.3.1
Positron Range and Photon Acollinearity
After the positron is emitted from the nucleus, it travels a short distance through the surrounding matter losing kinetic energy primarily through ionisation events with atoms and through inelastic scattering with nearby nuclei and atomic electrons. After the positron has reached thermal energies it then 11
Chapter 1. Introduction annihilates with an ordinary electron. The range of an individual positron (i.e. the distance between the point of emission and the annihilation site) can vary widely from one decay to the next because of the wide range of possible kinetic energies which can imparted to the positron and due to the high probability for wide angle deflections. To illustrate this effect, figure 1.3 shows the simulated distribution of annihilation sites in water for two commonly used PET isotopes. The data shown is for 18 F and 15 O with maximum positron energies, Emax β , equal to 634 keV and 1732 keV, respectively. Table 1.1 summarises a number of different properties of radionuclei commonly used in PET, including the average range for their emitted positrons. In conventional PET imaging, coincidence events only provide localisation of the annihilation event to anywhere along a line–of–response (LOR). As a result, another commonly used measure of the influence of positron range on spatial resolution is the effective positron range, defined as the average distance between the point of emission and the line defined by the trajectories of the annihilation photons. Table 1.1: Summary of the properties of some positron–emitting nuclei of interest for PET, based on a similar table in Bailey et al. [1] with updated values from the Evaluated Nuclear Structure Data Files [2]. Half–life Average Use in PET Emean Nuclide Emax β β Range in Water (keV) (keV) (min) (mm) 11 C 960 326 20.4 1.1 Labelled Organic Molecules 13 N 13 NH (ammonia) 1198 432 9.97 1.5 3 15 O 15 1732 696 2.04 2.5 O2 , H2 15 O, C15 O , C15 O2 18 F 634 202 109.8 0.6 [18 F]DG, 18 F− 68 Ga 1899 783 67.7 2.9 [68 Ga]EDTA, [68 Ga]PTSM 82 Rb 3400 1385 1.3 5.9 Perfusion Tracer Once the positron has lost a sufficient amount of energy, it combines with an electron, either by mutual annihilation, in which two anti-parallel photons are emitted, or in the formation of an intermediate state known a positronium. Positronium is a bound state of the positron and electron with proper12
Chapter 1. Introduction
18
8
8
6
6
4 2 0 −2 −4 −6
4 2 0 −2 −4 −6
−8
−8
−10 −10
−10 −10
−5 0 5 Displacement along X (mm)
O: Positron Range (X−Y)
10
Displacement along Y (mm)
Displacement along Y (mm)
15
F: Positron Range (X−Y)
10
10
−5 0 5 Displacement along X (mm)
18
15
F: Positron Range
18000
3000
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2500
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Number of Decays
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3500
16000
10000 8000 6000 4000
2000 1500 1000 500
2000 0 0
10
2
4 6 8 Distance From Origin (mm)
10
0 0
2
4 6 8 Distance From Origin (mm)
10
Figure 1.3: Simulated data showing the two–dimensional displacement from the origin (upper–plots) and total distance from the point of emission (lower– plots) for the positrons emitted by 18 F and 15 O in water. The maximum positron kinetic energies for 18 F and 15 O are 634 keV and 1732 keV, respectively. The data was generated using Monte–Carlo simulation software [8] and is based on a similar figure in Levin and Hoffman [9].
13
Chapter 1. Introduction ties similar to hydrogen. About one–third of the positrons emitted in water or muscle form positronium [49]. In its ground–state, positronium has two forms; ortho–positronium; in which the spins of the electron and positron are parallel (triplet state, total spin = 1); and para–positronium, where the spins are anti-parallel (singlet state, total spin = 0). Para–positronium decays by self-annihilation with a mean life–time of about 10 −7 seconds, generating two anti–parallel 511 keV photons to conserve spin and linear momentum. Ortho–positronium self–annihilates, with a mean life–time of about 10−9 seconds, by the emission of three photons [6]. This decay–mode is extremely rare (making up only 0.5% of the total annihilations in water) primarily because there is a high probability that the positron will annihilate with another electron outside of the ortho–positronium atom [49]. This is unfortunate, however, since detection of three photons could provide spatial localisation of the annihilation to a single point in space, instead of the localisation to a line–of–response used in conventional PET. In practice, the formation of positronium has negligible effect on PET imaging because of the short life–time of both species of positronium and due to the low probability of the three–photon decay mode. In the more probable mode of anti–parallel photon emission, the pair of photons are emitted with directions that are at 180 o with respect to each other in the centre of mass frame of the collision. Although the momentum of the positron is generally negligible at thermal energies, the orbital electrons can have non–zero momentum leading to deviations from 180 o for the two photons with an approximately Gaussian shaped distribution with a full– width at half–maximum of about 0.5o in pure water [49]. This leads to an uncertainty in spatial position, ∆x, which depends on the detector radius, D, where ∆x ≈ D/2 sin(0.5o /2) ≈ 0.0022 × D. For example, in a whole– body cylindrical PET imaging system with a typical diameter of 80 cm, this leads to an 2 mm error. This error in spatial localisation is referred to as photon acollinearity. The combined effects of positron range and photon acollinearity place a fundamental limit on the resolution of PET imaging.
14
Chapter 1. Introduction
1.3.2
The production of PET tracers
The wider use of clinical PET imaging has, necessarily, been accompanied by the development of self–shielded, low–energy (6-17 MeV) proton cyclotrons which can be located close to the hospital premises to facilitate the transport of relatively short lived PET nuclei [20] (see table 1.1). These units also contain an automated chemical synthesiser for producing labelled imaging probes which are controlled by a personal computer and often operated by a single technician. The development and implementation of these integrated PET radio-pharmacies have been reviewed by Satyamurthy et al. [50].
1.4
Photon Interactions in Matter
As the annihilation photons travel from their point of origin to the PET detector array there is a probability that they will interact with matter within the tissue or other materials present in the scanner. The reduction in the intensity of transmitted photon flux, that occurs as a result of photon interactions within matter, is referred to as attenuation. There are three main types of interactions that are important for the range of energies applicable to the work in this thesis. These are the photoelectric effect, Rayleigh scattering, and Compton scattering. For low Z organic materials and photons with energies in the range of interest for PET emission and transmission scanning (≈ 100–500 keV), Compton scattering is the dominant process.
1.4.1
Photoelectric Absorption
The photoelectric effect (PE) occurs when a photon is completely absorbed by an atom, which is ionised in the process. The PE effect can occur if the photon energy is sufficiently high to eject an orbital electron from the atom leaving a vacancy in the atomic shell. A transition then occurs in which an electron with a lower binding energy fills this vacancy. This process is accompanied by the emission of characteristic X–rays or Auger electrons. The energy of the characteristic X–ray corresponds to the difference between the binding energies of the two atomic shells involved in the transition. This 15
Chapter 1. Introduction energy can also be directly emitted in the form of an electron, called an Auger electron, ejected from another of the atomic shells. The probability τ , that photoelectric absorption occurs (or cross-section) per atom is approximately proportional to Z 3 , where Z is atomic number, for higher Z materials and to Z4.8 for low Z materials. The PE cross-section is also approximately proportional to E −3 γ , where Eγ is the incident photon energy. For biological materials and photon energies used in PET emission and transmission scanning, the contribution from PE to the total crosssection (probability for all interactions) is very small (e.g. 0.02% and 0.9% for water at 511 keV and 122 keV, respectively).
1.4.2
Rayleigh Scattering
In Rayleigh scattering, also referred to as coherent scattering, the oscillating electric field of an incident photon interacts with the electrons surrounding the nucleus. This causes the electrons to momentarily vibrate. These oscillating electrons emit radiation with the same wavelength as the incident photon. This radiation is emitted as a scattered photon with the same energy as the incident photon but moving in a different direction. The differential Rayleigh scattering cross-section represents the fraction of photons scattered per electron per unit solid angle, and can be expressed as: dσcoh r2 ¯ (θ) = 0 (1 + cos2 θ)F(x, Z), dΩ 2
(1.2)
where θ is the Rayleigh scattering angle (measured with respect to the direction of the incident photon), and r 0 is the classical electron radius. The total probability of Rayleigh interaction (total cross-section) per electron σcoh can be obtained by integrating equation 1.2 over the solid angle. The ¯ represents the coherent scattering atomic form factor and term F(x, Z), accounts for the fact that the electrons are bound to atomic nuclei. This ¯ the atomic number (or effective atomic number) of factor depends on Z, the scattering medium and on x, the momentum transfer of the scattered photon. The momentum transfer can be written:
16
Chapter 1. Introduction
x=
sin 2θ , λ
(1.3)
where λ is wavelength of the scattered photon. The contribution of Rayleigh scatter to the total cross-section for biological materials and photon energies used in PET is also small (e.g. 0.2% and 2.3% for water at 511 keV and 122 keV, respectively).
1.4.3
Compton Scattering
Compton or incoherent scattering is the most significant photon interaction for energies of interest in PET. For example, Compton scattering accounts for 99.8%, for 511 keV photons, and 96.3%, for 122 keV photons, of the total cross-section in water. In Compton scattering a photon interacts with a free electron or weakly bound orbital electron. Part of the photon energy is transferred to the electron and both the energy and direction of the scattered photon changes. The Klein-Nishina differential scattering cross-section for Compton scattering can be written as:
dσinc (θ) = dΩ
r02 1 (1 + cos2 θ) 2 1 + α(1 − cos θ)
2
×
!
α2 (1 − cos θ)2 ¯ S(x, Z), 1+ (1 + α[1 − cos θ])(1 + cos2 θ)
(1.4)
where: • θ is the Compton scattering angle, • α is energy of the incident photon divided by the rest mass of the electron, ¯ is the incoherent scattering atomic form factor and • S(x, Z) • r0 is the classical electron radius. The original Klein–Nishina derivation of the scattering cross-section was ¯ This for free electrons and did not include the atomic form factor S(x, Z). 17
Chapter 1. Introduction correction factor accounts for the fact that the electrons are bound to atomic ¯ the effective atomic number of the scattering nuclei and depends on Z, medium, and the momentum transfer of the scattered photon x. Integrating equation (1.4) over the solid angle yields σ inc , the total probability of Compton interaction per electron. The energy of the Compton scattered photon can be obtained from the principles of momentum and energy conservation. This energy can be written as: E0 =
E , 1 + α(1 − cosθ)
(1.5)
where E 0 and E are the energy of the scattered and incident photon respectively.
1.5
Photon Attenuation
If a mono–energetic beam of photon flux with intensity I 0 passes through a thickness dx of attenuating material, the change in flux dI that occurs as photons are removed or deflected from the beam (attenuated) is given by: dI = −µI0 dx,
(1.6)
where µ is a constant of proportionality referred to as the linear attenuation coefficient. Since attenuation occurs as a result of photon interactions µ represents the fraction of photons that interact per unit thickness of attenuating material. Each of the three photon interactions mentioned in the previous section can occur independently. The total µ for energies of interest in PET, for a given attenuating material, is the sum of the linear coefficients of each interaction. This coefficient can be written as: µ = µpe + µinc + µcoh ,
(1.7)
where µpe , µinc , and µcoh , are the linear attenuation coefficients for the photo–electric effect, Compton scattering and Rayleigh scattering respec18
Chapter 1. Introduction tively. These values are related to the total cross–sections discussed previously. For example, the linear attenuation coefficient of Compton scattering can be obtained from σinc , the total cross–section per electron, as follows: µinc = σinc × ρelec ,
(1.8)
where ρelec is the electron density per unit volume of the attenuating medium. Figure 1.4 shows the mass attenuation coefficients µ/ρ, where ρ is the mass density, along with the components due to each of the relevant photon interactions for water over a wide range of photon energies. The ratio µ/ρ tends to be more similar for different materials and is, therefore, usually tabulated instead of the linear attenuation coefficients. If equation 1.6 is integrated over thickness d of an attenuating medium, µ(x), the following expression is obtained for the fraction of photon flux, I(x), that is transmitted: I(x) = I0 exp(−
1.6
Z
d
µ(x)dx).
(1.9)
0
PET Instrumentation
Inorganic scintillation crystals are the most commonly used detector material for PET imaging systems. These crystals emit visible (scintillation) light photons when an incident γ–ray interacts within the crystal material. The amount of scintillation light produced is ideally proportional to the energy deposited in the crystal to allow for the discrimination of detected events based on energy. The characteristics of a scintillator material, which are most important for its use in PET imaging, are as follows: • Its ability to stop incident photons (i.e. its detection efficiency). This property can be characterised by the linear attenuation coefficient, µ, described in the previous section, its effective atomic number, Z ef f , and its mass density, ρ. This effect favours higher Z ef f materials. • Secondly, the decay constant, τ , is a measure of how long it takes for 19
Chapter 1. Introduction
Figure 1.4: Total mass attenuation coefficients (in units of cm 2 /g) for water as a function of energy between 0.001 MeV and 1.0 MeV. Also included is the individual contribution from the photoelectric effect, coherent and incoherent scattering. Data is from Berger et al. [10].
20
Chapter 1. Introduction all the scintillation light to be emitted following an incident photon interaction. A scintillator with a shorter decay time can more accurately identify true coincidence events (fewer “random” events discussed in the following sections) and are capable of higher count–rates (less detector dead–time also discussed later). • Another important characteristic is the conversion efficiency, which is usually given in units of average number of visible light photons per MeV of deposited incident γ–ray energy. Higher conversion efficiency provides better energy resolution and, therefore, better ability to reject Compton scattered γ–rays. The initial conversion rate, which is defined as the initial rate at which scintillation light is produced following a γ–ray interaction (given in units of number–of–photons/MeV/ns, for example) is a related property which influences the timing resolution of the PET imaging system [5]. • Some additional considerations include: energy range over which the yield of scintillation light is linearly proportional to the energy deposited; the wavelength of the emitted scintillation photons; the crystal’s transparency to its own scintillation light; the ease with which homogeneous crystals can be produced of the required size; and other handling considerations such as ideal temperature range, sensitivity to humidity and fragility. Table 1.2 summarises some of the properties of scintillator crystals which have been used for PET imaging. Many modern PET imaging systems now use LSO, because of its high stopping power, relatively fast timing capability and high light output. For imaging systems that use scintillators with high initial conversion rates and fast electronics, the timing resolution which can be achieved is in the range of hundreds of picoseconds (e.g. LSO, BaF 2 and LaBr3 ). For these systems, localisation of the annihilation event can be improved by employing time–of–flight (TOF) measurements, that is, measuring the difference, ∆t, between the arrival times for the two detected photons in a 21
Chapter 1. Introduction
Table 1.2: Properties of some of the inorganic scintillator crystals that have been used in PET based on the data provided in the following citations [3, 4, 5]. NaI:Tl Bi4 Ge3 O12 Lu2 SiO5 :Ce BaF2 LaBr3 Property (BGO) (LSO) ρ (g/cm3 ) 3.67 7.13 7.4 4.89 5.30 Zef f 50 74 66 54 46 −1 µ at 511 keV (cm ) 0.3411 0.9496 0.8658 0.4545 0.4690 Decay Constant (ns) 230 300 40 0.8, 620 † 35 Conversion–Efficiency 38,000 8200 25,000 11,800 60,000 (photons/MeV) Initial Conversion–Rate 165 37 676 2266 4000 (photons/MeV/ns) †: BaF2 has both fast and slow decay components given coincidence event. This time difference can be used to localise the point of emission of the annihilation photons. With the fastest scintillators and careful design of the acquisition electronics, time–of–flight PET can provide spatial localisation to within a few centimetres [51]. This data currently only enhances the information already provided by conventional PET (which localises annihilation events to lines–of–response). However, as faster scintillators and electronics are developed, the potential exists to localise the event to a smaller and smaller volume, which could eventually render obsolete tomographic reconstruction methods. Research into developing new materials continues, especially lanthanum scintillators such as LaBr 3 [52] which have a favourable combination of good timing resolution and high light output. These scintillators can be used to make PET systems capable of TOF with energy resolution in the range of about 3% (full–width at half maximum for a peak energy of 511 keV), thereby also greatly reducing errors due to scatter. Although inorganic scintillators are efficient at converting γ–rays into light photons, the amount of light produced is still quite small. The light signal must be amplified and converted to an electrical signal using, for example, a photomultiplier tube (PMT). As described earlier, PET imaging 22
Chapter 1. Introduction systems usually consist of an array of discrete scintillator elements arranged in cylindrical rings around the object being imaged. If each crystal element required its own PMT it would unacceptably increase the cost and complexity of a PET scanner. As a result, most PET imaging systems employ a block–detector design, first proposed in the 1980s [53] and shown schematically in figure 1.5. Each block consists of a piece of scintillator, segmented into smaller elements (generically referred to as crystals which represent the smallest subunit of the detector) using partial cuts through the scintillator with a precision saw. Since scintillation light is emitted isotropically, the spaces between the crystals are usually filled with an opaque reflective material (e.g. see Siegel et al. [54]) to optically isolate each crystal element. The amount of electrical signal produced by a given PMT is proportional to its distance from the photon absorption event in the block. A position in the X–Y plane (as shown figure 1.5) can, therefore, be determined by combining the electrical signals from each PMT in a logic circuit that weights the signals appropriately. More recently the use of a single position-sensitive PMT or semi–conductor photo–diodes have further simplified and improved the basic block detector design [1].
1.7
Resolution and the Depth of Interaction
Although fundamental limits in PET resolution are imposed by the positron range and photon acollinearity (discussed previously), in practice the resolution of the typical PET imaging system is constrained primarily by the size of its detectors (i.e. its crystal dimensions). In the typical configuration used for PET scanners consisting of fixed rings of discrete detector crystals, the resolution is at best (excluding resolution–losses due to positron range and photon acollinearity) equal to half the width of a crystal (see section 1.8). Another factor that limits the resolution of PET imaging is related to the fact that a finite thickness of crystal material is required to stop 511 keV photons. For PET imaging systems consisting of discrete crystals arranged in circular rings, the resolution is degraded for lines–of–response (LORs) that 23
Chapter 1. Introduction segmented block of scintillator crystal
array of four single−channel PMTs
y coordinate
reflective material between saw−cuts
x coordinate
Figure 1.5: Schematic diagram to illustrate a conventional PET detector block based on similar figures in Cherry et al. [3] and Bailey et al. [1]. intersect the crystal pairs at large angles with respect to the thickest part of the crystal (see figure 1.6). For photons incident on a given crystal, there is a characteristic depth, referred to as the depth–of–interaction (DOI), at which the photons are most likely to have deposited all there energy within the crystal. This DOI is related to the linear attenuation coefficient, µ, of the crystal material. However, photon absorption is a statistical process and one cannot, in general, know at which depth a given photon absorption took place within the crystal. This can create a spatial blurring and/or mispositioning of the LOR, sometimes refereed to as the parallax effect, for sources offset from the centre of the scanner (figure 1.6 (a)) and for LORs which span several different axial rings (figure 1.6 (b)). This effect is less severe for crystal materials with high stopping powers and for detector configurations in which the ring diameter is large relative to the crystal 24
Chapter 1. Introduction thickness. Conversely, the parallax effect can be very significant for small animal PET scanners. For example, the microPET R4 dedicated rodent scanner, which is used throughout this thesis, employs 10 mm thick crystals with a 148 mm ring diameter. Its resolution is about three times poorer for a source with a radial offset of 50 mm, than it is for a source in the centre of the scanner [55]. An interesting method that has been employed to reduce the parallax effect is the use of dual crystal layers (e.g. the Siemens HRRT scanner [56]). Each layer consists of a different scintillator material with different decay times. By monitoring the decay time of the pulse, the event can be localised into upper or lower crystal layers reducing the parallax effect and improving imaging system resolution.
(a)
(b)
Figure 1.6: Schematic diagram, based on a figure appearing in Bailey et al. [1], to illustrate the parallax effect. There is a degradation of the resolution in both the radial direction (a) and axial direction (b) because it is impossible to distinguish between the two annihilation–photon flight paths, shown as the solid and dashed arrows in each figure. PET detector crystals are shown as the grey rectangles.
25
Chapter 1. Introduction
1.8
Data Acquisition in PET
As described previously, PET scanners typically record events in pairs of detector crystals. The straight line connecting a pair of detectors are refereed to as a line–of–response (LOR). Once the PET data has been corrected for contamination from randoms, scatter and for several other effects to be discussed in the following sections, the total number of counts recorded in a pair of detectors is approximately proportional to a line integral (or more accurately a volume integral) along the appropriate LOR through the distribution of radio–tracer activity. The raw data in PET are stored as histograms which represent the total number of counts per LOR. With reference to figure 1.7, the raw data are usually organised into series of parallel LORs parameterised in terms of: the radial distance which represents the distance between an LOR and the centre of the detector ring; the projection angle which measures the orientation of the LORs in the transverse plane (figure 1.7 (a)); the oblique angle which corresponds to the orientation of the LOR with respect to the axial direction (figure 1.7 (b)); and the distance along the axial direction (figure 1.7 (b)). When the raw PET data are parameterised in this way, we refer to the two– dimensional binned histogram data (with bins corresponding to the radial distance and projection angle) for each plane (specified by the oblique angle and axial distance) as a sinogram. In practice, additional LORs with approximately the same projection and oblique angles are usually grouped together (offset by a pair of detector crystals in both the radial and axial directions) into the same sinogram. We illustrate this grouping schematically in figures 1.7 (c) and (d). Effectively, this increases the linear sampling (from d to d/2 where d is the crystal width) in both the radial and axial directions. It is appropriate to mention here, that older generations of PET scanners were operated in “2D mode”, in which the axial detector rings were separated by axial collimators or septa (shown schematically in figure 1.8 (a)) made of some dense photon–absorbing material such as lead. For these scanners, the sinogram data for the detector pairs within the same axial ring are 26
Chapter 1. Introduction
Co un t
s
Oblique Angle Axial Distance
D al di Ra
Projection Angle ist ce an
(a)
(b)
un
ts
Oblique Angle
Co
Axial Distance
al
di
Ra
Projection Angle
ce
an
ist
D
(c)
(d)
Figure 1.7: Schematic diagram to illustrate organisation of conventional PET sinograms. The figures shown in (a) and (c) show data that would be grouped into a particular projection angle in the sinogram data, with and without increased effective radial sampling. The figures shown in (b) and (d) show data that would be grouped into a particular oblique angle in the sinogram data, with and without increased effective axial sampling. 27
Chapter 1. Introduction referred to as the direct–plane sinograms. The data for detector pairs separated by one or more axial rings (as shown in figure 1.8 (a)) are refereed to as the cross–plane or oblique sinograms. Note that the sensitivity for events in the summed cross–plane data is approximately twice that of the direct planes. Although the septa used in 2D PET reduce randoms, dead–time and scatter (discussed in the next sections), it is clear that the sensitivity of such a scanner could be improved by removing the septa. The current generation of scanners can be operated in “3D–mode”, illustrated schematically in figure 1.8 (b), either by employing retractable septa or doing away with the septa altogether. When operated in 3D–mode coincidences between any two detector pairs can potentially be accepted. Depending on the ring diameter and axial extent of the PET scanner, 3D acquisition typically provides a four to eight times improvement in sensitivity over the same scanner operated in 2D–mode [3]. The axial sensitivity for a 3D PET scanner is much higher for the central regions of the scanner than it is for regions close to the edges of the scanner (as shown in figure 1.8 (b)). Usually a longer set of axial collimators are located on the outer axial edges of the scanner in both 2D and 3D mode. These collimators reduce the number of photons that can reach the detector from outside of the axial extent of the scanner, thereby, reducing scatter, random events and dead–time. A large increase in sensitivity was sufficient motivation for most modern commercial PET imaging systems to employ fully 3D data acquisition. At the same time, the move to 3D PET has not been without difficulties and has been a driving force in PET research over the past few decades. Without septa, scatter becomes a much more significant problem for PET scanners. To a lesser extent, random coincidences and dead–time also become more prominent for 3D data acquisition. The size of the acquired sinogram data can also become extremely large when many rings are operated in coincidence leading to data storage issues. The reconstruction algorithms used for the 3D PET data are also necessarily more complex than those employed for the other 2D tomographic imaging modalities (i.e. 2D PET, X–ray computed tomography, and single–photon computed tomography) where reconstructions are generally performed slice–by–slice and data through obliquely 28
Chapter 1. Introduction
(a)
(b)
Figure 1.8: Schematic diagram to illustrate 2D (a) and 3D (b) data acquisition in PET. The figures represent the arrangement of the detectors in the axial direction. The detector rings are shown in grey and collimators are shown in black. LORs are shown as the thin lines connecting the detectors. The acquisition shown in (a) allows for both direct and cross planes to contribute data. While figure (b) illustrates fully 3D data acquisition (only selected LORs are shown for clarity). This figure was adapted from a similar illustration in Cherry et al. [3]. tilted planes are unavailable.
1.9
Quantitative PET
One of the major goals of positron emission tomography has been to provide images whose intensities are proportional to the concentrations or amounts of activity present in the subject being scanned. PET imaging provides more useful information to clinicians and researchers when it is fully quantitative (calibrated to units of kBq/cm3 , for example) and is combined with the radio–tracer input function (from arterial sampling or based on a suitable reference region within the PET images) and with an accurate biological model (differential equations that reflect the delivery of the tracer and the 29
Chapter 1. Introduction individual steps in its biochemical metabolism [57]). The parameters fit to the dynamic PET data can provide calibrated biochemical measurements of flow, permeability, metabolism, flux and receptor density (e.g. in units of millilitres per minute per millilitre for flow, nanomoles per minute per millilitre for metabolism and picomoles per millilitre for receptor density [57]). There are a number of physical effects, discussed in the following sections, which must first be corrected for, in order to obtain quantitatively accurate PET images.
1.9.1
Attenuation Corrections
The largest correction, both in terms of its magnitude and its influence on reconstructed images, is the attenuation correction. If no corrections are applied for attenuation, the reconstructed intensities for regions lying deeper in the body will be underestimated relative to those closer to the surface. Fortunately, these corrections are conceptually simple to implement. For the detection of photons in coincidence, the probability of detecting both photons is the product of their individual transmission probabilities (the ratio I/I0 from equation (1.9)). With reference to the diagram shown in figure 1.9, consider the emission of the pair of annihilation photons (shown as ~ The probability, Patt , that arrows in the figure) from the vector position, B. both photons will be transmitted through an arbitrary spatial distribution of attenuating medium, µ(~r), is given by:
Patt = exp(−
Z
~ A ~ B
µ(~r)(−dr)) × exp(−
Z
~ C ~ B
µ(~r)dr) = exp(−
Z
~ C ~ A
µ(~r)dr, (1.10)
~ and C ~ are the vector positions of the pair of detectors and the intewhere A gral path, dr, is a straight line going from the left to the right in the figure. ~ and Note that this probability does not depend on the point of emission, B, rather only depends on the total distribution of attenuation along the line connecting the two detectors (the line–of–response). As a result, attenuation corrections can be applied as a relatively–simple single multiplicative
30
Chapter 1. Introduction correction factor for each line–of–response (LOR). For the LOR shown in figure 1.9 the attenuation correction factor is equal to: ACF = Patt −1 = exp(+
Z
~ C ~ A
µ(~r)dr),
(1.11)
In the next chapter, we will discuss in more detail the methods which have been employed to obtain these attenuation corrections or equivalently the methods which have been used to estimate the µ–map (i.e. µ(~r)),, representing the distribution of linear attenuation coefficients in the subject being imaged.
Detector 1
Detector 2 Point of Annihilation
A
B
C
µ (r)
Figure 1.9: Schematic diagram to illustrate the process of photon attenuation as it applies to PET coincidence measurements. A pair of annihilation ~ passing through an arbitrary attenuphotons are emitted from position B ~ and ating medium µ(~r) to be detected in the two detectors at positions A ~ which are operated in coincidence. C
1.9.2
Scatter Corrections
As mentioned earlier, for 511 keV photons in tissue, the predominant interaction that results in photon attenuation is Compton scattering. This interaction results in the emission of a scattered photon with a lower energy and different direction of propagation than the incident photon. There is a probability that the scattered photon will be detected in another crystal element of the PET imaging system. If the other annihilation photon is then detected in coincidence, it will be assigned to an incorrect LOR which 31
Chapter 1. Introduction does not pass through the original point where the annihilation γ–rays were emitted (as shown in figure 1.10 (a)). The contribution from scattered photons to the acquired PET data can be extremely high. For example, in whole–body PET imaging of the abdomen, scatter can account for as much as 60 to 70% of the detected counts [3]. Scatter fractions are smaller in small animal PET, where estimates are usually between 2 to 40% depending on the size of the object being imaged [46, 58]. Scatter can be more complex to account for in small animal PET imaging systems, since scatter from within other components of these scanners may also be very important [58]. This problem is further exacerbated by the fact that Compton scattering is also the most highly probable interactions for 511 keV γ–rays within the scintillation material (e.g. 62% of the total cross–section in LSO is the result of Compton scattering [10]). Scatter within the crystal can lead to events being rejected based on energy discrimination (if the deflected photon escapes) or to a form of spatial blurring if, for example, the scattered photon is subsequently absorbed in another crystal (as in figure 1.10 (b)). Many corrections have been proposed to account for scatter in PET data. In the next chapter, we review these correction techniques in greater detail.
1.9.3
Randoms Corrections
For a pair of detectors operated in coincidence, events are recorded whenever two photons are detected within a specified coincidence timing window. As a result, “random” events can occur when two uncorrelated photons (i.e. two photons which did not originate from the same annihilation event), are mistakenly identified as a coincidence event. The exact rate at which randoms are recorded depends on the details of the coincidence sorting, which is usually performed at the electronics level. For example, if each pulse (a single detected photon) opens a timing gate of duration, ∆t, and coincidences are recorded when two timing gates overlap, the random coincidence counting rate, RA,B , for detectors A and B, is given by: RA,B = 2∆tSA SB ,
(1.12)
32
Chapter 1. Introduction Scatter Event
Scatter Event Point of Emission
Assigned LOR
(a)
Point of Emission
Assigned LOR
(b)
Figure 1.10: Schematic diagram, based on a figure appearing in Zaidi and Koral [11], to illustrate the effects of scatter within the object being imaged (a) and from within the crystals (b). The paths of annihilation–photons are shown as solid arrows and the LORs to which they are assigned are shown as dashed lines. where SA and SB are the singles rates at detectors A and B respectively. Equation (1.12), indicates that the total randoms rate for the entire imaging system will, therefore, depend on square of the total singles rate. As a consequence, the randoms rate depends roughly on the square of the total activity (which may be located either inside or outside of the scanner field–of–view) that contributes to the singles–rate. The true coincidence– rate depends linearly on the activity inside the scanner. In practice random events can make up a large fraction of the total detected coincidences, particularly when the administered activity is high or large amounts of activity lie just outside the scanner. For example, typical values for human brain imaging are between 10–20% but can be as high as 80–100% for abdominal imaging [3]. It is clear from equation (1.12) that the detected randoms–rate can be reduced by using a shorter timing window. Typical values of ∆t are on the order of several nanoseconds, constrained by the timing resolution
33
Chapter 1. Introduction of the imaging system and also by the finite speed of light (light requires about ≈3.3 ns to travel one meter). The randoms rate can also be reduced by collimating the scanner from activity outside the field–of–view (FOV), as shown in figure 1.8. Equation (1.12) indicates that if S A and SB can be measured then RA,B can be calculated for each line–of–response (e.g. [59]). This method of randoms estimation has the advantage that it provides an estimate of R A,B which is relatively low–noise, since the singles–rates for individual crystals are generally a few orders of magnitude higher than the coincidence–rates for crystal pairs. Problems may occur if the singles–events are recorded before energy discrimination and because equation (1.12) does not take into account electronics dead–time (discussed subsequently) arising from the coincidence processing circuitry [1]. As a result, a more common method for randoms correction is the use of a “delayed” coincidence window [60]. In this method, a separate dataset is generated for each crystal pair by delaying the coincidence window by some length of time, T , which is much greater than the coincidence timing window and the amount of time required for photons to travel across the scanner FOV. The delayed coincidences are defined as pairs of detected single–events with time differences between T and T + ∆t. These events can only be a result of random events, since photons originating from the same annihilation will always arrive at the detectors within a few nanoseconds of each other. This method provides an accurate estimate of the randoms rates with the same dead–time characteristics as the coincidence rates. The randoms obtained using these techniques are usually subtracted from the acquired coincidence data, increasing the statistical noise. As a result, a number of methods have been proposed to improve the noise characteristics of the randoms estimates obtained using delayed coincidence windows [53, 61].
34
Chapter 1. Introduction
1.9.4
Normalisation
The current generation of PET scanners have tens of thousands of individual PET detectors and hundreds of millions of LORs (e.g. the microPET Focus rodent scanner has 13,824 detector crystals and 42.5 million LORs) and each LOR may have slightly different sensitivity for the detection in coincidence of photons. These sensitivity variations originate from a variety of sources. For example, the angle with which the LOR intersects the detector ring influences the effective depth of scintillator material in which incident photons can interact and, therefore, can change the sensitivity. Both differences in the crystal composition and/or dimensions due to imperfect manufacturing can also influence sensitivity. The position of a crystal within a block can change the fraction of scintillation light which is coupled to the PMT (called the block–effect). The gains for all PMTs are not identical and can also change over time. Many of the reconstruction methods used in PET, assume that the sensitivity for each LOR is the same. Normalisation refers to the process by which each LOR is corrected for this unwanted variation. This correction is usually applied as a multiplicative correction factor for each LOR (called normalisation coefficients). Conceptually, the most straightforward way of obtaining a normalisation is to expose each LOR the same intensity positron emitting source. The normalisation factors can then be obtained by dividing some ideal number of counts (often the average) by the actual number of counts recorded for that LOR [62]. This method is referred to as a direct–inversion normalisation. In order to obtain low–noise normalisation factors, this method require extremely large numbers of counts in each LOR. This condition necessitates extremely long normalisation scans with relatively moderate activity sources, since higher activity sources would result in distorted normalisation corrections due to dead–time. This poses a technical problem because these scans must also be performed frequently since factors such as the gains on PMTs can drift over time. To maximise the number of counts in each LOR of the normalisation scan, one can, alternatively, use a uniform cylindrical source (for example using a plastic with a uniformly distributed amount of
68 Ge
35
Chapter 1. Introduction activity). Unfortunately, this can introduce additional inconsistencies since these scans may suffer from large scatter fractions. Scattered photons do not necessarily originate from the correct LOR and the efficiencies of the crystals are different for lower–energy scattered photons. Alternatively, lower noise estimates of the normalisation factors can be obtained using component–based methods [63]. For these methods, the normalisation factors are modelled as the product of a number of components. Since there are many more LORs than there are individual crystals and many symmetries which can be exploited for circular PET detectors, the normalisation components can be computed using the average of many LORs thereby reducing the length of scan required. The first such method [63] used a simple two component model which included a geometric factor and a radial mispositioning factor (connected to the circular geometry of the scanner). Later normalisation methods [64] employed factors to account for the intrinsic detector efficiency, a geometric factor, the block–effect and dead–time. The normalisation of Badawi et al. [65], added a count–rate dependent block–effect factor as well as timing–alignment corrections to account for small errors in the temporal synchronisation between different crystals. Component–based methods can be extremely complex and require the estimation of multiple normalisation factors, often from different datasets, which can lead to image artifacts [66]. To simplify the normalisation procedure, a model–based normalisation has recently been proposed [66], which uses an iterative approach (based on a penalised reconstruction method [67]) in which the normalisation components are estimated using a known distribution of activity and attenuation. This method appears to provide artifact– free images, similar to those obtained using very high statistics direct– inversion normalisations, provided that the same models are employed in both reconstruction and normalisation.
36
Chapter 1. Introduction
1.9.5
Dead–Time Corrections
The rate at which events can be processed for all radiation counting systems, and for PET in particular, is limited by dead–time. For most PET imaging systems, the scintillation light is summed at the block level. If an additional photon interacts within a block before all the scintillation light from a previous photon interaction has been collected, the two signals will be summed (referred to as pulse pile–up) and only one pulse will be measured by the PMTs. The associated electronics also have a finite maximum rate at which they can process data (typically around 1 MHz). The loss of detected events that may occur as a result of these effects is referred to as dead–time. In PET dead–time losses are primarily dominated by pile-up within the scintillation crystals and are, therefore, strongly associated with the decay time, τ , of the scintillation crystal material. Dead-time can be characterised as either paralysable or non-paralysable [1]. Paralysable dead–time is applicable to systems which are unable to process another event (i.e. the system is “dead”) for a fixed amount of time, τ p , after each detection regardless of whether or not the system is dead at the time when the new event occurs. The dead–time of scintillation detectors have a strong paralysable component, because each time a photon interacts in the crystal, more electrons are raised into the conductance band, which must decay away before the detector can process the next event. If one fits a paralysable dead–time model to the response of one of these types of detectors, the parameter τp is usually similar in magnitude to the scintillation decay constant τ . For a paralysable system, the measured count–rate will eventually begin to decrease as the incident count–rate increases. The relationship between the measured event rate m, the actual event rate n and τp can be expressed as n = n exp(−τp n). Alternatively, a counting system’s dead–time may be non-paralysable. In this case, the system is again rendered “dead” for a time τnp after each event, but while the system is dead, further events have no effect. The dead–time of coincidence processing electronics are mostly non–paralysing, because events that arrive during processing are simply ignored. For this type of system, the measured count-
37
Chapter 1. Introduction −1 as the actual count–rate increases, rate approaches a limiting value of τ np
and the relationship between m, n and τ np is given by m = n/(1 − τnp n). To obtain quantitative results, PET data must be corrected for this effect. This is usually done by modelling the dead–time losses as a combination of paralysable and non-paralysable components and fitting the model parameters using repeated measurements of an initially high-activity, fast–decaying source [64]. Typical corrections are applied as a global multiplicative correction, while more sophisticated approaches apply dead–time corrections for each detector pair or at the block–level usually as part of the normalisation correction [64, 65].
1.9.6
Limits to the Absolute Quantification of PET images
If all of the corrections described above are applied accurately to the sinogram data, the intensity of the reconstructed images should be directly proportional to the amounts of radioactivity present in the object being imaged. In practice, another calibration is required to convert the reconstructed image into units of absolute radio–tracer concentrations. This is usually achieved by scanning a cylinder filled with a uniform concentration of activity and determining a calibration factor, in units of concentration (e.g. kBq/cm3 ) per reconstructed number of counts per voxel in the image. However, the images themselves are still subject to certain limitations due to the finite spatial resolution of the PET imaging system and due to patient motion. For structures smaller than the full–width at half–maximum (FWHM) characterising the spatial resolution of the imaging system (which includes all spatial blurring effects including the parallax effect, intrinsic scanner resolution, positron range, and annihilation–photon acollinearity) the partial volume effect (PVE) limits the quantitative accuracy of the reconstructed PET images. The PVE can lead to either an under or over–estimation of the radio–tracer present. An under–estimation occurs when a small volume with a high activity concentration is surrounded by lower activity concentration, while an overestimation can occur when a small volume of lower
38
Chapter 1. Introduction concentration is located in a higher concentration environment. In order to obtain accurate estimates of the activity present for structures of this relative size, it is necessary to correct the reconstructed images for the PVE, using a method such as that of Rousset et al. [68]. These corrections use co–registered images obtained from a higher–resolution anatomical imaging modality, such as magnetic resonance imaging (MRI) or X–ray computed tomography (CT), which have been segmented into anatomical regions. These corrections are derived by estimating the percentage of cross–contamination which is expected between different regions in the images based on a specific model of the spatial resolution of the PET imaging system. In many situations this approach will improve quantification, however, it requires very accurate image co–registration. Furthermore, it also assumes that each segmented anatomical region has a uniform activity concentration which may not be the case for non–uniform anatomical structures such as the lungs or tumours [1]. For the current generation of PET scanners, spatial resolution is approaching 2 to 3 mm FWHM in the centre of the scanner field–of–view. For these scanners, patient motion can also become a factor which limits the quantitative accuracy of PET images. For example, when patient motion (due to breathing) is not corrected for, an underestimation of 50% was observed in the concentration of activity within a lung tumor [69]. Several strategies have been investigated to reduce or compensate for the effects of motion including indirect approaches, such as increasing scanner sensitivity to allow for shorter scan times or physically restricting patient motion with head–holders in brain imaging. More direct approaches involve gating the acquired sinogram data into multiple time frames. The duration and starting point of each time frame is triggered using an external monitoring system which directly measures patient motion or the physiological process of interest (e.g. the cardiac and/or respiratory cycle). The frames are then spatially co–registered and, if the temporal changes in activity are not of interest, the frames are then summed [70, 71]. The gated images are often noisy, however, due to the small number of counts available in each frame, which can lead to co–registration errors. More sophisticated correc39
Chapter 1. Introduction tion techniques attempt to “undo” the effects of motion, by repositioning counts from individual LORs during the reconstruction process whenever motion is recorded [72, 73].
1.10
Image Reconstruction Methods
In PET, image reconstruction refers to the techniques used to obtain a three dimensional distribution of radio-tracer in the subject’s body from the acquired sinogram data. The techniques used to reconstruct PET data can be classified broadly into two categories. The first type use an analytical model of the reconstruction process, which does not take into account the noise which is present in the acquired data due to the statistical nature of photon decay and counting. The second group of reconstruction methods iteratively searches for the best estimate of the activity distribution taking into account the measured sinogram data and a particular model of the PET data acquisition process (called the system matrix or system model). The main advantage of these iterative methods is that effects such as spatial blurring, photon attenuation and scatter can be included directly in the system model. We shall also discuss some issues concerning the reconstruction of 3D PET data.
1.10.1
Filtered Back Projection Reconstruction
Filtered back projection (FBP) [74] is commonly used in the clinical environment and employs a very simple model of the PET imaging process. FBP is an analytical technique that assumes that the number photons recorded for each line–of–response (LOR) represent a one dimensional line integral through the activity distribution. This process is shown schematically in figure 1.11 (a). If we consider the two dimensional distribution of activity, λ(x, y), the total number of counts in a given LOR, Y (x 0 , φ), with a particular radial offset, x0 , and projection angle, φ, is assumed in FBP to be the line–integral given by:
40
Chapter 1. Introduction
0
Y (x , φ) =
Z
~l(x0 ,φ)
λ(x, y)dx0 ,
(1.13)
where x0 = x cosφ + y sinφ and ~l(x0 , φ) is a vector connecting the particular pair of detectors for the LOR under consideration. Back projection refers to the redistribution of these counts back along a line through the object. The points of intersection for the back-projected lines correspond to potential source locations. This process is shown schematically, for just three projection views, in figure 1.11 (b). Although the figure shows very few back–projected angular views (circular PET detectors usually provide Ncrystals /2 angular views, where Ncrystals is the number of crystals per axial ring), it demonstrates some typical features of back–projected images. The back–projected object contains low density streaks radiating out from the points of intersection with a characteristic 1/r distribution, where r is radial distance from the intersection point. The true activity distribution can be recovered only if projections are acquired for an infinite number of angular views between φ = 0 and φ = π, and the back–projected data are de–convolved to account for the 1/r blurring. The expression for the FBP reconstructed activity can be written concisely in terms of the inverse Fourier transform of the original projection data, Y if t (ν, φ), as:
λ(x, y) =
Z
π 0
Z
∞ −∞
|ν|Yif t (ν, φ) exp (2πiν(x cos θ + y sin θ)) dνdφ,
(1.14)
where ν is the polar spatial frequency and the term |ν| is the inverse Fourier transform of the 1/r filter. The main advantage of FBP is the speed with which a reconstruction can be performed. However, there are a number of problems associated with FBP reconstructions. One such practical issue is that an infinite number of projection angles is never available. As a result, streak artifacts are often quite prominent in reconstructed FBP images. Another difficulty is that statistical noise in data acquisition is not accounted for. The high frequency noise in the projection data is enhanced by the so–called ramp filter (the 41
Radial Offset
Counts
Counts
Chapter 1. Introduction
Radial Offset
λ (x,y)
(b)
(a)
Figure 1.11: A schematic diagram of the line integral data acquisition model is shown in figure (a). Figure (b) shows the back projection of this data. Intersection of back-projected lines through the object correspond to possible source locations. term |ν| in equation (1.14). This noise can be reduced by modifying the ramp filter to suppress high frequency components in the projections. However, these techniques degrade the resolution of the reconstructed images, since they also suppress high frequency components that are actually present in the imaged object.
1.10.2
Methods for 3D–PET
The FBP method, as described above, is inherently two–dimensional in its treatment of the reconstruction problem. With FBP it is assumed that the measured line integrals are without noise and that the data are sampled continuously. Under these assumptions, the direct–plane sinogram data are sufficient to uniquely reconstruct all activity tracer distribution within the axial extent of the scanner. From this point of view, the oblique sinogram data are redundant. In practice, however, noise in the sinogram data and finite sampling mean that improved images can be obtained if the oblique 42
Chapter 1. Introduction data can be exploited. A fully 3D version of FBP has been derived, differing from the 2D case mainly in that a more complex filter must be employed [75]. The limited axial extent of a cylindrical PET scanner implies that the entire activity object is not sampled in the oblique sinograms, as illustrated in figure 1.12. The derivation of Colsher [75], however, assumes non–truncated data. The reprojection and filtered back–projection (3D–RP) method [76], is a popular reconstruction technique which addresses this problem. First, a 2D reconstruction is performed on the direct plane subset of the 3D data. The resulting image is then used to obtain estimates of the missing data in the truncated oblique sinograms. The estimated data are then combined with the acquired data and reconstructed using the 3D FBP technique [75].
Truncated Region Viewed in Oblique Sinogram
Activity
Figure 1.12: A schematic diagram to illustrate the truncated portion of the total activity viewed in an oblique sinogram Although, the 3D–RP method provides improved image quality, it re43
Chapter 1. Introduction quires substantially more computation time than simple 2D–FBP. This has led to the development of several approximate algorithms for 3D data acquisition. The simplest approach is single–slice rebinning [77], in which oblique events are assigned to the direct plane sinogram with an axial position corresponding to the average position of the oblique detector pair. Another popular technique is Fourier rebinning [78] (FORE) which reduces the fully 3D sinogram to a 2D dataset using an approximate relationship relating the Fourier transforms of oblique and direct plane sinograms assuming a line integral model of data acquisition.
1.10.3
Iterative Reconstruction Algorithms
Iterative reconstruction algorithms provide an alternative to analytically formulated reconstruction procedures such as FBP. These methods iteratively search for the best estimate of the activity distribution taking into account the measured sinogram data and a particular system model of the PET scanner and data acquisition process. The main advantage of iterative methods are that effects such as spatially–varying spatial blurring, 3D data acquisition, photon attenuation and scatter can be accounted for more accurately when they are included directly in the system model. The PET data acquisition process can be thought of as a transformation that maps the three dimensional distribution of activity in the patient’s body to the distribution of counts within the sinogram data. We can write this transformation as: Y¯i =
J X
Pij λj ,
(1.15)
j=1
where λj is the amount of activity present in the j th discretized volume element (voxel) of the activity object and the summation over J is over all voxels in the object containing activity. In the model, Y¯i is the mean number of counts for the ith LOR, that would be obtained if one averaged over a infinite ensemble of identical scans. Obviously, Y¯i is not available, rather a single statistical realisation, Y i , is measured which is subject to Poisson 44
Chapter 1. Introduction noise. The term Pij is the system model matrix element which represents the probability that a photon emitted from object voxel j is detected in the ith LOR of the sinogram data. The system matrix can contain scanner dependent considerations such as spatial blurring and normalisation factors. It can also account for effects such as attenuation, scatter and dead-time which depend on the activity and attenuation distributions present in the subject being imaged. An example of a fairly complex system model, used for the microPET P4 dedicated– primate PET scanner, was described by Bai et al. [66]. In this work, the acquired data were modelled as: Y¯i = Pnorm Patt i i
J X
geom ¯ i + S¯i , Pblur λj + R ij Pij
(1.16)
j=1
where: Patt contains the object–dependent attenuation correction factors i for each detector pair; Pblur is a blurring kernel which models photon pair ij acollinearity, the positron range and inter-crystal scatter and crystal penis the geometric projection matrix deetration (parallax blurring); Pgeom ij scribing the probability that a photon pair reaches the front faces of the detector pair in the absence of attenuation and assuming perfect photon pair collinearity; and Pnorm are additional factors, derived from a component– i based normalisation, which model crystal sensitivity variations, dead–time, ¯ i and S¯i represent the avtiming alignment and block–effects. The terms R erage number of randoms and scatter detected for the i th LOR. Note that many of these factors depend only on the index i of the LOR and not on the position of the activity voxel. The system matrix mathematically describes a model of the data acquisition process. This model is then used in an iterative reconstruction algorithm to incrementally update the activity distribution. The iterations generally consist of two steps. The first is a forward projection step in which the data collection process is approximated using the system matrix model of data acquisition. A set of estimated projections is obtained using the current iteration of the activity distribution. The next step is often referred to
45
Chapter 1. Introduction as back–projection, where the differences between the estimated projection and measured projection data are then used to obtain a new estimate of the activity distribution [79].
1.10.4
Maximum Likelihood-Expectation Maximisation
Many of the iterative reconstruction algorithms currently in use are based upon the maximum likelihood–expectation maximisation (MLEM) algorithm, first proposed by Dempster et al. in 1977 [80] and soon after applied to emission tomography by Shepp and Vardi in 1982 [81]. This algorithm seeks the most likely activity distribution, that is consistent with the measured projections [1], via a statistical likelihood function given by: L(Y|λ) =
N Y exp(−Y¯i )Y¯iYi
i=1
(1.17)
Yi !
where the summation over i = 1 to N is over all measured LORs. The function, L(Y|λ), is a likelihood function which concerns the statistical probability of obtaining a particular set of measurements Y given a particular fixed activity distribution λ. L(Y|λ) is maximised when the difference between the estimated and measured projections is minimised. In the following discussion, λ and Y represent vectors consisting of particular activity values and measured number of counts for each LOR, respectively. The left–hand side of equation (1.17), is the appropriate likelihood function for Poisson distributed statistical noise in the measured Y. Equivalently, one can also find a solution, λ, which maximises the slightly more mathematically tractable log–likelihood function: `(Y|λ) = ln(L(Y|λ)) =
N X i=1
Yi ln(Y¯i ) − Y¯i + constant ,
(1.18)
where the constant, which is equal to ln(Y i !), does not involve the unknown, λ, and therefore can be ignored in the maximisation process. It can be shown [81], that for system matrices which can be modelled in the form of equation (1.15), a solution, λ, can be found which increases the log– 46
Chapter 1. Introduction likelihood function, `(Y|λ), monotonically at each iteration, n, and that converges as n → ∞ to the solution which maximises both equations (1.17) and (1.18). This solution can be written concisely as: λn+1 = j
λnj N X
Pij
N X i=1
i=1
Pij Yi M X
,
(1.19)
Pik λk
k=1
where Yi is the measured number of counts for the i th LOR, λnj is the nth iterative estimate of the activity in object voxel j, and P ij is the system matrix element for activity voxel j and detector pair i. The sum over M is over all activity voxels in the reconstructed object and the sum over N is over all LORs. The term
M X
Pik λk is the forward projection step and
k=1
represents the estimated number of counts in the i th LOR. The entire equation (1.19) represents a back–projection, in which a weighted ratio of Y i over the estimated projection pixel value is used to create a new estimate of the activity distribution. The solution described above was derived assuming the multiplicative system matrix of equation (1.15). For real data, additive factors should also be present to account for scatter and randoms as in equation (1.16). If one simply pre–subtracted the scatter and randoms estimate, the corrected data would no longer be distributed according to Poisson statistics and data reconstructed using equation (1.15) will be biased [82]. The data available to the end users of a PET scanner are frequently pre–subtracted for randoms, using the delayed coincidence window described previously. In this situation, the shifted–Poisson model should be used [82, 83], in which equation (1.15) is modified to include terms which account for the contribution from scatter and randoms to the counts in each LOR.
1.10.5
Ordered Subsets -Expectation Maximisation
A serious limitation of the MLEM algorithm is the time necessary to perform a reconstruction. The problem is exacerbated as the system matrix used
47
Chapter 1. Introduction becomes more complicated (less sparse) to better model the physics of the PET data acquisition process. A commonly used technique to accelerate MLEM is the ordered–subsets expectation maximisation (OSEM) algorithm [84]. The MLEM algorithm uses all the projection angles in each iterative step of the reconstruction process. The iterative process for OSEM is the same as MLEM with the exception that each iteration is accelerated by only employing a subset of the projection angles progressively using different subsets in each iteration. The OSEM algorithm converges to an image very similar to the MLEM result in a much shorter amount of time [84], provided that the subsets are ordered in such a way that differences between the information contained in subsequent subsets are maximised and the number of projections included in each subset is not too small.
1.10.6
Maximum a Posteriori (MAP) penalised reconstructions
Both OSEM and MLEM have a number of advantages over analytical methods such as FBP and 3D–RP, including the ability to better model the emission and detection processes and improvement of image noise properties due to its accurate modelling of the statistical noise in the measured data. One of the disadvantages of these techniques is that they do not converge to a unique solution in the presence of statistical noise in the sinogram data. Typically, when an iterative process is used to solve a problem, convergence is said to have been reached when some measure of the difference between subsequent iterations (such as the change in log–likelihood function) is less than some predetermined value. This definition of convergence does not seem to be appropriate for MLEM–based algorithms. As these methods iterate, the noise characteristics of the reconstructed image degrade as the image is fit to the noise in the measured data. As a result, it is not obvious at what number of iterations the reconstruction process should be stopped. This optimal number of iterations cannot be known prior to reconstruction because it depends on the spatial frequencies and intensities present in the unknown activity distribution as well as the exact form of the system model
48
Chapter 1. Introduction used. One of the proposed solutions to this problem is the use of Bayesian image reconstruction methods [85]. These techniques use Bayes’ theorem, named after the 18th century Reverend and statistician Thomas Bayes. In the context of the likelihood functions used in iterative image reconstruction, we can express Bayes’ theorem as: L(λ|Y) =
L(Y|λ)L(λ) , L(Y)
(1.20)
where L(Y|λ) is a likelihood function, such as that given for the Poisson model by equation (1.17), which is the statistical likelihood for obtaining Y given a fixed activity distribution, λ. Conversely, the term L(λ|Y) is the posterior probability which represents the likelihood for λ given a fixed set of measurements Y. The likelihood function L(λ) is some function which penalises solutions, a priori, which are considered unlikely. This term is often referred to as the prior, since it can contain any prior information that we might have about the expected distribution of activity and reflects our expectations concerning the relative probabilities of different images. The reconstruction task, is then formulated as determining the maximum a posteriori (MAP) estimate of λ which maximises the natural logarithm of the posterior probability given by: `(λ|Y) = ln(L(λ|Y)) = `(Y, λ) + `(λ) + constant,
(1.21)
where the constant, equal to − ln(L(Y)) in this case, does not influence the maximisation. The term `(Y, λ) is simply the log–likelihood of equation (1.18). A wide range of maximisation methods and priors have been proposed for PET imaging. Here we will neglect the mathematical details of maximisation (an excellent review of these topics was presented by Leahy and Byrne [86]) and focus instead on the characteristics of the most commonly used priors. Typically, the priors used for image reconstruction take the following form
49
Chapter 1. Introduction [87]: `(λ) = β
N k∈H Xi X
Vi,j (λi , λj ),
(1.22)
i=1 k6=i
where β is a parameter which influences the degree of smoothness of the estimated image, Vi,j (λi , λj ) are pair–wise potential functions defined for a specific “neighbourhood” of voxels, H i , which are spatially close to the ith voxel. The potential functions are ideally selected to encourage piece–wise smooth images without over-smoothing abrupt changes, which presumably should occur at the boundaries between different tissue types. One example of such a function is the Huber penalty function [88] given by: Vi,j (λi , λj ) =
(
|λi − λj |
if |λi − λj | > δ 2
(λi − λj ) /δ if |λi − λj | ≤ δ (1.23)
The Huber function changes from a quadratic (for heavier smoothing) to linear (for less smoothing) penalty at an intensity difference set by the δ parameter. This has the effect of reducing the penalisation for intensity changes greater than δ and, therefore, does not smooth the image excessively [87]. One of the primary difficulties for MAP reconstructions is selecting appropriate values for the parameters of the penalty function. Some of the parameters can be selected using one’s a priori expectations about the activity distribution. For example, the term δ in the Huber penalty function physically represents a cut–off parameter at which the heavier quadratic smoothing is switched to a more edge–preserving linear smoothing and, as such, can be selected intelligently based on a preliminary reconstruction. However, the selection of the smoothing term β is more difficult. A number of methods have been proposed for this purpose. The β parameter can simply be chosen heuristically to meet certain qualitative criteria (e.g. as β is increased the MAP image estimate becomes increasingly smooth). Alternatively, one could use statistical criteria to select β [89], however, the
50
Chapter 1. Introduction resulting images demonstrate non–uniform spatial resolution even for ideal simulated systems with spatially invariant resolution [90]. Alternatively, one can use a different β for each voxel of the image [91] determined to statistically optimise the local contrast–to–noise ratio. More complex methods have been proposed [90, 92] to yield uniform image resolution using leastsquares fitting of a parameterised local impulse response to a desired global response.
1.11
Purpose of this work
The primary focus of this research was to improve the correction for photon attenuation employed for the microPET series [46, 55] dedicated small animal PET scanners (see figure 1.1). Presently, attenuation corrections for these scanners are calculated using singles–mode transmission scans (discussed in greater detail in the next chapter), in which a radioactive source is rotated around the object being imaged and the transmitted photons are recorded. These data currently result in incorrect attenuation corrections (errors as high as 50% in the measured linear–attenuation coefficient for water) primarily because the effects of photon scatter are not accounted for properly. The purpose of this work is, therefore, the development of an analytical scatter correction for PET transmission scanning. We tested our correction method for two different transmission sources. The first was a parent–daughter source. Although the half–life of 68 Ge
decays by electron capture to
68 Ga
68 Ga
68 Ge/68 Ga
is only 68 minutes,
and has a half–life of 271 days al-
lowing the two species to exist in transient equilibrium [7]. As a result,
68 Ge
provides a source of positrons with a much longer effective half–life (i.e. 271 days) than that of 68 Ga alone. The second source was 57 Co, which decays by electron–capture to 57 Fe and has a 272 day half–life. Two major emission photons are produced during this decay with energies of 122.1 keV (85.6% intensity) and 136.5 keV (10.7% intensity) [93]. An accurate and reliable method to correct for scatter in transmissions scans will improve the quantitative accuracy and biological relevance of small animal PET imaging. Some 51
Chapter 1. Introduction common applications for these small animal scanners include neuroreceptor studies, blood–flow and glucose metabolism, and investigation of tumour hypoxia. All of these studies would benefit from improved quantification (i.e. having PET images that more accurately represent absolute radio–tracer concentrations). Quantitative PET imaging could also lead to new insights into physiology and improved efficacy of a medical treatment by improving the reliability with which one could compare images before and after medical interventions. This research has taken place in several stages. The first stage was the development of a Monte-Carlo (MC) based simulation tool to isolate the effects of scatter in the transmission data. Our simulations have also helped us identify two other important physical effects (detector dead–time and contamination from an intrinsic radioactivity present in the PET detector crystals) that can degrade the accuracy of PET attenuation corrections. The next stage was to implement an analytical scatter correction and validate the results by comparing our model to simulated data in which the true distribution of scattered photons and true attenuation–map is available. Since scatter depends strongly on the distribution of attenuating material, an unknown quantity we are trying to determine with the transmission scan, the next stage of this research was the incorporation of our validated scatter correction into an iterative reconstruction algorithm for PET attenuation– maps. We tested our scatter correction and reconstruction procedure using data from two different small animal PET scanners, one meant for rodents and another for primates. Finally, we also extended our methods for post– injection transmission scans (i.e. transmission scans in which non–negligible radio–tracer activity is present in the subject being imaged). For these data, we combined our scatter correction with a measured correction for the contamination to the transmission data from the emission activity.
1.12
Outline of Thesis
The work described in this dissertation has resulted in a number of publications [94, 95] and presentations [96, 97, 98]. In this thesis, we have 52
Chapter 1. Introduction rewritten, reorganised and expanded upon the material appearing in these publications and present additional results which have not yet been published. This manuscript consists of eight chapters. The following summary briefly describes the purpose and contents of each of the subsequent chapters in this thesis. Chapter 2: Literature Review The purpose of this work was to develop a technique to compensate for scatter in the transmission scans used to compute attenuation corrections. Hence, chapter 2 presents a review of the methods which have been used to obtain attenuation corrections in PET and the challenges associated with them. This chapter also briefly reviews the scatter correction techniques which have been employed in PET emission imaging. Chapter 3: Validation of the Simulation Data In this chapter, we describe the validation of our simulation software using experimental data from the microPET series small animal PET scanners. This chapter begins with a description of the PET imaging systems used here, followed by a discussion of our proposed correction for the contamination from the intrinsic radioactivity present in the PET detector crystals. In order to validate the parameters used in our simulations, we first compared simulated and experimental emission data (i.e. PET data obtained using conventional coincidence–mode acquisition). We then modified the simulation software in order to model transmission data acquisition. We compared simulated and experimental data for two different commonly used transmission sources (57 Co and
68 Ge)
and for three different sized water cylinders
which cover a wide range of possible applications for these small animal PET scanners. The purpose of this chapter was to verify that our simulations provide realistic PET data, which can be used to test and guide the development of our correction and reconstruction techniques that are employed in later chapters.
53
Chapter 1. Introduction Chapter 4: Development and Validation of the Scatter Correction This chapter begins with a description of our proposed scatter correction which determines the contribution from scattered photons to PET transmission data using an analytical model of photon transport through an arbitrary attenuating medium. We then discuss how we have validated our correction data using realistic simulated PET transmission data for each of the and
68 Ge
57 Co
transmission sources using the uniform water cylinder data of the
previous chapter. We also test our calculation for both sources using a more complex non–uniform phantom consisting of Teflon, water and air. These studies were undertaken in order to verify that, when provided with the correct distribution of attenuating material (true attenuation–map), our analytical calculation correctly determines what it is meant to calculate, that is, the distribution of scattered (and unscattered photons) in PET transmission data. Chapter 5: Development and Validation of the Reconstruction Procedure This chapter is concerned with the implementation of our scatter correction as part of a reconstruction algorithm for PET transmission data. The chapter begins with a description of our proposed reconstruction procedure. Since our scatter correction requires an estimate of the attenuation–map, which is precisely what we are trying to determine by reconstructing the transmission data, we require an iterative reconstruction procedure. We then proceeded to test our scatter correction and reconstruction procedure for the uniform and non–uniform phantom data using simulated and experimental data. In the case of the simulated data, we can isolate the artifacts associated with scatter and investigate in a step–by–step manner other features associated with these data (the influence of noise, data pre– processing, and scatter originating from outside the scanner). Finally we also applied our procedure to experimental data for two typical rodent studies (a mouse and a rat). The purpose of this chapter is to demonstrate that our 54
Chapter 1. Introduction reconstruction method provides the correct linear attenuation coefficients expected for each of the attenuation configurations considered. Chapter 6: Extension of the Scatter Correction for a Primate Scanner The purpose of this chapter is to demonstrate the general validity of our scatter correction and reconstruction procedure for a different imaging system and for larger objects. We applied our reconstruction and scatter correction procedure using 57 Co singles–mode transmission data acquired using a PET scanner suitable for primate studies. We reconstructed attenuation–maps and evaluated their accuracy with and without our scatter correction for a water cylinder and for a monkey study. Chapter 7: Post–Injection Transmission Scanning This chapter describes how our scatter correction and reconstruction methods can be applied to singles–mode transmission data which are acquired after the radio–tracer activity is administered to the imaged subject (post– injection). For post–injection transmission, contamination from the injected activity contributes to errors in the reconstructed attenuation–maps if appropriate corrections are not applied. In this chapter, we combine our analytical scatter correction with a measured emission–contamination correction. We evaluate the accuracy of attenuation–maps with different combination of corrections and analyse the reconstructed emission images which have been corrected using these data. Chapter 8: Conclusions and Future Work This thesis will conclude with a summary of the work done thus far. A discussion of future work that could further improve the utility of our reconstruction and scatter correction methods is also provided.
55
Chapter 2
Literature Review In this chapter, we present a review of attenuation and scatter correction techniques which have been employed for PET imaging with a focus on small animal applications.
2.1
Attenuation Corrections
Attenuation can introduce significant artifacts in reconstructed PET images. These effects are illustrated in figure 2.1 which demonstrates the influence of attenuation for a 30 mm radius water cylinder filled with a uniform concentration of activity. We generated simulated data using a line–integral model of PET data acquisition for an imaging system with similar characteristics as the microPET R4 scanner. Poisson noise was added to the simulated data, with a maximum of about 6000 counts per line–of–response (LOR) (without attenuation the maximum number of counts would be 10,000 in this case). The data was then reconstructed using FBP (using reconstruction software [99] developed by Dr. Jeff Fessler’s research group in the Department of Electrical Engineering and Computer Science at the University Michigan) with and without corrections for attenuation. As we can see from the figure, the reconstructed intensities for regions lying deeper in the object are underestimated if no correction is made for attenuation. Photons from regions deeper in the object will be attenuated more strongly than those closer to the surface (creating the “ring” artifact visible in the uncorrected reconstruction). For this particular set of data, there is an approximately factor of two difference between the corrected and uncorrected reconstructions at the centre of the object. The attenuation correction factor (ACF) for photons emitted along an LOR passing through 56
Chapter 2. Literature Review the centre of this cylinder can be computed as ACF= exp(+µD) = 1.78 where D is cylinder diameter and µ is the known linear attenuation coefficient for water at 511 keV (µ=0.096 cm −1 [10]). The effect of attenuation is much larger for whole–body human imaging where the ACFs can be as high as 70–100 for LORs which pass through the thickest parts of the human body [100]. The effects of attenuation are much more complex for non–uniform objects and, without accurate attenuation corrections, reconstructed images can exhibit geometrical distortion and artifacts which can be difficult to predict. As a result, it has long been recognised [100] that it is necessary to correct for photon attenuation to obtain quantitatively accurate PET images. For example, in regions of highly non–uniform attenuation, such as the thorax, it has been shown [101] that small tumour–like volumes of increased activity concentration can “disappear” in images reconstructed without attenuation correction. The critical tracer concentration at which the zero–contrast effect occurs depends on the size, location, and relative activity of the tumour. Below the critical value, the tumour appears to be of lower activity that its background. Examples of this effect were identified in simulated data, physical phantom studies, and in clinical oncology using FDG imaging [101]. A number of algorithms have been proposed for attenuation correction which assume uniform attenuation within the patient. In the brain and in regions of the abdomen where soft tissue is the dominant constituent this can be a reasonably accurate approximation. The body contour can be determined either manually or with automatic edge–detection methods from the emission data [102]. The regions within the contour can then be assigned the linear attenuation coefficient for water (or equivalently soft–tissue), thereby generating a patient specific distribution of linear attenuation coefficients within the body (attenuation–map). In many clinical and research applications, however, the attenuation distribution is not uniform (e.g. in the chest region of the thorax) and a more accurate method should be used to generate the attenuation–map. These include attenuation corrections derived from photon transmission scanning, rescaled CT images, or segmented MRI 57
Chapter 2. Literature Review
With AC
No AC
−50
−50
0
0
50 −50
0
50
50 −50
10000
5000
0
50
0
Average Profile
Counts
10000
With AC No AC
5000
0 −50
−30
−10 10 Distance (mm)
30
50
Figure 2.1: Comparison of reconstructed images (FBP) for simulated PET data for a 30 mm radius water cylinder with uniform activity concentration. data. These methods are described in more detail in the following sections.
2.2
Attenuation Corrections using Photon Transmission
PET attenuation corrections have traditionally been measured using external positron emitting sources [12]. The attenuation correction factors are obtained by comparing the coincidence count–rates for annihilation photons transmitted through the subject (the “transmission” scan) for each 58
Chapter 2. Literature Review LOR with those obtained from a reference acquisition (the “blank” scan) in which the subject is not present. The earliest types of photon transmission measurements [28] were obtained using thin rings or cylinders of positron– emitting activity (shown schematically in figure 2.2 (a)). The most common type of activity used were
68 Ge/68 Ga
parent–daughter sources which co–
exist in transient equilibrium with half–lives of 271 days and 68 minutes, respectively [7]. The sources could be mechanically moved into the scanner field–of–view (FOV) and were stored behind shielding when not needed. With the ring source geometry, however, the acquired data contained large fractions of detected counts from both randoms and scattered photons. Although accurate randoms corrections could be determined using the delayed window technique, described in the previous chapter, it was difficult to distinguish between scattered and unscattered photons using this geometry. This led to biased estimates of the attenuation correction factors (ACFs). One method to address this problem, first proposed by Carroll et al. [103] and later implemented by Thompson et al. [104], was to use single or multiple orbiting rod sources. This geometry is shown schematically in figure 2.2 (b). The scatter and randoms fractions were reduced for this geometry by “windowing” the transmission data, so that events were only accepted when the LOR recorded between two detectors was collinear with the known location(s) of the rod(s). Although rod sources were well suited to scanners which could be operated in 2D mode (with inter-plane septa in place), this geometry was less ideal for dedicated 3D PET scanners. When operated in 3D–mode, the photon–flux in the detector blocks closest to the rod source would be extremely high causing significant dead–time. Differences between the dead– time effects in the blank and transmission scans (which can be subject to very different count–rates) could cause errors in the ACFs. Currently, most photon transmission scans for 3D PET scanners are acquired in “singles–mode” [105] in which only the detectors further from the source (where photon flux is less) are used. The LOR associated with each event is determined using the known positions of the transmission source and the location of the crystal where the single–photon was recorded. In 59
Chapter 2. Literature Review order to determine unique LORs, one is limited to using a point–like source, which usually rotates in a circular orbit and translates axially to cover all possible LORs. This geometry is shown schematically in figure 2.2 (c). In addition to reducing problems associated with dead–time, in singles–mode one is no longer restricted to using positron–emitting sources [106, 107] allowing for the separation of the emission and transmission data based on energy discrimination. Two non–positron emitting sources currently used for singles-mode transmission data are
137 Cs
and 57 Co with photon energies
of 662 keV and 122 keV, respectively. For these sources the transmission data can be acquired in a separate energy window allowing for simultaneous or post–emission transmission scans. For singles–mode data acquisition “windowing” is no longer possible, and contamination from scattered photons can, therefore, be much higher [108] leading to errors in the estimated PET attenuation corrections. Some of the problems associated with singles–mode transmission acquisition have been addressed with the introduction of collimated
137 Cs
sources
[109, 110] which use a similar scanning geometry as that used in fan–beam X–ray computed tomography (CT). This geometry was evaluated using clinical data by Watson et al. [110]. The authors found that using a high activity (222 MBq)
137 Cs
source with a fan–beam tungsten collimator reduced scat-
ter and provided low–noise transmission data. A similar design has been implemented recently for the HRRT dedicated human brain PET tomograph [111]. Although scatter is reduced using collimated sources it cannot be completely eliminated [110] and contamination from emission photons when the scans are performed post–injection can also be a significant problem for these data [112]. As a result, it is recommended [110] that further post–processing of the transmission data (reconstruction of the attenuation distributions followed by segmentation as discussed in 2.2.1) should be performed before attenuation correction factors are calculated using these data. Several new approaches are being investigated to reduce emission contamination and scatter for point–like transmission sources. For example, in Watson et al. [113] coincidences were recorded between a small LSO detector mounted inside a collimated positron–emitting transmission source and 60
Chapter 2. Literature Review
µ−map
µ−map
Transmission Source
(a)
Transmission Source
(b)
µ−map
Transmission Source
(c)
Figure 2.2: Comparison of three different transmission geometries which have been employed for cylindrical PET detector designs. The figures (a) (b) and (c) respectively show: a stationary ring of positron emitting activity (the PET scanner is operated in coincidence–mode in this case); rotating rods of positron emitting activity (coincidence–mode); and a rotating source where the detectors are operated in singles–mode, that is, “coincidence”– events are made using the known source position and the recorded photons in the side of the detector ring opposite the source. The grey–scale image in the centre represents a possible attenuation distribution (µ–map) for the thorax. Examples of photon paths are shown as the solid arrows while the directions of source motion (if applicable) are shown as dashed arrows. This illustration is based on a similar figure appearing in Bailey et al. [12]. the detector ring opposite the source. Using similar principles, Camborde et al. [114] surrounded a positron emitting transmission source with a thin plastic scintillator dedicated for the detection of positrons and then recorded positron–gamma coincidences. Although, in principle, these techniques can reduce emission contamination and scatter, further investigation of technical problems such as randoms [113] and sub–optimal source geometries [114] need to be resolved before they can be used routinely.
2.2.1
Photon Transmission Post–Processing
As described in the introduction, the attenuation correction factors (ACFs), which are necessary for quantitatively accurate PET imaging, can be applied 61
Chapter 2. Literature Review as a simple multiplicative correction factor (the inverse of the transmission probability) for each line–of–response (LOR) of the acquired PET data. For the ith LOR of the measured PET data the ACF is given by: ACFi = exp(+
Z
~ `i
µ (~r, E)dr) ,
(2.1)
where the integral path, dr, is along a straight line with vector coordinates ~`i which connect the pair of detectors corresponding to the i th LOR and µ(~r, E) is the spatial distribution of linear attenuation coefficients inside the scanner for annihilation photons with an energy E = 511 keV. In order to obtain the ACFs, it is necessary to perform both blank scans (where the subject being imaged is not present inside the scanner) and transmission scans (with the subject in the scanner). The measured number of counts, Ti , for the ith LOR of the transmission data is approximately given by:
Ti ≈ Bi exp −
Z
~ `i
µ(~r, E)dr + Ci ,
(2.2)
where Bi is the corresponding number of counts for a blank scan of the same R
duration as the transmission scan; exp ~`i µ(~r, E)dr is the transmission probability along the ith LOR for photons with energies E which may or may not be equal to 511 keV depending on the transmission source radionuclide used; and Ci is a term to account for any background contamination which, depending on the transmission source geometry and acquisition settings, may include the contribution from scattered photons, randoms, emission
contamination and any other undesired source of counts. In principle, we can compute the required attenuation correction factors using the measured transmission and blank data (provided that a positron emitting transmission source is used and the contamination term C i is negligible or can be subtracted from the transmission data) using the following expression: ACFi ≈
Bi ≈ exp + Ti − C i
Z
~ `i
µ(~r, E)dr ,
(2.3)
The relationships of equations (2.2) and (2.3) are approximate ones, be-
62
Chapter 2. Literature Review cause radioactive decay is a stochastic process and the measured number of counts for both the blank and transmission scan are, to a good approximation for conventional scintillation crystal counters, Poisson distributed variables with a standard deviation equal to the square–root of the mean number of counts over an infinite ensemble of identical measurements [115]. As a consequence, direct application of measured ACFs, as given by equation (2.3), leads to substantial noise propagation into the corrected emission data [116]. This occurs because the denominator in the equation (2.3) becomes very small when either the attenuation coefficients are high (e.g. dense regions such as bone) or the thickness of tissue is large (e.g. through the thorax particularly if the patient’s arms are at their sides). To overcome this problem, the transmission and blank scans have traditionally been smoothed using low-pass filtering before calculating the ACFs [102]. This smoothing leads to a mismatch between the effective resolution of the ACFs and the measured emission data (which are not normally smoothed) and can cause bias and artifacts for reconstructed PET images in regions where the attenuation coefficients are non–uniform (at the interface between different tissue types and the edges of objects [117, 118]).
2.2.2
Reconstruction of Photon Transmission Data
Alternatively, one can use the blank and transmission data to reconstruct the distribution of linear–attenuation coefficients µ(~r, E) (which we refer to as an attenuation–map or µ–map throughout this thesis) and the ACFs can then be computed using equation (2.1). This method reduces the noise which is propagated into the emission data (depending on the quality of the reconstructed attenuation–map) and differences between the emission and transmission photon energies can be taken into account. One of the most commonly used methods for image reconstruction of transmission data is to simply reconstruct the natural–logarithm of blank data divided by the transmission data. Using the notation of the previous section, the data which is to be reconstructed has the following form:
63
Chapter 2. Literature Review
Z
Bi . µ(~r, E)dr ≈ ln ~ Ti − C i `i
(2.4)
The data in this form represent line integrals through a spatial distribution of attenuation values and are analogous to the emission reconstruction problem. One can, therefore, reconstruct these data using any of the reconstruction methods (e.g. FBP, MLEM, OSEM or MAP) which were discussed for emission data in the introduction. When the operation described by equation (2.4) is applied to the transmission and blank data, however, the statistical properties of the processed data differ from that of emission data. In particular, the logarithm function is only defined for positive numbers and does not exist for zero or negative values of the numerator. Moreover, for low–count data the logarithm skews the distribution of equation (2.4) leading to biased estimates of the line integrals and incorrect reconstructed linear attenuation coefficients [119, 120, 121]. Lange and Carson [122] first formulated a maximum likelihood expectation maximisation (MLEM) algorithm which uses the correct Poisson statistics of the measured blank and transmission data. The MLEM reconstruction algorithm for transmission data exhibits reduced bias, relative to emission reconstructions applied to logarithm data, but converges very slowly (many iterations are required) [122, 123]. It has long been recognised [124] that transmission tomography also provides an excellent opportunity to incorporate a prior information about the expected distribution of linear attenuation coefficients (µ–values) using Bayesian methods such as the maximum a posteriori (MAP) reconstruction algorithms. These methods have been widely found to converge faster than the original transmission maximum likelihood expectation maximisation (MLEM) algorithm [120]. The speed of convergence has been further improved with the introduction of several MAP reconstruction algorithms which apply the ordered–subsets principle (see the discussion in section 1.10.5) to transmission data [125, 126]. A common criticism of MAP for PET emission data is that the selection of the functional form of the penalty term and its associated parameters 64
Chapter 2. Literature Review can seem somewhat arbitrary. MAP is better suited for transmission tomography because the reconstruction problem is more constrained. That is, the attenuation–map should only contain a limited number of materials and tissue types. Furthermore, one expects that relatively sharp boundaries should exist between different regions, constrained only by the resolution of the PET imaging system. For example, in the thorax one expects to find muscle, adipose, soft–tissue, bone, the lungs and the patient–bed each with their own unique µ–values which have been tabulated in various sources (e.g. in Hubbell and Seltzer [13]). As a result, the penalty function and most of its parameters can be selected intelligently. The types of penalty functions which have been used for transmission tomography can be broadly classified into two categories. The first type use local smoothing functions [119, 120, 123, 125, 126, 127, 128] which penalise voxels which differ from their neighbours and preserve edges in the reconstructed image (e.g. the Huber penalty function as discussed in the introduction). Another interesting type of penalty which has been successfully applied to transmission data are “point–wise” priors [15, 129, 130], which do not enforce interactions between neighbouring voxels. Rather, these penalties draw each pixel toward certain local mean values which are selected from a small list of µ–values corresponding to the different materials and tissue types which are expected to be present in the scanner. Reconstructed images with many voxels that differ from the expected means are penalised, usually subject to some tolerance parameter which compensates for the partial volume effect (imperfect resolution and voxels which may contain a mixture of materials). Essentially, these methods perform an automated segmentation as part of the reconstruction procedure (routines which segment the image data after the reconstruction are discussed in the next section).
2.2.3
Segmentation of Attenuation–maps
When using singles–mode acquisition for PET attenuation corrections, it is not sufficient to simply reconstruct the data and then compute the attenuation correction factors using equation (2.1). Some additional post–
65
Chapter 2. Literature Review processing of the data is usually applied. The most important reason for this post–processing is the high number of scattered transmission photons which are present, especially for scintillators with poor energy resolution, such as BGO and LSO [131], and for uncollimated sources such as those used for the microPET series scanners [46, 55]. Singles–mode transmission data are not corrected for scattered photons, rather the data are simply reconstructed and then segmented. The data may also contain contamination from emission photons if the transmission data are acquired after the radio–tracer is administered. Many approaches for the segmentation of reconstructed transmission data have been proposed. Most of the earlier methods [118, 132] processed the attenuation–map into histograms of µ–values. These methods generally search for peaks in the histograms (e.g. in whole–body imaging three distinct peaks can usually be identified corresponding to air, lung and soft–tissue). Each voxel in the attenuation–map is classified as a particular tissue type, based on thresholds which are applied manually or derived by fitting the histogram, and then reassigned the “correct” µ–value for that tissue type. Another type of approach is a “fuzzy–clustering” segmentation technique [133, 134]. These methods are iterative procedures that minimise an objective function similar to the iterative image reconstruction methods discussed previously. They require as inputs the reconstructed attenuation–map and a user–specified number of “clusters” (corresponding to the number of tissue and material types expected to be present). For each voxel, the probability of belonging to a particular cluster is estimated. Obviously, the success of segmentation methods depends on the quality of the attenuation–maps and problems occur: (a) if the data are noisy; (b) or if few voxels are present in the image corresponding to a particular tissue type (such as bone [102]); (c) or if the bias in the attenuation values vary spatially in the attenuation– map, as is the case for transmission scans subject to high scatter fractions [108]. Even if the transmission data have been corrected for contamination from scattered photons and emission photons the linear attenuation coefficients (µ–values) may still be incorrect, since the energies of the photons emitted by 66
Chapter 2. Literature Review the transmission source are not typically equal to 511 keV (e.g. for
137 Cs
the
photon energy is 662 keV). The reconstructed µ–values must be corrected to account for this difference. For energies close to 511 keV, the dominant photon interaction for materials of biological interest is Compton scatter. As a result, a simple rescaling of the µ–values is sufficient. One can, for example, rescale the attenuation–map using the ratio: µ water (EE )/µwater (ET ), where µwater (EE ) and µwater (ET ) are the linear attenuation coefficients of water at the emission (511 keV) and transmission photon energies, respectively. For transmission sources which emit lower energy photons a simple rescaling can lead to errors, especially for materials with high effective atomic numbers such as bone where photo–electric absorption is more important [135]. To account for these differences, the reconstructed attenuation–maps should be processed using a bi–linear rescaling [136] or using a mixture of segmentation and rescaling [107, 137].
2.2.4
Post–injection Photon Transmission
There are many types of PET studies which require long radio–tracer uptake times, for example, cancer studies using fluoro–deoxy–glucose (FDG) usually require an uptake period of 40 minutes or more to enhance the contrast between the tumour and normal tissue [138]. Furthermore, there is some evidence that the uptake of FDG in tumours does not saturate until 2–3 hours post–injection while normal tissue saturates 1 hour following injection [139]. As a result, improved tumour to normal–tissue contrast can be obtained by simply increasing the uptake period. Conventional pre– injection transmission measurements require that the patient remain still on the imaging bed while the radio–tracer is taken up by tissues [140]. In the case of small animal imaging, the subject must be kept under anaesthesia during uptake which increases the biological stress on the animal and can eventually lead to its death [40]. Post-injection measurements can increase survival and allow for serial imaging of the same animal. For humans, postinjection transmission scanning allows the uptake period to be spent more comfortably in a waiting room. The patient is then only on the bed for
67
Chapter 2. Literature Review actual data acquisitions which increases throughput and also decreases the likelihood of patient motion [141]. Various approaches have been proposed to eliminate contamination from emission photons (due to the injected activity) in transmission data acquisition performed post–injection [102]. The details of how these corrections have been implemented depend on the transmission source geometry used. For the previous generation of 2D PET scanners, transmission measurements were acquired in coincidence–mode using rotating rods of activity. For this geometry rod–windowing was used [103] to reduce scatter and emission contamination. Various methods were proposed to account for the residual emission contamination present in the transmission data with rod–windowing. Since these data were acquired in coincidence–mode, the emission contamination had a very similar distribution as the ordinary emission sinograms. As a result, a correction could be derived using the emission data rescaled to account for radioactive decay and for the additional attenuation due to the presence of the rod–sources [141, 142]. Alternatively, the contamination has also been estimated by acquiring a separate dataset, concurrently with the transmission data, such that all LORs were collinear with an imaginary source offset from the physical source by some small angular offset [138]. This contamination scan, however, was subject to a different distribution of scattered events than the transmission data. Thereby causing errors in the attenuation corrections which could be difficult to predict [140]. Another correction method was also proposed [140], in which the emission contamination was obtained directly by acquiring a “mock” transmission scan, in which the transmission source was removed or shielded from the detectors. A drawback of this correction was that it may not correctly represent the true distribution of emission contamination, since the mock scan was subject to different dead–time because the high activity transmission source was no longer present. For the current generation of PET scanners in which transmission data are acquired in singles–mode fewer options are available for the correction of post–injection emission contamination. Of the three methods described 68
Chapter 2. Literature Review above, only the mock scan correction can be used to obtain an accurate estimate of the emission contamination for these data. Although conceptually simple and potentially very accurate, the mock acquisition increase the overall scan time. It has been noted by several authors [112, 143, 144] that the shape of the emission photon contamination sinograms in singles–mode are spatially very smooth. These observations have prompted the use of very short duration mock scans, which are either smoothed to reduce noise [145] or averaged to obtain a constant estimate of the contamination for all LORs [112]. The accuracy of such estimates is not always sufficient, however, to obtain unbiased attenuation corrections [112]. One major advantage of singles–mode data, is the ability to distinguish between emission and transmission photons based on energy discrimination. For the acquisition of the microPET scanner transmission data using
57 Co
sources (with a photon energy of 122 keV), there are minimal amounts of emission contamination from 511 keV photons [107]. For human studies, however, these low–energy single photon sources are not penetrating enough and, therefore, the numbers of transmitted photons are insufficient to obtain good quality attenuation–maps. Consequently, 137 Cs transmission sources (with photon energies of 662 keV) are in more common use for human scanners. For scintillators with poor energy resolution (e.g. LSO and BGO) there can be significant contamination from emission photons (with energies of 511 keV) in these data [113].
2.3
Attenuation Corrections using X–ray transmission
The process of acquiring transmission scans for PET emission data using a photon–emitting source is conceptually very similar to the acquisition used for X–ray computed tomography (CT). In CT imaging, radiation emitted by a rotating X–ray tube is transmitted through the patients body and then recorded by an array of detector elements opposite to the tube and patient. It is not surprising, therefore, that attenuation corrections for PET
69
Chapter 2. Literature Review can be derived from CT images [135]. This can be achieved using sequential scanning of a patient on separate dedicated PET and CT scanners or by using a combined PET/CT scanner. In PET/CT imaging systems, the two modalities typically use separate hardware which is housed in a single gantry and a shared bed moves axially while the patient first receives a CT scan and then a PET scan. For separate stand–alone PET and CT scanners, a software–based image registration must be performed with or without the help of external fiducial markers [146]. For neurological studies, the skull provides a rigid structure that greatly simplifies the co–registration problem. Whole-body imaging can present difficulties because the organs within the torso can move and be more easily deformed [102]. Less deformation and co– registration errors occur for dual–modality PET/CT imaging systems [147], since the functional PET and anatomical CT images are obtained without moving the patient (other than the axial motion of the bed). CT–based attenuation corrections have become particularly important with the recent increase in use of combined PET/CT scanners in clinical oncology [26] and, more recently, in small animal PET imaging [148]. There are, however, some differences between the measurements obtained using X–ray CT and those from conventional photon transmission sources. For example, the photon flux which is typically used for X–ray CT imaging is considerably higher than that used for photon transmission, the emitted X–rays are of much lower energy than 511 keV, and are not mono–energetic (typically X–ray spectra have energies which are distributed between 30 to 130 keV [149]). The reconstructed intensities for CT images are usually given in terms of “Hounsfield units” (HU), which represent the linear attenuation for the effective energy of the X–ray spectrum and are water normalised to water using the following relationship: HU= µ−µ µwater × 1000
where µ and µwater are the effective attenuation coefficients for the particular voxel and for water respectively. With this definition, air and water have CT numbers of -1000 and 0 HU, respectively. Figure 2.3 shows the relationship between photon energy and the linear attenuation coefficient (µ) and mass attenuation coefficients (µ/ρ where ρ is the mass density) for several materials which are of particular interest for 70
Chapter 2. Literature Review PET and CT imaging. Photoelectric absorption and Compton scatter are the dominant interactions for lower (X–ray) energies and the linear attenuation coefficients depend on both the mass density and composition of the material (i.e. the effective atomic number). At higher energies, Compton scattering dominates and the variation between linear attenuation coefficients depend mainly on the mass density of the material. The preferential attenuation at lower energies forces the spectra of transmitted X–rays to shift toward higher energies as they are transmitted through increasing thicknesses of attenuating media, resulting in the well–known “beam–hardening” effect. Corrections for beam–hardening are typically provided for in commercial CT imaging systems, however, an accurate conversion between HU, which are applicable for low–energy poly–energetic X–ray spectra, to linear attenuation coefficients at 511 keV can still be challenging to derive. Several methods have been proposed for the conversion of HU to the appropriate linear attenuation coefficients for PET imaging. A comparison of simple rescaling, segmentation and combinations thereof was performed by Kinahan et al. [135]. Bi–linear rescaling is another commonly used conversion method which uses two different linear relationships between HU and µ–values, one for low and one for high values of HU [136, 150]. Another method uses dual CT–scans each with a different tube voltage setting (and hence different X-ray energy spectra) to decompose the Compton and photoelectric components of X–ray attenuation [151]. The use of CT–images for attenuation correction in PET can cause a variety of artifacts and degrade the quantitative accuracy of the PET images. Since the duration of a typical PET scan (20–60 minutes) can be so much longer than that of the CT scan (1–2 minutes), patient movement and respiratory motion can cause image co–registration errors, leading to incorrect anatomical localisation and errors in the CT–based attenuation corrections [152, 153]. Respiratory gating during the PET emission scans has been investigated as a way to minimise these problems [154]. The use of common CT contrast agents, such as iodine, can also cause “hot–spot” PET image artifacts [155]. This error occurs because the methods used to convert linear attenuation coefficients from low–energy to 511 keV which are applicable to 71
Chapter 2. Literature Review
X−ray Energy
4
10
511 662 keV keV
Iodine Cortical Bone Soft Tissue Lung Tissue
3
10
2
µ (cm−1)
10
1
10
0
10
−1
10
−2
10
1
10
2
X−ray Energy
3
10
3
10 Energy (keV)
10
511 662 keV keV Iodine Cortical Bone Soft Tissue Lung Tissue
2
µ/ρ (g/cm2)
10
1
10
0
10
−1
10
−2
10
1
10
2
10 Energy (keV)
3
10
Figure 2.3: Comparison of linear attenuation coefficient (µ) and mass attenuation coefficient (µ/ρ) for several materials of biological interest in the energy range applicable to PET/CT data. Data shown was compiled by Hubbell and Seltzer [13]. Note that, in the lower figure, the data for lung and soft tissue are very similar and cannot be clearly distinguished.
72
Chapter 2. Literature Review tissue overestimate the attenuation coefficients of contrast agents such as iodine [156] (see figure 2.3). Additional artifacts can occur for dense metallic objects such as prostheses [157] and dental implants [158]. For these data, CT images tend to exhibit “streak” artifacts around the metal objects. If the CT images are not corrected for this effect [159], errors propagate into the attenuation corrections and reconstructed PET images. Since the field of view for a CT scanner is smaller than that of a PET imaging system, image truncation can also cause errors in CT–based attenuation corrections [160]. Another source of error for PET/CT scanners is X–ray scatter in the acquired CT transmission data. Scatter is more pronounced in the latest generation of compact CT scanners, which use flat–panel detectors with a cone–beam source geometry, than it was for the previous generation of larger fan–beam CT scanners [161]. Perhaps the most serious criticism of PET/CT systems is related to the dose of harmful ionising radiation which is delivered during the CT portion of the acquisition. For conventional whole body PET/CT scanning, it has been shown [162] that the dose delivered during the CT scan is orders of magnitude larger than that delivered during a typical FDG–PET scan. Furthermore, Wu et al. [162] also found that conventional 68 Ge photon transmission scans delivered negligible dose compared with either PET or CT scans. For microPET/CT imaging of small animals [148], the problem can be even more severe, where it has been shown that radiation levels are high enough to induce changes in the immune response and other biological pathways which may alter the experimental outcomes [163].
2.4
Segmented MRI–based Attenuation Corrections
The concerns regarding CT dose and the poor soft–tissue contrast for neurological and small animal applications using CT imaging, have spurred considerable recent interest in simultaneous multi-modality PET/MRI. In order to combine PET and MRI in the same imaging device a number of techno-
73
Chapter 2. Literature Review logical challenges have had to be addressed. The most significant challenge is that conventional PET detector blocks use photomultiplier tubes (PMTs), which simply do not work correctly in the presence of the magnetic fields and gradients used in MRI [102]. Two different approaches have been used to overcome this problem. The first feeds light from the scintillator crystals (which are not magnetic and can therefore be located inside the MRI magnet) through long optical fibres to shielded PMTs outside the magnet [164]. The quality of the PET images tend to be poor, however, because a significant loss of scintillation light occurs as it travel along the fibres. Another group of researchers [165] have replaced the PMTs with avalanche photo-diodes (APDs) which are solid–state photon detectors that are much less sensitive to magnetic fields. Interference can still occur, however, between the APDs and the rapidly switching magnetic field gradients and radio-frequency (RF) signals used in MRI. Although combined PET/MRI imaging systems are clearly a rich topic for research, the likelihood of such scanners becoming widely used depends on many factors. For example, the cost of PET/MRI scanners is anticipated to be very high and it is not clear that it will provide diagnostic information which is superior to co-registered images from separate standalone PET and MRI scanners or from PET/CT systems. It is also more difficult to derive attenuation–maps from anatomical MRI images due to the absence of MR signal from bone. Complex and unconventional segmentation routines [166, 167] are, therefore, required to obtain attenuation corrections from MRI images.
2.5
Scatter–Corrections in PET
As described in the introduction photon scatter can cause significant errors in reconstructed PET images if not corrected for. In order to demonstrate some of the typical energy and spatial characteristics of scatter in PET emission data, figure 2.4 shows the results of a Monte–Carlo simulation study for a line source inside a 30 mm radius water cylinder. The line source and cylinder were centred radially inside a simulated scanner based on the specifications of the microPET R4 scanner [55]. In our simulations, we only 74
Chapter 2. Literature Review recorded scatter interactions which took place within the water cylinder. In the energy spectra, shown in the left–hand plot, certain unscattered photons can be observed at energies much lower than 511 keV. This occurs as a result of scatter interactions within the scintillation crystals in which the full photon energy is not deposited in the detector. For the radial profile data, shown in the right–hand plot, a typical energy discrimination window between 350 and 750 keV was used and data were summed axially and over all projection and oblique angles to reduce noise. The total scatter fraction (number of detected photons which scattered within the cylinder divided by the total number of counts) for the simulated data using an energy discrimination window between 350 and 750 keV was 27%. In whole– body PET imaging with 3D data acquisition, it is estimated that scatter contributes as much as 60 to 70% of the total detected counts [3]. 5
8000
4000
Total Unscattered Scatter
4
10 Counts
Counts
6000
3
10
2
2000
0 0
10
Total Unscattered Scatter
10
200
400 600 Energy (keV)
800
1000
1
10 −50
−30
−10 10 Radial Offset (mm)
30
50
Figure 2.4: Simulated data showing the energy (left–plot) and radial sinogram profile (right–plot) for scattered and unscattered photons in PET emission data. The simulation modelled a 30 mm radius water cylinder, a positron emitting line source centred radially in a scanner based on the microPET R4. Data are shown separately for the total events (scatter+unscattered), unscattered photons and scattered photons. The data were generated using Monte–Carlo simulation software [8]. Many corrections have been proposed to account for scatter in PET data (recently reviewed by Zaidi and Koral [11]). These scatter compensation techniques utilise one or more of the following properties of scattered radiation in PET: (1) the randoms–corrected counts for the LORs which 75
Chapter 2. Literature Review lie outside of the object boundaries (which can be determined using a reconstructed µ–map) can only be the result of scatter; (2) the contribution from scattered photons to the acquired PET data is low–frequency and less structured in the spatial domain than the unscattered photon data; and (3) Compton scattered photons have lower energy than 511 keV. All of these characteristics are visible in the data shown in figure 2.4. The first and second characteristics are exploited in a class of methods which fit slowly varying functions (e.g. Gaussian [168] or polynomial [169] functions) to regions of the sinogram data that correspond to LORs which do not intersect the object (usually the outer edges of the radial profiles). For example, in the radial profile data illustrated in figure 2.4 the LORs with radial offsets less than -30 mm and greater than +30 mm lie outside the 30 mm radius of the cylindrical water phantom (information which would be available from a reconstructed µ–map). Counts in these LORs can, therefore, only be the result of scatter. Although these methods provide reasonably accurate scatter corrections when the activity and scatter medium are relatively uniform (such as brain scanning [168]), they can fail for objects with asymmetric scatter distributions (any non-uniform object) and for larger objects (such as whole body scans) where fewer LORs outside of the object are available for fitting. Convolution–subtraction techniques [170, 171, 172] model the scattered data as an integral transformation of the acquired data (which includes contributions from both scattered and unscattered photons). For example, Bailey and Meikle [172] obtained a scatter estimate by applying a two dimensional convolution on the measured PET data, using exponential functions with spatially varying parameters fit to experimental phantom data. The complexity and empirical nature of the scatter estimate makes these methods seem arbitrary and can lead to errors in more complex non-uniform objects such as the human thorax [173]. Other methods exploit the fact that scattered photons which have undergone Compton interactions have lower energy than unscattered photons by acquiring data in additional energy widows [174, 175, 176]. Scatter corrections have been proposed [174, 175] which estimate the scatter component 76
Chapter 2. Literature Review in the photo-peak window from events recorded in either one [175] or two [174] additional lower energy windows. The data collected in the lower energy window(s) are rescaled and then used as a subtractive scatter correction. The rescaling is based on empirical measurements using a source in air and measurements using the same source subject to scatter within a phantom. Energy–window based scatter corrections are no longer widely used in PET, especially for 3D acquisition where scatter fractions are high, since the scatter estimates contain statistical noise which propagates into the scatter corrected data and reconstructed images [11]. These methods also assume that photons detected in the lower energy window have the same spatial distribution as the scattered photons in the photo-peak window. This is not the case, however, since photons recorded in the lower energy window(s), are more likely to have scattered through larger angles and undergone multiple-scatter interactions. The multi–spectral method [176] combines the ideas of convolution– subtraction and energy–window based methods. For this correction, data is acquired in a very large number of windows each with a constant upper energy threshold (645 keV) but with a variable lower energy threshold (between 129–516 keV). Data is acquired using 16 different energy window settings for each detector making a total of 16×16=256 measurements for each detector crystal. The spatial distribution of scattered and unscattered components for each energy window is estimated by fitting phantom data using mono–exponential functions [176]. This method also attempts to differentiate between events corresponding to scatter in the detector (which are potentially useful for image formation) and scatter from within the patient (which should be subtracted from the acquired PET data). However, the hardware and software capabilities for multi–spectral acquisition is a major obstacle for implementation of this correction using commercial PET scanners [11].
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Chapter 2. Literature Review
2.6
Theoretically–Based Scatter Corrections
Another approach to model the spatial distribution of scattered photons is to use a theoretically–based description of the interactions of photons with matter (both in the patient and in the detector). Since the formulae that describe these phenomena are well known, one can, in principle, very accurately determine the scatter distribution for a given PET scanner and particular activity and attenuation distribution. However, the equations that describe the propagation of photons through the scattering medium and their detection do not in general have analytical solutions. As a result, many of the proposed theoretically–based methods reduce the complexity of the problem by making some physically–justifiable approximations and then use statistical simulation or numerical evaluation techniques to obtain solutions. Many such methods have been proposed [177, 178, 179, 180], most of which compute the distribution of “single–scatter”, that is, photon pairs detected in coincidence for which just one of the photons have undergone a Compton scattering interaction. It has been shown that between 75 to 80% of the total detected scatter for a typical PET scan consists of single–scatter events [177]. With reference to figure 2.5 and using the formulation of Watson et al. [179], the single–scatter coincidence rate, S AB , for the LOR con~ and B, ~ is necting the detector crystals, located at the vector positions A approximately given by: SAB =
Z
Vs
dVs
σAS σBS 2 R2 4πRAS BS
!
µ dσinc [IA + IB ] , σinc dΩ
(2.5)
where IA =
AS 0BS
IB =
0AS BS
exp(− exp(−
Z
~ S
~ A Z S~ ~ A
µds 0
Z
µ ds
~ B
µ ds)
~ S
Z
0
~ B ~ S
µds)
Z
~ S
~ A Z B~ ~ S
λds, λds. (2.6)
78
Chapter 2. Literature Review
A S
µ−map/ λ−map B
Figure 2.5: Schematic diagram to illustrate the single–scatter simulation scatter correction [14] as given by equation (2.5). The correction uses estimates of the distribution of attenuation (µ–map) and activity (λ–map) to compute the contribution from single–scattered photons. For the detector pair labelled A and B, one possible path for scattered photons is indicated by the dashed lines in the diagram. The line–of–response to which the scatter events would be assigned is shown as a solid line. Note that the energy of a Compton scattered photon depends on the scatter angle, as given by equation (1.5). Many of the processes involved with the propagation and detection of photons are energy dependent. In equation (2.5), all primed variables correspond to the scattered photon energy while unprimed variables are evaluated at 511 keV. To understand each of the terms given in equation (2.5), one must consider two types of photon paths. In the first type of path, a pair of annihilation photons are emit~ and S. ~ The unscattered ted from some point along the line connecting A ~ One of the photons undergoes Compton scatter photon is detected at A. 79
Chapter 2. Literature Review ~ and the scattered photon is recorded by the detector located at the point S ~ This path contributes the term IA to the above equation. For the at B. second type of path, which contributes I B , the emission takes place between ~ and S ~ and the scattered and unscattered photons are detected at A ~ and B ~ respectively. B, We will describe in detail only the first type of path. Any emission that ~ and S ~ can contribute to this path. The photon flux takes place between A ~ is, therefore, proportional to which is incident on the scatter position S the line integral:
R S~ ~ A
λds, where λ is the spatial distribution of annihilation
photon emissions (λ–map). The λ–map is typically estimated using a preliminary reconstruction of the emission data (which is usually not corrected for scatter). The probability that the unscattered photons will not be attenuated as they travel through the object is given by: exp(−
R S~ ~ A
µds) where
µ is the distribution of attenuation in the object being imaged (µ–map). Similarly, the probability that the scattered photons will not be attenuR ~ to B ~ is given by: exp(− ~B~ µ0 ds) where µ0 is ated as they travel from S S the distribution of attenuation at the scattered photon energy (obtained by
appropriately rescaling the µ–map). The integral over VS is over the entire volume of possible scattering positions. For most materials of biological interest, the linear attenuation coefficient, µ, at 511 keV is dominated by Compton scattering. Using equation (1.8), the approximate number of electrons present in infinitesimal volume dVs is approximately given by dVs × µ/σinc where σinc is the total cross-section for Compton scatter per electron (e.g. see Johns and Cunningham [181]). If we multiply the Klein–Nishina scattering cross–section,
dσinc dΩ ,
by the total number of electrons in dV s , we then have the scatter flux per ~ unit solid angle, divided by the incident flux per unit area at the position S. ~ we multiply by the solid To compute the total scatter flux incident at B, ~ for photons emitted from S. ~ The solid angle subtended by the detector at B 2 where σBS is a geometrical efficiency term angle term is given by σBS /RBS ~ and S. ~ A similar expression (σAS /R2 ) and RBS is the distance between B AS
gives the approximate flux of unscattered photons incident upon the detector ~ Finally we must multiply the single–scatter count–rate by located at A. 80
Chapter 2. Literature Review AS and 0SB which represent the efficiencies of the two detectors, at the unscattered and scattered energies respectively. The efficiencies take into account the angle of incidence of each photon, the stopping power of the crystals, and the energy resolution and discrimination settings used. We discuss the formulation of a similar expression for scatter in singles–mode transmission data, in greater detail in chapter 4. Methods which compute the single–scatter coincidence rate have been shown to be reasonably accurate under most circumstances [1]. They would be very computationally demanding, however, if it were necessary to compute the volume integral over every possible scatter point and for each pair of detectors in the PET scanner. Fortunately due to the typically broad distribution of scatter in 3D PET data, it has been shown that it is necessary to sample the µ–map and λ–map on only a regular grid of sparsely spaced points and to calculate the scatter for only a subset of all LORs [14, 182, 183, 184]. For example, the single–scatter calculation described in [14] is reported to require only 22 seconds of computation time using a 0.5 GHz processor, a relatively slow processor by today’s standards, with a sparse sampling of 4096 LORs and 1580 scatter points within the µ–map. These types of single–scatter models implicitly assume that the distribution of scatter from outside the field–of–view (FOV) and multiple scatter have the same shape as that of single–scatter from inside the FOV. The contribution of scatter from outside the FOV remains a challenging issue, especially for whole–body PET imaging with 3D acquisition with axially large scanners [11]. Scatter from outside the FOV could be directly taken into account by translating the bed axially and imaging the entire patient. This method is not typical in clinical studies, however, due to the increased scan time which would be required. This method could also be problematic if the half–life of the radio–tracer is short or if the tracer is not in metabolic equilibrium (i.e. the distribution of activity changes between the emission and auxiliary scan). Multiple scattering can also be a source of error, since it often has a very different spatial distribution from single scatter [185] and can contribute as much as 30% of the total scatter for simulated whole–body PET [186]. 81
Chapter 2. Literature Review Although accurate methods have been proposed to analytically compute the distribution of multiple scatter [185], the computation time has been reported to be 200 times longer than the single scatter calculation (5 hours for a single point source using a 3 GHz processor). With extremely sparse sampling, another multiple scatter calculation was found to require more reasonable computation times (less than 5 minutes using a similar processor) but for certain activity and attenuation distributions the multiple scatter distribution did not agree well with simulated data [186].
2.7
Monte Carlo-Based Scatter Corrections
Another way to model the spatial distribution of scattered photons is by using Monte–Carlo (MC) simulations. These simulators stochastically propagate individual photons through a scattering media and record their interactions. MC is a statistical simulation method that utilise sequences of random numbers to solve problems or model physical systems. These random numbers are used to sample probability distribution functions (PDFs) which describe the behaviour of the system. Due to the statistical nature of MC methods, typically many individual simulations must be performed, often called “trials” or “histories”, to obtain a result that accurately reflects the behaviour of the system under investigation. MC methods are an extremely useful tool for PET imaging, since it allows one to have complete control over how the data are generated. One can, for example, switch on and off different effects such as photon scatter and attenuation and investigate the properties of novel scintillation materials. Examples of research areas which benefit from Monte–Carlo simulation are the design of new medical imaging devices, the optimisation of acquisition protocols and the development and assessment of image reconstruction algorithms and correction techniques [8]. Since the simulated data can be easily separated into scattered and unscattered components, scatter corrections are an obvious application for MC simulations. An early MC–based scatter correction for 3D PET data was found to provide accurate scatter corrections [187]. The computation times, 82
Chapter 2. Literature Review however, were unacceptably long with one simulation requiring between 2 to 25 hours on a single Sparc10 CPU, depending on the sampling used for the activity and attenuation–map images. Since that time, the speed of computer processors have increased substantially. Variance reduction methods have also been introduced to decrease the number of photon–histories required to obtain a given variance in the simulation data and hence reduce the required computation time. Holdsworth et al. [188] employed a number of variance reduction techniques including stratification, photon–splitting and the use of look–up tables. In stratification the initial starting position of the photon is not selected at random. Rather locations that have a higher probability of resulting in photon detection (based on a short preliminary simulation) are selected with greater frequency. They also employed photon–splitting in which the photon history is split at each site where more than one type of interaction is possible. At each interaction point, multiple copies of the original photon are created and weighted to appropriately account for the probability of each interaction. In order to reduce memory access and computation time, certain functions which depend on common variables (e.g. the energy of the Compton scattered photon, the Klein–Nishina formula and energy response of the system can all be parameterised in terms of the scatter angle) can be evaluated at the beginning of the simulation and stored in RAM as a look–up table. With all these simplifications, the required execution time to acquire 10 million scatter coincidence events for a 3D thorax PET scan was only 4 minutes on a 300 MHz Sun dual–processor workstation [188].
2.8
Conclusions
As mentioned previously, scatter in transmission data was a much less significant problem for the previous generation of 2D PET scanners which used coincidence–mode transmission acquisition with rotating rod sources. For those data, the presence of inter-plane septa and rod–windowing techniques greatly reduced the detected scatter. In spite of this, a number of accurate scatter corrections were proposed [128, 178, 189] for coincidence–mode 83
Chapter 2. Literature Review transmission sources which used a theoretically–based model similar to that described in the previous sections. It is widely recognised that photon scatter is much more significant source of contamination for the singles–mode transmission acquisition used for the current generation of 3D PET scanners, however, very few corrections have been proposed to directly account for it. One of the earliest papers on singles–mode transmission proposed the use of an energy window–based scatter correction for these data [190]. However, these types of corrections are not currently applied because of their questionable accuracy and because they increase the noise in the acquired transmission data. Currently scatter in the transmission data is simply ignored and the data are reconstructed to form a µ–map which is subsequently segmented. As we shall see in chapter 5, if scatter is not corrected for in these data, the errors in the reconstructed µ–maps depend on position within the image. This local variation in µ–values due to scatter can lead to segmentation errors, since it can be comparable to the difference between the attenuation coefficients for muscle and bone. For singles–mode transmission data using
137 Cs,
the development of a
theoretically–based scatter calculation and its comparison to MC simulation data was reported in a conference proceeding [191]. A similar comparison was also reported by Wegmann et al. [192]. These calculations were never applied as a correction, however, and Accorsi et al. [191] did not pursue the transmission scatter correction further when portions of the same work were published in a peer–reviewed journal [184]. In this thesis, we also describe the development of a theoretical scatter calculation for singles–mode PET transmission scanning and its validation using Monte–Carlo data. We pursue our calculation further, however, testing for transmission sources which emit different energy photons (57 Co and
68 Ge)
and by incorporating it as a
scatter correction for the iterative reconstruction of PET attenuation–maps using both simulated and experimental data. The latter step is non–trivial, as scatter depends strongly the attenuation–map, precisely the unknown quantity we are trying to determine with the transmission scan.
84
Chapter 3
Validation of the Simulation Data Monte–Carlo simulations can be used to model accurately the data acquisition process and photon interactions involved with PET imaging systems. We modified the well–validated Monte–Carlo simulation tool, GATE, the Geant4 Application for Tomographic Emission [8], to simulate singles–mode transmission scanning. The purpose of this chapter 5A version of this chapter has also been published in is to describe how we validated this Monte– Carlo simulation software. This was achieved by comparing simulated and experimental data obtained using the Siemens microPET R4 [55] and microPET Focus 120 [46] small animal PET scanners. Before modifying GATE to model transmission data, we first validated the parameters used in our simulations using conventional coincidence–mode emission data. This analysis was performed by comparing experimental and simulated resolution and scatter fractions (SFs) for PET sinogram data. The next stage was to extend GATE to model singles–mode transmission data. For these data, we have compared simulated and experimental scatter fractions for transmission data acquired using different sized water cylinders (25, 30 and 45 mm radii) using
57 Co
and 68 Ge transmission sources. Due to
scanner availability, our validation was first started using a R4 scanner and then continued on a Focus 120. This did not, however, impact the development of the simulation software or the final results, since the two scanners have very similar characteristics, differing essentially only in terms of crystal size. During the course of this analysis, we identified a large contamination in the 68 Ge transmission data due to the intrinsic 176 Lu radioactivity present
85
Chapter 3. Validation of the Simulation Data in the PET detector crystals. In this chapter, we also propose a simple new correction method to account for this background.
3.1 3.1.1
Methods and Analysis The microPET R4 Tomograph
The microPET R4 [111] is a dedicated small animal PET scanner that uses LSO scintillator blocks each consisting of an 8×8 crystal array (crystal size: 2.1×2.1×10 mm3 ). Each block is coupled to a position–sensitive photomultiplier tube (PS–PMT) by a fibre–optic light guide. Four of these blocks make up an axial detector module. A total of 24 detector modules are arranged in a ring, providing 32 crystal rings and 192 crystals per ring. The ring diameter of the scanner is 148 mm and the axial length is 78 mm. Coincidence data are acquired using a 6.0 ns timing window. The detectors are shielded on either axial end of the scanner by 25 mm thick lead disks with an opening diameter of 120 mm. Transmission data for this work were acquired in 3D singles–mode using two different rotating transmission sources. The first source was a 2.0 MBq positron emitter ( 68 Ge, with a 271 day half–life [7]) and the second was an 89 MBq photon emitter ( 57 Co, 272 day half–life) with two major emission photons with energies of 122.1 keV (85.6% intensity) and 136.5 keV (10.7% intensity) [93]. Both transmission sources were uncollimated, were rotated with a 124 mm diameter circular orbit, and translated axially across the entire scanner field–of–view. No emission activity was present in the scanner at the time of the transmission scans (i.e. they were performed pre–injection). For the annihilation photons emitted by the
68 Ge
source, an energy discrimination window between 350
and 750 keV was used. For photons emitted by the
57 Co
source a much nar-
rower energy window between 120 and 125 keV was used to reduce scatter. The average measured energy resolution of the system was found to be 24% at 511 keV and 35% for the
57 Co
photon energies.
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Chapter 3. Validation of the Simulation Data
3.1.2
The microPET Focus 120 Tomograph
The microPET Focus 120 [46] is a third generation small animal PET scanner with similar characteristics (the use of LSO scintillation crystals, axial lead shielding, 140 mm ring diameter, 77 mm axial length, and the same transmission hardware and data acquisition parameters) as the R4. The primary advance over the previous generation of scanners being a smaller crystal size (1.5×1.5×10 mm3 ) and, therefore, improved spatial resolution. The LSO scintillator blocks for the Focus 120 consist of 12×12 crystal arrays with 48 crystal rings and 288 crystals per ring. The
68 Ge
and
57 Co
trans-
mission sources that were used had activities of 0.9 MBq and 165 MBq, respectively. The average measured energy resolution (characterised by the FWHM of the photo–peak divided by the peak energy) of the system was 18% at 511 keV and 35% for the
3.1.3
57 Co
photon energies.
Simulations
As mentioned previously, we compared simulated and experimental data for coincidence–mode emission and singles–mode transmission data acquisition. All experimental emission data were acquired using the transmission data were acquired using either
68 Ge
68 Ge
or
line sources and
57 Co
point sources.
The details of the emission and transmission experiments are provided in sections 3.1.4 and 3.1.5. In this section, we describe more generally the physical models which were used for each type of simulation. For all coincidence and singles–mode simulations presented in this work, only the individual LSO crystals were modelled. Although we modelled the spaces between individual crystals within a block (as air), other materials present in the scanner (e.g. axial lead shielding, detector gantry and light guides) were not included in our simulations. The simulated response of the data acquisition electronics was based only upon the amount of energy deposited in the LSO crystals by photon interactions and the energetic charged particles subsequently produced as a result of these interactions. Effects associated with the propagation and detection of scintillation light (e.g. crystal cross–talk, PMT resolution and quantum efficiency) were not 87
Chapter 3. Validation of the Simulation Data explicitly modelled. Detector dead–time and the
176 Lu
background were
also not included in our simulations. Coincidence–Mode Simulations For all coincidence–mode simulations, three levels of complexity were modelled for simulated emission sources: • γ No Casing: The first modelled only the emission of two perfectly collinear annihilation γ–rays. The simulated γ–rays were emitted from a volume with the same dimensions as the physical line source but the source material (the line source casing) was not simulated (i.e. it was set to vacuum). • γ With Casing: The second type of simulations also only modelled perfectly collinear annihilation γ–rays but these simulations also included an accurate model of the line source material and the photon interactions within this source material. • β + With Casing: The third type of simulation modelled β + interactions (i.e. positron range and annihilation photon acollinearity) as well as the composition of the line source. Specifications provided by the line source manufacturer indicate that the line source casing consists of a stainless steel shell (inner radius 1.0 mm and outer radius of 1.5 mm) surrounding an active plaster in which the
68 Ge
atoms are located. For both types of γ simulations, photons were equally likely to be emitted from anywhere within the entire volume of the source material (within the outer radius of 1.5 mm). For the β + simulations, the positrons were emitted from within the plaster material (within the inner radius of 1.0 mm), to interact within the source material and eventually annihilate releasing γ–rays.
88
Chapter 3. Validation of the Simulation Data Singles–Mode Simulations For the singles–mode transmission data we simulated two levels of complexity for the transmission sources: • γ:
In the first type of simulation we modelled the source as a thin
ring of γ-emitting activity 124 mm in diameter (see right image in figure 3.1), rather than explicitly modelling a source that rotates over time. For the
57 Co
transmission source, we simulated the emission of
the 122.1 keV and 136.5 keV γ–rays with activities set to match the experimentally observed intensities [93]. • β+:
The second type of simulations (performed only for the
68 Ge
source) modelled a rotating transmission source and included all positron and photon interactions within the transmission source material. In order to model singles–mode transmission data, it was necessary to modify the existing GATE software. The most significant of these changes was the development of an efficient method for the binary output of data relevant to transmission scanning. We required information that identified the crystal in which each photon was detected, details about the photon scattering history, and each photon’s point of emission (or equivalently the time at which the emission occurred for the rotating point–like source). It was then necessary to develop an analysis program which could process these data (off–line) into PET sinograms.
3.1.4
Coincidence–Mode Studies
We compared experimental and simulated data for two different source geometries using coincidence–mode acquisition. The first source geometry was meant to investigate the sinogram spatial resolution. The second was used to compare simulated and experimental sinogram data scatter fractions (SFs). This validation was only performed for the microPET R4 scanner. A separate validation of the Focus 120 was not deemed to be necessary because of the large number of similarities between the R4 and Focus 120 scanners. 89
Chapter 3. Validation of the Simulation Data
Figure 3.1: Graphics showing the objects included in the GATE simulations for the line sources (left image), the scatter phantom (centre image) and the transmission scan using the 30 mm radius cylinder (right image). The simulated scanner model shown here is the microPET R4. In addition, similar validation studies for the Focus 220 dedicated primate scanner have already been published in [193]. The Focus 220 has the same block structure, crystal size, acquisition hardware and software as the R4 differing only in its ring diameter and the number of crystals per ring. All experimental coincidence–mode data presented in this chapter have been corrected for randoms by subtracting data acquired using a delayed coincidence window (as described in section 1.9.3). For the sinogram resolution studies, we performed six scans, each 5 minutes long, using a 100 mm long, 3 mm diameter line source ( 68 Ge, 2 MBq) lying along the axial direction and positioned at 10 mm radial intervals (see left hand image in figure 3.1). Experimental and simulated sinogram data were first rebinned using Fourier Rebinning (FORE) [78] (see section 1.10.2) and reconstructed using two–dimensional filtered back–projection (FBP). No corrections were applied for normalisation, scatter or attenuation. We then summed all image slices and estimated, for each line source position, the full–width at half maximum (FWHM) and full–width at one tenth of the maximum (FWTM) in both the horizontal and vertical direction. Certain PET scanners have been found to exhibit asymmetric line–spread functions for sources offset from the centre of the scanner [194]. This effect is especially common for high–resolution circular PET scanners and is associated with
90
Chapter 3. Validation of the Simulation Data the parallax effect (discussed in section 1.7). As a result, we determined these widths using linear interpolation rather than fitting a symmetric function to the profiles. For the SF data, we acquired a 30 minute long scan using another line source (68 Ge, 7 MBq) attached to the outside of a 60 mm diameter water bottle (see the middle image in figure 3.1). To model the experiment as realistically as possible, portions of the water bottle that were outside the scanner field–of–view (FOV), such as the bottle–cap, were also simulated. To reduce noise and emphasise potentially small systematic differences between these data, we analysed radial profiles averaged over all sinograms corresponding to detector pairs within the same axial ring (ring difference equal to zero using the typical 3D–PET terminology) and pairs separated by just one ring (ring difference equal to ±1). The SFs were calculated for the radial profile at each angular view of the average sinogram. The SFs were computed by estimating the total number of coincidences due to scattered photons and dividing by the sum of all (scattered and unscattered) coincidences. To determine the SFs first the radial bin with the maximum number of counts (peak) was found for each profile. The contribution from scatter under the peak was estimated as the area under a straight line. This line was found using linear interpolation based upon the number of counts in the two bins located seven bins away (approximately ±8.5 mm from the peak) to the left and to the right of the peak. All counts outside of this peak region were also considered to be part of the scatter background. This method to estimate scatter fractions is based on performance evaluation methods defined by the National Electrical Manufacturers Association (NEMA) which are in common use for human PET scanners [195]. For the simulations, we can separate the data into the true unscattered and scattered components and compare the estimated SFs with the true SFs. The true SF is the sum of the counts in which at least one of the simulated photons had undergone a scatter interaction divided by the total sum of the counts.
91
Chapter 3. Validation of the Simulation Data
3.1.5
Singles–Mode Studies
In order to evaluate GATE for singles–mode transmission data, we used three cylindrical water bottles with radii of 25, 30 and 45 mm (axial lengths of 65, 85 and 145 mm, respectively) as the attenuating media. For these data, we compared the effective attenuation correction factors (ACFs) for the simulated and experimental transmission sinograms. As discussed in section 2.2.1, the attenuation correction factors (ACFs), necessary for quantitatively accurate PET images, are typically obtained from the ratio of counts in the blank scan relative to the transmission scan. For any spatial distribution of attenuating medium, µ(~r), the measured ACF can be written as: ACFi =
Bi B R i ≈ , Ti Bi exp − ~`i µ(~r)dr + Si
(3.1)
where Ti and Bi are the number of counts in the ith bin of the transmission and blank scan sinogram, respectively. The term S i represents the contamination from photon scatter in the ith sinogram bin of the transmission data and ~`i represents the vector coordinates of the line–of–response (LOR) corresponding to the ith sinogram bin. It is clear from equation (3.1) that scatter lowers the apparent attenuation correction factors (i.e. there appears to be less photon attenuation if scatter is not corrected for). We also estimated the transmission SFs in both experimental and simulated data. Similar to previous studies [131], we can use the approximate relationship of equation (3.1) to estimate transmission data SF for a uniform attenuation distribution as: SF ≈
P
Ti −
P Bi exp (−µ`i ) P ,
Ti
(3.2)
where `i represents the intersection of the attenuating media and the LOR corresponding to the ith sinogram bin and µ is the linear attenuation coefficient (for water in this case) at the appropriate photon energy, (values from Berger et al. [10]). We computed `i assuming that the cylinders used in our studies were centred perfectly in the scanner. For the simulated transmission 92
Chapter 3. Validation of the Simulation Data data, we can also compare the estimated SFs computed using equation (3.2) with the true SFs given by: SFtrue
P Si =P ,
Ti
(3.3)
where Si is the contribution from photon scatter in the i th sinogram bin of the simulated transmission data. Correction for the
176 Lu
Background
The Siemens microPET scanners all use LSO scintillation crystals, containing natural lutetium with a 2.6% abundance of the long–lived isotope (half–life of 4 × 1010 years).
176 Lu
176 Lu
decays by β − emission (average energy
of 182 keV) followed by a cascade of X–rays, γ–rays and secondary electrons emitted over a wide range of energies. Some of the higher intensity photon emissions include X–rays with energies of 8 and 54 keV and γ–rays with energies of 88, 202 and 307 keV (with absolute intensities of 23%, 26%, 14%, 78% and 94%, respectively) [196]. In singles–mode, the only way to distinguish these intrinsic events from detected photons originating from the transmission source is by energy discrimination. The influence of this contamination will, therefore, depend on the energy window used, the energy resolution and integration time of the system. This effect will also be more prominent for lower activity transmission sources. Using a combination of analytic computation and measured values, it was found [197] that the
176 Lu
background could significantly degrade the accuracy of attenuation corrections obtained from singles–mode transmission data. To our knowledge, however, no correction method has been proposed to account for it and subsequent experimental studies [198] have failed to detect any effects from this background. For all transmission experiments using the R4 scanner, no attempt was made to compensate for the
176 Lu
experiments using the 0.9 MBq
background. In the initial Focus 120
68 Ge
source, the ACFs were found to be
considerably lower than those obtained with the R4 scanner (that used a
93
Chapter 3. Validation of the Simulation Data higher activity 2.0 MBq 68 Ge source). This was consistent with the expected effects of the 176 Lu background [197]. In order to estimate the contamination from this background, we performed a “mock” scan without any source in the transmission mechanism for both the 57 Co and 68 Ge acquisition settings. The mock scans were performed both with and without the water cylinders inside the scanner FOV. The background sinograms were then subtracted from both the blank and transmission data before computing the SFs and ACFs. Unfortunately, by this time, the R4 was no longer available to us and we could not perform these corrections for the R4 data. Therefore, the Focus 120 data were also analysed without the background subtraction in order to determine if the results were consistent with the uncorrected R4 data. Table 3.1 summarises some of the more important details about each simulated dataset (both emission and transmission). Included in the table are details concerning: which simulated source models were used (see section 3.1.3); which PET imaging system was employed for the experiments; and information, where applicable, about the
176 Lu
background corrections
applied to the experimental data.
3.2 3.2.1
Results Coincidence–Mode Data
A comparison of the experimental and simulated sinogram (summed over all axial sinograms with ring difference zero and ±1) for the line source data is shown in figure 3.2. Data have been combined for all six line source positions. All the simulated sinogram images appear very similar, therefore, only the experimental and β + sinograms are shown. The simulation data were rescaled to have the same total number of counts as the experimental data. Profiles through the sinogram data for the experimental and all the simulation data are shown on linear (middle plots) and logarithmic scales (lower plots) in figure 3.2. The position of the profiles are indicated as dashed and dotted lines on the sinogram images. Reconstructed images
94
Chapter 3. Validation of the Simulation Data
Table 3.1: Summary of the source models used for each simulated dataset (emission and transmission) and details concerning the experimental data to which the simulated data were compared. Dataset Simulated Source microPET Imaging 176 Lu Background Models Used System Used in Corrections Applied Experiments to Experimental Data Emission: • γ No Casing • R4 • Not Necessary † Line Sources • γ With Casing • β + With Casing Emission: • γ No Casing • R4 • Not Necessary † Scatter • γ With Casing Phantoms • β + With Casing Transmission: •γ • R4 • No 176 Lu 68 Ge Source • β+ Correction 25, 30 and 45 mm Radius Water Cylinders Transmission: • β+ • Focus 120 • With and Without 68 Ge Source 176 Lu Correction 25, 30 and 45 mm Radius Water Cylinders Transmission: •γ • R4 • Not Necessary ‡ 57 Co Source 25, 30 and 45 mm Radius Water Cylinders Transmission: •γ • Focus 120 • Not Necessary ‡ 57 Co Source 25, 30 and 45 mm Radius Water Cylinders † Based on coincidence–mode background rates measured using the R4 scanner. ‡ Based on singles–mode background rates measured using the Focus 120 scanner.
95
Chapter 3. Validation of the Simulation Data (FORE–FBP) for the experimental and simulated sinogram data are shown in the images in the top of figure 3.3. Once again, only the experimental and β + data are shown. The calculated FWHMs and FWTMs are shown in the middle and lower plots of figure 3.3, respectively. The top images in figure 3.4 show the emission scatter phantom sinograms (experimental and β + data) for data summed over all sinograms with ring difference zero and ±1. The middle images in figure 3.4 show profiles through the sinogram data and the lower plot shows the estimated SF for the profile at each angular view of the sinogram. The estimated SFs for each simulated dataset were also compared to the true scatter fractions in these data (data not shown). For all three types of simulations, the estimated and true SFs were within 4% of each other (and on average differed by less than 1.5%) for all projection angles. The intermittent “spikes” observed in the estimated SF as a function of projection angle were also present in the true SFs and correspond to lowered detection efficiency for unscattered photons associated with the block structure of the PET scanner.
3.2.2
Singles–Mode Data
The ACF profiles, for the experimental and simulated singles–mode transmission data and for all cylinder sizes, are shown in figure. 3.5 and 3.6 for the 68 Ge
and
57 Co
transmission sources, respectively. Data have been summed
over all projection views and all sinograms with ring difference zero and ±1 to reduce noise. The left–hand portion of figures 3.5 and 3.6 show data from the R4 scanner, while the right–hand plots are from the Focus 120. For the experimental transmission data, only the Focus 120 data were corrected for the was about
176 Lu
3.7×104
background. The measured background count–rate
histogrammed counts per second (cps) for the
68 Ge
data
acquisition settings. For these data about 40% of the total events originated from the background using a 0.9 MBq transmission source. We also found (data not shown) that the count–rate and distribution of background counts did not change significantly with and without the water cylinders inside the scanner (less than 1% change in the background count–rates). For the
96
Chapter 3. Validation of the Simulation Data
0
40 60 80 100 120 140
Beta Simulation
20 Projection Angle (degrees)
Projection Angle (degrees)
0
Experiment
20
160
10000
40
8000
60 80
6000
100
4000
120 140
2000
160 −40
−20 0 20 Radial Offset (mm)
40
−40
−20 0 20 Radial Offset (mm)
Dotted Profiles
Dashed Profiles Experiment Simulation: γ: No Casing Simulation: γ: With Casing + Simulation: β With Casing
7000 6000
40
Experiment Simulation: γ: No Casing Simulation: γ: With Casing + Simulation: β With Casing
2500 2000
Counts
Counts
5000 4000 3000
1500 1000
2000 500
1000 0 −10
−5
0 Radial Offset (mm)
5
0 −10
10
Dotted Profiles
0
10 20 30 Radial Offset (mm)
40
Dashed Profiles 3 Experiment 10 Simulation: γ: No Casing Simulation: γ: With Casing + Simulation: β With Casing
3
10
2
Counts
Counts
10 2
10
0
10
0
10
−50
1
10
1
10
0 Radial Offset (mm)
50
−50
Experiment Simulation: γ: No Casing Simulation: γ: With Casing + Simulation: β With Casing
0 Radial Offset (mm)
50
Figure 3.2: Comparison of experimental and β + simulation sinograms (top images) for the line source data summed over all sinograms with ring difference zero and ±1. Middle plots show profiles through the simulated and experimental data for two different angular views. Lower plots show the same profile data on a logarithmic scale.
97
Chapter 3. Validation of the Simulation Data
Simulation: β+ With Casing
Experiment −50
Activity (Arbitary Units)
−50
Vertical Distance (mm)
Vertical Distance (mm)
2
0
50 −50
0 Horizontal Distance (mm)
1.5 0
1 0.5
50 −50
50
FWHMs: Horizontal Profile
Experiment Simulation: γ: No Casing Simulation: γ: With Casing Simulation: β+ With Casing
4 Width (mm)
Width (mm)
3.5
3
2
Experiment Simulation: γ: No Casing Simulation: γ: With Casing Simulation: β+ With Casing −50
−40
−30 −20 −10 Horizontal Position (mm)
3.5 3 2.5 2
0
FWTMs: Horizontal Profile
7.5
Width (mm)
Width (mm)
6 Experiment Simulation: γ: No Casing Simulation: γ: With Casing + Simulation: β With Casing
5 −50
−40
−30 −20 −10 Horizontal Position (mm)
−40 −30 −20 −10 Horizontal Position (mm)
0
FWTMs: Vertical Profile Experiment Simulation: γ: No Casing Simulation: γ: With Casing + Simulation: β With Casing
12
6.5
4.5
−50
14
7
5.5
50
FWHMs: Vertical Profile 4.5
4
2.5
0 0 Horizontal Distance (mm)
10 8 6
0
4
−50
−40 −30 −20 −10 Horizontal Position (mm)
0
Figure 3.3: Comparison of experimental and β + simulation reconstructions (top images) for the line source data. Middle plots show the horizontal and vertical full-widths at half–maximum (FWHM) and lower plots show full–widths at one–tenth maximum (FWTM) for the reconstructed data.
98
Chapter 3. Validation of the Simulation Data
0
60 80 100 120 140 160 −20 0 20 Radial Offset (mm)
2.5
100
2
120
1.5
140
1 0.5 −40
−20 0 20 Radial Offset (mm)
0
40
Dashed Profiles Experiment Simulation: γ No Casing Simulation: γ With Casing Simulation: β+ With Casing
4
10
3
3
10
Counts
Counts
3
80
40
Experiment Simulation: γ No Casing Simulation: γ With Casing + Simulation: β With Casing
4
10
2
10
2
10
1
3.5
60
Dotted Profiles
10
4.5 4
40
160 −40
10
Beta Simulation:
20
40
Projection Angle (degrees)
Projection Angle (degrees)
20
4
x 10
0
Experiment
−40
−20
0 20 Radial Offset (mm)
40
1
10
−40
−20
0 20 Radial Offset (mm)
40
60 Experiment Simulation: γ No Casing Simulation: γ With Casing Simulation: β+ With Casing
Scatter Fraction (%)
50 40 30 20 10 0 0
20
40
60 80 100 120 Projection Angular (degrees)
140
160
Figure 3.4: Comparison of simulated and experimental profiles, summed over all sinograms with ring difference zero and ±1, for the scatter phantom data. Middle plots show profiles through the simulated and experimental data for two different angular views. The bottom plot shows the estimated scatter fractions for each angular view for the experimental and all simulated data.
99
Chapter 3. Validation of the Simulation Data Focus 120 experiments that used the 57 Co source, we detected about 250 cps from the background. This background did not contribute significantly to the singles–mode data and accounted for less than 0.03% of the total events in these scans using a 165 MBq source. 68
Ge: R4 25mm Radius Water Cylinder Experiment: 176 No Lu Corr. Simulation: γ + Simulation: β
1.4 1.3 1.2
1.3 1.2
1.1
1.1
1
1
0.9 −50
−30
−10 10 30 Radial Offset (mm)
0.9 −50
50
68
Ge: R4 30mm Radius Water Cylinder 1.5
1.4
1.4
1.3
1.3
1.2 1.1
0.9 −50
−30
−10 10 Radial Offset (mm)
0.9 −50
50
68
1.6
1.6
1.5
1.5
1.4
1.4
1.3
1.3
1.2 1 0.8 −50
−30
−10 10 Radial Offset (mm)
−30
1.2 1 0.9
30
−10 10 Radial Offset (mm)
30
50
Experiment: With 176Lu Corr. Experiment: No 176Lu Corr. Simulation: β+
1.1
Experiment: No 176Lu Corr. Simulation: γ Simulation: β+
0.9
Experiment: With 176Lu Corr. Experiment: No 176Lu Corr. Simulation: β+
68 Ge: Focus 120 45mm Radius Water Cylinder
ACF
ACF
Ge: R4 45mm Radius Water Cylinder
1.1
50
1.2
1 30
−10 10 30 Radial Offset (mm)
Ge: Focus 120 30mm Radius Water Cylinder
1.1
Experiment: No 176Lu Corr. Simulation: γ Simulation: β+
1
−30
68
ACF
ACF
1.5
Experiment: 176 With Lu Corr. Experiment: 176 No Lu Corr. + Simulation: β
1.4
ACF
ACF
68
Ge: Focus 120 25mm Radius Water Cylinder
50
0.8 −50
−30
−10 10 Radial Offset (mm)
30
50
Figure 3.5: Comparison of experimental and simulated effective ACFs for the R4 (left–hand plots) and Focus 120 (right–hand plots). Data shown is for the transmission source.
68
Ge
100
Chapter 3. Validation of the Simulation Data
57
2
Co: R4 25mm Radius Water Cylinder
57
2
Experiment Simulation: γ
1.8
ACF
ACF
1.6
1.4
1.4
1.2
1.2
1
1 −30
−10 10 Radial Offset (mm)
30
0.8 −50
50
57
Co: R4 30mm Radius Water Cylinder 2
1.8
1.8
1.6
1.6
1.4 1.2
50
Co: Focus 120 30mm Radius Water Cylinder
1
Experiment Simulation: γ −30
−10 10 Radial Offset (mm)
30
0.8 −50
50
Experiment Simulation: γ −30
−10 10 Radial Offset (mm)
30
50
57 Co: Focus 120 45mm Radius Water Cylinder
2.4
2.4
2.2
2.2
2
2
1.8
1.8
ACF
ACF
30
1.4
57 Co: R4 45mm Radius Water Cylinder
1.6 1.4
1.6 1.4
1.2
1.2
Experiment Simulation: γ
1 0.8 −50
−10 10 Radial Offset (mm)
1.2
1 0.8 −50
−30
57
ACF
ACF
2
Experiment Simulation: γ
1.8
1.6
0.8 −50
Co: Focus 120 25mm Radius Water Cylinder
−30
−10 10 Radial Offset (mm)
Experiment Simulation: γ
1 30
50
0.8 −50
−30
−10 10 Radial Offset (mm)
30
50
Figure 3.6: Comparison of experimental and simulated effective ACFs for the R4 (left–hand plots) and Focus 120 (right–hand plots). Data shown is for the transmission source.
57
Co
101
Chapter 3. Validation of the Simulation Data Table 3.2 summarises the estimated SFs for the
68 Ge
source for both
scanners. For the R4 scanner we performed simulations using both the annular γ source and the rotating β + source models. Since we observed very little difference between the two types of simulations, only the β + simulations were analysed for the Focus 120 scanner. The average SFs were computed using equation (3.2) for just the central region of each profile (within a radial distance of ±12 mm). The estimated SFs for the
57 Co
source are summarised in table 3.3. The estimated simulation data SFs, computed using equation (3.2), were also found to agree with the true SFs, given by equation (3.3), to within 1% and 2.5% for the
68 Ge
and
57 Co
data,
respectively (data not shown). Table 3.2: Comparison of the estimated scatter fraction (in percent) for the simulated and experimental transmission data using the 68 Ge source.
Exp.: R4, No 176 Lu Corr. Sim.: R4, γ source Sim.: R4, β + source
Cylinder Radius 25 mm 30 mm 45 mm 21.50 28.14 42.08 14.70 20.45 34.45 14.12 19.97 33.95
Exp.: Focus 120, No 176 Lu Corr. Exp.: Focus 120, With 176 Lu Corr. Sim.: Focus 120, β + source
26.48 14.42 14.90
3.3 3.3.1
32.17 19.09 20.08
46.89 32.29 33.83
Discussion Coincidence-Mode Data
For the line source sinogram images and profiles shown in figure 3.2, the qualitative appearance of all three types of simulations and the experimental data are very similar. Although it is difficult to distinguish between the different radial profiles when plotted on a linear scale some differences become apparent on a logarithmic scale. As one would expect, as the simulations 102
Chapter 3. Validation of the Simulation Data
Table 3.3: Comparison of the estimated scatter fraction (in percent) for the simulated and experimental transmission data using the 57 Co sources. Cylinder Radius 25 mm 30 mm 45 mm Exp.: R4 Sim.: R4
20.46 19.31
26.50 26.58
43.54 45.23
Exp.: Focus 120 Sim.: Focus 120
21.35 19.08
26.59 25.97
42.64 45.18
become more sophisticated they begin to more closely resemble the experimental data. In particular the shape of the low count background is better matched for the simulations that include both β + interactions (positron range and annihilation photon acollinearity) and photon scatter within the source material. One unexpected feature observed in the line source profiles was a relatively–flat, low–count background for the β + simulations located at large distances (as far as ±50 mm) from the line source radial position (most visible in the logarithmic plots shown in figure 3.2). This background also appears in the experimental data and cannot be explained by positron interactions (i.e. positron–range and acollinearity) within the source material. Further analysis of our simulations revealed that this background occurs because approximately 3% of all simulated annihilations take place outside the line source material (in the detector crystals). These results suggest that the thickness of stainless steel used in these line sources (inner radius 1.0 mm, outer radius 1.5 mm) may not be sufficient for the positrons emitted by
68 Ge.
For the reconstructed data shown in figure 3.3, the FWHMs and FWTMs fit to simulated data agree with experimental values to within 0.7 mm (less than 32% of the crystal width) for all three types of simulations. In general, the γ–source simulations have slightly larger widths than the experimental data and the β + simulated widths are slightly narrower. Since crystal 103
Chapter 3. Validation of the Simulation Data cross–talk and PMT resolution were not simulated, it was expected that the simulated resolution should be better than the experimental values (i.e. narrower widths). This was observed only for the β + simulations. As described in section 3.1.3, the simulated positrons were only emitted within the inner radius (1.0 mm) of the line source corresponding to the activity carrying plaster. In the γ simulations, photons could be emitted from anywhere within the outer radius (1.5 mm) corresponding to both the plaster and the stainless steel casing. By choosing the source volume to be slightly larger in the γ simulations than the β + simulations, we seem to have partially (over) compensated for the effects of the positron range. The scatter phantom data (shown in figure 3.4) indicates that more scatter is present in the experimental data than in the simulations. However, the average difference in SFs is much smaller for the simulations in which the source material is accurately modelled. The shape of the sinogram profiles, particularly for the data shown in the centre–left plot of figure 3.4, are also better matched for the simulations that include the composition of the line source. For the SFs as a function of projection angle shown in the lower plot of figure 3.4, the average difference between the estimated SFs for the experimental and simulated data were 7.5% and 4.1% for the γ simulations with and without the source material, respectively. This difference was only 3.1% for the β + simulations, indicating that, once again, the more complex simulations agree better with the experimental data. It has been hypothesised [58] that for small animal PET systems, a large portion of the estimated scatter fractions (computed using NEMA–based methods) may be the result of scatter from within other components of the scanner (e.g. light guides, lead shielding and supporting structures), rather than from the object itself or from the detector crystals. It is possible that including these additional components in our simulations could further reduce the discrepancy between the simulated and experimental data.
104
Chapter 3. Validation of the Simulation Data
3.3.2
Singles-Mode Data 176 Lu
back-
ground has a large effect on the ACF profiles for the experimental
68 Ge
The data shown in figure 3.5 indicate that the correction for the
transmission data using the Focus 120 scanner. Qualitatively, the shape of the simulated ACF profiles appear to agree well with Focus 120 experimental data that has been corrected for the background. The SFs for the background–corrected Focus 120 data in table 3.2 agree with the simulations to within 1.5%. We estimate that a similar amount of
176 Lu
activity
(approximately 0.1 MBq) is present in both R4 and Focus 120 scanners. It is reasonable to assume, therefore, that the R4 is subject to similar background count–rates as the Focus 120 scanner and that this effect accounts for the discrepancies observed for the R4 data in figure 3.5 and table 3.2. It should be noted, however, that the 68 Ge source used here was considerably weaker than what is normally used for these scanners. Siemens provides 68 Ge
transmission sources with an initial activity of about 20 MBq and rec-
ommends that they be used for a year (final activity of about 8 MBq). Even for these higher activities, the
176 Lu
background is still significant, initially
contributing 4% of the total counts in the blank scan and growing to 10% over a year. If this background is not corrected for, errors in the ACFs or reconstructed attenuation–maps will increase over time, which could cause a variety of artifacts. For example, this could lead to errors in automated segmentation routines for attenuation–maps (see section 2.2.3) and erroneous long–term trends in the longitudinal studies which are common in small animal PET (as discussed in section 1.2.4). For the
57 Co
singles–mode data, we did not detect a large
176 Lu
back-
ground. Boellaard et al. [198] did not find a significant effect due to the 176 Lu
background using a 740 MBq
Incidentally, we also measured the
137 Cs
176 Lu
(662 keV) transmission source.
background using the data acquisi-
tion settings used for singles–mode transmission with a
137 Cs
source (energy
window between 450 and 950 keV) and found similar count–rates (2.7 × 10 4 cps) as those for the
68 Ge
source. It is, therefore, clear that this effect is
dependent on the source strength, the specifications of the scanner and the
105
Chapter 3. Validation of the Simulation Data data acquisition settings. To determine if this background is significant one can simply perform a “mock” transmission scan without any transmission source in the point source mechanism. Then, if necessary, the mock scan can be subtracted from the blank and transmission data to correct for the background. Our results also indicate that this background is not affected by the presence of the object in scanner. This should be true, in general, because the
176 Lu
emissions (particularly the electrons and lower energy
photons) are more likely to deposit energy close to their point of emission inside the detector crystals and, therefore, very few of these particles will ever interact with the subject inside the scanner. This fact, combined with the long half–life of
176 Lu,
suggests that the mock scan does not need to
be performed each time transmission data are acquired, but rather could be incorporated as part of longer term quality control and scanner upkeep. Unlike the emission case, there was very little difference between the 68 Ge
simulations that modelled a simple annular γ emitting source and those
that modelled a rotating β + source and the various interactions within the source material. For the ACFs shown in the left–hand plots of figure 3.5, it is difficult to distinguish between the two types of simulations. Similarly, the SFs for the two types of simulations in table 3.2 agree to within 0.5% of each other. Since very little difference was observed between the two types of simulations, we analysed only the Focus 120 simulations for the using the more complex rotating
β+
68 Ge
source
source model. For this particular
68 Ge
source, we do not anticipate that β + interactions should have a significant effect on the singles–mode data, however photon scatter within the source material may be important. Our simulations indicate that about 5.5% of the annihilation photons undergo Compton scatter (and therefore are emitted with lower energy) before escaping the transmission source. Additionally, this model better represents the instantaneous singles count–rates than the annular γ source simulations and could, therefore, lend itself more easily to future work such as modelling detector dead–time (already available with the current release of GATE) and errors in the transmission source position. The experimental and simulated ACF profiles appear very similar for both scanners using the
57 Co
source (figure 3.6). The experimental and 106
Chapter 3. Validation of the Simulation Data simulated SFs for these data (table 3.3) agree to within 2.5% for both scanners and all cylinder sizes. In contrast to the emission studies, we found that for certain phantoms (45 mm radius water cylinder) our transmission simulations predict higher scatter fractions than those estimated using experimental data. This is due to the fact that, in the simulations, we do not model the axial lead shielding. This shielding reduces the number of photons which scatter from within parts of the phantom outside of the field–of–view. This effect is most prominent for larger phantoms with lengths much longer than the axial length of scanner. We have also observed that the experimental SFs can change by as much as 4% depending on the strength of the
57 Co
source used (data not shown).
In general we find that the experimental SFs are lower for weaker sources. One possible explanation for this effect is detector dead–time. Dead–time is not modelled in our simulations and not corrected for in our experimental transmission data and its effects depend on both the source strength and object size. For a transmission source with sufficiently high activity, dead–time will reduce the apparent count–rates in the blank scan. For smaller objects dead–time losses in the transmission scans will be comparable to those in the blank scan and will, therefore, have little affect on the ACFs and SFs as given by equations (3.1) and (3.2). For more highly attenuating objects, the count–rates in the transmission scan may be considerably lower than the blank scan leading to greater dead–time losses in the blank relative to the transmission scan, thereby, lowering the apparent ACFs and increasing the estimated SFs. and
68 Ge
sources. In [107] it was previously noted that the use of a high activity
57 Co
It is also interesting to compare our results using the
57 Co
source with a narrow energy window provided better signal to noise ratios in reconstructed attenuation–maps than conventional
68 Ge
sources but sug-
gested that scatter and dead–time may cause problems in these data. If we compare the SFs in tables 3.2 and 3.3 we can see that estimated scatter fractions are indeed higher using the 57 Co source (7 to 10% higher SFs depending on the phantom size). We also found that dead–time influenced our results for
57 Co,
changing our estimated SFs by as much as 4%. We expect that 107
Chapter 3. Validation of the Simulation Data much of our results and observations for the to those that would be obtained using a
68 Ge
137 Cs
source should be similar
source (662 keV), which has
a photon energy close to 511 keV and is also commonly used in PET transmission tomography. Although these higher energy sources, such as and
137 Cs,
68 Ge
are well suited for larger primate and human studies [55], there
are many advantages to using
57 Co
for small animal PET [107]. The pri-
mary advantage is the higher contrast between the attenuation–coefficients for different tissue types at lower energies and hence improved ability to distinguish small structures. Additionally, for the
57 Co
transmission source
there is reduced emission photon contamination for post–emission transmission scanning (as we shall see in chapter 7) and lower
176 Lu
background
count–rates (as shown in this chapter). These advantages may outweigh the problems associated with scatter and dead–time for this source.
3.4
Conclusion
For emission data, the GATE simulations agreed well with microPET R4 data in terms of the shape of the sinogram profiles, the line source resolution and scatter fractions for the two source geometries investigated. The simulations that included positron interactions and an accurate model of the source material provided the best overall agreement with the experimental data. When compared with experimental values, the simulated FWHMs and FWTMs (fit to the reconstructed images of the line source data) differed by less than 0.7 mm and scatter fractions (SFs) agreed to within 3.1%. Our results indicate that the microPET R4 tomograph (and by extension the microPET Focus 120) can be well modelled using the existing GATE software for emission data. For singles–mode transmission data, we have compared simulated and experimental data for three different sized water phantoms using 57 Co (122 keV photon emitter) and
68 Ge
(positron emitter) transmission sources. We
have analysed data from both the microPET R4 and Focus 120 small animal PET scanners. We focused primarily on estimating the influence of scatter on the transmission data. We have also investigated the effects of a contam108
Chapter 3. Validation of the Simulation Data ination, present in the
68 Ge
transmission data, due to the intrinsic
radioactivity present in the PET detector crystals. For the found a significant
176 Lu
background (≈ 3.7 ×
104
68 Ge
176 Lu
source, we
counts per second). The
SFs for the background–corrected experimental data agree with the simulations to within 1.5%. These background count–rates remain constant over time (due to the long half–life of
176 Lu)
and do not appear to be influenced
by the object inside the scanner. This background can be easily measured and should be subtracted from the experimental data to correct for this effect. For the
57 Co
transmission source the background count–rates were
much lower (250 cps). For the
57 Co
data, our simulated and experimental
SFs agreed to within 2.5% for both scanners and all phantom sizes. However, in the future it may be useful to include detector dead–time to better model data acquisition for higher activity transmission sources. Our simulations accurately model the photon interactions and data acquisition for singles–mode transmission tomography in PET. In subsequent chapters of this thesis we describe how we used this simulation tool to aide in the development of our corrections for scatter and emission contamination in singles–mode transmission data. It was also used to provide data with which we could test different reconstruction and correction procedures. In the future, this software could also help optimise transmission hardware and optimise scanning protocols.
109
Chapter 4
Development and Validation of the Scatter Correction In this chapter, we describe the validation of our analytical scatter correction for singles–mode transmission data. We evaluated the accuracy of the scatter correction data using the simulation software discussed in the previous chapter. We have tested our correction using simulated data for two different transmission source radionuclei ( 57 Co and
68 Ge)
and for a number of
different attenuating media. These media include three different sized water cylinders (25, 30 and 45 mm radii) and a non-uniform phantom consisting of Teflon, water and air. We conclude the chapter with a brief investigation of the influence of multiple and Rayleigh scattering and scattering from outside the scanner field–of–view. As in Watson et al. [14], our correction predicts only the contribution from single–scattered photons (i.e. those which have undergone only one Compton scattering interaction) from material within the scanner field–of–view. We, therefore, propose a simple rescaling method which compensates for these additional sources of scatter.
4.1
The Scatter Correction
Our scatter correction uses a similar approach to that of the single scatter simulation (SSS) [14] for PET emission data, with some modifications to account for singles–mode data acquisition. Since we acquire our transmission data in singles–mode, our correction is also partially based on similar work for SPECT [199, 200]. First we assume that the transmission source can be well modelled as a thin ring of uniformly distributed γ–emitting activity (rather than explicitly modelling a source that rotates over time). 110
Chapter 4. Development and Validation of the Scatter Correction We then use the known transmission–source activity concentration, λ (equal to the known source activity divided by the annulus volume), an estimate of the attenuation–map, µ(~r, E), at the unscattered photon energy E, and the Klein–Nishina scattering cross–section,
dσ dΩ
[181], to compute the con-
tribution from scattered photons, S k , detected by the k th line–of–response (LORk ), corresponding to a particular pair of crystals within the PET detector array. As in Watson et al. [14], we only explicitly calculate the contribution from single–scattered photons. With reference to the diagram shown in figure 4.1, we can write an approximate expression for S k as:
Sk ≈
2 X N X
λVλi ∆t ~λ − R ~ µ |2 4π|R j i
i=1 j=1
dσ
exp −
Z
ρelec j Vµ
!
exp − !
×
Ai cos Ψ (θ) ~µ − R ~ A |2 dΩ |R j i
~A R i
~µ R j
0
~µ R j
Z
~λ R i
!
µ(~r, E)dr ×
!
µ(~r, E )dr PE (E 0 )(Ψ, E 0 ). (4.1)
The summation from j = 1 to N in equation (4.1) is over all attenuation– map voxels. The summation from i = 1 to 2 takes into account both source positions that contribute to LORk . The geometry shown in figure 4.1 (a) refers to i = 1, for which the rotating transmission source is closest to the crystal to the left (i.e. when the left–most crystal would be “switched–off” and the right most crystal would become “switched–on”). Figure 4.1 (b) refers to i = 2, when the two crystals reverse roles. The individual terms in equation (4.1) can be broken down as follows. If we ignore photon attenuation for the moment and assume that all photons originating from the source volume element, V λi , are emitted from the centre of the volume, the total photon flux per unit area that reaches the centre ~λ − R ~ µ |2 ), where ∆t of the j th µ–map voxel is given by: (λVλ ∆t)/(4π|R i
i
j
~λ − R ~ µ | is the distance between the centres of is the scan duration and |R j i 111
Chapter 4. Development and Validation of the Scatter Correction Off Crystal
On Crystal
Vλ i
Ψ’ Ψ
Transmission Source Orbit
Vµ
Ai
j
Transmission Source Orbit
On Crystal
Ai
Ψ’
Vλ i
Ψ Vµ
θ
µj
Off Crystal
θ µ (r,E)
j µ j
µ (r,E)
(a)
(b)
Figure 4.1: Schematic diagram illustrating the assumed geometry used to compute the distribution of scattered and unscattered photons given by equations (4.1) to (4.3). One possible path for scattered photons is indicated by the thick solid arrows in each diagram. The unscattered path is shown as a dashed arrow. Figures (a) and (b) show the two source positions, corresponding to i = 1 and i = 2 in the equations, which contribute to each line–of–response defined by the crystal pair shown in the diagram. ~ λ and R ~µ , Vλi and the j th µ–map voxel, located at the vector positions R j i respectively. The volume of activity, V λi , that contributes to LORk is defined by the positions of the two crystals, and is equal to the same value for both source positions (i = 1 and 2) by symmetry. To take into account photon ~µ RR attenuation, we must multiply the incident flux by exp − ~ j µ(~r, E)dr , R λi
the probability that an unscattered photon (with energy E) will not be
attenuated as it travels along the path between the source volume element and the attenuation voxel. For our scatter correction, all line–integrals of this form are calculated using nearest–neighbour interpolation within the µ–map data array. The total number of electrons present in the j th attenuation–map voxel is elec is the electron density (expressed in electrons given by (ρelec j Vµ ), where ρj
per unit volume) for the j th voxel of the µ–map and Vµ is the voxel volume. If we multiply the Klein–Nishina scattering cross–section, total number of electrons in the
j th
dσ dΩ (θ),
by the
µ–map voxel, we then have the scatter
flux per unit solid angle (scattered at an angle θ), divided by the incident flux per unit area at the centre of the µ–map voxel. For scattered photons 112
Chapter 4. Development and Validation of the Scatter Correction emitted from the j th µ–map voxel, the solid angle subtended by the detector ~µ − R ~ A |2 where crystal with area Ai is approximately given by Ai cos Ψ/|R j
i
~µ − R ~ A | is the distance between the centres of the scatter voxel (located |R j i ~ µ ) and face of the detector crystal (located at R ~ A ) and Ψ is the angle at R j
i
of incidence for the scattered photons. The probability that the scattered photons (with energy E 0 ) will not be attenuated along the path betweenthe µ–map voxel to the detector crystal is given by exp −
~A RR i ~µ R j
µ(~r, E 0 )dr .
The term PE (E 0 ) represents the probability that a photon, with energy
E 0 , will be recorded within the energy discrimination window used by the scanner. For this work, the energy response of the detector was modelled as a Gaussian function with an experimentally determined full–width at half– maximum (FWHM). The probability, P E (E 0 ), is determined by integrating the Gaussian function between the lower and upper level discriminators (LLD and ULD) used by the scanner for energy discrimination. The function (Ψ, E 0 ) represents the intrinsic detection efficiency, that is, the probability that a scattered photon with energy E 0 and angle of incidence Ψ, will interact in the detector crystals and deposit a sufficient amount of energy within the crystal to be accepted within the energy discrimination window of the detector. For simplicity, we used an analytical function for similar to that described by Cherry et al. [3], in which (Ψ, E 0 ) = (1 − exp(−µen (E 0 )t/ cos Ψ)), where µen (E 0 ) is the linear energy– absorption coefficient1 for LSO (values from Hubbell and Seltzer [13]) at the scattered photon energy and t is the crystal thickness (10 mm). This function assumes that the PET scanner can be approximated as a cylindrical ring of LSO with thickness t. We do not currently include block–effects, 1
The linear energy–absorption coefficient, µen , is a dosimetry quantity defined for mono–energetic photon beams by Kc = φ(µen /ρ) were φ is the energy fluence, ρ is the mass density and Kc is the collisional kerma, defined as the expectation value of net energy per unit mass which is converted at a point of interest into the kinetic energy of charged particles (excluding the energy of photons which escape from the material) [13]. Note that our use of µen in (Ψ, E 0 ) is not rigorously consistent with the definition above but it can be thought of as assuming that the fraction of absorbed photons are proportional to the fraction of energy deposited locally in the kinetic energy of charged particles. This approximation is meant to partially compensate incomplete energy deposition which can occur when photons scatter within the detector crystals.
113
Chapter 4. Development and Validation of the Scatter Correction effects due to the gaps between crystals, the small experimentally–observed variations in efficiency for individual detectors or dead–time in our efficiency model. We evaluate the accuracy of our intrinsic efficiency function in section 4.3.1. The electron density, ρelec j , is not directly measured in PET transmission scans, therefore, it must be estimated from the µ–map data (µ j ). It is reasonable to assume that, for most materials of biological interest, elec and µ are the electron density and linear ρelec ≈ (ρelec w w /µw )µj where ρw j
attenuation coefficient of water, respectively. The electron densities computed using this approximation are accurate to within about 3% for muscle and fat using the µ–values given in Hubbell and Seltzer [13] between 100 and 500 keV. However, these approximations can lead to slightly larger errors at lower energy and for materials with higher effective Z (e.g. the error for bone at 100 keV is about 10%). Since the estimated µ–maps are assumed to represent the correct linear attenuation coefficients at the unscattered photon energy (E), the attenu~A RR ation probability for scattered photons: exp − ~ i µ(~r, E 0 )dr , can also R µj
only be determined approximately. We therefore assume that ≈ (µ0w /µw )
~A RR i ~µ R j
~A RR i ~µ R j
µ(~r, E 0 )dr
µ(~r, E)dr where µ0w and µw are the linear attenuation coeffi-
cients for water at the scattered and unscattered photon energy, respectively (using values from Berger et al. [10]). For an unscattered photon energy of 511 keV, this approximation is valid to within 1% for fat, muscle, lung and bone tissue, over the range of energies (300–511 keV) where the probability of being accepted within the energy discrimination window for the Focus 120 scanner is greater than 1%. Similar agreement (to within 4%) is found for muscle, lung and fat tissue in the energy range appropriate for the detection of
57 Co
photons (80–122 keV for the Focus 120). However, errors as high as
14% occur at the lower end of the
57 Co
energy range for bone. These error
calculations use the µ–values for different tissue types given as a function of photon energy in Hubbell and Seltzer [13]. We expect that the errors in the approximate values of µ(E 0 ) and ρelec j
114
Chapter 4. Development and Validation of the Scatter Correction for bone are unlikely to have a large effect on the overall contribution from scattered photons, since the line integrals and summation in equation (4.1) depend on many voxels within the attenuation map, of which bone is usually only a small component. Introducing these approximations and simplifying equation (4.1) gives the following expression for Sk : Sk ≈
2 X N X µj × F (θ, Ψ) λVλ ∆tρelec w AVµ × ~ ~ 2 ~ ~ 2 4πµw i=1 j=1 |Rλi − Rµj | |Rµj − RAi |
exp −
Z
~µ R j
~λ R i
µ0 µ(~r, E)dr − w µw
Z
~A R i
~µ R j
!
µ(~r, E)dr , (4.2)
where F (θ, Ψ) =
dσ dΩ
(θ) PE (E 0 (θ)) (Ψ, θ) cos Ψ. Note that F (θ, Ψ) only
depends on the scatter angle, θ, and angle of incidence, Ψ. We, therefore, pre-calculate this function as a look–up table and interpolate values as needed during the scatter calculation. All the terms outside of the summation in equation (4.2) are constant or, in the case of V λ , depend only on the LORk , which is defined by the positions of the crystal pair, and the transmission source orbit. In order to reduce computation time, we found that the scatter calculation could be performed using lower resolution µ–maps and using interpolation between a smaller subset of sampled sinogram elements. The calculation can also be performed in “2D–mode” where only the scatter distribution for the direct plane sinograms (sinograms for which the detector pairs are in the same axial ring) is calculated explicitly. The oblique scatter sinograms are approximated as the nearest direct–plane scatter sinogram (i.e. corresponding to the average axial position of the oblique pair of detectors). This approximation is similar to those made for single–slice rebinning which was discussed in greater detail in section 1.10.2. For example, in all of the experimental studies discussed in this chapter, the scatter corrections were calculated in “2D–mode” with 32×32 sized attenuation–map images (reduced from the typical microPET image dimen115
Chapter 4. Development and Validation of the Scatter Correction sions of 128×128) and the sinogram data were sampled at 26×29×10 elements within the full 128×144×48 direct plane sinogram size. Although we have not performed a thorough optimisation, we found that a good compromise between computation time and accuracy could be achieved using these settings. We can derive a similar approximate expression for the contribution from unscattered photons to LORk during the transmission scan as: Pk ≈
2 cos Ψ0 (Ψ0 , E) PE (E)λVλ ∆tAi X × ~ ~ 2 4π i=1 |Rλi − RAi |
exp −
Z
~A R i
~λ R i
!
µ(~r, E)dr ,
(4.3)
~λ − R ~A | where Ψ0 is the angle of incidence for unscattered photons and | R i i is the distance between the source volume and detector crystal. For the unscattered photon calculations, all line–integrals are calculated using tri– linear interpolation within the full resolution µ–map data array (128×128 image size) and for every bin of the direct plane sinogram. Since there is no object within the scanner field–of–view during the blank scan there is also no object scatter or attenuation. An estimate of the total number of counts in for LORk during the blank scan, Bk , is therefore given by: Bk ≈
4.2
2 cos Ψ0 (Ψ0 , E) PE (E)λVλ ∆tA X ~ ~ 2 4π i=1 |Rλi − RAi |
(4.4)
Simulations
Using the Monte–Carlo software discussed in the previous chapter, we generated eight different datasets. We simulated singles–mode transmission data using
68 Ge
and 57 Co radionuclei for the three uniform water cylinders (with
radii of 25, 30 and 45 mm and axial lengths of 65, 85 and 145 mm, respectively) and for a non–uniform phantom consisting of the 30 mm radius 116
Chapter 4. Development and Validation of the Scatter Correction water cylinder with two cylindrical inserts made of Teflon (12.5 mm radius and 85 mm length) and air (7.5 mm radius and 85 mm length). These phantom configurations were selected to match as closely as possible our experimental studies. For this analysis, we used the simulation parameters for the microPET Focus 120 scanner. In chapter 3, very little difference was observed between simulated transmission data in which only the emission of γ–rays were modelled and those which included photon and, for the
68 Ge
source, positron interactions within the transmission source material. We, therefore, only modelled the emission of γ–rays for all simulated transmission data considered in this chapter.
4.3
Results
Figures 4.2 and 4.3 show the mean radial profiles (summed over all projection angles and over all sinograms with ring difference zero and ±1) for the transmission (top–row) and single–scatter sinogram data (bottom–row) obtained using the uniform phantoms. Figures 4.4 and 4.5 show similar data for the non–uniform phantom using the
68 Ge
and
57 Co
transmission
sources, respectively. Figures 4.6 shows the single–scatter fractions (SFs) per sinogram for all phantoms and both transmission sources. The purpose of the analysis, shown in figures 4.2 to 4.6, was to validate our single–scatter calculation. We, therefore, directly compared the contribution from single–scatter (i.e. only one Compton scattering interaction) to the sinogram data. Both the simulations and the scatter calculation also included the effects of scatter from outside of the scanner FOV. In our SC calculation, this was done by using larger µ-maps extended to include portions of the phantom outside of the scanner. It is important to note that the distribution of attenuation outside of the scanner is not normally available in experimental studies. We used this additional information in order to make our scatter calculation as similar as possible to what was modelled in the simulations. For the same reason, we also computed the scatter contribution from both γ–rays (122.1 keV and 136.5 keV) for the
57 Co
source. The
only rescaling applied to the calculated data was to match the total sum of 117
Chapter 4. Development and Validation of the Scatter Correction the counts (unscattered + single–scatter) with that of the simulated data. This allowed us to determine if our SC predicts the correct magnitude of the single–scatter component relative to the unscattered photon component in the transmission data. 68
Ge: 25mm Radius: Transmission
Summed Counts
10000
Simulation Model
8000
68
Ge: 30mm Radius: Transmission
10000
Simulation Model
8000
68
10000
Ge: 45mm Radius: Transmission Simulation Model
8000
6000
6000
6000
4000
4000
4000
2000 2000 2000 −50 −30 −10 10 30 50 −50 −30 −10 10 30 50 −50 −30 −10 10 30 50 Radial Distance (mm) Radial Distance (mm) Radial Distance (mm) 68
68
Summed Counts
Ge: 25mm Radius: Scatter
Ge: 45mm Radius: Scatter
600
400
800
500
300
400
700
300
200 100
68
Ge: 30mm Radius: Scatter
Simulation Model
−50 −30 −10 10 30 50 Radial Distance (mm)
200
600
Simulation Model 500 −50 −30 −10 10 30 50 −50 −30 −10 10 30 50 Radial Distance (mm) Radial Distance (mm)
100
Simulation Model
Figure 4.2: Average radial profiles for the simulated and calculated transmission data (top–row) and single–scatter sinogram data (bottom–row). Data shown is for the uniform water cylinders using the 68 Ge transmission source.
4.3.1
Validation of the Efficiency Model
In order to estimate how well our analytical intrinsic efficiency function fits the simulated data, we performed a simple simulation of a point source located inside the Focus 120 scanner with no attenuating media. We varied the energy, E, of the point source and obtained an estimate of the simulation data intrinsic efficiency by dividing the detected number of counts by the estimated number of photons incident on the i th detector, Nincident . We used i 118
Chapter 4. Development and Validation of the Scatter Correction
57
Co: 25mm Radius: Transmission
Summed Counts
8000
Simulation Model
6000
57
8000
Co: 30mm Radius: Transmission Simulation Model
6000
57
8000
4000
4000
2000
2000
2000
−50 −30 −10 10 30 50 Radial Distance (mm)
57
Summed Counts
400
Co: 25mm Radius: Scatter
300 200
500 400
Simulation Model
−50 −30 −10 10 30 50 Radial Distance (mm)
57
Co: 30mm Radius: Scatter
300
57
800 700
Simulation Model
Simulation Model
6000
4000
−50 −30 −10 10 30 50 Radial Distance (mm)
Co: 45mm Radius: Transmission
Co: 45mm Radius: Scatter Simulation Model
600
100 200 500 −50 −30 −10 10 30 50 −50 −30 −10 10 30 50 −50 −30 −10 10 30 50 Radial Distance (mm) Radial Distance (mm) Radial Distance (mm)
Figure 4.3: Average radial profiles for the simulated and calculated transmission data (top–row) and single–scatter sinogram data (bottom–row) for the 57 Co transmission source.
119
Chapter 4. Development and Validation of the Scatter Correction
Simulation
50
100
150
1000
1000 −40
500
−20 0 20 40 Radial Distance (mm)
Summed Dashed Profile
Summed Dotted Profile Simulation Model
200
200
150
150
100 50
Simulation Model −40
−20 0 20 40 Radial Distance (mm)
−40
−20 0 20 40 Radial Distance (mm) Summed Dotted Profile
250
Counts
Counts
Counts
1500
Counts
0
2000
1500
0
0 50 Radial Distance (mm)
2500
2000
250
50
150
3000
2500
500
100
3500
Simulation Model
3000
150
100
−50
0 50 Radial Distance (mm) Summed Dashed Profile
200
50
−50
3500
Model
0 Angular View (degrees)
Angular View (degrees)
0
100 50 0
Simulation Model −40
−20 0 20 40 Radial Distance (mm)
Figure 4.4: Comparison of the simulated and calculated transmission sinogram data (top-row), transmission profiles (middle-row) and single-scatter sinogram data (bottom-row) for the non-uniform phantom using the 68 Ge transmission source. The profile data in the plots have been summed over the dashed and dotted lines shown in the sinogram images to reduce noise.
120
Chapter 4. Development and Validation of the Scatter Correction
Simulation
50
100
150
2500
Counts
Counts
1000
Summed Dotted Profile Simulation Model
1000
−40
500
−20 0 20 40 Radial Distance (mm)
Summed Dashed Profile
160
140
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120 100
−40
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180
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80 60
0
1500
Counts
Counts
50
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2000
1500
180
100
100
2500
Simulation Model
200 150
50
−40 −20 0 20 40 Radial Distance (mm)
−40 −20 0 20 40 Radial Distance (mm) Summed Dashed Profile
2000
500
Model
0 Angular View (degrees)
Angular View (degrees)
0
Simulation Model −40
−20 0 20 40 Radial Distance (mm)
80 60
Simulation Model −40
−20 0 20 40 Radial Distance (mm)
Figure 4.5: Comparison of the simulated and calculated transmission sinogram data (top-row), transmission profiles (middle-row) and single-scatter sinogram data (bottom-row) for the non–uniform phantom using the 57 Co transmission source. The profile data in the plots have been summed over the dashed and dotted lines shown in the sinogram images to reduce noise.
121
Chapter 4. Development and Validation of the Scatter Correction
68
20
Ge: Scatter Fractions
57
30
Co: Scatter Fractions
18
SF (%)
14 12
25
Model; o Simulation; 25 mm Model; x Simulation; 30 mm Model; + Simulation; 45 mm Model; * Simulation; Non−Uniform
20 SF (%)
16
10 8
10
6 4 2
15
Model; o Simulation; 25 mm Model; x Simulation; 30 mm Model; + Simulation; 45 mm Model; * Simulation; Non−Uniform
5
−20 0 20 Axial Distance (mm)
−20 0 20 Axial Distance (mm)
Figure 4.6: Percent scatter fractions per sinogram for the simulated and calculated single–scatter sinogram data. Data shown is for all three uniform and the non–uniform phantoms and for both transmission sources. the following expression for Nincident : i Nincident ≈ Nemitted i
Ai cos Ψ P (E), ~λ − R ~ A |2 E 4π|R i i
(4.5)
where Nemitted is the total number of photons emitted by the point source during the simulation and all other variables have been previously defined. The left hand plots in figure 4.7 compare the simulated efficiency with the analytical model: (Ψ, E 0 ) = C (1 − exp(−µen (E 0 )t/ cos Ψ)), where C is scaling factor which is normalised to the simulated efficiency at the unscattered photon energy (511 keV). For reference, we also show the same analysis using an alternative intrinsic efficiency function which we briefly considered: (Ψ, E 0 ) = C (1 − exp(−µ(E 0 )t/ cos Ψ)), where µ is the linear attenuation coefficient (µ–values are more commonly used for this type of detector efficiency modelling). For energies lower than about 200 keV (e.g. for
57 Co),
the simulated and model efficiencies become relatively constant for all angles (the simulated efficiency and (Ψ, E 0 ) are both equal to approximately one) and it is no longer necessary to model the intrinsic efficiency in our scatter 122
Chapter 4. Development and Validation of the Scatter Correction correction. 1.2
1.2
1
1 0.8 Efficiency
Efficiency
0.8 300 keV
0.6 0.4
400 keV
En
Simulation
0.2 0
0.4
ε=C[1−exp(−µ (E)t/cosΨ)]
500 keV
0.6
20
40 Ψ (degrees)
300 keV
400 keV 500 keV
ε=C[1−exp(−µ(E)t/cosΨ)] Simulation
0.2 60
80
0
20
40 Ψ (degrees)
60
80
Figure 4.7: Comparison of the intrinsic efficiency estimated using simulated point source data with two different efficiency models one which uses the energy absorption coefficients µen (left–hand plot) and one which used the linear attenuation coefficients µ (right–hand plot).
4.3.2
Influence of the 136.5 keV γ–ray on
57
Co data
We also investigated the influence that the lower intensity (10.5%) 136.5 keV γ–ray has on the simulated 57 Co data. For this analysis we compared the simulated sinogram data with and without the contribution from the 136.5 keV γ–ray. We found that, on average, 11.0% of the detected unscattered photons and 12.5% of the detected scattered photons originated from the lower intensity γ–ray. However, the shapes of the scatter profiles and the SFs per sinogram did not change significantly (SFs differed by at most 1.2% and on average differed by less than 0.5%) when the 136.5 keV γ–ray contribution was removed from the sinogram data. Based on these findings, for all subsequent scatter corrections discussed in this work, we computed the scatter contribution only from the 122.1 keV γ–ray for the 57 Co
transmission data.
4.3.3
Effects of multiple and out of FOV scatter
Our simulated data were also used to assess the contribution from multiple and Rayleigh scatter. Our simulations indicated that for the
68 Ge
123
Chapter 4. Development and Validation of the Scatter Correction data, single–scatter contributes between 93% to 87% of the total detected scatter in the transmission data for the 25 and 45 mm radius cylinders, respectively. Multiple and Rayleigh scatter is more significant for the
57 Co
source, where single–scatter only accounts for between 84% to 74% of the total detected scatter. For the non–uniform phantom simulations, single– scatter contributed 91% and 80% of the total scatter for the
68 Ge
and
57 Co
transmission sources, respectively. The high contribution from single scatter suggests that a simple scatter sinogram rescaling, will therefore be sufficient to compensate for scatter from outside the field–of–view and for Rayleigh and multiple scatter. To test this hypothesis, we performed our model–based scatter calculation using the correct µ–maps which were limited axially to regions inside the scanner. We then compared radial profiles and SFs for the simulated total–scatter sinograms and the calculated single–scatter sinograms which were rescaled using the following expression:
scale factor =
P
Tsim k
−
P
Bsim k exp P
−
Smodel k
~A RR i ~λ R i
µ(~r, E)dr
,
(4.6)
are the number of counts for the k th sinogram bin and Bsim where Tsim k k of the simulated transmission and blank scan sinogram, respectively; and represents the calculated contribution from single scatter forthe k th Smodel k
sinogram bin. The transmission probability: exp −
~A RR i ~λ R i
µ(~r, E)dr
is cal-
culated using the axially–truncated (but otherwise correct) µ–map data.
We compared radial profiles and total SFs for the simulated total–scatter (single + multiple scatter) sinograms and the calculated single–scatter sinograms which were rescaled using equation (4.6). Figures 4.8 and 4.9 evaluate the accuracy of the rescaling method using the
68 Ge
and
57 Co
sources, re-
spectively, for each of the uniform water cylinders. In the upper plots of these figures, we compare the total simulated transmission profiles (unscattered + all orders of scatter) with the analytically–calculated data (unscattered + rescaled scatter). For reference, we also show the corresponding experimental data from the microPET Focus 120 (the details of these experiments 124
Chapter 4. Development and Validation of the Scatter Correction were discussed in chapter 3). In the lower plots, we compare the simulated and rescaled calculated scatter sinograms (the experimental scatter is obviously not available). Data were summed over all projection angles and all sinograms with ring difference zero and ±1. Similar results were obtained for the non–uniform phantom studies. The total SF data also agreed well for all phantoms and transmission source radionuclei (data not shown). For example, the rescaled analytical–calculation and simulation total SFs per sinogram differed by less than 1.5%, on average, for all phantoms and transmission sources considered here.
4.4
Discussion
The radial profiles shown in figures 4.2 and 4.3 demonstrate that our scatter correction correctly predicts the magnitude and spatial distribution of single scatter in the sinogram data for the uniform water cylinders. These simulations were also previously found to agree well with experimental data (see chapter 3). The profiles for the non–uniform phantom (shown in figures 4.4 and 4.5) demonstrate similar agreement between the calculated and simulated data. Although the simulated and calculated scatter profiles agree extremely well, a small discrepancy is visible in the transmission profiles in the regions of the profiles where the LORs pass through the Teflon insert. These differences are the result of a small discrepancy between the linear attenuation coefficients used in our calculation (values from Berger et al. [10]) and the interaction cross–sections from the Geant4 low–energy physics package used by the GATE simulation software. Since the scatter distributions agree well, we do not expect that this will influence scatter corrections computed using our method. The differences between the analytically–calculated and simulated SFs per sinogram (figure 4.6) were less than 1.5% for all phantoms using the 68 Ge
source. There are, however, some discrepancies (maximum difference
of 7%) between the simulated and calculated SFs per sinogram in the edge plane sinograms for the 45 mm radius water cylinder using the
57 Co
source.
As the phantom radius and axial length become larger the probability for 125
Chapter 4. Development and Validation of the Scatter Correction
68
Summed Counts
8000
Ge: 25mm Radius: Transmission Simulation Model Experiment
6000 4000
68
8000
Ge: 30mm Radius: Transmission Simulation Model Experiment
6000 4000
68
8000
Ge: 45mm Radius: Transmission Simulation Model Experiment
6000 4000
2000 2000 2000 −50 −30 −10 10 30 50 −50 −30 −10 10 30 50 −50 −30 −10 10 30 50 Radial Distance (mm) Radial Distance (mm) Radial Distance (mm)
68
Summed Counts
500
Ge: 25mm Radius: Scatter
400 300 200 100
Simulation Model 0 −50 −30 −10 10 30 Radial Distance (mm)
68
800
Ge: 30mm Radius: Scatter
68
1200
600
1000
400
800
200
600
Simulation Model 0 50 −50 −30 −10 10 30 50 Radial Distance (mm)
400 −50
Ge: 45mm Radius: Scatter
Simulation Model −30 −10 10 30 Radial Distance (mm)
Figure 4.8:
Average radial profiles for the experimental, simulated and analytically calculated transmission data (top–row). The simulated and calculated total–scatter sinogram data are shown in the bottom–row. Data shown are for the uniform water cylinders using the 68 Ge transmission source.
126
50
Chapter 4. Development and Validation of the Scatter Correction
57
Co: 25mm Radius: Transmission
Summed Counts
6000
Simulation Model Experiment
5000 4000
57
Co: 30mm Radius: Transmission
6000
Simulation Model Experiment
5000 4000
57
Co: 45mm Radius: Transmission
6000
Simulation Model Experiment
5000 4000
3000
3000
3000
2000
2000
2000
1000 1000 1000 −50 −30 −10 10 30 50 −50 −30 −10 10 30 50 −50 −30 −10 10 30 50 Radial Distance (mm) Radial Distance (mm) Radial Distance (mm)
57
Summed Counts
500
Co: 25mm Radius: Scatter
57
600
400
500
300
400
200
300
100
200
Simulation Model 0 −50 −30 −10 10 30 Radial Distance (mm)
57
Co: 30mm Radius: Scatter
Co: 45mm Radius: Scatter
900
Simulation Model
800
Simulation Model
100 50 −50 −30 −10 10 30 50 Radial Distance (mm)
700 600 −50
−30 −10 10 30 Radial Distance (mm)
Figure 4.9:
Average radial profiles for the experimental, simulated and analytically calculated transmission data (top–row). The simulated and calculated total–scatter sinogram data are shown in the bottom–row. Data shown are for the uniform water cylinders using the 57 Co transmission source.
127
50
Chapter 4. Development and Validation of the Scatter Correction detecting scattered photons increases in the edge crystals. This is due to the fact that, in the simulations, photons which scatter from within parts of the phantom just outside of the field–of–view can strike anywhere along the lateral side of the end–plane crystals. This effect is more prominent for
57 Co
because its requires very little LSO (only about one to two mm thickness) to stop photons in the 100 keV range. The increased scatter in the edge planes does not appear to occur in experimental data. We have performed a short simulation which includes axial lead shielding (which is present in the experimental scanner) and found that this effect no longer occurs. If one excludes the edge planes, the difference between simulated and calculated SFs were less than 2% for all phantoms using the
57 Co
source. Although
we could explicitly model these axial edge effects, this was not found to be necessary because the scatter rescaling –which is given by equation (5.1) and is required for experimental data to account for multiple scatter and scatter from outside the scanner FOV– compensates for these small differences in the scatter estimate. It is widely acknowledged [3] that modelling the intrinsic efficiency of PET detectors is a complex process, especially at 511 keV, where the dominant photon interaction in LSO is Compton scattering [10]. It is clear from the right–hand plots in figure 4.7 that the use of the commonly used linear attenuation coefficient, µ, in our efficiency model does not agree well with the simulated data. We used µen because we found that it provided a better agreement with simulated data over a wide range of energies without requiring the introduction of additional empirically fit parameters. For example, for photon energies between 350 to 750 keV, corresponding to the energy discrimination window used for the acquisition of the
68 Ge
transmission data,
our model of (Ψ, E 0 ) agrees with efficiencies estimated from simulation data to within about 6%. Although it is beyond the scope of the current work, our calculation could easily be extended to include a more complex efficiency function with parameters fit to simulated or experimental data (e.g. one that includes the effects of crystal scatter, the gaps between crystals and block effects). This should not be necessary because these efficiency effects have been found to “average–out” for scattered photons, since many 128
Chapter 4. Development and Validation of the Scatter Correction scattered photons can contribute to each LOR each with different energies and different incident angles [201]. One of the underlying assumptions for our scatter correction (and other similar corrections [14, 201]) is that the total distribution of scattered photons (including multiple and Rayleigh scatter) can be well modelled by a simple rescaling of the contribution from single–scattered photons. We found that single–scatter makes up between 93% and 87% of the total detected scatter in our simulated
68 Ge
data. Multiple and Rayleigh scatter is slightly
more significant for the
57 Co
source, where single–scatter only accounts for
between 84% to 74% of the total detected scatter. The analysis for the total simulated scatter and the analytically calculated data for the uniform water cylinder data (shown in figures 4.8 and 4.9) demonstrates that the scatter sinogram rescaling, given by equation (5.1), accurately compensates for scatter from outside the field–of–view and for Rayleigh and multiple scatter for most phantoms considered. Not surprisingly, some minor differences between the shape of the total scatter and rescaled single–scatter data were observed for the 45 mm radius water cylinder (where the contribution from multiple and/or Rayleigh scatter was the greatest). It is conceivable that these errors could negatively influence the uniformity of reconstructed µ–values for larger objects. It should be stressed, however, that the 45 mm radius cylinder is an extreme case (it fills about 90% of the microPET field–of–view) and no such non–uniformity was detected in the reconstructed images for the simulated or experimental data for this cylinder using either transmission source (discussed in the next chapter). An unrelated bias was also observed for the 57 Co
transmission source which increased with phantom size. These dif-
ferences are the result of the discrepancy, discussed in the first paragraph of this section, between the µ–values used in our calculation [10] and the low–energy cross–sections used by the GATE simulation software. We expect that the µ–values from Berger et al. [10] are more accurate and this discrepancy should not, therefore, negatively effect our experimental results. Finally, we also found, in our analysis of simulated
57 Co
data, that the
scatter profiles and the SFs per sinogram did not change significantly (SFs 129
Chapter 4. Development and Validation of the Scatter Correction differed by at most 1.2% and on average differed by less than 0.5%) when the 136.5 keV γ–ray contribution was removed from the sinogram data. As a result, we do not expect that it is necessary to model the 136.5 keV γ–ray in our scatter correction model.
4.5
Conclusion
We have presented a new scatter correction for singles–mode transmission data. We have compared our correction with simulated data for three different uniform water cylinders and for a non–uniform phantom consisting of water, Teflon and air, using two transmission sources ( 68 Ge and
57 Co).
The results of this chapter confirm that our analytical calculation correctly determines the spatial distribution of scattered photons which contribute to singles–mode PET transmission data. There were, however, some minor differences in the calculated single scatter fraction (SF) per sinogram, particularly, for the edge planes of the sinogram data using the
57 Co
source and
the largest cylinder size. We believe that the simulations overestimate the scatter in these edge–planes because we did not model the axial lead shielding which is present in the experimental PET imaging system. Excluding the edge planes, the calculated single scatter fraction (SF) per sinogram agreed with simulated values to within 2% for all phantom sizes and both transmission isotopes. In this chapter, we have demonstrated that our scatter calculation is accurate when it is provided with the true attenuation–maps (i.e. the distribution of correct linear attenuation coefficients for the material(s) present within the scanner). In experimental PET studies, however, the attenuation– map is precisely the unknown quantity which we are trying to determine with a transmission scan. In the next chapter, we describe and validate a reconstruction and correction procedure for the more realistic situation in which we use a reconstructed attenuation–map to compute our scatter corrections.
130
Chapter 5
Development and Validation of the Reconstruction Procedure In this chapter, we describe a reconstruction and scatter correction procedure applicable to singles–mode transmission data. We begin with a description of our proposed method which incorporates our scatter correction, discussed in the previous chapter, as part of an iterative reconstruction algorithm [125]. We have tested our procedure using simulated and experimental data for the uniform water cylinders and for the non–uniform phantom using 68 Ge
and
57 Co
transmission radionuclei. The simulated data were used to
isolate the effects of scatter and examine a number of issues related to the reconstruction and transmission data in a step–by–step manner not possible with experimental data. We also applied our procedure to two typical experimental rodent studies (a mouse and a rat). This analysis has focused on demonstrating that our correction and reconstruction method provides the correct linear attenuation coefficients (µ–values) for each set of data considered. For the reconstructed attenuation–maps obtained using simulated and experimental phantom data, we compared the reconstructed µ–values with the true values corresponding to the material(s) present in each phantom. For the rodent studies, we performed a region of interest (ROI) analysis and qualitatively examined the corrected and uncorrected µ–map data.
131
Chapter 5. Development and Validation of the Reconstruction Procedure
5.1
Reconstructions and Scatter Rescaling
Unless stated otherwise, all data were reconstructed using ordered–subset transmission (OSTR) [125], a maximum a posteriori (MAP) reconstruction algorithm for transmission tomography. The OSTR algorithm is distributed as part of the reconstruction software library ASPIRE (a sparse iterative reconstruction) [99, 202]. For all OSTR reconstructions, we used 20 iterations with 4 subsets, a smoothing intensity parameter β = 2 8 and a Huber penalty function [88] with a cutoff parameter δ = 0.5µ water , where µwater is the linear attenuation coefficient for water at the appropriate photon energy. Recall from section 1.10.6, that this penalty applies heavier quadratic smoothing for neighbouring voxels with reconstructed µ–values which differ by less than δ. For differences greater than δ, linear smoothing is applied to discourage over-smoothing of abrupt changes which should occur at the boundaries between different tissue types. All of the transmission and blank data (both simulated and experimental) were acquired in 3D–mode and rebinned using single–slice rebinning (SSRB). The rebinning used in SSRB corresponded to an oblique angle (i.e. the maximum angle between the LORs in an oblique sinogram and the direct plane sinogram to which they are assigned in SSRB) of 13.7o . Since our scatter correction requires an estimate of the µ–map for its calculation, our reconstructions proceed in five distinct steps: 1. In the first step, we reconstruct the µ–map with no scatter correction. 2. In the second step, the attenuation–map is processed into a histogram of µ–values for each slice of the reconstruction. Examples of µ–value histograms are shown in the results section of this chapter in figures 5.14 and 5.15. We then search for a peak in each histogram (a bin containing a relatively large number of voxels). Since the majority of image slices in a typical study consist of air and soft–tissue we assume that this peak corresponds to the linear attenuation coefficient of soft–tissue (water). We also only classify a particular bin as a peak if we also find that, on average, the slopes in the histogram distribution 132
Chapter 5. Development and Validation of the Reconstruction Procedure before the peak bin are positive (i.e. the differences between adjacent bins in the µ–value histogram are positive for several bins approaching the peak) and that the slopes are negative after the peak. The µ–map is then automatically rescaled, slice–by–slice, so that the peak in each histogram corresponds to the correct linear attenuation coefficient for water. If no peak is found that meets the criteria described above, then no rescaling is performed for that particular slice. 3. In the third stage, the rescaled attenuation–map is used to compute our scatter correction (i.e. we compute the estimated contribution from unscattered and scattered photons to the singles–mode transmission data using equations (4.1) to (4.3). 4. In the fourth stage, each of the calculated scatter sinograms is rescaled to match the acquired data. This step is necessary because the data contain multiple scatter and the reconstructed µ–maps are truncated, that is, they only include portions of the phantom which are inside the scanner field–of–view (FOV). We use the same approach for scatter sinogram rescaling described in the previous chapter, with the following expression for the scale factor:
scale factor =
P
Texp k
−
P
Bexp k
exp −
P
Skmodel
~A RR i ~λ R i
µ(~r, E)dr
,
(5.1)
where µ(~r, E) is the estimated µ-map data at the unscattered photon energy; Texp and Bexp are the number of counts for the k th sinok k gram bin of the experimental transmission and blank scan sinogram, respectively; Smodel represents the calculated contribution from sink gle scatter for the k th sinogram bin. The transmission probability: ~A RR exp − ~ i µ(~r, E)dr , is obtained from the ratio of our analytically– R λi
calculated unscattered and blank data (equations (4.3) and (4.4)).
5. In the fifth and final stage, the rescaled scatter sinogram is subtracted from the transmission data and the corrected data is reconstructed to 133
Chapter 5. Development and Validation of the Reconstruction Procedure generate a scatter corrected (SC) µ–map. The implicit assumption here is that the rescaled attenuation–map (obtained in step 2 of the above procedure) is not accurate enough to calculate the correct attenuation corrections, but may be sufficient to compute the transmission data scatter correction given by equation (4.2). The justification for this assumption comes from the fact that the line integrals and summation in equation (4.2) depend on many voxels within the µ–map and small errors should, on average, cancel out. If this is not the case, it may also be necessary to iteratively repeat our scatter correction (i.e. repeat steps 2 to 5 starting from the previous scatter corrected µ–map) in order to obtain more accurate µ–values. During step 2, the developed software automatically reports the peak–values found for each slice of the µ–value histogram. In order to determine if additional scatter iterations are necessary, the user can verify that these peaks occur at the true linear attenuation coefficient for soft–tissue (water). It should be noted here that, in certain circumstances, the µ–map rescaling (step 2) may fail to find a distinct peak in the histogram data (corresponding to water or soft–tissue) in a given slice. We found that this can occur in “empty” slices that consist mostly of air and for smaller objects with noisier attenuation–maps (e.g. mice). This could potentially lead to problems in the scatter sinogram rescaling (step 4), which also uses the rescaled µ–map data. For those slices where the µ–map rescaling failed we, therefore, apply a simplified version of step 4. For these data each analytically calculated scatter sinogram is multiplied by the ratio of the experimentally–measured and analytically–calculated sum of the counts in the blank sinogram. This alternative scatter sinogram rescaling will compensate for single–scatter from within the scanner FOV, but does not take into account multiple and Rayleigh scatter or scatter from regions outside the scanner. Fortunately, this occurs primarily for smaller sized objects, such as mice (see sections 5.4.2 and 5.5.2), where this additional scatter is expected to be less significant.
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Chapter 5. Development and Validation of the Reconstruction Procedure
5.2
Simulations
Before we applied our correction and reconstruction procedure to experimental data, we first tested our algorithm using simulated data. As in the previous chapter, we simulated singles–mode transmission data for 57 Co
68 Ge
and
radionuclei for the three uniform water cylinders and the non–uniform
phantom discussed in the previous chapters. For this analysis, we used the parameters for the microPET R4 imaging system. We chose to reconstruct simulated R4 data (instead of the simulated Focus 120 data from the previous chapters) simply because our simulations for this scanner had higher counts per line–of–response (LOR) with values more similar to those encountered in our experimental transmission scans. The simulation data were used to investigate the µ–map reconstruction procedure (described in section 5.1) in ways which are not possible with experimental data. For example, we could isolate the effects of scatter from reconstruction issues by comparing our scatter corrected reconstructions to those reconstructed from simulated data containing no scatter (scatter–free). The simulated data were also used to investigate three additional features of our reconstruction method: (1) to determine if it is necessary to iteratively repeat our scatter correction; (2) to investigate the influence of noise in the transmission data; and (3) to evaluate the errors introduced by processing the 3D PET data using single–slice rebinning prior to reconstruction.
5.3
Experiments
All experimental data for this chapter were acquired using the microPET Focus 120 scanner using the two rotating source geometries and acquisition parameters discussed in previous chapters. At the time of the experiments, the
68 Ge
and
57 Co
sources had activities of 10 MBq and 165 MBq, respec-
tively. Using both sources, we acquired experimental data for the three different sized water cylinders and for the non–uniform phantom. We also obtained transmission data using the dents (a rat and mouse). All
57 Co
57 Co
source for two anaesthetised ro-
transmission and blank data were acquired 135
Chapter 5. Development and Validation of the Reconstruction Procedure for 17 minutes and all
68 Ge
data were collected for 25 minutes. Typical sin-
gles count rates for transmission and blank scans were approximately 2500 and 27,000 counts/second per block for the
68 Ge
and
57 Co
sources, respec-
tively. The microPET acquisition software does not currently provide corrections for detector dead–time in singles–mode, but we do not anticipate large dead–time effects with these count–rates. Transmission and blank data were pre–corrected for contamination due to the naturally occurring
176 Lu
radioactivity present in the crystals as described in chapter 3. No emission activity was present in the scanner at the time of the transmission scans (i.e. they were performed pre–injection).
5.4 5.4.1
Results Simulations
Figures 5.1 and 5.2 show µ–maps with and without scatter corrections (SC) for each uniform phantom reconstructed using the simulated
68 Ge
and 57 Co
data, respectively. These figures show transverse slices and profiles for each phantom that have been summed over 10 planes in the central region of the scanner to decrease noise. The positions of the profiles are indicated by the dashed lines on the scatter corrected (SC) µ–map images. We also determined the average linear attenuation values in a circular region of interest (ROI) as a function of the axial position for each µ–map slice (shown in figure 5.3). The location of the ROI for each phantom is shown as a dashed circle in the µ–map images with no scatter correction (No SC) in figures 5.1 and 5.2. Figures 5.4 and 5.5 show slices and profiles for the non–uniform phantom data. We also determined the average reconstructed µ–value for three regions of interest (ROIs) corresponding to the Teflon, water and air regions of the phantom. The positions of the three ROIs are shown on the No SC image in figures 5.4 and 5.5. Figures 5.6 shows the average value in each ROI as a function of axial position for each µ–map slice for
68 Ge
(top–row) and 57 Co (bottom–row) transmission sources. In all plots, we also compare our reconstructed images with ideal scatter–free images (i.e. those
136
Chapter 5. Development and Validation of the Reconstruction Procedure reconstructed from only the unscattered photon component of the simulated transmission data). Tables 5.1 and 5.2 show a summary of the axial ROI analysis for the uniform and non–uniform phantom data, respectively. The values given in these tables have been averaged over all axial slices of the µ–map excluding five outer slices at either end of the µ–maps corresponding to the outer edges of scanner. These outer slices were excluded from the analysis because the sinogram data for these regions have much higher noise and were found to be biased toward higher µ–values. A discussion of this effects is included in the following section. Both tables also include the linear attenuation coefficients (values from Berger et al. [10]) for the material(s) present in each phantom at the appropriate energy for each transmission source. Please note that, in the simulations for the 57 Co transmission source, we did not model Rayleigh scatter, therefore, the true attenuation coefficients for the simulation data do not include the effects of this interaction. The amounts of CPU time (2.2 GHz processor) for the calculation of the scatter correction for each dataset depended on the isotope and the phantom size. These amounts ranged from 6 minutes ( 68 Ge: 25 mm radius) to 20 minutes (57 Co: 45 mm radius). The OSTR reconstructions required about 2 minutes for 20 iterations using the same processor. The Effects of Transmission Data Noise In the ROI analysis, the results of which are shown in figures 5.3 and 5.6, we observed that, in the outer slices of the reconstructed µ–maps, the average µ–values were much higher than in the central regions. We found that this was true for all reconstructions considered, including the ideal scatter– free reconstruction, implying that this effect is not connected to the scatter correction method. We attribute this bias to difficulties in reconstructing accurate µ–values using noisy transmission data. To isolate the effects of noise, we reconstructed analytically calculated blank and unscattered transmission data generated using equations (4.4) and (4.3), respectively. For this study we used digital phantoms corresponding
137
Chapter 5. Development and Validation of the Reconstruction Procedure
68
68
Ge: 25 mm Rad.
68
Ge: 30 mm Rad.
Ge: 45 mm Rad.
0.1
0.05 SC
SC
68
SC
68
Ge: 25 mm Rad.
0
68
Ge: 30 mm Rad.
Ge: 45 mm Rad.
0.1
0.05 No SC 68
Ge: 25 mm Rad.: Horizontal Profile
0.1
No SC 68
µ (cm−1)
Ge: 45 mm Rad.: Horizontal Profile
0.1
µ
Water
Water
0.08
0.08
0.06
0.06
0.06
0.02 0 −50
0.04
No SC SC: Iter. 1 SC: Iter. 2 Scatter Free −30
−10 10 30 Distance (mm)
0 −50
0.04
No SC SC: Iter. 1 SC: Iter. 2 Scatter Free
0.02 50
µ
Water
0.08
0.04
−30
−10 10 30 Distance (mm)
No SC SC: Iter. 1 SC: Iter. 2 Scatter Free
0.02 50
0
68
Ge: 30 mm Rad.: Horizontal Profile
0.1
µ
No SC
0 −50
−30
−10 10 30 Distance (mm)
50
Figure 5.1: Slices through the µ–map images with (upper images) and without scatter corrections (middle images) for the uniform water cylinders reconstructed using simulated 68 Ge transmission data. Profiles through the reconstructed data are shown in the lower plots.
138
Chapter 5. Development and Validation of the Reconstruction Procedure
57
57
Co: 25 mm Rad.
57
Co: 30 mm Rad.
Co: 45 mm Rad.
0.15 0.1 0.05
SC
SC
57
SC
57
Co: 25 mm Rad.
57
Co: 30 mm Rad.
Co: 45 mm Rad.
0.15 0.1 0.05
No SC 57
Co: 25 mm Rad.: Horizontal Profile µ
µ (cm−1)
0.15
Water
0.1
No SC
57
Co: 30 mm Rad.: Horizontal Profile µ
0.15
Water
0.1
No SC SC: Iter. 1 SC: Iter. 2 Scatter Free
0.05
0 −50
No SC
−30
−10 10 30 Distance (mm)
0 −50
Co: 45 mm Rad.: Horizontal Profile µWater
0.15
0.1
No SC SC: Iter. 1 SC: Iter. 2 Scatter Free
0.05
50
57
−30
−10 10 30 Distance (mm)
No SC SC: Iter. 1 SC: Iter. 2 Scatter Free
0.05
50
0 −50
−30
−10 10 30 Distance (mm)
50
Figure 5.2: Slices through the µ–map images with (upper images) and without scatter corrections (middle images) for the uniform water cylinders reconstructed using simulated 57 Co transmission data. Profiles through the reconstructed data are shown in the lower plots.
139
Chapter 5. Development and Validation of the Reconstruction Procedure 68
68
Ge: 25 mm Rad.: Average in ROI
0.14 0.12 µ (cm−1)
0.14
No SC SC: Iter. 1 SC: Iter. 2 Scatter Free
0.1
0.1
0.08
0.06
0.06
0.06
−10 10 30 Axial Distance (mm)
50
−30
0.2 0.18 0.16
−30
No SC SC: Iter. 1 SC: Iter. 2 Scatter Free
0.18 0.16
0.16 0.14
0.12
0.12
0.12
0.1
0.1
0.1
50
−50
−30 −10 10 30 Axial Distance (mm)
µ
Water
0.14
−10 10 30 Axial Distance (mm)
No SC SC: Iter. 1 SC: Iter. 2 Scatter Free
0.18
0.14
−30
50
Co: 45 mm Rad.: Average in ROI
0.2
µWater
Water
−10 10 30 Axial Distance (mm)
57
Co: 30 mm Rad.: Average in ROI
0.2
µ
50
57
Co: 25 mm Rad.: Average in ROI No SC SC: Iter. 1 SC: Iter. 2 Scatter Free
−10 10 30 Axial Distance (mm)
µ
Water
0.08
57
µ (cm−1)
0.1
0.08
−30
No SC SC: Iter. 1 SC: Iter. 2 Scatter Free
0.12
µWater
Water
Ge: 45 mm Rad.: Average in ROI
0.14
No SC SC: Iter. 1 SC: Iter. 2 Scatter Free
0.12
µ
68
Ge: 30 mm Rad.: Average in ROI
−30
50
−10 10 30 Axial Distance (mm)
Figure 5.3: Average value of attenuation coefficients (in cm −1 ) in ROIs for each axial slice of the uniform water cylinders reconstructed using simulated 68 Ge (top–row) and 57 Co (bottom–row) transmission data. Table 5.1: Reconstructed attenuation coefficients (in cm −1 ) averaged over all axial slices (excluding edges) of the uniform water cylinder attenuation– maps for the simulated 68 Ge and 57 Co transmission data.
68 Ge:
No SC SC: Iter.1 SC: Iter.2 SC: Iter.3 Scatter–Free
57 Co:
µTrue = 0.0957 25 mm 30 mm 45 mm
25 mm
µTrue =0.1558 30 mm 45 mm
0.0697 0.0942 0.0990 0.1002 0.1000
0.1282 0.1548 0.1593 0.1601 0.1607
0.1190 0.1562 0.1591 0.1596 0.1613
0.0614 0.0949 0.0983 0.0988 0.1009
0.0528 0.0933 0.0975 0.0977 0.1013
0.1081 0.1565 0.1604 0.1606 0.1621
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Ge: Non−Uniform
Ge: Non−Uniform
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No SC
SC 68
68
Ge: Non−Uniform: Horizontal Profile
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Ge: Non−Uniform: Vertical Profile
µWater
µ (cm−1)
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0.1 µWater
No SC SC: Iter. 1 SC: Iter. 2 Scatter Free
0.02 0 −50
No SC SC: Iter. 1 SC: Iter. 2 Scatter Free
0.2 µTeflon
0.08
0.04
−30
−10 10 Distance (mm)
0.05
0.05 30
50
0 −50
−30
−10 10 Distance (mm)
30
50
Figure 5.4: Slices through the µ–map images with and without scatter corrections (upper images) for the non–uniform phantom reconstructed using simulated 68 Ge data. Profiles through the reconstructed data are shown in the lower plots.
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0.3
57
Co: Non−Uniform
Co: Non−Uniform
0.2 0.1 SC
No SC
57
57
Co: Non−Uniform: Horizontal Profile
µ (cm−1)
0.15
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µWater
0.3
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0.2
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−30
−10 10 Distance (mm)
No SC SC: Iter. 1 SC: Iter. 2 Scatter Free
µ
Teflon
µ
Water
No SC SC: Iter. 1 SC: Iter. 2 Scatter Free
0.05
0
0.1
30
50
0 −50
−30
−10 10 Distance (mm)
30
50
Figure 5.5: Slices through the µ–map images with and without scatter corrections (upper images) for the non–uniform phantom reconstructed using simulated 57 Co data. Profiles through the reconstructed data are shown in the lower plots.
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68
Ge: Non−Uniform: Teflon ROI No SC SC: Iter. 1 SC: Iter. 2 Scatter Free
µ (cm−1)
0.25 0.2
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50
57
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Co: Non−Uniform: Teflon ROI No SC SC: Iter. 1 SC: Iter. 2 Scatter Free
0.35 µ (cm−1)
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Water
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−30
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Air
57
0.1 µWater
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0
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No SC SC: Iter. 1 SC: Iter. 2 Scatter Free
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−10 10 30 Axial Distance (mm)
57
0.22
µ
0.04 0.02
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No SC SC: Iter. 1 SC: Iter. 2 Scatter Free
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0.08
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No SC SC: Iter. 1 SC: Iter. 2 Scatter Free
0.12 µTeflon
−30
68
Ge: Non−Uniform: Water ROI
0.14
0
µ
Air
−30
−10 10 30 Axial Distance (mm)
50
Figure 5.6: Average attenuation coefficients (in cm −1 ) for three ROIs (Teflon, water and air) for each axial slice of the non–uniform phantom reconstructed using simulated 68 Ge (top–row) and 57 Co (bottom–row) transmission data.
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Table 5.2: Reconstructed attenuation coefficients (in cm −1 ) for each of the three ROIs (Teflon, water and air) averaged over all axial slices (excluding edges) of the non–uniform phantoms reconstructed using simulated 68 Ge and 57 Co transmission data. 68 Ge
Truth No SC SC:Iter.1 SC:Iter.2 SC:Iter.3 Scatter–Free
57 Co
Water
Teflon
Air
Water
Teflon
Air
0.0957 0.0586 0.0964 0.0978 0.0985 0.0992
0.1825 0.1293 0.1854 0.1878 0.1888 0.1874
0.0000 0.0026 0.0128 0.0135 0.0138 0.0144
0.1558 0.1215 0.1555 0.1583 0.1588 0.1650
0.2963 0.2439 0.3016 0.3068 0.3076 0.3160
0.0000 0.0091 0.0173 0.0179 0.0181 0.0195
to the 30 mm radius water cylinder and the correct µ–values for 511 and 122 keV photon energies. Since we used our expression for the unscattered photon contribution to the transmission data, we expect that the average µ–values for these reconstructions should correspond to the correct linear attenuation coefficient for water. We added Poisson-distributed noise to the blank and transmission data for a wide range of noise levels. We reconstructed the data using several different algorithms which are also distributed as part of the ASPIRE reconstruction library [99, 202], these include: filtered back–projection for emission and for transmission data (FBPem and FBPtr); maximum–likelihood expectation–maximisation (MLEM) for transmission data; and the ordered–subsets maximum a posteriori (MAP) transmission data reconstruction described previously (OSTR). The line–integral data used by the FBPem reconstruction consisted of the logarithm of the blank divided by the unscattered transmission counts. We did not remove any negative values from the logarithm data (i.e. where the blank counts were less than the transmission counts) and if either the transmission or blank data were equal to zero, we also set the corresponding 144
Chapter 5. Development and Validation of the Reconstruction Procedure line integral to zero. The MLEM reconstruction used 100 iterations (to ensure convergence) while OSTR used 20 iterations and used the same penalty function and parameters described earlier. Figure 5.7 shows the average µ– value in a circular ROI for these data as a function of the average number of counts per sinogram bin in the blank data. The Effects of Single–Slice Rebinning Even when the noise levels were low (i.e. in the central slices of the µ– maps), we observed that the µ–values for scatter–corrected and scatter–free reconstructions were slightly higher than their expected values. All of the transmission and blank data were acquired in 3D–mode and then rebinned using single–slice rebinning (SSRB). We contend that the biases observed in the central planes of our images are a result of using a 2D reconstruction algorithm on the SSRB data. To isolate the effects of SSRB, for all uniform water cylinders we rebinned the scatter–free Monte–Carlo sinogram data using different maximum oblique angles (i.e. the maximum angle between the oblique sinogram and the direct plane sinogram to which they are assigned in SSRB) of 2.8, 13.7 and 26.7 degrees. In order to avoid the influence of statistical noise, we summed all the axial planes of the rebinned sinogram data prior to reconstruction. We reconstructed the data using OSTR and computed the average µ–value in a circular ROI for each image. The results are summarised in table 5.3.
5.4.2
Experimental Data
Phantom Data We performed the same analysis for the reconstructed images obtained using the experimental phantom data. Example transverse slices and profiles through the reconstructed images for the uniform water cylinder reconstructions using experimental
68 Ge
and
57 Co
transmission data are shown in fig-
ures 5.8 and 5.9, respectively. Figures 5.11 and 5.12 show slices and profiles for the non–uniform phantom data. For the
57 Co
data, the true µ–values 145
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0.6
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Ge
Ge
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OSTR
57
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0.16
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0
57
0.2
68
FBPem FBPtr MLEM OSTR
0.2
0.15 0
µWater
200 400 600 800 1000 Average Counts Per Sinogram Bin
Figure 5.7: Top rows show images for three of the reconstruction methods applied to the analytically calculated scatter–free data. The images shown used blank data with an average of 500 counts per LOR. Lower plots show average attenuation coefficients (in cm −1 ) in ROIs for data with different levels of statistical noise. Data shown is for the 30 mm radius uniform water cylinder.
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Table 5.3: Average reconstructed attenuation coefficients (in cm −1 ) in ROIs for the uniform water cylinders and both transmission sources. The data shown here is for the scatter–free simulation data and is compared for different maximum oblique angles used in the single–slice rebinning (SSRB). All the axial sinograms were summed prior to reconstruction to reduce noise.
68 Ge:
57 Co:
Max. Oblique Angle (degrees)
µTrue = 0.0958 25 mm 30 mm 45 mm
µTrue =0.1560 25 mm 30 mm 45 mm
2.3 13.7 26.7
0.0962 0.0979 0.1002
0.1555 0.1586 0.1625
0.0977 0.0989 0.1011
0.0981 0.0994 0.1014
0.1568 0.1588 0.1624
0.1552 0.1570 0.1604
from Berger et al. [10] include the effects of Rayleigh scatter and are, therefore, slightly higher than the true–values given for the simulation data. The amounts of CPU time (2.2 GHz processor) required for one iteration of our scatter correction ranged from 9 minutes ( 68 Ge: 25 mm radius) to 27 minutes (57 Co: 45 mm radius) and the OSTR reconstructions took about 3 minutes for 20 iterations using the same processor. Rodent Studies Images of the reconstructed µ–maps for the mouse and rat scans are shown in figures 5.14 and 5.15, respectively. The scatter corrected data shown in the images are for the third SC iteration. We performed a simple ROI analysis in three perpendicular planes (transverse, coronal and sagittal) for the images shown. The average reconstructed µ–value and its standard deviation are shown below each image. We expect that the correct µ–values for soft–tissue should be approximately 0.16 cm−1 . In the lower part of each figure, we also show µ–value histograms for all voxels in the reconstructed data both with and without scatter corrections. Using a 2.2 GHz processor, both studies
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68
Ge: 25 mm Rad.
68
Ge: 30 mm Rad.
Ge: 45 mm Rad.
0.1
0.05 SC
SC
SC
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68
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Ge: 30 mm Rad.
Ge: 25 mm Rad.
Ge: 45 mm Rad.
0 0.1
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Ge: 25 mm Rad.: Horizontal Profile
−1
µ (cm )
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No SC 68
Ge: 30 mm Rad.: Horizontal Profile
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µ
No SC 68
Ge: 45 mm Rad.: Horizontal Profile
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Water
0.04 No SC SC: Iter. 1 SC: Iter. 2
0.02 50
0
−30
−10 10 30 Distance (mm)
No SC SC: Iter. 1 SC: Iter. 2
0.02 50
0 −50
−30
−10 10 30 Distance (mm)
50
Figure 5.8: Slices through the µ–map images with (upper images) and without scatter corrections (middle images) for the uniform water cylinders reconstructed using 68 Ge transmission data. Profiles through the reconstructed data are shown in the lower plots. Note that these data were reconstructed using experimental transmission data, unlike those shown in figure 5.1 which were reconstructed from simulated data. The half circular structure beneath each cylinder is the animal bed.
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57
57
57
Co: 25 mm Rad.
Co: 45 mm Rad.
Co: 30 mm Rad.
0.15 0.1 0.05
SC
SC
57
SC
57
Co: 25 mm Rad.
57
Co: 30 mm Rad.
Co: 45 mm Rad.
0.15 0.1 0.05
No SC 57
Co: 25 mm Rad.: Horizontal Profile
No SC 57
57
Co: 30 mm Rad.: Horizontal Profile
µ
Co: 45 mm Rad.: Horizontal Profile
µ
Water
µ (cm−1)
No SC
µ
Water
Water
0.15
0.15
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No SC SC: Iter. 1 SC: Iter. 2 −30
−10 10 30 Distance (mm)
50
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No SC SC: Iter. 1 SC: Iter. 2
0 −50 −30 −10 10 30 Distance (mm)
50
0 −50
No SC SC: Iter. 1 SC: Iter. 2 −30
−10 10 30 Distance (mm)
50
Figure 5.9: Slices through the µ–map images with (upper images) and without scatter corrections (middle images) for the uniform water cylinders reconstructed using 57 Co transmission data. Profiles through the reconstructed data are shown in the lower plots. Note that these data were reconstructed using experimental transmission data, unlike those shown in figure 5.2 which were reconstructed from simulated data. The half circular structure beneath each cylinder is the animal bed.
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68
Ge: 25 mm Rad.: Average in ROI
0.14
No SC SC: Iter. 1 SC: Iter. 2
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µ (cm )
0.12
µ
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50
−30
Co: 25 mm Rad.: Average in ROI No SC SC: Iter. 1 SC: Iter. 2
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−30
No SC SC: Iter. 1 SC: Iter. 2
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Water
0.15
0.15
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0.1
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50
No SC SC: Iter. 1 SC: Iter. 2
µ
Water
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Co: 45 mm Rad.: Average in ROI
0.15
−30
−10 10 30 Axial Distance (mm)
57
Co: 30 mm Rad.: Average in ROI
µ
Water
50
57
0.2
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0.1
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0.08
−30
No SC SC: Iter. 1 SC: Iter. 2
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57
µ (cm−1)
Ge: 45 mm Rad.: Average in ROI
0.14
No SC SC: Iter. 1 SC: Iter. 2
0.1
Water
68
Ge: 30 mm Rad.: Average in ROI
0.12 µ
0.1
68
0.14
50
−30
−10 10 30 Axial Distance (mm)
50
Figure 5.10: Average value of attenuation coefficients (in cm −1 ) in ROIs for each axial slice of the uniform water cylinders reconstructed using experimental 68 Ge (top–row) and 57 Co (bottom–row) transmission data.
Table 5.4: Reconstructed attenuation coefficients (in cm −1 ) averaged over all axial slices (excluding edges) of the uniform water cylinder attenuation– maps for the experimental 68 Ge and 57 Co transmission data.
68 Ge:
No SC SC: Iter. 1 SC: Iter. 2 SC: Iter. 3
57 Co:
µTrue = 0.0957 25 mm 30 mm 45 mm
25 mm
µTrue =0.1600 30 mm 45 mm
0.0683 0.0925 0.0961 0.0965
0.1140 0.1543 0.1612 0.1623
0.1080 0.1580 0.1618 0.1622
0.0619 0.0940 0.0964 0.0967
0.0557 0.0934 0.0959 0.0960
0.1013 0.1592 0.1628 0.1629
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Ge: Non−Uniform
68
Ge: Non−Uniform
0.15 0.1
SC
No SC Ge: Non−Uniform: Vertical Profile
Ge: Non−Uniform: Horizontal Profile
µWater
−1
µ (cm )
0.15
0.06
0.1
0.04 No SC SC: Iter. 1 SC: Iter. 2
0.02 0 −50
No SC SC: Iter. 1 SC: Iter. 2
0.2 µTeflon
0.08
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−10 10 30 Distance (mm)
0
68
68
0.1
0.05
µWater
0.05 50
0 −50
−30
−10 10 30 Distance (mm)
50
Figure 5.11: Slices through the µ–map images with and without scatter corrections (upper images) for the non–uniform phantom reconstructed using experimental 68 Ge data. Profiles through the reconstructed data are shown in the lower plots.
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Co: Non−Uniform
0.3
57
Co: Non−Uniform
0.2 0.1 SC
No SC 57
57
Co: Non−Uniform: Vertical Profile
Co: Non−Uniform: Horizontal Profile
µ (cm−1)
0.15
µWater
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0.2 µ Water
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No SC SC: Iter. 1 SC: Iter. 2
µ
0.1
No SC SC: Iter. 1 SC: Iter. 2 −30
−10 10 30 Distance (mm)
50
0 −50
−30
−10 10 30 Distance (mm)
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Figure 5.12: Slices through the µ–map images with and without scatter corrections (upper images) for the non–uniform phantom reconstructed using experimental 57 Co data. Profiles through the reconstructed data are shown in the lower plots.
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0.11
−1
µ (cm )
µ
Teflon
0.18 0.16 0.14
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50
Co: Non−Uniform: Teflon ROI
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No SC SC: Iter. 1 SC: Iter. 2
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Air
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50
57
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0.16
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−30
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57
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0.25
µ
0
0.18
µ
0.3
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No SC SC: Iter. 1 SC: Iter. 2
0.07
57
−1
Water
0.06 −30
µ (cm )
µ
0.08
0.12
No SC SC: Iter. 1 SC: Iter. 2
0.08
0.09
No SC SC: Iter. 1 SC: Iter. 2
Ge: Non−Uniform: Air ROI
Ge: Non−Uniform: Water ROI
Ge: Non−Uniform: Teflon ROI
0.2
68
68
68
50
−30
−10 10 30 Axial Distance (mm)
50
Figure 5.13: Average attenuation coefficients (in cm −1 ) for three ROIs (Teflon, water and air) for each axial slice of the non–uniform phantom reconstructed using experimental 68 Ge (top–row) and 57 Co (bottom–row) transmission data. Table 5.5: Reconstructed attenuation coefficients (in cm −1 ) for each of the three ROIs (Teflon, water and air) averaged over all axial slices (excluding edges) of the non–uniform phantoms reconstructed using experimental 68 Ge and 57 Co transmission data. 68 Ge
Truth No SC SC: Iter. 1 SC: Iter. 2 SC: Iter. 3
57 Co
Water
Teflon
Air
Water
Teflon
Air
0.0957 0.0588 0.0950 0.0960 0.0960
0.1825 0.1259 0.1777 0.1795 0.1796
0.0000 0.0021 0.0131 0.0134 0.0134
0.1600 0.1053 0.1568 0.1592 0.1595
0.3058 0.2124 0.3005 0.3063 0.3068
0.0000 0.0028 0.0109 0.0110 0.0111
153
Chapter 5. Development and Validation of the Reconstruction Procedure required about 20 minutes of CPU time for each scatter correction iteration and 3 minutes for each reconstruction. 57
Co: Rat: SC: Transverse (X−Y)
57
57
Co: Rat: SC: Coronal (X−Z)
Co: Rat: SC: Sagittal (Y−Z)
0.3 0.2 0.1
µ=0.15985+/− 0.014152
0
µ=0.16161+/− 0.014676
µ=0.16012+/− 0.015083 (X−Y) Co: Rat: No SC: Transverse 57 Co: Rat: No SC: Coronal (X−Z)
57
57
Co: Rat: No SC: Sagittal (Y−Z)
0.3 0.2 0.1
µ=0.13911+/− 0.013955
0
µ=0.12477+/− 0.014218
µ=0.14081+/− 0.01444
57
µWater
10000 5000 0 0.05
0.1 0.15 0.2 µ−value (cm−1)
57
Co: Rat: No SC: µ−value Histogram
Number of Voxels
Number of Voxels
Co: Rat: SC: µ−value Histogram
0.25
µ
Water
10000 5000 0 0.05
0.1 0.15 0.2 µ−value (cm−1)
0.25
Figure 5.14: Slices through the µ–map images with (upper images) and without scatter corrections (middle images) for the rat study reconstructed using experimental 57 Co data. The average value and standard deviation for all voxels within an ROI (the boundaries of which are indicated by the dashed lines in the images) are shown below each image. The lower plots show µ–value histograms for the reconstructed data with and without scatter corrections (lower left and right–hand plots, respectively). Please refer to section 5.5.2 for a detailed description of each of the objects visible in the image.
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57
Co: Mouse: SC: Transverse (X−Y)
57
Co: Mouse: SC: Coronal (X−Z)
57
Co: Mouse: SC: Sagittal (Y−Z) 0.3 0.2 0.1 µ=0.15534+/− 0.014695
µ=0.15505+/− 0.014978
0
µ=0.15383+/− 0.01606 57
Co: Mouse: No SC: Transverse (X−Y)
57
Co: Mouse: No SC: Coronal (X−Z)57Co: Mouse: No SC: Sagittal (Y−Z)
0.3 0.2 0.1
µ=0.13135+/− 0.014606
µ=0.13198+/− 0.014202
0
µ=0.12967+/− 0.015208
57
10000
Water
5000
0 0.05
57
Co: Mouse: No SC: µ−value Histogram
µ
Number of Voxels
Number of Voxels
Co: Mouse: SC: µ−value Histogram
0.1 0.15 0.2 µ−value (cm−1)
0.25
8000
µWater
6000 4000 2000 0 0.05
0.1 0.15 0.2 µ−value (cm−1)
0.25
Figure 5.15: Slices through the µ–map images with (upper images) and without scatter corrections (middle images) for the mouse study reconstructed using experimental 57 Co data. The average value and standard deviation for all voxels within an ROI (the boundaries of which are indicated by the dashed lines in the images) are shown below each image. The lower plots show µ–value histograms for the reconstructed data with and without scatter corrections (lower left and right–hand plots, respectively). Please refer to section 5.5.2 for a detailed description of each of the objects visible in the image.
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5.5
Discussion
5.5.1
Simulations
The µ–map data for the simulated uniform water cylinder data, shown in figures 5.1 and 5.2, indicate that when no scatter correction is applied (No SC), the errors in the reconstructed µ–values depend strongly on the phantom size and the energy of the transmission photons. For the ROI analysis, the results of which are summarised in tables 5.1 and 5.2, the differences between the average µ–value for the reconstructions without scatter corrections (No SC) and the true values were between 27 and 45% for the source and between 18 and 31% using the
57 Co
68 Ge
source. The errors also vary
with radial and axial position within the µ–map image (as seen in the µ–map profiles in figures 5.1 and 5.2 and in the ROI analysis of figure 5.3). For the uncorrected image of the 45 mm radius water cylinder using the 68 Ge source, the radial variation is comparable to the difference between the attenuation coefficients for muscle and bone. This would make segmentation (as discussed in section 2.2.3) very difficult and prone to error for these data. In the ROI analysis of the non–uniform phantom, errors between 18 and 39% were observed for the Teflon and water ROIs. For the air insert, however, the µ–values were closer to the truth with no SC than in either the scatter corrected or scatter–free reconstructions. Since a similar sized bias occurs for both scatter corrected and the ideal scatter– free images, this problem is not related to the scatter correction and must be connected to the reconstruction or transmission data itself. These results may be related to the imperfect global convergence properties of the OSTR reconstruction algorithm which could potentially create biases in the reconstructed µ–maps [125]. This effect may also be connected to over-smoothing of the image due to the quadratic term in the Huber penalty function. We have found (data not shown) that the average µ–value for the air ROI, reconstructed using scatter–free simulated data, can be reduced by using a lower value for δ in equation (1.23) for less quadratic smoothing. However, these µ–map images contained streak–artifacts and noise–correlated clusters of high µ–values which did not correspond to any physical boundaries in the 156
Chapter 5. Development and Validation of the Reconstruction Procedure phantom. The attenuation coefficients for the first iteration of the scatter corrected reconstructions (SC: Iter. 1), given in table 5.1, were within 3% of the expected value for water for both isotopes and all cylinder sizes. For the non–uniform phantom (table 5.2) similar agreement was obtained for the Teflon and water ROIs (SC: Iter. 1 values were within 1.5% of the truth for both isotopes). As the scatter correction procedure is repeated, the average µ–values for all ROIs increase slowly with each iteration eventually exceeding their true µ–values, and finally begin to approach the scatter–free values (which are also slightly higher than the true values). By the third scatter iteration (SC: Iter. 3), our reconstructed µ–values agree with the scatter–free data to within about 4% for all the ROIs considered excluding only air. The scatter–free data represent the best one can expect to do with any scatter correction, since no scatter was present in the simulated sinogram data for these reconstructions. However, both the scatter–free and scatter corrected reconstructions exhibit a small bias toward higher µ–values (3 to 6% higher than the true values) due to the effects of single–slice rebinning (SSRB) and noise in the sinogram data, both of which are to be discussed subsequently. The Effects of Transmission Data Noise Figure 5.7 indicates that all of the reconstruction algorithms considered here demonstrate a bias toward higher µ–values for lower–count transmission data. Since the data shown in figure 5.7 was generated using our expression for the unscattered photon data, we expect that the reconstructed µ–values should be close to their true values. Any bias in these values should only be the result of problems associated with the reconstruction algorithm and sinogram noise. These include the non–negativity constraint for EM and MAP reconstructions or, in the case of FBP, pre–processing of the sinogram data prior to reconstruction. The average number of counts per bin in the simulated blank data generated using our simulation software (GATE), was between a minimum of 10 counts in the outer slices to a maximum of 230 (for
68 Ge)
and 360 (for
57 Co)
counts in the central planes of the single–slice
157
Chapter 5. Development and Validation of the Reconstruction Procedure rebinned (SSRB) sinogram data. The errors in the reconstructed µ–values that were obtained using the scatter–free GATE data, were of roughly the same magnitude as those observed in the 10 to 30 count region of figure 5.7. In the analysis of FBPem, less bias was observed for low count transmission data than for FBPtr (which had similar bias for low–count transmission data as OSTR). According to the ASPIRE users guide [99], the transmission FBPtr algorithm has a “fix negative” operation which checks for any residual non–positive logarithm values and tries to interpolate neighbouring positive values to “fill–in” these values. The log value is only set to zero if all 4 neighbours in the sinogram are non–positive. This pre–processing is the cause of biases seen for low count transmission data with FBPtr. Unfortunately, we do not have access to the source code of OSTR algorithm and can only speculate about how it deals with the problem of negative line integrals. It seems reasonable to infer that it does something similar to FBPtr, since OSTR is distributed as part of the ASPIRE reconstruction package. Although FBPem provided better agreement with the true µ–values for water for extremely low numbers of counts per LOR, these reconstructions were considerably noisier than those obtained using the penalised MAP reconstructions and contained non–physical negative µ–values (see upper images in figure 5.7) similar to the results found in many previous studies (e.g. see Erdogan and Fessler [125]). The Effects of Single–Slice Rebinning Even when the noise levels were low (i.e. in the central slices of the µ– maps), we observed that the µ–values for scatter–corrected and scatter–free reconstructions were slightly higher than their expected values. The transmission and blank data for these reconstructions were acquired in 3D–mode, with a maximum oblique angle of 13.7 o , and then rebinned using single–slice rebinning (SSRB). These errors occur because this procedure does not correctly take into account the data in the oblique LORs (e.g. our cylindrical phantoms appear elliptical and, therefore, more highly attenuating in an obliquely tilted plane).
158
Chapter 5. Development and Validation of the Reconstruction Procedure The ROI analysis for the scatter–free µ–maps given in table 5.3 demonstrates the size of the errors associated with this effect. Recall that all data in this chapter were reconstructed using a maximum oblique angle of 13.7o . From the values in table 5.3, the errors for this maximum angle were between 1 and 4% depending on the transmission photon energy and phantom considered. Clearly, using a smaller maximum angle reduces this error, however the bias associated with noise in the sinogram data would offset this improvement. Recall that the sinogram data, used for the analysis summarised in table 5.3, were summed over all axial planes prior to reconstruction to reduce the effects of noise. It should be noted here, however, that our scatter correction data and rescaling is computed in 2D mode (i.e. as though the transmission data were only acquired for the direct planes). This reduces the computation time and also partially compensates for the SSRB error in the sinogram data. The system matrices used by the OSTR reconstruction also assume the transmission data are acquired in 2D mode. This may be the reason that the values in tables 5.1 and 5.2 for the SC µ–maps are slightly lower (and closer to the true attenuation coefficients) than the scatter–free values even after 3 scatter iterations.
5.5.2
Experimental Data
Phantom Data The results of the experimental phantom studies were similar to those of the simulations. In the ROI analysis of the phantom data, summarised in tables 5.4 and 5.5, the errors in the µ–values for the reconstructions with no SC were between 29 and 42% for the for the
57 Co
68 Ge
source and between 29 and 37%
source (excluding the air ROIs). The average µ–values for the
first iteration of the scatter corrected reconstructions (SC: Iter. 1), given in tables 5.4 and 5.5, were within 4% of the expected value for all ROIs (excluding air) and within 2% by the third scatter iteration (SC: Iter. 3). As in the simulation data, the scatter corrected µ–maps obtained using experimental data provide accurate µ–values and reduced radial and axial 159
Chapter 5. Development and Validation of the Reconstruction Procedure variation (see figures 5.8 to 5.12). In the axial ROI analysis, the average µ–values for the SC reconstructions using experimental data agreed with the true–values better than those using the simulation data. This is likely because these experiments used higher–count transmission data, with an average number of counts per bin in the blank sinograms of between 32 to 800 and between 20 to 600 for the
68 Ge
and
57 Co
transmission sources,
respectively. For these data, we expect much less bias in our µ–values due to effects associated with statistical noise in the sinogram data (see the higher count region in figure 5.7). None the less, a bias toward higher µ–values was still observed in the edge slices of all µ–maps (see figures 5.3 and 5.13) where the sinogram data had fewer counts. Rodent Studies In addition to the animal bed and rodent being scanned, one also observes a number of additional objects in the images shown in Figures 5.14 and 5.15. In the µ–map images of the rat, one also observes a plastic head holder (with density similar to water). In the mouse images, a sheet of perspex (with a density approximately 1.1 to 1.2 g/cm 3 ) is visible, that sits on top of the animal bed during these scans, on which the mouse is fixed using an adhesive putty of unknown composition (the darker material below the mouse’s torso). The hexagonal artifact visible in the transverse mouse images is caused by variations in the mechanical switching on and off of detector blocks (triggered by the motion of the point source) during singles– mode data acquisition. This can lead to artificial differences between the blank and transmission sinograms in the outer edges of the radial profiles and, hence, artifacts in the edges of the reconstructed µ–map images. Although it is slightly more difficult to quantify our results for the rodent studies (where the true µ–values are unknown), we can make a number of general observations. In the ROI analysis of the reconstructed µ–map for the rat with no SC (middle images in figure 5.14), the average µ–values are lower in the neck region where scatter from the rat’s torso is more significant. The images with no SC also exhibit radial variations in profiles through
160
Chapter 5. Development and Validation of the Reconstruction Procedure soft–tissue regions (data not shown) similar to that of the phantom data. Although a distinct soft–tissue peak is visible in the µ–value histograms for the image without SC, simple rescaling of the uncorrected µ–map would not correct for these axial and radial variations. These errors are corrected for in the SC images, where the average µ–values in the ROI analysis were within 1% of the expected values for soft tissue. The reconstructed µ–map data for the mouse study, shown in figure 5.15, were noisier than those of the rat and it was not, therefore, possible to find a distinct peak in the µ–value histograms for these data. In situations where no peak is found, no µ–map rescaling is applied and our algorithm automatically uses a simpler scatter sinogram rescaling which does not account for multiple and Rayleigh scatter or for scatter from outside the FOV (see section 5.1). Fortunately, the small size of mice implies that these additional types of scatter should be minimal. In spite of these potential difficulties, the ROI analysis resulted in µ–values within 4% of the expected value for soft– tissue. The slight underestimation of µ–values observed here may indicate that further scatter iterations are required.
5.6
Conclusion
We have applied our scatter correction as part of an iterative reconstruction algorithm using simulated and experimental transmission data using 68 Ge
and
57 Co
transmission sources for three uniform water cylinders and
a non–uniform phantom. We have also tested our reconstruction and SC procedure for two experimental rodent studies. For both simulated and experimental data, the reconstructed linear attenuation coefficients (µ–values) agreed with expected values to within 4% when scatter corrections were applied, for both transmission sources. We have also tested our reconstruction and scatter correction procedure for two experimental rodent studies. For these studies, we found that the average µ–values for soft–tissue regions of interest agreed with expected values to within 4%. Our analysis indicates that, for most cases, one to two scatter correction iterations are necessary to obtain quantitatively accurate linear attenuation coefficients. 161
Chapter 5. Development and Validation of the Reconstruction Procedure In this chapter we have also investigated errors which occur because of noise in the transmission data and single–slice rebinning (SSRB) data pre–processing. One simple way to reduce the former problem would be to use higher activity sources or a modified transmission source orbit where a few additional rotations (or slower rotations) are performed close to edges of the scanner to increase the counts in the sinogram data corresponding to these regions. The latter problem could be overcome simply by using a fully 3D reconstruction method directly on the 3D transmission data, instead of SSRB followed by a 2D reconstruction method. Implementation of this solution is beyond the scope of this thesis, however, and awaits future releases of ASPIRE [99, 202] which are expected to support MAP reconstruction algorithms for 3D transmission data. Alternatively, one could use higher–activity transmission sources with axial collimation [113] which would provide higher–quality transmission data with reduced scatter and an inherently 2D geometry. Our scatter correction could be easily modified to account for the scatter present with collimated transmission sources. In this chapter, we have introduced an iterative procedure for the reconstruction and scatter correction of singles–mode transmission data for PET. This correction could improve the absolute quantitative accuracy of PET emission imaging, since attenuation and scatter corrections depend critically on the attenuation–maps obtained from transmission scans. In the next chapter, we test our reconstruction and scatter correction procedure using data acquired with the microPET Focus 220 scanner [46]. The Focus 220 is a larger diameter PET scanner dedicated for primate studies.
162
Chapter 6
Extension of the Scatter Correction for a Primate Scanner In this chapter, we apply our reconstruction and scatter correction procedure to singles–mode transmission data acquired with the microPET Focus 220 [46], a PET scanner suitable for primate studies. We have reconstructed attenuation–maps using experimental singles–mode transmission data for a uniform water cylinder (with a 50 mm radius) placed in two different positions within the scanner and for a primate study (a nemestrina monkey of the genus Macaca). The purpose of these studies was to verify that the assumptions made as part of our correction and reconstruction procedure are valid for the imaging of larger objects and for a PET scanner with different physical properties. We can anticipate, for example, that larger objects will exhibit more photon attenuation and higher amounts of both single and multiple scatter. The larger ring diameter of the Focus 220, relative to the Focus 120, will result in decreased solid–angle for the detection of both scattered and unscattered photons. We also expect that the negative effects of single–slice rebinning (SSRB) (i.e. errors associated with using a 2D reconstruction on transmission data which have been processed using SSRB) will be reduced for this scanner, since its axial length is small relative to its ring diameter. As in chapter 5, we focus our analysis on showing that our procedure provides the correct µ–values without any user intervention and removes the systematic effects of scatter.
163
Chapter 6. Extension of the Scatter Correction for a Primate Scanner
6.1
Experiments
All experimental data for this chapter were acquired using the microPET Focus 220 dedicated primate PET scanner [46]. As mentioned previously, the Focus 220 and Focus 120 are very similar scanners which share hardware and software. For example, they both use the same block detector modules, have the same crystal size (1.5×1.5×10 mm 3 ), the same number of axial crystal rings (48), have the same axial length (77 mm) and share the same acquisition software. The primary difference is that the Focus 220 has a larger number of crystals per axial ring (504 in total) which is achieved by arranging more detector modules (46) in a larger diameter (261 mm) ring. A larger sinogram size is also used for the Focus 220 with 288 radial samples per projection angle and 252 projection angles per sinogram. As a result the acquired datasets for this scanner are larger than the Focus 120 2 . All transmission data were acquired in singles–mode using a
57 Co
transmission
source with similar source and acquisition specifications as those discussed for the Focus 120 in previous chapters (e.g. roughly the same activity
57 Co
source and the same energy resolution and energy discrimination settings). Singles–mode transmission data were acquired for a 50 mm radius water cylinder placed in two positions: (1) centred in the scanner field of view and (2) offset vertically from the centre by 27 mm. For both these studies, no activity was present in the cylinder, the transmission data were acquired for 27 minutes and the duration of the blank scans were 2 hours. The primate transmission data were acquired prior to injection (no emission contamination was present in the scanner) using a 6.5 kg nemestrina monkey. For this study, the transmission and blank data were acquired for 45 minutes and 2 hours, respectively. The uniform cylinder data and primate data were kindly provided by Dr. Joel Perlmutter and Dr. Tom Videen, professors with the Departments of Neurology and Radiology, at Washington University in St. Louis. Unlike the data discussed in previous chapters, it was necessary to cor2
Recall from chapter 3, that the Focus 120 dedicated rodent scanner has 288 crystals per ring, 24 detector modules per ring, a 140 mm ring diameter and uses a 128×144 sinogram size.
164
Chapter 6. Extension of the Scatter Correction for a Primate Scanner rect the blank and transmission data for the fact that they had different scan durations. This type of acquisition is becoming common in PET facilities which use statistical attenuation–map reconstruction methods, many of which assume noiseless blank data. These longer blank scans are typically acquired less frequently as part of longer term maintenance of the scanner and are reused for different transmission scans. For all studies discussed in this chapter, the blank and transmission scans occurred approximately one month apart. It was, therefore, also necessary to correct the blank data for radioactive decay, since
57 Co
has a half–life of 272 days. In our centre, we
perform a blank scan each day in which transmission data are acquired, usually of the same duration as the transmission scan. We discuss the reasons for these differences in more detail in the discussion section of this chapter.
6.2
Reconstructions and Scatter Corrections
All Focus 220 data were reconstructed using OSTR [125] with the same parameters (20 iterations, 4 subsets, the Huber penalty function with β = 2 8 and δ = 0.5µwater ) as those used in the studies described in previous chapters. All of the transmission and blank data were rebinned using SSRB with a maximum oblique angle of 13.7o . Since the Focus 220 provides more than twice as many radial samples per sinogram than the Focus 120, we elected to use a larger matrix size for these reconstructions (256×256) than we had used previously. We also employed a scatter correction and reconstruction procedure similar to that discussed in chapter 5, with the following exceptions: for the scatter correction, the size of the attenuation–map images were reduced from 256×256 to 32×32; and the scatter sinogram data were sampled sparsely for only 29×32×10 elements within the full 288×256×48 direct plane sinogram size.
6.3
Results
We performed analyses very similar to those described in chapter 5, the results of which are shown in figures 6.1, 6.2 and 6.3. Please refer to the figure 165
Chapter 6. Extension of the Scatter Correction for a Primate Scanner captions for a more detailed description of how the reconstruction data were analysed. For all datasets considered in this chapter, distinct peaks could be found in all µ–value histograms. Based on the results of chapter 5, we therefore concluded that only a single scatter correction iteration was necessary for each study. The amounts of CPU time required for the calculation of the scatter correction were between 17 to 26 minutes using a 2.2 GHz processor. Each OSTR reconstructions required about 21 minutes of CPU time using the same processor. Note that this is about 10 times more CPU time than that for the reconstruction of the Focus 120 data and is due to the large sizes of the sinograms and images employed for these studies.
6.4
Discussion
Our results indicate that our reconstruction and scatter procedure recovers linear attenuation coefficients (µ–values) within a few percent of their expected values for singles–mode transmission data acquired using the Focus 220 scanner. For example, with scatter corrections (SC) the average reconstructed µ–values for the ROI analysis shown on the attenuation–map images were within 3% of their expected values for both cylinder experiments and for the primate study. As in the previous chapter, the percent errors in the reconstructed µ–values without SC varied for different sized objects (errors were 19 to 22% for the primate study and 25 to 26% for the cylinder study) and also depended on the location of the ROI (for the primate study the errors become larger closer to monkey’s body where there is more scatter from outside the field–of–view). The profiles shown in the lower plots indicate that our SC reduces the radial variation due to scatter for the uniform cylinder located in both positions within the scanner. The profile for the primate data passes through several types of tissue including the skull, the brain, a nasal sinus, and parts of the monkey’s snout. For the soft tissue regions of this profile (brain and parts of the snout), we obtain reconstructed µ–values close to their expected values when the SC is applied. We also obtain low reconstructed µ–values (as low as those without SC) for the regions of the profile passing through 166
Chapter 6. Extension of the Scatter Correction for a Primate Scanner
Coronal (X−Z)
Transverse (X−Y)
µ=0.15867+/− 0.011305 Sagittal (Y−Z)
µ=0.15795+/− 0.011027
µ=0.15864+/− 0.011249
Transverse (X−Y)
Coronal (X−Z)
µ=0.11859+/− 0.010024 Sagittal (Y−Z)
µ=0.12014+/− 0.0096352
µ (cm−1)
µ (cm−1)
0.1 0.05 0 −100
0.2 µ
Water
With SC No SC −50 0 50 Distance (mm)
0.25
10 8
µ−value Histogram µWater With SC No SC
6 4 2
0.15 100
12 Number of Voxels
Water
4
x 10
With SC No SC
0.3
µ 0.15
µ=0.11811+/− 0.0097413
Average in Transverse ROI
Average Dashed Profile
−20 0 20 Axial Distance (mm)
0
0.1 0.2 µ−value (cm−1)
0.3
Figure 6.1: Slices through the reconstructed µ–map images with (upper images) and without scatter corrections (middle images) for the water cylinder (50 mm radius) reconstructed using experimental transmission data acquired using a 57 Co transmission source. Profiles through the reconstructed data (averaged over 10 axial planes), the results of an axial ROI analysis (for the ROI shown in the transverse images), and the µ–value histograms for both µ–maps are shown in the lower plots. For these data, the cylinder was centred vertically in the scanner and rests above the animal–bed (the half–circular structure visible below the cylinder).
167
Chapter 6. Extension of the Scatter Correction for a Primate Scanner
Transverse (X−Y)
Coronal (X−Z)
µ=0.15782+/− 0.011305 Sagittal (Y−Z)
µ=0.15667+/− 0.01045
µ=0.15805+/− 0.011779
Transverse (X−Y)
Coronal (X−Z)
µ=0.11896+/− 0.0096248 Sagittal (Y−Z)
µ=0.11809+/− 0.0099362
µ=0.11941+/− 0.0092493
Average Dashed Profile
Average in Transverse ROI
Water
0.15 µ (cm−1)
−1
µ (cm )
0.25 0.1
0.05
0 −100
0.2 µ
Water
With SC No SC −50 0 50 Distance (mm)
0.15 100
4
x 10
With SC No SC
0.3
µ−value Histogram µ
Water
12 Number of Voxels
µ
With SC No SC
10 8 6 4 2
−20 0 20 Axial Distance (mm)
0
0.05
0.1 0.15 0.2 0.25 −1 µ−value (cm )
0.3
Figure 6.2: Slices through the reconstructed µ–map images with (upper images) and without scatter corrections (middle images) for the shifted uniform water cylinder (50 mm radius) reconstructed using experimental transmission data acquired using a 57 Co transmission source. Profiles through the reconstructed data (averaged over 10 axial planes), the results of an axial ROI analysis (for the ROI shown in the transverse images), and the µ–value histograms for both µ–maps are shown in the lower plots. For these data, the cylinder has been shifted by 27 mm in the vertical direction.
168
Chapter 6. Extension of the Scatter Correction for a Primate Scanner
Transverse (X−Y)
Coronal (X−Z)
µ=0.15547+/− 0.011254 Sagittal (Y−Z)
µ=0.15655+/− 0.012201
µ=0.15751+/− 0.012825
Transverse (X−Y)
Coronal (X−Z)
µ=0.12847+/− 0.0099683 Sagittal (Y−Z)
µ=0.1253+/− 0.010042
µ=0.12757+/− 0.01024
4
With SC No SC
0.15
µ
Water
0.1 0.05 0 −100
7
x 10
µ−value Histogram µ
Water
6 Number of Voxels
0.2
−1
µ (cm )
Average Dashed Profile
With SC No SC
5 4 3 2 1
−50
0 Distance (mm)
50
100
0
0.05
0.1
0.15 0.2 µ−value (cm−1)
0.25
0.3
Figure 6.3: Slices through the reconstructed µ–map images with (upper images) and without scatter corrections (middle images) for the primate transmission scan acquired using the microPET Focus 220 with a 57 Co transmission source. Profiles through the reconstructed data (averaged over 3 axial planes) and the µ–value histogram for both µ–maps are shown in the lower plots. The profile shown passes through the skull, brain, the sinus and the monkey’s snout. Note the improved contrast between tissue and sinus for the With SC µ-maps relative to No SC.
169
Chapter 6. Extension of the Scatter Correction for a Primate Scanner the sinus. This differs from the results of chapter 5, where the reconstructed µ–values for low attenuation regions (air) of the non–uniform phantom were closer to the truth without scatter corrections. We hypothesise that this difference is connected to the relatively large size of the sinus cavity and due to the properties of the OSTR reconstruction. In these studies, we have used a larger image size (256×256) with the same penalty function parameters that we used for the 128×128 image size in the previous chapter. For the primate study, we have compared reconstructed data with the 256×256 image size to those using a 128×128 image size and found (data not shown) that the use of the larger image size reduces the relative amount of image smoothing and enhances contrast with only a small increase in image noise. In the edge planes of all µ–map images, we observed large errors in the reconstructed µ–values both with and without SCs. As discussed in chapter 5, these planes correspond to low count transmission data for which we have observed errors in the OSTR reconstructed µ–values (see figure 5.7). The errors for these data (see the axial ROI analysis shown in figures 6.1 and 6.2) were larger than those observed in chapter 5. We believe that this effect may be associated with the practice of acquiring long duration blanks at a different time than the transmission scan. We do not currently acquire long duration blanks in our research centre because we have observed that small changes occur from scan to scan and over time due to the mechanical switching on and off of detector blocks (triggered by the motion of the point source). We hypothesise that the precise position of the point source can be slightly different in different scans since, when not in use, the transmission source is physically removed from the rotation mechanism and shielded. The rotation mechanism itself is also frequently removed from and reattached to the scanner during maintenance. We believe that this can lead to artificial differences between the blank and transmission data, especially for the sinograms corresponding to the outer axial edges of the scanner, which cause errors in the edge planes of the reconstructed µ–map images.
170
Chapter 6. Extension of the Scatter Correction for a Primate Scanner
6.5
Conclusion
To adapt our scatter correction for the microPET Focus 220 imaging system, it was necessary to change only a single parameter file, which is read at the time of execution and contains all the relevant details about the scanner geometry. The results of this chapter indicate that our reconstruction and scatter correction procedure can be easily adapted to different PET scanners. The scatter corrected attenuation–maps provide the correct quantitative µ–values (within 3% for water and soft-tissue regions–of–interest) for larger objects acquired using a PET scanner with a larger ring diameter (relative to the objects and scanner used in the previous chapter). As described in the section 2.2.4, if singles–mode transmission scans are performed post–injection, contamination due to the radio–tracer activity can also be a significant problem. In the next chapter we investigate this problem in greater detail.
171
Chapter 7
Post–Injection Transmission Scanning In this chapter, 5Data for Small–Animal PET. we investigate how emission photon contamination influences attenuation–maps reconstructed from singles–mode transmission data acquired post–injection. As mentioned in section 2.2.4, a correction for the contamination from emission photons can be obtained directly by acquiring a “mock” transmission scan [143], in which the transmission source is removed or shielded from the detectors. In this chapter, we combine our analytical scatter correction (SC) with an emission– contamination correction (EC) obtained from mock transmission scan data. As in previous chapters, we then evaluate the accuracy of the reconstructed µ–maps obtained with different combinations of SCs and ECs to determine if we obtain accurate linear attenuation coefficients. Unlike previous chapters, in this analysis we also investigate the quantitative accuracy of reconstructed emission images in terms of absolute quantitative units (MBq) within a region of interest. For this analysis, we used the software provided by the microPET manufacturer to reconstruct the emission images with corrections for randoms, normalisation, dead–time, scatter and attenuation. For the emission photon attenuation and scatter corrections, we use µ–maps reconstructed with and without our corrections for the scattered photons and emission contamination present in the singles–mode transmission data.
7.1
Experiments
All experimental data for this chapter were acquired using the microPET Focus 120 scanner with the singles–mode transmission hardware and acqui172
Chapter 7. Post–Injection Transmission Scanning sition settings discussed in previous chapters. We acquired post–injection data for a 30 mm radius water cylinder with an approximately 1.3 MBq
68 Ge
line source (one of the same line sources used in chapter 3) attached to its side. The length of the line source was 115 mm which is larger than the axial length of the scanner (77 mm), we therefore estimate that approximately 0.9 MBq of the line source activity was present inside the scanner during the emission scan. Two sets of transmission and blank scans were acquired using a 14 MBq
68 Ge
source and a 100 MBq
57 Co
source. The duration of
all blank, transmission, and mock scans were 17 minutes. We also acquired transmission and blank data for a healthy rodent injected with fluoro–deoxy–glucose (FDG). We attached a small glass vial (diameter of 10 mm and length of 34 mm) to the rat’s shoulder which was filled with a solution of 18 F (volume of 2.0 cm3 ). This was meant to approximate an external tumour similar to oncology studies in which human cancer cells are transplanted close to the surface of a rodent’s skin [203, 204]. At the beginning of the emission scan, we estimate that 6 MBq of
18 F
activity
was present in the simulated tumour and about 24 MBq of FDG activity were distributed throughout the rat. At the time of the rodent studies, the 68 Ge and 57 Co transmission sources had activities of 10 MBq and 72 MBq, respectively. For this study, all blank and transmission scans were acquired for 17 minutes while the mock scans were acquired for only 8.5 minutes. This is more typical of clinical studies where the mock acquisitions are usually of short duration to minimise the total scan times.
7.2
Transmission Data Reconstructions and Correction Procedures
All transmission data were reconstructed using OSTR [125] with the same parameters as those used in previous chapters. For this analysis, we compared reconstructions with three different combinations of emission contamination corrections (EC) and scatter corrections (SC).
173
Chapter 7. Post–Injection Transmission Scanning • No EC or SC: Here we reconstruct the data with no corrections for scatter or for the emission contamination, in order to determine the combined importance of these effects. • EC Only: For this reconstruction we subtract the experimentally measured mock scan sinograms (corrected for the radioactive decay of the emission tracer) from the transmission data. We then reconstruct the data with no corrections for transmission photon scatter. • EC+SC: This correction also subtracts the decay–corrected mock scans from the transmission data but then applies the scatter correction and reconstruction procedure described in chapter 5. All EC+SC attenuation–maps shown or analysed in this chapter correspond to the second scatter correction iteration.
7.3
Emission Data Reconstructions and Correction Procedures
The acquired PET emission data were sorted into 3D sinograms. The emission sinogram data were corrected for random events by subtracting data acquired in a delayed coincidence window, as described in section 1.9.3. The sinogram data were then corrected for detector normalisation (see section 1.9.4) using a component–based method [65]. All emission data were corrected for dead–time using a global correction (a single scaling factor for the entire sinogram) which takes into account elements of both paralyzable and non–paralyzable detector response models, as discussed in section 1.9.5. Using Fourier Rebinning (FORE), which was described in section 1.10.2, the 3D sinograms were rebinned into a smaller set of 2D sinograms and then reconstructed using 2D–FBP with a ramp filter cut–off at the Nyquist frequency (see section 1.10.1). For each emission sinogram, a total of six reconstructions were performed differing only in which µ–map was used for the emission photon attenuation and scatter corrections. These correspond to each of the three attenuation– maps (No EC or SC, EC Only, and EC+SC) and transmission sources ( 68 Ge 174
Chapter 7. Post–Injection Transmission Scanning and
57 Co).
All emission data were corrected for attenuation using the mul-
tiplicative attenuation correction factors computed for each µ–map using equation (2.1), as described in section 2.2.1. Before computing the attenuation correction factors from data acquired using the
57 Co
transmission
source (with a photon energy of approximately 122 keV), it was first necessary to appropriately convert the µ–values for the annihilation photon energy (511 keV). This was done using a piece–wise linear rescaling similar to the bi–linear interpolation methods [136, 150] proposed for CT–based attenuation corrections (see section 2.3). Our method differs from those cited previously in that, instead of using just two linear rescalings (one for low and high attenuation coefficients), we use several different linear rescalings based on the linear attenuation coefficients at 122 and 511 keV for several different material types (vacuum, lung, water, bone, aluminium, iron, and lead). All reconstructions were corrected for emission photon scatter using the single–scatter simulation method [113]. Calibration to absolute quantitative reconstruction units was achieved using a 0.3 MBq
68 Ge
point source and matching the reconstructed counts
(after all appropriate corrections) to a well counter measurement (see section 1.9.6). For structures smaller than the reconstructed voxel size and those with dimensions comparable to the full–width at half maximum characterising the spatial resolution of the imaging system, the partial volume effect (PVE) can lead to errors in activity concentrations determined from reconstructed PET images. We did not apply any corrections for the PVE. We, therefore, focused our analysis on determining the total amount of reconstructed activity present in a large region of interest (ROI) within each image and then compared these values with well–counter measurements. We expect that this type of measurement should avoid complications which could be caused by the PVE.
175
Chapter 7. Post–Injection Transmission Scanning
7.4
Results
7.4.1
Reconstructed Attenuation–Maps
Figures 7.1 and 7.2 show examples of transverse slices through the reconstructed µ–map images for the cylinder and rodent studies, respectively. The images shown in the upper–row of each figure correspond to those reconstructed using the sponds to the
57 Co
68 Ge
transmission data while the lower–row corre-
data which have been rescaled to the appropriate values
for 511 keV photons. Below the images in each figure, we show the µ–value histograms for the
68 Ge, 57 Co
and
57 Co
attenuation–maps which have been
rescaled for a photon energy of 511 keV. Using a 2.2 GHz processor, each SC iteration for the rodent study required about 23 minutes of CPU time and each SC iteration for the cylinder data required approximately 16 minutes. Each OSTR reconstruction required about 3.5 minutes of CPU time using the same processor.
7.4.2
Image Quantification
Figures 7.3 and 7.4 show transverse slices and profiles through the FBP reconstructed activity concentration images for the cylinder and rodent studies, respectively. The images shown in this figure correspond to the same transverse slices shown in figures 7.1 and 7.2. The emission reconstructions and corrections required about 2 minutes of CPU time on the microPET acquisition and processing computer. We converted the concentration images to units of total activity per voxel (by simply multiplying by the voxel volume) and summed over all transverse planes in which either the line source or simulated tumour were visible. We then placed circular regions–of–interest (ROIs) (with radii of 4.3 and 7.8 mm for the line–source and tumour ROIs, respectively) on each summed image and determined the total amounts of activity within each ROI. Table 7.1 compares the total reconstructed activity within the ROIs with the measured activity estimated using a well counter.
176
Chapter 7. Post–Injection Transmission Scanning
−1
Transverse (X−Y)
Transverse (X−Y)
cm 0.3
Transverse (X−Y)
0.25 0.2 0.15 0.1 68
68
Ge: No EC or EC
0.05
68
Ge: EC only
Ge: EC+SC
0 −1
Transverse (X−Y)
Transverse (X−Y)
cm
Transverse (X−Y)
0.25 0.2 0.15 0.1
57
57
Co: No EC or EC
Ge: µ−value Histogram
4
4
µ Number of Voxels
Water
15000
No EC or SC EC Only EC + SC
10000
3
x 10
µWater
0.2
0
4
x 10 4
No EC or SC EC Only 3 EC + SC
Co: µ−value Histogram Rescaled to 511 keV µWater No EC or SC EC Only EC + SC
2
1
0.05 0.1 0.15 µ−value (cm−1)
0
57
Co: µ−value Histogram
2
5000
0
Co: EC+SC
57
68
0.05
57
Co: EC only
1
0.1 0.2 µ−value (cm−1)
0.3
0
0.05 0.1 0.15 µ−value (cm−1)
0.2
Figure 7.1: Comparison of transverse attenuation–map images (averaged over 3 axial planes) reconstructed using 68 Ge (top–row) and 57 Co (centre– row) transmission data. The lower–row shows µ–value histograms corresponding to the 68 Ge, 57 Co and 57 Co attenuation–maps which have been rescaled for 511 keV photons. Data shown are for the line source attached to a water cylinder and demonstrates the influence of emission contamination (EC) and scatter corrections (SC). 177
Chapter 7. Post–Injection Transmission Scanning −1
Transverse (X−Y)
Transverse (X−Y)
cm 0.4
Transverse (X−Y)
0.3 0.2 0.1 68
68
Ge: No EC or EC
68
Ge: EC only
Ge: EC+SC
0 −1
Transverse (X−Y)
Transverse (X−Y)
cm
Transverse (X−Y)
0.2 0.15 0.1 0.05
57
57
Co: No EC or EC
Co: EC only
Number of Voxels
µWater
10000
µWater
No EC or SC 8000 EC Only EC + SC 6000 4000
4000
2000
2000
2000 0
Co: µ−value Histogram Rescaled to 511 keV
10000 No EC or SC 8000 EC Only EC + SC 6000
4000
0.05 0.1 0.15 µ−value (cm−1)
0.2
0
0
57
Co: µ−value Histogram
Ge: µ−value Histogram
6000
Co: EC+SC
57
68
8000
57
0.1 0.2 µ−value (cm−1)
0.3
0
µWater No EC or SC EC Only EC + SC
0.05 0.1 0.15 µ−value (cm−1)
0.2
Figure 7.2: Comparison of transverse attenuation–map images (averaged over 3 axial planes) reconstructed using 68 Ge (top–row) and 57 Co (centre– row) transmission data. The lower–row shows µ–value histograms corresponding to the 68 Ge, 57 Co and 57 Co attenuation–maps which have been rescaled for 511 keV photons. Data shown are for the rodent experiment and demonstrates the influence of emission contamination (EC) and scatter corrections (SC). These images correspond to a transverse slice of the rat’s torso and include parts of its lungs and a circular cross–section of the external tumour on the left hand side of the image. 178
Chapter 7. Post–Injection Transmission Scanning
3
Transverse (X−Y)
Transverse (X−Y)
MBq/cm 1.2
Transverse (X−Y)
1 0.8 0.6 0.4 0.2 68
68
Ge: No EC or EC
Transverse (X−Y)
0
68
Ge: EC only
Ge: EC+SC
Transverse (X−Y)
MBq/cm3 1.2
Transverse (X−Y)
1 0.8 0.6 0.4 0.2
57
57
Co: No EC or EC
57
Co: EC only
Activity Profiles: 68Ge µ−maps No EC or EC EC only EC+SC
1 3
0.8
Activity Profiles: 57Co µ−maps
Concentration (MBq/cm )
Concentration (MBq/cm3)
1
0.6 0.4 0.2 0 −10
0
10 Distance (mm)
20
0
Co: EC+SC
30
0.8
No EC or EC EC only EC+SC
0.6 0.4 0.2 0 −10
0
10 Distance (mm)
20
30
Figure 7.3: Comparison of transverse activity–concentration images (averaged over 3 axial planes) reconstructed using attenuation and scatter corrections computed using the attenuation–maps corresponding to the 68 Ge (top–row) and 57 Co (centre–row) transmission data. Profile data through the reconstructed activity images are also shown in the bottom–row plots. Data shown are for the line source attached to a water cylinder.
179
Chapter 7. Post–Injection Transmission Scanning
MBq/cm3 Transverse (X−Y)
Transverse (X−Y)
Transverse (X−Y)
2.5 2 1.5 1 0.5
68
Ge: No EC or EC
68
0
68
Ge: EC only
Ge: EC+SC
3
MBq/cm Transverse (X−Y)
Transverse (X−Y)
2.5
Transverse (X−Y)
2 1.5 1 0.5 57
Co: No EC or EC
57
Co: EC+SC
Activity Profiles:
2 1.5 1 0.5 0 −50
0 Distance (mm)
50
3
No EC or EC EC only EC+SC
Concentration (MBq/cm )
3
Concentration (MBq/cm )
Activity Profiles: 68Ge µ−maps 2.5
0
57
Co: EC only
57
Co µ−maps No EC or EC EC only EC+SC
2.5 2 1.5 1 0.5 0 −50
0 Distance (mm)
50
Figure 7.4: Comparison of transverse activity–concentration images (averaged over 3 axial planes) reconstructed using attenuation and scatter corrections computed using the attenuation-maps corresponding to the 68 Ge (top-row) and 57 Co (centre-row) transmission data. Profile data through the reconstructed activity are also shown in the bottom-row plots. Data shown are for the rodent study with a simulated external tumour.
180
Chapter 7. Post–Injection Transmission Scanning
Table 7.1: Comparison of total amounts of activity within a circular regions– of–interest (ROIs) for emission images reconstructed using attenuation– maps with different combinations of emission contamination (EC) and scatter corrections (SC). Based on well-counter measurements, we estimate that approximately 0.9 MBq and 6.0 MBq of activity were present within the ROIs for the cylinder and rodent studies, respectively.
Cylinder Study 57 Co
Rodent Study 68 Ge 57 Co
1.03 0.84 0.66
5.84 5.20 4.20
68 Ge
EC+SC EC only No EC or SC
7.5
1.02 0.88 0.87
5.94 5.53 5.26
Discussion
7.5.1
Reconstructed Attenuation–Maps
For the histograms shown in figures 7.1 and 7.2, we expect to observe a peak corresponding to the reconstructed µ-value of water (or equivalently soft–tissue). For the cylinder study using the
68 Ge
transmission data and
reconstructed with no emission–contamination or scatter correction (No EC or SC), no discernible peak can be found in the histograms and the µ–map images (see figure 7.1) are clearly distorted because of the presence of the line source (located on the lower right–hand side of the cylinder). This situation was even worse for the rodent study, where the µ–value for almost every voxel of the
68 Ge
attenuation–map with no EC or SC was equal to zero.
This implies that for most lines–of–response the number of counts in the 68 Ge
transmission data were greater than the blank data (which would give
physically incorrect negative µ–values if the non–negativity constraint were not enforced during reconstruction). Based on our mock scan measurements, we estimate that the emission contamination contributed 19 and 78% of the counts in the
68 Ge
transmission data for the cylinder and rodent studies,
181
Chapter 7. Post–Injection Transmission Scanning respectively. For the
57 Co
transmission data, we estimate that the emission
contamination only contributed 0.6 and 5.7% of the total counts for cylinder and rodent transmission scans. Obviously the contribution from emission photons depends on the amount of emission activity relative to that of the transmission source, the energy of the transmission source and the energy discrimination settings used. It is worth noting that the rodent study represents an extreme case, where much higher emission activity (5–10 times) was employed than we would typically use. Our results indicate that it is not sufficient to perform only the EC correction and that, in order to obtain accurate µ–values, one must also apply scatter corrections. When corrections were applied only for emission contamination (EC only) in the cylinder study, the µ–value histogram peak positions were 36 and 30% lower than the expected values for water for the
68 Ge
and
57 Co
transmission data, respectively. For the EC+SC
attenuation–maps, however, the histogram peak positions were within 2% of their expected values for water for the cylinder study using each transmission source. For the rodent study using the 57 Co transmission source, the peaks in the µ–value histograms were 25% (No EC) and 16% (EC only) lower than the expected value for soft–tissue when transmission scatter corrections were not applied. When both scatter and emission contamination corrections were applied (EC+SC), the histogram peak value was within 1% of its expected value for the
57 Co
transmission data. The attenuation–maps with emission
contamination corrections (EC only and EC+SC) for the rodent study using the
68 Ge
transmission data were extremely noisy and no discernible peaks
could be found in either of their µ–value histograms. As a result, our scatter correction and reconstruction procedure automatically used the simplified scatter sinogram rescaling which does not account for multiple and Rayleigh scatter or for scatter from outside the field–of–view (see section 5.1). We performed a simple ROI analysis (data not shown) in three perpendicular planes (transverse, coronal and sagittal) for the EC only and EC+SC attenuation–maps reconstructed from the 68 Ge transmission data. We found that, in spite of their poor image quality, the EC+SC reconstructed images 182
Chapter 7. Post–Injection Transmission Scanning had average µ–values within about ±13% of their expected values for soft– tissue ROIs. The average values, for the same ROIs placed on the EC only images, were between 30 and 34% lower than the expected value. The piece–wise linear rescaling we have proposed here correctly converts the linear attenuation coefficients for water from the values expected for 122 keV photons to those for 511 keV photons. For the histograms shown in in figures 7.1 and 7.2, the water peaks are correctly shifted to the appropriate µ–values for the EC+SC attenuation–maps. For the
68 Ge
line source, the
imperfect spatial resolution of the microPET imaging system does not allow us to distinguish between the different materials within the line source. Based on values given in Berger et al. [10] we estimate that the effective linear attenuation coefficient for the line source material is approximately 0.27 cm−1 at 511 keV. We placed ROIs inside the vicinity of the line source for the 57 Co attenuation–map (corrected for EC and SC and rescaled for 511 keV photons) and we found that the average reconstructed µ–values were between 0.25 and 0.28 cm−1 (data not shown).
7.5.2
Image Quantification
The images and profiles shown in figures 7.3 and 7.4 indicate that the reconstructed activity concentrations are strongly influenced by the corrections applied during the attenuation–map reconstructions. If we examine the results of the ROI analysis summarised in table 7.1, we find that the total activities for the EC only images (i.e. the emission images reconstructed using the EC only attenuation–maps for scatter and attenuation corrections) were between 7 and 18% lower than the corresponding values for the EC+SC images. The total reconstructed activities for the no EC or SC images were between 15 and 35% lower than those obtained for the EC+SC images. We also found that the total activity within the ROIs for each study were consistent (agreed to within 2%) for images reconstructed using either 57 Co
68 Ge
or
provided that both ECs and SCs were applied.
If we compare the total reconstructed activities in table 7.1 with the total activities estimated from the well counter measurements, we find that
183
Chapter 7. Post–Injection Transmission Scanning values for the EC+SC images were within about 3 to 4% of the measured values for the rodent study but were slightly overestimated for the line– source and cylinder data. It is important to note that the well–counter measurements are only an approximate estimate of the total activity for the studies discussed here. There may be some uncertainties associated with measuring the activity of an extended line–source using a well–counter, since it is calibrated for point sources centred in the bottom of the well. For example, we have observed a 6% variation in measured activity using this detector for certain extended sources (data not shown) depending on their orientation and position within the well–counter. Another source of uncertainty occurs because the well–counter measurement for the line–source used in the phantom study was subject to photon attenuation, while the reconstructed image has been corrected for attenuation. In the measured activity values given in table 7.1, we have compensated for this effect by multiplying the measured line–source activity by an estimated attenuation correction factor. The correction factor we used was equal to exp(+µss ∆xss + µp ∆xp ) = 1.09 where µss and µp are the linear attenuation coefficients (values from Berger et al. [10]) for stainless steel and plaster, respectively. The terms ∆x ss and ∆xp are the assumed thicknesses of stainless steel and plaster that the annihilation photons must pass through in order to be detected. Recall from chapter 3, that the line source consists of a stainless steel shell (inner radius 1.0 and outer radius of 1.5 mm) encasing an activity carrying plaster. Based on these specifications, we assumed ∆xss =1.0 mm and ∆xp =2.0 mm. This approximation assumes that all photons are emitted from the centre of the line source in a plane perpendicular to the length of the line source. It is possible that we have underestimated the attenuation effect, since photons emitted from the centre of the line–source in any non–perpendicular plane will be subject to more photon attenuation. The well–counter measurements for the line– source are further complicated because most of the photon attenuation is a result of Compton scatter interactions. These scattered photons could be detected, due to the high solid angle of the well–counter, thereby reducing these attenuation effects. 184
Chapter 7. Post–Injection Transmission Scanning
7.6
Conclusions
The results of our attenuation–map analysis indicate how important the emission contamination correction is for
68 Ge
singles–mode transmission
data. Emission contamination can be reduced significantly simply by using energy discrimination with a lower–energy γ–emitting transmission source such as
57 Co.
Furthermore, these data demonstrate that the
57 Co
trans-
mission source provides lower–noise attenuation–map images than those obtained using the the
68 Ge
and
68 Ge
57 Co
source. For the two studies analysed in this chapter,
blank data (which had the same duration as the trans-
mission data) happened to have approximately the same average number of counts (about 350 per bin) in each bin of the single–slice rebinned sinogram data. This suggests that the improved quality of the
57 Co
images
is largely due to the lower amounts of emission contamination relative to those for the
68 Ge
data and the better contrast between µ–values at lower
energy. In spite of these differences in image quality, for all studies using either transmission source, accurate µ–values could be obtained when both emission–contamination (EC) and scatter corrections (SC) were applied. Even though the relative size of the emission–contamination was smaller for the 57 Co transmission data, there was still a noticeable increase in image noise for EC only and EC+SC attenuation–maps relative to No EC or SC. It is well known that any subtractive correction will increase image–noise in emission tomography especially if the subtracted data is subject to noise. This increase in noise after EC was most noticeable for the rodent study, where the duration of the mock scan was only 7.5 minutes (half the duration of the transmission scan). This is typical of clinical mock scan acquisitions which are usually of shorter duration than the transmission scan to reduce overall scan times. As a result, noise from the mock scans can propagate into the attenuation corrections and subsequent emission reconstructions. In the Future Work and Conclusions section of this thesis, we briefly investigate the feasibility of an analytical (noiseless) EC correction, which could reduce the attenuation–map image noise for singles–mode transmission data acquired post–injection. 185
Chapter 7. Post–Injection Transmission Scanning We also analysed reconstructed emission images, which had been corrected for randoms, normalisation, dead–time, scatter and attenuation and then calibrated to absolute quantitative units (MBq/cm 3 ) using a point source with known activity. For the attenuation and scatter corrections, we used µ–maps reconstructed with and without corrections for scatter and emission–contamination. We found that the reconstructed emission activity concentrations depend strongly on which attenuation–map was used. For the rodent study, we obtained total activities closest to values measured using a well–counter for images corresponding to the EC+SC attenuation– maps. For the line–source and water cylinder experiment, we found some minor discrepancies between the measured and reconstructed total activities for the EC+SC image. These differences may be connected to attenuation effects and other uncertainties in the well–counter measurement which we anticipate would be more prominent for this particular source.
186
Chapter 8
Future Work and Conclusions In this work, we presented an analytical scatter correction method for the singles–mode transmission scans which are used to obtain the photon attenuation corrections for small animal PET. We begin this chapter with an overview of the work performed thus far. We then provide some general comments about how the methods proposed here compare with alternative strategies for attenuation correction. The chapter continues with a description of future work which could improve our method’s computational efficiency. We proceed to describe ways in which the calculation could be extended for other transmission scanning protocols and for other types of corrections. The chapter ends with some concluding comments regarding the scope of the current work.
8.1
Summary
We began our investigation by using Monte–Carlo software to simulate singles–mode transmission data acquisition. We obtained good agreement between simulated and experimental transmission data for three different sized water cylinders (representative of a wide range of possible small animal PET applications) acquired using
68 Ge
(a positron emitter) and
57 Co
(an approximately 122 keV photon emitter) transmission sources. During this analysis, we also identified a large previously–ignored contamination in the
68 Ge
transmission data due to the intrinsic
176 Lu
radioactivity present
in the PET detector crystals. We proposed a measured correction method which could be used to account for this background. In the work which fol187
Chapter 8. Future Work and Conclusions lowed, this simulation tool was used to guide the development of our scatter correction and to provide realistic simulated data which could be used to validate our reconstruction and scatter correction procedure. The next stage in this research was to develop and validate an analytical scatter correction, based on the Klein–Nishina scattering cross–section, which calculates the contribution from scattered (and unscattered) photons to singles–mode transmission data for any arbitrary distribution of attenuating material (attenuation–map). We confirmed that, when provided with the correct attenuation–maps, our scatter correction data agreed well with the scattered photon component in the simulated
68 Ge
and
57 Co
transmis-
sion data for uniform and non–uniform phantoms. We observed some minor discrepancies, however, between our model and the simulated data for one of the 57 Co datasets. Further investigation revealed that the simulated scatter was overestimated for this particular dataset because we did not model the axial lead shielding, which is present in the experimental PET imaging system. We then proposed a reconstruction and scatter correction procedure for singles–mode transmission data. We tested the procedure for uniform and non–uniform phantom data using both simulated and experimental 68 Ge and 57 Co transmission data. We also applied our procedure to experimental transmission data for two rodent studies. For most cases, we found that the average reconstructed linear attenuation coefficients (µ–values) agreed with expected values to within a few percent after just one or two iterations of our reconstruction and scatter correction procedure. For smaller objects such as mice, however, the attenuation–maps were noisier and required more iterations to obtain the correct µ–values. Our results also indicated that biases in the reconstructed µ–values can occur for low–count transmission data. We demonstrated that these errors were associated with the transmission reconstruction algorithms and were not connected to the accuracy of our scatter correction method. The general validity of our reconstruction and scatter correction procedure was then tested using transmission data for larger objects (a water cylinder and a monkey) acquired with a PET imaging system dedicated 188
Chapter 8. Future Work and Conclusions for primate studies. For these data, we obtained accurate reconstructed attenuation–maps indicating that our algorithms are flexible and applicable to imaging systems with characteristics that differ from those of the dedicated rodent PET devices employed previously. We also investigated how contamination due to the radio-tracer activity influences attenuation–maps reconstructed from singles–mode transmission data acquired post–injection. In these studies we saw how important the emission contamination corrections could be for the
68 Ge
data and that
lower–noise, higher–quality attenuation–maps could be obtained using
57 Co,
largely due to the lower amounts of emission contamination and better contrast between µ–values at lower energy. We also investigated the quantitative accuracy of reconstructed emission images. For the scatter and attenuation corrections applied to the emission data, we used attenuationmaps reconstructed with and without transmission scatter corrections and emission–contamination corrections. For these data, we found that the reconstructed emission activity concentrations depend strongly on the which attenuation–map was used.
8.2
Comparison of the Proposed Method With Potential Alternatives
As described in the literature review section of this thesis (chapter 2), singles–mode transmission data are not currently corrected for scatter. Most typically, these data are reconstructed and the resulting attenuation–map images are segmented using one of the methods described in section 2.2.3. Alternatively, the reconstructed attenuation–map could simply be rescaled using the knowledge that most biological materials have attenuation coefficients close to that of water. These rescaling methods usually require that the user define one or more region–of–interest (ROI) that correspond to soft–tissue in the attenuation–map images. The attenuation–map data are then rescaled so that the average value in the ROI(s) is equal to the known linear attenuation coefficient of water. In this section, we examine these two
189
Chapter 8. Future Work and Conclusions alternatives to scatter correction in greater detail. Throughout this thesis, we have examined the errors in attenuation coefficients obtained from singles–mode transmission scans which have not been corrected for scatter. We have not, thus far, directly examined the effects of using an uncorrected attenuation–map which has been rescaled to have the correct average linear attenuation coefficient for water. To illustrate the systematic errors which are caused by using this rescaling method, we have reconstructed experimental emission data (using FORE-FBP with all the necessary corrections) from the microPET Focus 120 for a water cylinder filled with a uniform concentration of activity. We acquired singles–mode transmission data using a
57 Co
source and reconstructed the data with and
without our scatter corrections. We also rescaled the attenuation–map without scatter correction as described above. In figure 8.1, we compare the images and profiles for each of these attenuation–maps and for the emission reconstructions which were corrected using them. In the profiles we also show the attenuation coefficient for water (µwater ) and the average reconstructed activity for the emission reconstruction using the scatter corrected attenuation–map (λ With SC ). These data demonstrate that the erroneous radial variation, which is visible in the uncorrected attenuation–map data and is a result of photon scatter, causes systematic errors in the reconstructed emission images. Rescaling the non– corrected attenuation–maps does not eliminate these effects. The most uniform activity concentration profiles are obtained using our scatter correction method. More commonly, the attenuation–map images without scatter correction are segmented. Since our scatter correction and reconstruction procedure recovers the correct quantitative µ–values for different materials, there may be little need for additional post–processing of the µ–maps before the attenuation correction factors are calculated. One may still wish, however, to employ µ–map segmentation to reduce noise propagation from the transmission scans to the PET emission data. Using our scatter corrected attenuation– maps, an automated segmentation routine could easily be developed which uses fixed thresholds around the true linear attenuation coefficients for dif190
Chapter 8. Future Work and Conclusions
Transverse (X−Y)
Transverse (X−Y)
cm−1
Transverse (X−Y)
0.1 0.08 0.06 0.04 0.02
57
57
Co: With SC
57
Co: No SC
Co: No SC, rescaled
Transverse (X−Y)
Transverse (X−Y)
0 MBq/cm3
Transverse (X−Y)
0.02 0.015 0.01 0.005 57
57
Co: With SC µwater
Co: No SC, rescaled
µwater
0.1 0.08
0.08
0.06
0.06
0.06
0.04
0.04
0.04
0.02
0.02
0.02
50
λ
With SC
−50
0.02
0 Distance (mm) 57 Co: No SC
50
λWith SC
−50
0.02
0 50 Distance (mm) Co: No SC, rescaled
57
λ
With SC
3
0.02
0 Distance (mm) 57 Co: With SC
µwater
0.1
0.08
−50
0
57
Co: No SC
0.1 µ−value (cm−1)
Co: No SC, rescaled
Co: No SC
57
Concentration (MBq/cm )
57
57
Co: With SC
0.015
0.015
0.015
0.01
0.01
0.01
0.005
0.005
0.005
0 −50
0 Distance (mm)
50
0 −50
0 Distance (mm)
50
0 −50
0 Distance (mm)
50
Figure 8.1: Comparison of transverse attenuation images (top–row) and activity concentration images (second–row) for a water cylinder filled with a uniform concentration of 18 F activity. Data were averaged over all axial planes to reduce noise and illustrate the systematic effects of scatter in the transmission data. The third and forth rows show profiles through the attenuation and activity images, respectively. Note that the small regions of lower attenuation and activity at the top of the images correspond to an air bubble in the cylinder.
191
Chapter 8. Future Work and Conclusions ferent biological materials. Our scatter correction could eliminate the need for user–intervention and greatly simplify these segmentation routines. The results of this thesis also suggest that segmentation of uncorrected data could be quite difficult, since errors in the reconstructed µ–values vary widely depending on the phantom size and transmission source used but also on the radial and axial position within each µ–map image. For example, in chapter 5, we observed an approximately 36% radial variation in µ–values across the image profile for the 45 mm radius water cylinder using the
68 Ge
source (see figure 5.8). To illustrate the types of segmentation errors which are caused by transmission scatter, we applied a reconstruction–based automated segmentation routine [15] (described in section 2.2.2) to these data. In figure 8.2, we show images and profiles for the attenuation–map data with and without scatter corrections. In the profiles we show the true attenuation coefficients for water and lung (µwater and µlung , respectively). As we can see from the figure, with scatter correction there are very few segmentation errors (most voxels are assigned the correct µ–value for water). Without scatter corrections, however, many voxels are incorrectly assigned the linear attenuation coefficient for lung tissue. Although we did not acquire emission data for this particular study, we can expect that, in general, this type of segmentation error will cause an underestimation of the reconstructed activity for regions lying deeper inside the object relative to those closer to the surface. For animal studies, the errors which could be caused by using rescaled or segmented attenuation–maps without scatter corrections may be difficult to predict (particularly for regions of highly non–uniform attenuation) and could result in misleading reconstructed emission images and errors in the biological parameters derived from them. For example, the measurement of neuro–receptor binding potentials in small–animal PET can be particularly sensitive to attenuation errors, since this measurement depends critically on the relative amounts of activity present in the striatum relative to that in the cerebellum [205]. In rats, the cerebellum is located in a region which is subject to higher attenuation (and, as a result, higher transmission photon scattering) than that of the striatum. The rescaling and segmentation errors 192
Chapter 8. Future Work and Conclusions
Transverse (X−Y)
cm−1
Transverse (X−Y)
0.15
0.1
0.05 68
68
Ge: With SC
Ge: No SC
68
µ−value (cm−1)
Ge: No SC
µwater
0.1
0.08
0.06
0.06
0.04
µwater
0.1
0.08
0.04 µlung
0.02 0 −50
0 68
Ge: With SC
0 Distance (mm)
µlung
0.02 50
0 −50
0 Distance (mm)
50
Figure 8.2: Transverse attenuation images (top–row) and profiles (bottom– row) for a central slice of the 45 mm radius water cylinder using 68 Ge transmission data. Data was reconstructed using the automated reconstruction– based segmentation method of Nuyts et al. [15]. for attenuation–maps without scatter corrections, as illustrated in figures 8.1 and 8.2, would lead to errors in this type of activity ratio. In our own centre, we have evaluated the user–variability of manual segmentation routines (distributed as part of the microPET acquisition and analysis software) applied to uncorrected attenuation–maps. For experimental rodent studies, we observed variations in the thresholds that different users selected and found that the binding potentials derived from emission images corresponding to attenuation–maps segmented by different users varied between 4 and 5% (unpublished data). These differences were the results that initially motivated this research, since the biological changes we hope to measure with these studies are of a similar size to these attenuation–related errors.
193
Chapter 8. Future Work and Conclusions
8.3
Future Work
The first area of future work which we shall consider is the optimisation of the scatter correction software. Most of the computation time required for our calculation is spent performing line integrals through the attenuation– map. One way which we could optimise our SC is by pre-computing and storing the attenuation lengths corresponding to each scatter voxel and some subset of detector crystals. Currently, our calculation performs two summations over all attenuation voxels and determines the attenuation length from each voxel to each of the pair of crystals corresponding to a particular line–of–response (LOR). However, the number of possible LORs is much larger than the number of crystals and if these attenuation lengths could be reused, a great deal of time could be saved. We estimate that the size of such a dataset could be as small as 27 MBytes for the Focus 120 and could, therefore, easily be stored in the RAM of a typical personal computer. Here we have assumed the values are stored as 4 byte floats using a coarse resolution 32×32×24 image size, that line–integral data are only stored for one sinogram at a time, and that the correction is computed in 2D mode (data are therefore only required for a single axial ring consisting of 288 detector crystals). Other methods which are becoming more widely used for the acceleration of 3D PET image reconstruction could also be easily adapted for the optimisation of our software. Some of these methods include the use of parallel processors [206] and modified personal computer graphics hardware for line–integral operations [207]. Although there are many ways that we could improve the computational efficiency of our scatter correction software, it is worth noting that our current computation times, without optimisation, were between 6 and 27 minutes and would be considered by many to be clinically feasible (these amounts of CPU time are comparable to those required for many 3D image reconstruction methods). A potential future application of the scatter correction developed here, would be its application for high–activity collimated
137 Cs
sources [109, 110]
which are in use for many whole–body human PET scanners. Although the 194
Chapter 8. Future Work and Conclusions collimator reduces scatter for this scanning geometry, it has been recognised that scatter can still contribute to errors in the reconstructed attenuation– maps [110]. In order to apply our scatter correction for these data, the only changes which would be required are trivial changes to the parameter file (detector and acquisition parameters) and some minor software modifications to account for the collimation (limiting the possible angles at which photons are emitted from the
137 Cs
source). Similarly, our software could
be modified to correct for the high amounts of scatter acquired using the cone–beam CT scanners [161, 208] which are becoming more common for PET/CT and image–guided radiation therapy applications. This would require more extensive modification of our software, however, in order to model poly–energetic X–ray spectra and the imaging hardware used in CT scanners. Another potential future application of this work is its extension to improve emission contamination corrections for post–injection transmission scanning. We saw in chapter 7 that subtracting the measured mock–scan correction data increases the noise in reconstructed µ–maps, particularly if the mock scans are of short duration. We have begun the initial stages of testing and validation for an analytical (noiseless) emission contamination correction (EC), based on the photon transport model described in chapter 4. We present here a few promising initial results from this new work and directions in which it could be pursued further. We first tested this correction using the post–injection transmission study in which a line–source was attached to a water cylinder (discussed in chapter 7). Figure 8.3 demonstrates that, for the
68 Ge
transmission source, our
model–based estimate of the emission contamination agrees well with the experimentally measured mock scan data. Although it was difficult to visually detect a difference between images reconstructed using model–based and mock–scan corrections, the µ–value histograms (shown in figure 8.4) indicate that the model–based EC provides lower noise (i.e. narrower histogram peaks) attenuation–maps. In spite of favourable preliminary results with this analytical EC, we concluded that the microPET series scanners are not ideal imaging systems 195
Chapter 8. Future Work and Conclusions
Angular View (degrees)
0
Experiment:
68
Ge Mock
10000 8000
50
Model:
0
68
Ge Mock 8000
50 6000
6000 100
100
4000
4000 150
2000
−50
10
4
x 10
8 Counts
0 50 Radial Distance (mm)
2000
−50
Summed Dashed Profile
10
68
Experiment: Ge Mock 68 Model: Ge Mock
0 50 Radial Distance (mm) 4
x 10
9
Summed Dotted Profile Experiment: 68Ge Mock 68 Model: Ge Mock
8
6
7 6
4 2 −50
150
5 0 Radial Distance (mm)
50
4 −50
0 Radial Distance (mm)
50
Figure 8.3: Comparison of the experimental and analytically calculated (model) emission contamination sinograms (summed over all axial planes) for the cylinder study using the 68 Ge transmission acquisition settings. Profiles are shown in the lower plots, summed between the dashed and dotted lines shown on the sinograms.
196
Chapter 8. Future Work and Conclusions µ−value Histogram
4
Number of Voxels
2
x 10
µ
Water
1.5
Experiment: 68 Ge Mock Model: 68 Ge Mock
1 0.5 0
0.05
0.1 −1 µ−value (cm )
0.15
Figure 8.4: Comparison of µ-value histograms for reconstructed attenuation-maps corrected using either the measured or analytically calculated emission contamination sinograms for the cylinder study. Data shown has also been corrected for scatter and was acquired using the 68 Ge transmission acquisition settings. for its application. In chapter 7, we concluded that 57 Co is the preferable source for post–injection transmission scanning. Our simulation software indicates that, for the acquisition settings used for the Focus 120 scanner, the emission contamination in the
57 Co
data consists mainly of photons which
have undergone multiple–scattering interactions in the imaged object and photons which have deposited only part of their energy within the detector crystals via Compton scattering interactions. Our correction does not currently take into account either of these effects directly. Furthermore, an analytical approach may not be ideal for
57 Co
emission–contamination cor-
rections, since it requires long computation times to model multiple scatter accurately using photon transport equations [185]. Our analytical EC correction is most suitable for the emission contamination present in transmission data acquired using energy sources such as
137 Cs
68 Ge
or other higher
(which emits 662 keV photons). For these
data, our simulations indicate that the emission contamination is primar197
Chapter 8. Future Work and Conclusions ily due to photons which were not scattered or scattered only once within the imaged object (which is more consistent with our analytical model). Currently, most human scanners equipped with single–photon transmission systems use
137 Cs
sources. Lower energy sources (such as
57 Co)
would not
be appropriate for these scanners because an insufficient amount of photon flux would be transmitted through the patient (especially for the torso) due to the higher µ–values for tissue at lower energy. A specific scanner which would benefit from an analytical EC correction is the HRRT, a PET scanner for human (and other large primate) brain imaging [111]. It uses a collimated
137 Cs
source but its current design does
not allow for mock scan acquisition [112]. Since this scanner employs two layers of scintillator crystals and uses a hexagonal array of planar detectors, our efficiency model would need to be modified significantly and then thoroughly validated for this scanner. Unfortunately, this application was beyond the scope of the current work.
8.4
Conclusions
The work presented in this dissertation was motivated by the need for more accurate attenuation corrections for small animal PET. This is particularly the case for the microPET series dedicated small animal imaging systems, which obtain their attenuation corrections from uncollimated photon– emitting sources using singles–mode transmission data acquisition. These transmission data are subject to high amounts of contamination from scattered photons leading to errors in the emission images (and the biological parameters derived from them) which are corrected using these data. We proposed a scatter correction, based upon the Klein–Nishina formula, and studied its accuracy using simulation data. We implemented our scatter correction as part of a reconstruction algorithm for transmission tomography and validated the procedure for experimental and simulated transmission data for uniform and non–uniform phantoms. We also tested our reconstruction and scatter correction procedure for several animal studies (a mouse, two rats and a monkey), for different imaging systems and for different 198
Chapter 8. Future Work and Conclusions scanning protocols (pre– and post–injection). For all studies considered, we found good agreement between the expected linear attenuation coefficients and those reconstructed with our scatter correction. This is the first scatter correction to be proposed and validated for singles–mode transmission data. Our algorithm provides improved accuracy and reduced operator– introduced variability compared to the existing alternatives (segmentation or rescaling of uncorrected attenuation–maps).
199
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