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Statistical list-mode image reconstruction and motion compensation techniques in high-resolution positron… Rahmim, Arman 2005

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STATISTICAL LIST-MODE IMAGE RECONSTRUCTION AND MOTION COMPENSATION TECHNIQUES IN HIGH-RESOLUTION POSITRON EMISSION T O M O G R A P H Y (PET) by A R M A N R A H M I M B.Sc. (Comb Hons), The University of British Columbia, 1999 M.Sc, The University of British Columbia, 2001 A THESIS SUBMITTED IN PARTIAL F U L F I L M E N T OF T H E REQUIREMENTS FOR T H E D E G R E E OF DOCTOR OF PHILOSOPHY in T H E FACULTY OF G R A D U A T E STUDIES (Physics) T H E UNIVERSITY OF BRITISH COLUMBIA April 2005 (c) Arman Rahmim, 2005 Abstract A b s t r a c t The work presented here is devoted to the proposal and investigation of 3D image reconstruction algorithms suitable for high resolution positron emission tomography (PET). In particular, we have studied imaging tech-niques applicable to the high resolution research tomograph (HRRT): a 3D-only state-of-the-art dedicated brain tomograph. The HRRT poses a num-ber of unique challenges, most significant of which include presence of gaps in-between the detector heads, as well as the very large number of lines-of-response (LORs) which it is able to measure (~4.5xl0 9), exceeding most modern P E T scanners by 2-3 orders of magnitude. To address the existing issues, we have developed and implemented statis-tical list-mode image reconstruction as a powerful technique applicable to the high resolution data obtained by the HRRT. We have furthermore verified applicability of this technique to dynamic (4D) P E T imaging, thus quali-fying the technique as viable and accurate for the research intended to be performed on the scanner. We have paid particular attention to the study of convergent algorithms; i.e. iterative algorithms which (with further itera-tions) consistently improve such figures of merit as resolution and contrast, relevant to research and clinical tasks. With the spatial resolution in modern high resolution tomographs (in-cluding the HRRT) reaching the 2-3mm F W H M range, small patient move-ments during PET imaging can become a significant source of resolution degradation. We have thus devoted a portion of this dissertation to the pro-posal of new, accurate and practical motion-compensation techniques, and Abstract iii studied them on the HRRT. We have theoretically proposed and experimen-tally validated the benefits of modeling the motion into the reconstruction task, thus signaling the way beyond the existing purely event-driven motion-compensation techniques. Table of Contents , V Table of Contents Abstract i i Table of Contents iv List of Tables and Figures ix Glossary x ix Acknowledgements xxi i . Introduction to PET Imaging 1 1.1 Nuclear Medicine Imaging 1 1.1.1 Single Photon Imaging 1 1.1.2 Positron Annihilation Photon Imaging 3 1.2 Positron Emission Tomographs 6 1.2.1 Applications of P E T Imaging 7 1.2.2 Annihilation Coincidence Detection 8 1.2.3 Scintillator Detectors 10 1.2.4 Detector to P M T Coupling 12 1.2.5 Avalanche Photodiodes (APDs) 13 1.3 2D vs. 3D P E T Imaging 14 1.4 Sinograms in the Projection-space 16 Table of Contents v 1.5 Dynamic P E T Imaging 18 1.6 Causes of Degradation in Image quality and Quantitative Ac-curacy 19 1.6.1 Attenuation of Annihilation Photons 21 1.6.2 Detection of Scattered Events 24 1.6.3 Detection of Random (or Accidental) Coincidences . . 26 1.6.4 Variations in Detector Pair Sensitivity 28 1.6.5 Detection Deadtime 30 1.6.6 Decay of Radioactivity '. 30 1.6.7 Detector Blurring 31 1.6.8 Positron Range and Photon Non-collinearity 32 1.6.9 Patient Motion 32 1.7 The High Resolution Research Tomograph 33 1.7.1 Challenges 33 1.7.2 More details for the Scanner 35 1.8 The Aim of This Work and Contributions 37 2. 3D Image Reconstruction 39 2.1 Introduction 39 2.2 Analytic Image Reconstruction 40 2.2.1 Reconstruction from Non-Truncated, 2D Parallel Pro-jections 42 2.2.2 Central Section Theorem 42 2.2.3 Orlov's Sufficiency Condition 43 2.2.4 Image Reconstruction from Redundant Data 45 2.2.5 Direct Fourier Reconstruction 45 2.2.6 Filtered Back-Projection (FBP) 47 2.2.7 Backprojection followed by Filtering (BPF) 49 2.2.8 Reconstruction from Truncated, 2D Parallel Projections 51 Table of Contents vi 2.3 Rebinning Algorithms 51 2.3.1 The SSRB and MSRB Algorithms 53 2.3.2 The F O R E X Algorithm 54 2.3.3 The FORE Algorithm 56 2.3.4 The FOREJ Algorithm 58 2.4 Statistical Image Reconstruction 59 2.4.1 Object Parametrization 60 2.4.2 The System Matrix 61 2.4.3 Statistical Modeling and Objective Functions 63 2.4.4 Reconstruction Algorithms 69 2.5 Problems with Expectation Maximization Algorithms 73 2.5.1 Importance of Convergent Reconstruction 73 2.5.2 Tackling Noise Artifacts: Regularization 74 2.5.3 Accelerated E M Reconstruction Algorithms 77 2.5.4 Summary 80 3. List-mode Image Reconstruction 82 3.1 Introduction 82 3.2 List-mode Acquisition 83 3.3 List-mode Reconstruction 84 3.3.1 Benefits of List-mode Image Reconstruction 85 3.3.2 Analytic List-mode Image Reconstruction 88 3.3.3 Statistical List-mode Image Reconstruction 89 3.4 Accelerated List-mode Reconstruction Algorithms 91 3.4.1 Ordinary Subsetized List-mode E M Algorithm 91 3.4.2 Convergent OSEM Reconstruction 92 3.4.3 Convergent Subsetized List-mode E M Algorithm . . . . 94 3.4.4 Hybrid S/CS List-mode E M Algorithm 96 3.5 Random Correction in List-mode Reconstruction 96 Table of Contents vii 3.5.1 Ordinary Poisson E M Reconstruction . 97 3.5.2 The Delayed Events Subtraction Technique 100 3.6 Scatter Correction in List-mode Reconstruction 100 3.7 Summary 102 4. Reconstruction Implementation and Comparison 103 4.1 List-mode Geometric Back- and Forward-Projection Techniquesl04 4.1.1 Projection Techniques 104 4.2 Subtraction of Delayed Events and Non-negativity Constraints 107 4.3 Methods 109 4.4 Results and Discussion 115 4.5 Dynamic P E T List-mode Image Reconstruction 127 4.6 Methods and Results 129 4.7 Conclusion 132 5. Motion Compensation: Beyond the Event-Driven Approach . . . . 135 5.1 Introduction 135 5.2 Motion Correction in Histogram-mode E M Reconstruction . . 137 5.2.1 Modification of the Histogram-mode E M Algorithm . . 138 5.2.2 A Note on Convenience and Accuracy 143 5.2.3 An Alternate, Fast Method for Calculation of Sensitiv-ity Factors 144 5.3 Motion Correction in List-mode E M Reconstruction 147 5.4 Methods 148 5.5 Results and Discussion 153 5.6 Conclusion 156 6. Summary, Opinion and Conclusions 159 6.1 Summary 159 6.2 Opinion and Suggested Future Directions 161 Table of Contents viii 6.3 Conclusion "'. 164 Bibliography 165 Appendices A. Analytic Image Reconstruction from Truncated Projections 182 B. Space-variant and Anisotropic Resolution Modeling 185 C. Printed Sources for PET 195 D. Derivation of the List-Mode EM Reconstruction Algorithm with Mo-tion Compensation 204 List of Tables ix List of Tables 1.1 Examples of Biomedical Applications of Positron Emitting Tracers 4 1.2 Table of Commonly Used Positron Emitting Isotopes 6 1.3 Table of Commonly Used Scintillation Materials (BGO: bis-muth germanate; GSO: gadolinium oxyorthosilicate; LSO: lutetium oxyorthosilicate; Nal: sodium iodide). BGO, GSO and LSO are not hygroscopic (unlike Nal), making packaging of the de-tectors easier 13 1.4 Specifications for Two PET Scanners 36 4.1 Table of Statistics for Frames used in the analysis 111 4.2 Image Bias for 3D-OSEM reconstruction of Frames 1 and 5 in table (4.1) (10M counts) - 117 4.3 Table of random fractions 119 B . l Characterization of Space-variance and Anisotropicity 189 List of Figures x List of Figures 1.1 A n example of a gamma camera used in single photon imaging. 2 1.2 Reconstruction of tomographic data allow non-invasive imag-ing of three-dimensional objects 3 1.3 A n emitted positron interacts with the surrounding medium until it reaches thermal energies, after which it annihilates into gamma rays. 5 1.4 In single photon imaging, collimators are used in order to limit the incident angles at which events will enter the detectors. This lowers the sensitivity of the camera, but is necessary for image formation. Use of physical collimators may however be avoided in positron annihilation photon imaging systems, as described in text (courtesy of Barry Pointon, B C Institute of Technology) 9 1.5 In positron annihilation photon imaging systems, coincidence de-tection may be utilized, in which two events arriving within a cer-tain coincidence time window are used to trace the path along which the annihilation occurred 10 1.6 The energy distribution for an LSO (lutetium oxyorthosilicate) de-tector is shown. The energy resolution of a detector (also depicted) determines the extent and accuracy to which the incident gamma rays may be filtered so as to reject events that do not correspond to 511 keV gamma rays 11 1.7 An 8x8 crystal block is coupled to four PMTs 14 List of Figures xi 1.8 3D P E T imaging can be achieved by using scanners with retractable septa or no septa at all. It can greatly increase the sensitivity of the scanner, thus reducing noise in the final reconstructed images, while posing new challenges in computation as well as data inver-sion (courtesy of Dr. C-H. Chen, with modifications) 15 1.9 Geometry of a cylindrical P E T scanner. Coordinates (s,4>,z,6) are used to parametrize the L O R connecting detectors A and B . The direction along z corresponds to the axis of the scanner. 17 1.10 (a) A 2D plane across an object (in image-space) containing four fixed points results in (b) sinogram data containing four different curves (projection-space). 18 1.11 Plots indicating annihilated photons from an event detected with (a) true coincidence, (b) scattering, and (c) random coincidence. . The last two clearly result in mis-positioning the L O R along which the event is detected 24 1.12 Three possible ways, as described in text, by which only a single event will be detected 26 1.13 An open display of the high resolution research tomograph (HRRT). 33 1.14 Sinogram data for a uniform cylinder (direct plane #110 is shown): the gaps comprise over 10% of the sinogram space. This sinogram can be thought of as a superposition of corre-sponding sinusoidal curves (Fig. 1.10) for the points which the cylinder consists of. The observed non-uniformities are due to noticeable variations in detector pair sensitivities (Sec. 1.6.4). 34 2.1 A P E T event consists of detecting two coincident gamma rays. 40 List of Figures xii 2.2 (a) A spherical PET scanner, with an LOR parametrized by (u, s) passing through it. (b) A cylindrical P E T scanner, with two LORs (u, S i ) and (u, s 2) passing through it, one measured and one unmeasured. Therefore the measured data are trun-cated since for certain directions, e.g. along u shown, the line integrals are not measured for all values of s 41 2.3 Each point on the sphere S2 represents one 2D parallel projec-tion. The set of measured projections Q is in-between the two parallels shown. The set Q is intersected by the great circle normal to any frequency v 44 2.4 The backprojection of a ID parallel projection onto a 2D image. 48 2.5 The rebinning procedure is summarized (courtesy of Dr. M . Defrise, Vrije Universiteit Brussel) 52 2.6 (a) Single-slice rebinning (SSRB): Each oblique (9 > 0) event is assigned to the direct (9 — 0) sinogram centrally located be-tween the two detectors, (b) Multiple-slice rebinning (MSRB): Each oblique event is fractionally allocated to direct sinograms that it traverses 54 2.7 Fourier rebinning (FORE): In the frequency domain, data from an oblique sinogram are allocated to a direct sinogram axially located kS/w below the central z value, as described by Eq. (2.39) 58 3.1 Moore's Law vs. Nutt's Law, as described in text. The ver-sion of the HRRT depicted is the single layer (~60K crystals), whereas most existing HRRT scanners are dual layer (~120K crystals) (courtesy of Dr. R. Nutt) 83 List of Figures xiii 3.2 With conventional reconstruction (left) voxels along the LOR are incremented regardless of position along the LOR. With TOF reconstruction (right), each voxel is incremented by the probability (as measured by the TOF measurement) that the source originated at that voxel 87 4.1 Projection algorithms employing (a) the Siddon method, (b) trilinear, and (c) bilinear interpolation techniques are drawn, as elaborated in text 106 4.2 Results of back-projecting constant values along a projection direction onto a 64x64 image at axial 0=0 and transaxial <j6=30° 106 4.3 (a) Plots of horizontal profiles across the back-projection im-ages depicted in Fig.( 4.2). (b) Projection vector obtained with forward-projection of a uniform circle along #=0 and 0=15°. . 107 4.4 A sample reconstruction of the radioactive paper source (3 iterations of the S-LMEM algorithm). The lower row of points sources as well as the rectangular box were utilized for analysis of resolution and noise properties 114 List of Figures xiv 4.5 Calculated ratios between total counts for images reconstructed using 3D-0SEM and those obtained using F0RE-2D-FBP, for the hot (top), background (middle) and cold (bottom) re-gions of the phantom, when (a) no random correction, (b) the sinogram non-negativity constraint and (c) the image non-negativity constraint have been imposed. For better visibility, the plots have been shifted by ten units with respect to one another. The case of reading 10M counts is shown, and qual-itatively similar results have been observed in the 40M and 80M cases. The error bars indicate statistical variation for the five transaxial planes selected 116 4.6 Calculated ratios between total counts for images reconstructed using S-LMEM and those obtained using FORE-2D-FBP, for the hot (top), background (middle) and cold (bottom) regions of the phantom. For better visibility, the plots have been shifted by ten units with respect to one another. The case of 10M counts is shown, and qualitatively similar results have been observed in the 40M and 80M cases. The error bars indi-cate statistical variation for the five transaxial planes selected. 118 4.7 Contrast vs. noise plots for images reconstructed with S-LMEM and OSEM. Three iterations are performed with 16 subsets, with results shown every 4 subsets. Results are shown for (a) 20M and (b) 40M total counts. Presence of limit cycles is noticeable in both algorithms 120 4.8 Contrast vs. noise plots for images reconstructed using the S-LMEM (dotted), CS-LMEM (- -) and hybrid S / C S - L M E M (solid) algorithms with scan durations containing (a) 10M and (b) 5M total counts. Three iterations are shown 121 List of Figures xv 4.9 Plots of measured resolution vs. radial position for the S-L M E M algorithm implemented using the Siddon algorithm as well as bilinear and trilinear interpolation techniques. For comparison, results of output-driven (as described in text) histogram-mode OSEM reconstruction are also shown. Three iterations were performed 121 4.10 Plots of noise vs. iteration for the reconstruction schemes described in the caption of Fig. (4.9) 122 4.11 Plots of reconstructed F W H M width vs. Iteration are shown for the 3D-OSEM, S-LMEM, and the C S - L M E M and hybrid algorithms for the point source located at (a) 1 cm and (b) 5 cm from center of the FOV. (c) Final F W H M values are shown for all the reconstructed points sources using the afore-mentioned four schemes (after 3 iterations) 124 4.12 Plots of reconstructed F W H M width vs. Iteration are shown for the 3D-OSEM, S-LMEM, and the C S - L M E M and hybrid algorithms for the point source located at (a) 1 cm and (b) 5 cm from center of the FOV. (c) Final F W H M values are shown for all the reconstructed points sources using the afore-mentioned four schemes (after 3 iterations) 125 4.13 Plots of reconstructed F W H M width vs. Iteration are shown for the 3D-OSEM, S-LMEM, and the C S - L M E M and hybrid algorithms for the point source located at (a) 1 cm and (b) 5 cm from center of the FOV. (c) Final F W H M values are shown for all the reconstructed points sources using the afore-mentioned four schemes (after 3 iterations) 125 < List of Figures xvi 4.14 (a) Overall F W H M vs. iteration for images reconstructed using the hybrid S / C S - L M E M algorithm with axial spans of 3 and 9: point sources located at 3 cm and 5 cm from center of F O V are shown, (b) Degradation in F W H M (along Z-direction) vs. radial position as one switches from axial span 3 to 9: all the point sources reconstructed using the S - L M E M , CS-L M E M and hybrid schemes (after 3 iterations) are shown. . . 126 4.15 Time activity curves (TACs) for 18 frames each 300 s in du-ration. First frame has a random fraction of 25%, last frame 5%. One iteration and 16 subsets are used in list-mode and O S E M reconstructions. . 130 4.16 A x i a l uniformity comparisons: profiles were drawn through the hot (top), cold (bottom) and background (middle) regions in each of the reconstructed images 131 4.17 Plots of percentage contrast for the various dynamic frames. . 133 5.1 A n event detected at an L O R i' generated in voxel j, which has been translated to voxel j' = M ( j ) at time of detection, would have been detected at L O R i = L if the object had not moved 139 5.2 A n event detected at time t2 along an L O R i', which would have been detected along an L O R i if the object had not moved, has the same attenuation correction factor as mea-sured for A{ at time t\. The normalization correction factor however is LOR-specific and changes with motion. 142 List of Figures xvii 5.3 (Top) Sinograms of a moving line source (a) without and (b) with motion-compensated histogramming. Clearly, upon ap-propriate histogramming, some counts are histogrammed into bins corresponding to detector gaps (shown by arrows), signi-fying that they would not have been detected had the object not moved. (Middle) Images reconstructed using histogram-mode algorithm applied to (c) data with no motion and (d) data for motion when reconstructed without motion correc-tion. (Bottom) (e) Image resulting from the proposed re-construction algorithm given by Eq. (5.13). Radial profiles through this image as well as the one resulting from no mo-tion data are shown in (f). Three iterations were applied to the data. Direct plane #110 is shown (randomly selected). . 152 5.4 Images reconstructed using list-mode algorithm applied to a) data with no motion, as well as data for motion when re-constructed b) without and c) with motion correction, (d) Radial profiles through imaged line source in cases a) and c) are shown. Three iterations were applied to the data. Plane 110 is shown (randomly selected) 153 5.5 Images reconstructed for a radioactive printed "W"-sign (coro-nal view) when (a) no motion is introduced; and with motion (three axial positions: 1 min - 3 min - 3 min) when (b) no motion compensation, (c) the purely event-driven approach (conventional motion correction) and (d) the proposed algo-rithm given by Eq. (5.20) are applied to the data. Plots of uniformity profiles drawn through images in (c) and (d) are shown in (e). Five iterations were applied to the data 157 List of Figures xviii B . l Resolution relations implied by the common case of 90°-transaxial-rotation symmetry 190 B.2 Observed transaxial (x/y)-resolution values as functions of space-invariant, isotropic kernel a values. Points at which the curves are each minimized are indicated using circles 192 B. 3 Variation of A as a function of modeled space-invariant, isotropic a values, as described in text. Values achieved by using expo-nential and inverse-Gaussian modeling of degradation of reso-lution are also shown 194 C. l Left: Side View, cut lines on printer ink cartridge. Right: Side View, section and assembly of modified ink cartridge 197 C.2 Axial (squares) and radial (diamonds) resolution measured for the HRRT 200 C.3 An example of a source profile imaged with (solid) and without (dashed) one layer of A l foil. The two profiles nearly coincide with one another 201 Glossary xix Glossary Abbreviations and Acronyms 3 D R P 3 D Reprojection A P D Avalanche photodiode ART Algebraic Reconstruction Technique BGO Bismuth Germanate B P F Backprojection followed by filtering C S - L M E M Convergent Subsetized L M E M DOI Depth of Interaction E M Expectation Maximization F B P Filtered backprojection FORE Fourier Rebinning F W H M Full width at half maximum GSO Gadolinium Oxyorthosilicate HRRT High resolution research tomograph L M E M List-Mode Expectation Maximization LOR Line of Response LSO Lutetium Oxyorthosilicate LYSO Lutetium Yttrium Oxyorthosilicate M A P Maximum a posteriori M L Maximum likelihood MSRB Multi-Slice Rebinning Nai Sodium Iodide NC Normalization coefficient OSEM Ordinary Subset E M reconstruction algorithm hardware reconstruction algorithm scintillator reconstruction algorithm reconstruction algorithm hardware reconstruction algorithm reconstruction algorithm rebinning algorithm resolution measure scintillator scanner reconstruction algorithm geometry scintillator scintillator objective function objective function rebinning algorithm scintillator correction factor reconstruction algorithm Glossary xx P E T Positron Emission Tomography P M T Photomultiplier tube ROI Region of Interest SIR Statistical image reconstruction S-LMEM Subsetized L M E M SPECT Single Photon Emission Computed Tomography SSRB Single Slice Rebinning T T R True three-dimensional reconstruction imaging technique hardware image analysis reconstruction technique reconstruction algorithm imaging technique rebinning algorithm reconstruction filter Acknowledgements xxi Acknowledgements He who does not express gratitude to creatures, has not expressed gratitude to the Creator. -Prophet Mohammad Many individuals have contributed to my success in the course of completing this work. I would like to deeply thank my supervisor, Dr. Vesna Sossi, whose contributions to my work are many and diverse. Her very thorough and broad knowledge of P E T imaging combined with her extraordinary enthusiasm have been truly inspirational. She has granted me much room for exploring my own ideas, while providing me with extremely valuable guidance throughout this work. I would also like to thank my supervisory committee members, Drs. Ruth, MacKay and Michal for their support and constructive feedback. In particular, I am in-debted to Dr. Ruth for instructing me on various topics, particularly in the field of nuclear chemistry. I would also like to thank Drs. Kinahan, Kiefl, McKeown and Schrack for taking the time to review my dissertation and for their valuable comments during and following my oral examination. A very special thanks is due to Dr. Stephan Blinder, Kevin Cheng, Marie-Laure Camborde, Ken Buckley and Paul Piccioni for their much needed help in completing this work, and also to Barry Pointon, Eric Vandervoort, Siobhan McCormick and Dr. Doris Doudet for many interesting and vibrant discussions. I would also like to thank Christian Michel, Mark Lenox, Ziad Burbar, Merence Sibomana, Andrew Reader and Peter Bloomfield for their collaboration and feedback. I am simply unable to recite my family's contributions to my life and work. My accomplishments are direct results of their love and support. My cousin and 'carpool buddy', Amir, been very supportive, and continues being a role model of kindness for me. My brother and friend, Iman, has also been incredibly kind to me. I am grateful for his words of advice in the past many years, and particularly during my job interview process. My wife, Zahra, has been truly supportive of my academic and extracurricular involvements, and has made the last year of my graduate studies one to remember for the rest of my life. My parents have sacrificed many things throughout their lives for my sake and I owe them all that I have. As a humblest gesture of appreciation, this dissertation is dedicated to them. 1. Introduction to PET Imaging 1 1. Introduction to PET Imaging 1.1 Nuclear Medicine Imaging Nuclear medicine imaging is a non-invasive technique which allows dynamic examination of large areas of the body in a scanning session, producing im-ages of human body functions unobtainable by other imaging techniques. It involves the use of radioactively labeled pharmaceuticals (radiopharmaceuti-cals) for functional (as compared to structural) imaging of the human body. The choice of pharmaceuticals, which can be intravenously injected, inhaled or ingested, depends on the function under investigation. The obtained im-ages capture biochemical processes, such as early cancer tumor activity, that cannot be revealed by anatomical imaging with conventional X-ray, C T or MRI , and can therefore provide unique information on metabolic processes in healthy and diseased states. As further elaborated in Sec. (1.5), nuclear medicine imaging, combined with the application of appropriate mathemati-cal models, can be used to investigate and quantify physiologic or metabolic processes in vivo. Modern nuclear medicine imaging mainly consists of two main approaches: i) single photon imaging, and ii) positron annihilation photon imaging in which two photons are detected in coincidence. 1.1.1 Single Photon Imaging Single photon imaging modality is based on detection of two-dimensional projections of the three dimensional radiopharmaceutical distribution. The 1. Introduction to PET Imaging 2 Fig. 1.1: An example of a gamma camera used in single photon imaging. radiopharmaceuticals are formed using radioisotopes that emit single photons upon emission. A typical imaging system suitable for this task is the gamma camera, as shown in Fig. ( 1 . 1 ) . Upon rotating the gamma camera around the patient, a series of 2D projections can be obtained which can be combined to determine depth in-formation. This technique is commonly referred to as single photon emission computed tomography (SPECT). The word ''tomography1 is derived from the Greek words 'tomo) meaning to slice and ''grapK meaning image. The basic idea is that if a sample is imaged several times in different orientations, three-dimensional (volume) information on the sample structure can be obtained using mathematical algorithms. This is called a tomographic reconstruction or tomography1. It 1 Tomography is often perceived as an imaging tool for medical examination purposes. It has to be emphasized, however, that the concept of tomography and its non-invasive way of imaging are not restricted to the medical field. Tomography has been developed, over the 1. Introduction to PET Imaging 3 Experiment Raw data Processed data X-ray beam bulk sample 2-d detector Pro jecSon radiographs 3-d volume data ' for every angle ' on the interior of the sample Fig. 1.2: Reconstruction of tomographic data allow non-invasive imaging of three-dimensional objects. enables one to look at slices of the investigated object without physically cutting it, as depicted in Fig. (1.2). Modern S P E C T systems consist of two or three camera heads mounted on a single rotating gantry, thus allowing detection of a larger fraction of emitted photons (i.e. increased sensitivity) and therefore increasing the signal-to-noise ratio. 1.1.2 Positron Annihilation Photon Imaging Positron-emitting isotopes can be also used for labeling of pharmaceuticals. As elaborated below, upon annihilation of an emitted positron, two gamma rays are produced which can be subsequently detected by the photon imaging system. One advantage of this technique (over SPECT) is that a number of elements which are fundamentally used by compounds of biological interest (C, N , 0 , and F) can be positron-emitters, allowing such positron-emitters to be more readily incorporated into a wide variety of useful radiopharma-ceuticals (than is the case with single photon emitters). Some examples of last decade, into a reliable tool for imaging numerous industrial applications (e.g. electrical capacitance or impedance tomography). This field of application is commonly known as Industrial Process Tomography (IPT) or simply Process Tomography (PT). 1. Introduction to PET Imaging 4 Radiotracer Radiopharmacutical Examples of Biomedical Applications 0-15 oxygen oxygen metabolism 0-15 carbon monoxide blood volume 0-15 carbon dioxide blood flow 0-15 water blood flow N-13 ammonia blood flow F-18 F D G glucose metabolism F-18 F D O P A pre-synaptic dopaminergic activity F-18 FMISO hypoxic cell tracer C - l l D T B Z vesicular monoamine transporter V M A T 2 C - l l Raclopride dopamine D2 receptor C - l l methylphenidate (MP) dopamine membrane transporter DAT C - l l SCH23390 dopamine D l receptor C - l l flumazenil benzodiazepine receptor Tab. 2.1: Examples of Biomedical Applications of Positron Emitting Tracers biomedical application for radiopharmaceuticals with positron-emitting ra-diotracers are shown in table (1.1). Positron Emission: A neutron-deficient nucleus can become stable via electron capture, whereby an atomic electron is captured by a proton (thus transforming the proton into a neutron), or alternatively via positron emis-sion which arises from proton (p). decay, according to p -»• n + (3+ + ve (1.1) where n is a neutron, f3+ is a positron and ue is an electron neutrino. For a general radionuclide, X, the following process occurs: M X ^ z M x + p + + U e ( L 2 ) Due to emission of a neutrino along with the positron, the positron can be emitted with a continuous range of energies (up to a maximum energy; 1. Introduction to PET Imaging 5 Unstable parent tiudtous V t m i l n w En I H J < U - I I S -p t M i t r t i f t »twl i t M i t i ' l n u « n i t t « i Fig. 1.3: An emitted positron interacts with the surrounding medium until it reaches thermal energies, after which it annihilates into gamma rays. see table 1.2). Positrons give up their kinetic energy principally by Coulomb interactions with electrons within the surrounding medium. As the rest mass of the positron is the same as that of the electron, the positrons may undergo large deviations in direction with each Coulomb interaction, as is depicted in Fig. (1.3). When the positrons reach thermal energies, they start to interact with electrons: either (i) by annihilation, which produces two anti-parallel 511 keV photons, or (ii) by the formation of a hydrogen-like orbiting couple called positronium. In its ground-state, positronium has two forms: (i) ortho-positronium (o-Ps), where the spins of the electron and positron are parallel, and (ii) para-positronium (p-Ps), where the spins are anti-parallel. Para-positronium again decays by self-annihilation, generating two anti-parallel 511 keV photons. Ortho-positronium, on the other hand, self-annihilates by the emission of three photons, but composes a negligible fraction of total 1. Introduction to PET Imaging 6 Ti/2 •^ave E F W H M Range F W H M Range (min) (MeV) (MeV) in water (mm) in water (mm) F-18 109.8 0.24 0.64 0.6 1.8 C - l l 20.3 0.39 0.96 0.9 2.8 N-13 9.97 0.49 1.19 1.0 3.5 0-15 2.03 0.73 1.70 1.6 5.2 Tab. 1.2: Table of Commonly Used Positron Emitting Isotopes positron annihilations2 and can be safely ignored. Higher energy positrons, upon being emitted, require to traverse a larger distance, on the average, in the surrounding medium before they can reach thermal energies in order to be annihilated: this distance is referred to as positron range. One notes that in positron annihilation photon imaging, the presence of positron range imposes a fundamental limit on spatial localiza-tion of the points from which positrons are emitted. We also note that since different positron-emitting isotopes exhibit distinct energy distributions, dif-ferent isotopes also exhibit distinct positron range values. Table (1.2) sum-marizes such properties of commonly employed positron-emitting isotopes, as extensively elaborated in [2]. 1.2 Positron Emission Tomographs In what follows we provide a brief overview of the important and unique clinical applications of positron emission tomograpy (PET), followed by an explanation of the main components of a modern P E T scanner and the prin-2 This is because, despite the larger amount (3:1) of o-Ps to p-Ps initially formed, the two gamma lifetime for p-Ps is 125 ps (in vacuum or in liquids). On the other hand, o-Ps has a three gamma lifetime of 142 ns (vacuum), yet in liquids, due to a "pick-off" process in which a second electron with opposed spin reacts with the positron in the o-Ps atom resulting in a two photon annihilation, the lifetime of o-Ps in liquids is considerably shorter (1800 ps in water) [1]. 1. Introduction to PET Imaging 7 ciples employed to achieve high sensitivity and resolution. 1.2.1 Applications of PET Imaging P E T is a camera that produces powerful images of the human body's biolog-ical functions and entire organ system with one image. It reveals metastatic diseases other imaging techniques may not be able to detect, enabling the physicians to potentially diagnose disease before it shows up in other imaging modalities. P E T can also diagnose cancer, cardiac disease, neurological brain disorders, as outlined below, and help guide physicians to the most beneficial therapies. Tumors: P E T imaging is very accurate in differentiating malignant from benign growths, as well as showing the spread of malignant tumors. P E T imaging can help detect recurrent brain tumors and tumors of the lung, colon, breast, lymph nodes, skin, and other organs. Information from P E T imaging can be used to determine what combination of treatment is most likely to be successful in managing a patient's tumor. Cardiac PET: P E T allows quantitative assessment of myocardial perfu-sion and metabolism. It can be used not only to diagnose coronary artery disease in patients with equivocal studies from conventional diagnostic tech-niques, but also to evaluate myocardial viability (the ability of the heart to recover after revascularization) in patients with heart failure. Diseases of the Brain: P E T imaging can provide information to pin-point and evaluate diseases of the brain. P E T imaging can show the region of the brain that is causing a patient's seizures and is useful in evaluating degenerative brain diseases such as Alzheimer's, Huntington's, and Parkin-son's. Within the first few hours of a stroke, P E T imaging may be useful in determining treatment therapies. P E T is an important research tool for the assessment of cerebral tunc-1. Introduction to PET Imaging 8 tion, metabolism and receptor ligand systems. Dynamic P E T imaging (see Sec. 1.5 for more detail) is particularly suited for research into physiologic and metabolic processes in the healthy and diseased states. In particular, in our P E T centre at the University of British Columbia 3, the main clinical research focus is the investigation of the neurodegeneration as it manifests itself in Parkinson's disease. The causes, origin and detailed disease progres-sion mechanisms are still unknown. At present there is no cure, and only symptomatic treatment is available, which often causes disabling side-effects. P E T has however been an invaluable tool in providing information on some aspects of the disease such as providing some evidence for early compensatory changes [3,4], and possible mechanisms contributing to treatment induced motor complications [5,6]. Furthermore, P E T imaging has also proven to be a powerful tool for understanding the in vivo kinetics of new drugs, and thus is actively employed in drug discovery and development tasks. 1.2.2 Annihilation Coincidence Detection In single photon imaging, collimators (as shown in Fig. 1.1) are used in order to enable the scanner to have knowledge of the angle at which photons are incident on the detectors. In other words, in such systems, without use of a collimator, the detected photon could have arrived from any region in the field of view, and therefore use of collimators is necessary for image formation (Fig. 1.4). In the imaging of positron emitters, gamma cameras may again be utilized for measurement of events. Nevertheless, much greater sensitivity can be gained by noting that collimation can instead be performed electronically, as illustrated in Fig. (1.5): if two events arrive within a certain coincidence 3 http://www.pet.ubc.ca 1. Introduction to PET Imaging 9 D t i II(e 11:it 5 2 • J iii[e f cI i t i Fig. 1.4: In single photon imaging, collimators are used in order to limit the in-cident angles at which events will enter the detectors. This lowers the sensitivity of the camera, but is necessary for image formation. Use of physical collimators may however be avoided in positron annihilation photon imaging systems, as described in text (courtesy of Barry Pointon, BC Institute of Technology). time window, they are recorded as dually emitted gamma rays from a single positron annihilation. The location at which the positron annihilation has taken place can subsequently be traced to within the line-of-response (LOR) in between the two detectors. Positron emission scanners (commonly known as positron emission tomographs or P E T scanners) therefore consist of a series of detector rings, each ring having a series of crystals, such that image-domain information may be extracted from tomographic measurement of data. This technique introduces two further considerations: (i) problem of random coincidences, in which the two events detected within the coinci-dence timing window originate actually from different positron annihilations, wherein their duals have not been detected. This is an important issue in P E T imaging, and is further elaborated in Sec. 1.6.3; (ii) photon non-collinearity effect: since the positron and electron are never exactly at rest prior annihilation, conservation of momentum dictates a deviation from 180° between the trajectories of the two annihilation photons. This deviation is 1. Introduction to PET Imaging 10 Coincidence Fig. 1.5: In positron annihilation photon imaging systems, coincidence detection may be utilized, in which two events arriving within a certain coincidence time window are used to trace the path along which the annihilation occurred. around 0.5° full width at half maximum, which corresponds to a resolution blurring of ~1.1 mm for two detectors separated by 0.5 m. Nevertheless, the absence of the need for physical collimation in P E T (replaced by electronic collimation) results in an approximately three-orders-of-magnitude increased detection sensitivity compared to SPECT. 1.2.3 Scintillator Detectors In positron emission tomography, scintillator detectors are most commonly used to detect incoming gamma rays. The incoming radiation excites elec-trons in the scintillating material (molecular excitations) through Compton scattering and/or photoelectric absorption. The scintillator material subse-quently "glows" as molecules return to their ground state, emitting scintilla-tor photons. The number of emitted photons is (ideally) proportional to the energy deposited in the crystal. The scintillator photons next create electrons in the photocathode of the photomultiplier tubes (PMTs), which are subse-1. Introduction to PET Imaging 11 energy Fig. 1.6: The energy distribution for an LSO (lutetium oxyorthosilicate) detector is shown. The energy resolution of a detector (also depicted) determines the extent and accuracy to which the incident gamma rays may be filtered so as to reject events that do not correspond to 511 keV gamma rays. quently amplified by about 6-8 orders of magnitude and finally measured at the exit nodes of the PMTs. Each scintillator material has its own characteristic properties. Generally, these can be categorized as: i) Stopping Power (linear attenuation coefficient): the density and ef-fective atomic number of a scintillator determine the relative probability by which Compton scattering and/or photoelectric absorption will occur as the processes by which incoming radiation is attenuated. Presence of Compton scattering is less preferred as it can result in the incoming radiation to be deflected to neighboring detectors, thus blurring the detection process, as elaborated in Sec. (1.6.7). ii) Decay Time: this represents the time it takes for the electrons excited by the incoming radiation to return to their ground state, after which they are available to be excited by more incoming photons. Scintillators with shorter decay times allow higher count rates to be processed. 1. Introduction to PET Imaging 12 iii) Spectral Distribution: for good detection efficiency, a scintillator must give off light at wavelengths that are efficiently detected by the PMTs . It affects the energy resolution, as defined in Fig. (1.6), which will subsequently affect the ability to reject detected events that do not correspond to 511 keV gamma rays (e.g. scattered gamma rays as discussed in Sec. 1.6.2). iv) Linearity: the amount of light produced by the scintillating material should be proportional to the energy deposited by the radiation. v) Conversion Efficiency: this is the fraction of the radiation energy con-verted to detectable scintillator light. It is directly related to the photon yield (which is the number of scintillator photons produced per keV of radiation energy), and thus to energy resolution. Table (1.3) compares properties of some scintillators commonly used in P E T . B G O scanners have been very common in the past. However, with the advent of LSO (Lutetium Oxyorthosilicate) in 1991 [7] as a scintillator mate-rial which provided notable improvements in terms of decay time, conversion efficiency (therefore light output) and energy resolution, LSO (or L Y S O 4 ) scintillators have gradually become more popular. 1.2.4 Detector to PMT Coupling P E T scanners were originally designed for a one-to-one coupling of detector crystals and PMTs. In 1985, the scintillating crystal block technology was introduced by Casey and Nutt [8], followed nowadays in most P E T scanners. A n example of this implementation is shown in Fig. (1.7). In this case, an 8x8 block of crystals is coupled to four PMTs, with the basic idea being that an event detected at a particular crystal would dispense light in the four PMTs according to its distance from them, and the particular crystal geometric pattern. The crystal position can therefore be identified by linear 4 LSO doped with Yttrium. 1. Introduction to PET Imaging 13 Nai B G O GSO LSO Density (g/cm 3) 3.67 7.13 6.7 7.4 Effective Z 51 74 61 66 Attenuation Length (cm) 2.88 1.05 1.43 1.16 Decay Time (nsec) 230 300 30-60 35-45 Conversion Efficiency (%NaI) 100 15 25 75 \L (cm - 1 ) at 511 keV 0.34 0.91 0.72 0.79 Energy Resolution 12 23 7.6 11.4 Hygroscopic Yes No No No Tab. 1.3: Table of Commonly Used Scintillation Materials (BGO: bismuth ger-manate; GSO: gadolinium oxyorthosilicate; LSO: lutetium oxyorthosil-icate; Nai: sodium iodide). BGO, GSO and LSO are not hygroscopic (unlike Nai), making packaging of the detectors easier. averaging of light output in the four PMTs. With the considerable expense of PMTs and limitations on how small they can be made, the aforementioned scheme has allowed cost-effective manufacturing of P E T scanners with small crystal sizes (few millimeters), thereby notably increasing resolution of P E T scanners in the past two decades. 1.2.5 Avalanche Photodiodes (APDs) In recent years, there has been a growing interest in exploring the possibility of replacing the PMTs with light-sensitive semiconductor detectors, such as Si photodiodes. These detectors exhibit a higher quantum efficiency (QE) compared to PMTs , yet in their original form, do not internally amplify the signal, unlike PMTs . Avalanche photodiodes (APDs), which are composed of Si photodiodes with internal gain, have therefore been proposed combin-ing the two advantages. This has enabled detection of low light levels with good signal-to-noise ratios [9,10]. APDs have the additional advantage that they can be made very small in area and that they are typically only a 1. Introduction to PET Imaging 14 • • | : :j: Fig. 1.7: A n 8x8 crystal block is coupled to four P M T s . few millimeters thick, including the packaging5. It must be noted that one major concern using APDs for P E T applications is the limited timing reso-lution [12]. Furthermore, amidst a higher QE, they still provide 2-3 orders of magnitude less gain compared to PMTs. Nevertheless, these limitations are currently under continuous investigation and improvement, and APDs are likely going to be increasingly popular in the upcoming years. 1.3 2D vs. 3D PET Imaging Originally, P E T data were acquired in so-called septa-in or 2D mode. Metal septa were positioned between each detector ring in order to effectively absorb photons not being detected in the same ring, as shown in Fig. (1.8). For a scanner with N detector rings, the LORs were assigned to one of 2^-1 unique planes. With N of these planes corresponding to the detector ring planes, the other N-l rings were taken to lie between the direct planes, corresponding to 5 Two types of APDs are commonly under investigation: (i) pixelated APDs, and (ii) position-sensitive (PS) APDs. The latter works by collecting signals from four contacts placed at the corners of the anode side of the PS-APDs, and using them to determine the position of the photon's interaction within the APD. The main advantage of this latter approach is the reduction in the number of channels that need to be read out, which results in a cost reduction and more compact design [11]. 1. Introduction to PET Imaging 15 Fig. 1.8: 3D PET imaging can be achieved by using scanners with retractable septa or no septa at all. It can greatly increase the sensitivity of the scanner, thus reducing noise in the final reconstructed images, while posing new challenges in computation as well as data inversion (courtesy of Dr. C-H. Chen, with modifications) events detected with a ring difference A = l (the septa did not absorb photons at this minimal ring difference). Later developments of 3D P E T were motivated by a need to increase sen-sitivity of P E T scanners, which would enable enhancement of image signal-to-noise ratios. This was achieved by use of retractable septa (or no septa at all) and thus allowing the acquisition of those events where the gamma rays were emitted at an oblique angle relative to the tomograph axis. Typically, this leads to a fourfold to eightfold improvement in sensitivity [13]. How-ever, 3D P E T imaging poses new challenges: an important consideration has been that of additional computation burden caused by a large increase in the number of possible LORs. As an example, in the case of the high reso-lution research tomograph (Sec. 1.7), a system with 104 detector rings, one would have 2x104-1=207 direct (2D) data planes, whereas in fully 3D mode, interaction between all planes would be considered and thus one would have 1. Introduction to PET Imaging 16 1042=10816 possible planes, a 50-fold increase in the size of the data. One method to tackle this computational burden is to use rebinning algorithms to convert the 3D-acquired data into a stack of ordinary 2D data sets, which can subsequently be reconstructed using 2D reconstruction algorithms. Sec. (2.3) discusses this approach in detail. Another consideration is the application of suitable 3D reconstruction algorithms. This is especially complicated due to presence of incomplete data: exact analytic 3D reconstruction algorithms require access to data in all angles covering the field-of-view (FOV), yet practical scanners do not detect events along all possible directions. This has been discussed in more detail in Sec. (2.2.8). A third difficulty with 3D P E T imaging is the substantial increase in the scatter fraction (typically by a factor of ~3), as explained in Sec. (1.6.2), which renders application of accurate and suitable scatter correction algo-rithms very important. 1.4 Sinograms in the Projection-space We shall restrict our attention to the general case of 3D P E T imaging. As elaborated in Sec. (1.2.2), detection of two photons in coincidence defines an L O R joining the corresponding detector-pairs. The acquired LORs, corre-sponding to ID lines in 3D space, can be parametrized using four parameters. We shall do so using the coordinates (s, (fi, z, 9) where s and (f) are the radial and azimuthal angular coordinates, z is the axial coordinate of the point midway between the two detectors, and 9 is the copolar angle between the L O R and the transaxial planes. This is illustrated in Fig. (1.9). Defining p(s, (f>, z, 9) as the number of measured events along an L O R defined by the coordinates, it measures the projection of the image-space distribution onto the particular L O R (i.e. it represents those events whose annihilation gamma 1. Introduction to PET Imaging 17 Fig. 1.9: Geometry of a cylindrical PET scanner. Coordinates (s,(f>,z,6) are used to parametrize the LOR connecting detectors A and B. The direction along z corresponds to the axis of the scanner. 1. Introduction to PET Imaging IS (a) Imaged points (b) Corresponding sinogram Fig. 1.10: (a) A 2D plane across an object (in image-space) containing four fixed points results in (b) sinogram data containing four different curves (projection-space). pairs are emitted along the ID line-of-response passing through the object). 3D P E T imaging of an object (in image-space), therefore, produces 4D data in the so-called projection-space). In conventional P E T imaging, the measured LORs are binned (or his-togrammed) onto 2D data sets, each defined by a particular z and 6. A given 2D data stack p(s,4>,.,.) (fixed z and 6) is referred to as a sinogram and is typically shown using a 2D plot with the vertical column representing 4> and the horizontal column s. The sinogram data corresponding to a fixed point describes a sinusoidal curve, hence the origin of the term sinogram. A n example of this is shown in Fig. (1.10). 1.5 Dynamic PET Imaging The quantity that is measured in P E T is the in vivo regional or local tis-sue concentration of the positron-emitting radiotracer. This quantity can be related to a physiologic or metabolic process through the application of an 1. Introduction to PET Imaging 19 appropriate mathematical model6 of the process to dynamic P E T data from the patient. As compared to static imaging, used for instance in tumor detec-tion, dynamic imaging necessarily requires quantification (i.e. reconstructed images based on which meaningful quantitative measures can be made, as compared to, for instance, a simply qualitative detection of tumor contrast). One major issue that renders P E T an inherently quantitative imaging method (allowing the measurement of regional concentrations of the radio-pharmaceutical injected after proper calibration) is the fact that the proba-bility of survival of a pair of annihilation gamma rays is independent of the position of the annihilation along the L O R (more discussion in Sec. 1.6). To achieve quantitative detection, several problems still have to be overcome, which we review next. The work presented in this dissertation has been focused on the need for practical quantitative image reconstruction procedures compatible with the high resolution research tomograph (HRRT; see Sec. 1.7 for details), and its requirements (e.g. a very large number of LORs, presence of gaps in-between the heads, etc.). In what follows, we first review general issues that need to be considered in P E T in order to yield accurate and quantitative P E T images, followed by a description of the design and properties of the HRRT. 1.6 Causes of Degradation in Image quality and Quantitative Accuracy There are several physical phenomena that complicate P E T imaging. It is important to understand these issues since lack of correction for them can re-6 There exist various kinds of mathematical models, commonly referred to as tracer kinetic modeling techniques, of widely different mathematical characteristics - determinis-tic vs. stochastic, distributed vs. non-distributed, compartmental vs. non-compartmental, and linear vs. non-linear. In biomedical applications, linear compartmental models are most frequently used, because of their attractive mathematical properties. The reader is referred to Refs. [13-15] for more discussion. 1. Introduction to PET Imaging 20 suit in degradation of the image quality and/or quantitative accuracy. They are: i) Attenuation of Annihilation Photons. ii) Detection of Scattered events. iii) Detection of Random (or accidental) coincidences. iv) Variations in detector pair sensitivity. v) Detection Deadtime. vi) Decay of radioactivity. vii) Detector blurring. viii) Positron range ix) Photon non-collinearity. x) Patient Motion. Of these, random coincidences, positron range and photon non-collinearity are specific to P E T . The other effects also appear in SPECT. Furthermore, (corrections for) detector blurring, positron range and photon non-collinearity have been traditionally ignored since the camera resolution was not sufficient to be sensitive to these effects. However, with the arrival of high-resolution scanners, it has become more important to account for these factors in the reconstruction tasks, as we discuss later. Compared to S P E C T imaging, attenuation fractions are larger in P E T yet corrections for them are much easier, as described next (thus relatively easing the task of quantitative imaging using PET) . However, detection of scattered events is much more prevalent in 3D P E T as discussed in Sec. (1.6.2), thus complicating quantitative reconstruction using P E T . We discuss these phe-nomena in the following subsections and briefly review correction techniques used to compensate for their presence. 1. Introduction to PET Imaging 21 1.6.1 Attenuation of Annihilation Photons One or both of the 511 keV annihilation photons may undergo photoelectric or Compton scattering prior to being detected. The incidence of photoelectric absorption is negligible (less than 1%) for 511-keV photons in body tissues. In a Compton interaction, the photon interacts with a free.or outer shell electron. Subsequently, in every Compton scatter the direction of the gamma photon changes and its energy is decreased. These interaction can result in the scattered photon (i) to be deflected out of the field of view and never be detected, (ii) or to be deflected such that it is detected at another detector. Either of these cases results in non-detection of the L O R which would have been detected had no scattering occurred: this phenomenon is referred to as attenuation7. Defining the survival probability Aa as the probability of a photon not interacting as it propagates along a path a, it is given by A a = e~ •/>(*)*< (1.3) where .//(x) is the linear attenuation coefficient (which provides an indica-tion of how effective a given material is, per unit thickness, in promoting photon interactions at position x), increasing with higher matter densities, and decreasing with higher photon energies8. Similarly, the probability of attenuation, often referred to as the attenuation factor (AF), is easily given by 1 - Aa. At first thought, one might then conclude that since P E T is based on higher energy photons compared to SPECT, the attenuation effect is less pronounced with P E T . However, this is hot the case due to the dual-photon nature of this modality. In other words, the survival probability At of an 7 Case (ii) also results in incorrect LOR attribution, i.e. detection of scattered events, as we discuss in Sec. (1.6.2) 8 This is the case for energy levels in which the photoelectric effect and/or Compton scattering are dominant [16], which is the case in nuclear medicine imaging. 1. Introduction to PET Imaging 22 L O R is the product of the the probabilities that each of the two photons do not interact as they propagate along their paths a and b: At = e~ L ^ ( x ) d x x e" J > M d x = e - Ji ^ W d x . (1.4) where I refers to the union of a and b, or in other words, the entire LOR. In P E T imaging, typically more than 60% of all emitted photons interact with tissue [17]. However, a very significant conclusion may be drawn from Eq. (1.4): the attenuation along an L O R in P E T is independent of the po-sition along the L O R at which the annihilation event is generated. This is unlike S P E C T in which attenuation exhibits depth-dependence (i.e. atten-uation of gamma rays is dependent on the distance-to-the-detector of the location at which they are generated). This key observation renders correc-tions for attenuation in P E T simpler and more quantitatively correct than in SPECT. The Determination of the Attenuation Factors The AFs need to be known in order to appropriately model the detection process and to produce quantitatively accurate reconstructed images. There are three main methods of obtaining the attenuation factors (when one makes use of the P E T modality only 9). (i) Calculated attenuation factors: In cases when the object being scanned consists of uniformly attenuating material, and the location of the object is known, one can simply calculate the AFs using Eq. (1.4) applied to a body contour of constant \x values (recall: A F / = 1 — A\). This technique is more 9 In cases when PET scanning is combined with the CT (computed tomography) modal-ity, the exact attenuation map /z(x) (i.e. attenuation values as a function of location) can be obtained using CT, and subsequently, one can calculate the AFs from the known atten-uation map. This technique is complicated by the fact that p values have a strong energy dependence, and the various components in the attenuation maps obtained using x-rays in C T need to be rescaled to correspond to p values for positron-annihilation gamma rays. 1. Introduction to PET Imaging 23 applicable in imaging of the brain, but fails in whole body imaging, due to notable mixing of different attenuating material (e.g. bones, lungs). (ii) Use of blank and transmission scans: Since the AFs are independent of the position of the annihilation along the LOR, one can consider a trans-mission source located outside of the object and performing measurements without and with the object in the FOV. The former is referred to as a blank scan and the latter, a transmission scan. Denoting P>i and 2] as the measured blank and transmission counts along an L O R I, the ratio between them is therefore a measure of the probability that generated annihilation photons are not attenuated along the L O R I. We therefore have: In other words, without knowledge or reconstruction of attenuation maps, one is able to measure the AFs by performing a transmission source along with a reference scan, namely the blank, combination of which allows a measure of the probability of attenuation. The major problem with this approach is that presence of noise in the AFs can further amplify the present noise in the emission scans. This can be reduced by acquiring longer attenuation scans (i.e. more statistics), but this adds to the total scan time. One approach to reduce the noise is described next. (iii) Transmission image segmentation: This is a technique, combining the first two approaches, intended to reduce the noise in the measured AFs. Combining Eqs. (1.4) and (1.5), we arrive at which is the projection of the attenuation map (defined earlier) onto the L O R I. One can then reconstruct ,u(x) values from their projections using well-known analytic reconstruction techniques, e.g. filtered back-projection (described in detail in the next chapter). Alternatively, it is possible to (1.5) (1.6) 1. Introduction to PET Imaging 24 * = Annihilation event *> m CiuniM raj * Aligned LOR (a) True Coincidence (b) Scattered Event (c) Random Coincidence Fig. 1.11: Plots indicating annihilated photons from an event detected with (a) true coincidence, (b) scattering, and (c) random coincidence. The last two clearly result in mis-positioning the LOR along which the event is detected. reconstruct /^(x) values using statistical techniques, to further suppress noise effects, as investigated in Refs. [18-20]. Once the typically-noisy attenuation map is reconstructed, it can be used to determine the boundaries of anatomical structures. Subsequently, the / i -map can be segmented by assigning to regions within the boundaries the corresponding values of [i (which are known a priori for anatomical struc-tures). The segmented /x-map may subsequently be used to calculate the AFs as given by Eq. (1.4). In cases when very long transmission scans are not practical, this approach allows an improved estimate of the AFs. 1.6.2 Detection of Scattered Events Those annihilations for which one or both photons are scattered, but both are still detected, are termed scattered events. Scattering can result in assign-ing the detected photons to an incorrect LOR, as depicted in Fig. (1.11b). 1. Introduction to PET Imaging 25 Corrections for attenuated events as compared to those for scattered events differ in that for a given LOR, the former corrects the L O R for counts which it should have received had no Compton interaction taken place, whereas the latter seeks to correct the L O R for counts incorrectly attributed to it due to scattering. In 2D P E T , the scatter fractions (ratio of scattered events to all measured counts) are in the 10-20% range (depending on scanner geometry, object size and the energy threshold used). This increases by about a factor of three to 30-55% in 3D P E T imaging [17,21,22]. This renders use of scat-ter correction techniques even more critical in 3D imaging, and has been an important challenge in the transition from 2D to 3D P E T imaging. Since photons lose a fraction of their energy when they undergo a Comp-ton interaction, a significant portion of the scattered events can be rejected by narrowing the range of photon energies accepted by the coincidence detec-tion system (termed energy discrimination). Nevertheless, given the approx-imately 10% energy resolution on most current scanners, the threshold can-not be made too narrow so as not to reject true coincidences, and additional scatter correction techniques need to be employed for accurate quantitative reconstruction. Various scatter correction techniques have been proposed: (i) use of convolution-subtraction with exponential functions [23,24], (ii) comparison of 2D and 3D distributions [25], (iii) dual energy-window acquisition meth-ods [26], (iv) direct calculation of the scatter distribution using the Klein-Nishina formula [27,28]. The latter type of calculation is attractive since it treats the scatter using basic physical principles, and is gradually becoming a practical alternative due to the increasing computing power. 1. Introduction to PET Imaging 26 1.6.3 Detection of Random (or Accidental) Coincidences A detected annihilation photon is termed a single if the other photon is not detected. This can take place in three ways: (a) The orientation of the annihilation is such that the other photon does not pass through the detectors. (b) The other photon is scattered out of the FOV. (c) The other photon passes through the detectors but is not detected1 0. These are depicted in Fig. (1.12). Now, as explained in Sec. (1.2.2), annihilation coincidence detection works by recording two events arriving within a certain coincidence time window as dually emitted gamma rays from a single positron annihilation. However, it is possible for two singles arising from separate annihilations to be detected within the same coincidence timing window. A n example of this has been shown in Fig. (1.11c). The rate of random coincidences along a certain L O R connecting detec-tors i and j is given by Rr = 2rSiSj (1.7) 1 0 There is actually a fourth negligible possibility, and that is when the other photon undergoes photoelectric absorption. All of the three aforementioned mechanisms (but not this latter one) will be eliminated in a theoretical 4ir scanner (completely covering the FOV) with 100%-sensitive detectors. 1. Introduction to PET Imaging 27 where r is the coincidence timing window, and Si and Sj refer to the singles rates at the two detectors11. Examination of this relation shows that: (i) reducing the coincidence timing window reduces the count rate of random coincidences, and (ii) since singles rates are proportional to the activity in the scanned object, random rates vary as square of the activity. Subsequently, the fraction of detected random events (unlike the attenuation and scatter counter-parts) depends on the injected activity, and can therefore be an important factor in determination of the maximum injected activity. Contribution of random coincidences to the measurement process has in the past been estimated and taken into account in a number of different ap-proaches depending on the capabilities of the P E T scanners being used. Such methods include direct use of the emission data for random correction, an instance of which is the image-based convolution-subtraction technique [24], which estimates the randoms in the object by fitting a function to the ran-dom tail outside the object. Work is also currently in progress to incorporate randoms into the system matrix of the E M algorithm by estimation of the spatial distribution of randoms contribution [29]. For P E T systems in which singles rates can be measured during the co-incidence measurement process, the singles count rate at the detectors can be used to estimate the randoms distribution [74]. For scanners without the capability to measure singles rates along with the coincidences, work is currently in progress to analytically calculate the singles rates from the re-constructed emission and transmission data [30]. This latter approach may require iterations since the initial emission image, using which singles rates are calculated, is not random-corrected. Alternatively, a technique to correct for random coincidences is to acquire 1 1 The number 2 in the equation can be attributed to the observation that the annihi-lation photon at detector i may be detected up to r seconds before or after detection of the other photon at detector j in order for the coincidence detection system to accept and label the random pair as a coincidence event. 1. Introduction to PET Imaging 28 events arriving within a delayed coincidence window, delayed such that the probability of a true coincidence is zero. This modality is available in most modern P E T scanners. The spatial distribution of the events within the delayed window is the same as those acquired in the true coincidence window. Thus, subtraction of the events acquired within the delayed window from the events acquired in the coincidence window (prompts) effectively corrects for the bias introduced by random coincidences. However, this subtraction technique can be a source of potential problems, as discussed in Sec. (2.4.3). 1.6.4 Variations in Detector Pair Sensitivity LORs in P E T scanners exhibit varying degrees of sensitivity due to a number of reasons: with changing incident angles to a crystal, the cross section (or effective surface area) of the crystal as well as its effective depth-of-interaction change, thus altering the efficiency. Furthermore, crystal imperfections, light guide variations, differences in P M T gains and variations in the electronics used to detect P M T signals also result in variations in efficiency, not only from block detector to block detector, but also across the elements of a block detector. In order to account for these variations, current scanners typically acquire a normalization scan by illuminating all LORs with a uniform source. The normalization coefficients (NCs) are then proportional to the reciprocal of the number of counts obtained in each LOR. This process is known as direct normalization, from which the detector efficiencies can be calculated. However, direct normalization can be confounded by three principal fac-tors: i) Very long acquisition times are required to obtain sufficient counts in high-sensitivity scanner-modes (i.e. non-compression of data) since low-1. Introduction to PET Imaging 29 activity sources must be used to avoid problems with system dead-time1 2. ii) A further complication is that for any given L O R along which the events are attributed, the sensitivity is different depending on whether the events are trues or scattered. This is the case because: (1) for a given LOR, the scattered photons may have arrived from a wide range of angles. (2a) scattered events have different photon energies compared to un-scattered events (2b) detector efficiencies are energy-dependent since interactions and energy depositions inside the crystals depend on the initial photon energy. Lack of consideration of this issue can lead to both high- and low-frequency artifacts in the images [31]. iii) When using the block-detector technology, the degree of event mispo-sitioning due to pulse pile-up (i.e. interference amongst pulses generated from separate gamma rays detected within the same block and within the same coincidence timing window) changes with count-rate, and this variation is not accounted for by direct normalization which is performed at a particular count-rate only [32]. To this end, component-based normalization approaches have been pro-posed that can take the above factors into consideration [33]. These tech-niques are based on separately measuring and/or calculating the various normalization components (e.g. geometric and detector efficiency factors). Self-normalization of emission data, without acquiring normalization data, has also been proposed [34]: but it is an approximate technique (it hinges on a limiting assumption) and does not address the second aforementioned complication. 1 2 In fact, this is one important reason why direct normalization is currently not being utilized for span zero (i.e. no compression at all) in the HRRT scanner. 1. Introduction to P E T Imaging 30 1.6.5 Detection Deadtime The time required to process an event limits the counting rate of a P E T scanner. With higher count-rates, a larger portion of the incoming counts are lost. The degree to which the deadtime effect is uniform within the tomograph is determined by the tomograph design and the activity distribu-tion. This can be another limiting factor in determination of the maximum injected activity. Depending on whether the bottleneck of the processing is in the scintilla-tor decay time (Sec. 1.2.3), crystal identification (Sec. 1.2.4) and/or energy discrimination (Sec. 1.6.2), or the overall coincidence detection (Sec. 1.2.2), the deadtime effect will be at the level of the crystal, block or the entire sys-tem. In a system where the coincidence detection is the limiting factor, one can use the singles rates (which are not as much affected by the deadtime) to measure the global deadtime correction factors. However, other non-global deadtime correction schemes would be needed for other tomographs. 1.6.6 Decay of Radioactivity Positron-emitting isotopes decompose and release radioactivity, enabling emis-sion of gamma rays upon annihilation of the decayed positrons. As radioac-tive isotopes decay with time, the number of decays per second also decrease. Subsequently, scanning of the same object at a later time will result in fewer annihilation photons, and therefore, fewer reconstructed counts in the im-age. In order to enable the dynamic P E T modality to be only sensitive to the biological process, one must correct the reconstructed images for decay. Relative to the other corrections required in P E T imaging, this one is the most trivial. One is simply required to scale the reconstructed image counts, depending on the radionuclide used in the scan. One complication can occur: scintillation detectors can themselves exhibit very low radioactive 1. Introduction to PET Imaging 31 background counts, which are constant in time and unrelated to the injected activity 1 3. In cases when the activity is close to background counts, scaling the entire image will amplify the background effect. One method to take this issue into account is to subtract an estimate of the background counts from the data prior to application of reconstruction and decay correction. 1.6.7 Detector Blurring As described in Sec. (1.2.3), the incoming gamma rays excite electrons in the scintillating crystals through Compton scattering and/or photoelectric absorption. If these processes all occur in the crystal on which the radiation was incident, the event will be properly positioned. However, the photon detection process can exhibit two complications: (i) It is quite likely for an incident photon to be scattered off to adjacent crystals, with this especially being the case in the detector-block technology. This is referred to as inter-crystal scattering. This explains the fact that the edge crystals typically receive less counts relative to the center crystals: this is because a photon scattering in a central element, may scatter to adjacent crystals but will eventually be detected (though it will be mis-positioned), however, it is quite likely for a photon entering an edge crystal to be scattered out of the entire block and not being detected at all. (ii) It is also possible for gamma rays entering a crystal at an angle to simply pass through the crystal undetected and to be only detected in an adjacent crystal. As radiation occurs in voxels increasingly distant from the center of the FOV, it is more likely for the radiation to reach crystal fronts at higher angles of incidence, and to subsequently penetrate and be recorded in nearby crystals, degrading image resolution for such voxels. This effect 1 3 For instance, LSO has an intrinsic radioactivity of about 280 Bq/mL with single-photon emissions in the range of 88 keV to 400 keV. On the HRRT scanner, one observes ~2K counts/sec (in coincidence) of LSO background using a 400-650 keV energy window. 1. Introduction to PET Imaging 32 is commonly referred to as the parallax effect). Measurement of depth-of-interaction (DOI) within the crystals is known to minimize this problem, but its implementation has not achieved complete spatial invariance for resolution (see for e.g. [35]). The above considerations are not commonly taken into account in recon-struction tasks due to computation difficulties. Furthermore, they cannot be accurately accounted for in analytic reconstruction algorithms. However, it is possible to incorporate these effects in statistical reconstruction tasks, as described in Sec. (2.4.2), though they can complicate the computation. 1.6.8 Positron Range and Photon Non-collinearity These two effects were elaborated in sections (1.1.2) and (1.2.2). They are not often discussed as physical phenomenon that can be corrected for, rather they are often seen as limitations of P E T imaging. However, with the arrival of statistical reconstruction algorithms (Sec. 2.4), it is now possible to model these effects. This is because, while it is not possible to determine, for a particular detected event, the positron range and photon non-collinearity, it is possible to calculate and incorporate the probability distributions for these effects into the system matrix, as elaborated in Sec. (2.4.2). 1.6.9 Patient Motion Recent developments in 3D positron emission tomography (PET) systems have enabled the spatial resolution to reach the 2-3mm F W H M range. With such improvements in spatial resolution, small patient movements during P E T imaging have become a significant source of resolution degradation. Chapter 5 provides an overview of existing motion compensation techniques. Accurate and practical motion correction algorithms, proposed in the course of this work, have also been described and tested experimentally. 1. Introduction to PET Imaging 33 Fig. 1.13: An open display of the high resolution research tomograph (HRRT). 1.7 The High Resolution Research Tomograph The Vancouver high resolution research tomograph (HRRT), as shown in Fig. (1.13), is a state-of-the-art high sensitivity, high resolution scanner. It is a 3D-only dedicated brain tomograph employing the scintillators L S O / L Y S O and using depth-of-interaction (DOI) information (see below). 1.7.1 Challenges The HRRT is built of eight panel detector heads arranged in an octagon. Each head is separated by 46.9 cm from the opposing head and by 1.7 cm from the two neighboring heads. This presence of gaps in-between the detectors is clearly manifested in the sinogram data, as is shown in Fig. (1.14), and poses new challenges to image reconstruction from the H R R T data. To see this, one notes that in analytic reconstruction algorithms (see Sec. 2.2), one requires access to the complete data (i.e. data measured along all possible 1. Introduction to PET Imaging 34 • •A ..." A .-" -r ".""">,,*•''<"* ,.• \,-""" '"••••, ••".'•3c •< * **»'. • "• . ' "-: it Fig. 1.14: Sinogram data for a uniform cylinder (direct plane #110 is shown): the gaps comprise over 10% of the sinogram space. This sinogram can be thought of as a superposition of corresponding sinusoidal curves (Fig. 1.10) for the points which the cylinder consists of. The observed non-uniformities are due to noticeable variations in detector pair sensi-tivities (Sec. 1.6.4). LORs passing through the FOV). This means that the gaps manifested in the sinograms need to be filled using approximate techniques14. On the other hand, gap-filling is not required in statistical reconstruction algorithms (see Sec. 2.4), which has been one key motivation for employing such methods in this work. Another challenging property of the H R R T is the very large (namely 4.49xl0 9 ) number of LORs which it measures, exceeding most modern P E T scanners by 2-3 orders of magnitude. For data stored and/or reconstructed in histogram-mode, one is expected to commonly measure (especially in dy-namic studies wherein a single study is divided into multiple frames), the total counts per frame to be significantly less than the total number of sino-1 4 Such gap-filling techniques used in analytic reconstruction include (i) direct interpola-tion of the sinogram data, (ii) forward-projection of preliminary image estimates , or (iii) a constrained Fourier space method [36]. 1. Introduction to PET Imaging 35 gram bins. The L O R data are acquired in list-mode (see Sec. 3.2), i.e. they are stored as they are detected one-by-one in the form of a list. The data may subsequently be histogrammed into appropriate sinogram frames. In this work, we have explored taking the list-mode approach one step fur-ther, namely skipping the histogramming step, and reconstructing the data directly from the list-mode data. This is elaborated in chapter 3. The DOI information in the HRRT is obtained by pulse shape discrimi-nation (PSD) to distinguish between events in the front layer from those in the back layer [37,38]. This DOI modality was introduced for the H R R T to reduce the parallax effect: space-variant degradation in resolution of the scanner caused by increasing inter-crystal penetration of gamma rays with increasing angles of incidence. Nevertheless, the DOI information (due to its discrete nature) is not able to completely eliminate the parallax effect. In Sec. (B), we have investigated modeling of this effect into feasible statistical reconstruction of the H R R T data. 1.7.2 More details for the Scanner A detector head consists of 9(transaxial)xl3(axial) crystal blocks, which are viewed by 10x14 P M tubes. The H R R T therefore contains 8x9=72 blocks per ring and 8x9x13=936 total blocks. Each block consists of two 10-mm-thick layers (LSO and LYSO). Each double-layer block is cut into an 8x8 matrix of crystals: each crystal is (2.438 mm) 2 . In total therefore, the H R R T contains 936x8x8x2~120K total crystal elements. Each head is in coincidence with five opposing heads, which results in 4.49B possible LORs. These properties are summarized in table (1.4), along with comparisons with another scanner currently in use by our group, namely the Concorde microPET R4 scanner for rodents [39]. The events are initially stored as 64-bit list-mode data, containing the in-1. Introduction to PET Imaging 36 Specifications microPET R4 HRRT Detector Diameter 14.8 cm 46.9 cm Radial Field-of-View 10.0 cm 31.2 cm Axial Field-of-View 7.5 cm 25.1 cm No. of Blocks per Ring 24 72 Total No. of Blocks 96 936 No. of Crystal Rings 32 104 Depth-of-Interaction Encoding NO Y E S Total No. of Crystal Elements 6,144 . 119,808(1!) No. of Lines-of-Response 8.26M 4.49B(H) Absolute System Sensitivity - 2 % - 6 % Resolution at Center of F O V 1.8 mm 2.4 mm Tab. 1.4: Specifications for Two PET Scanners formation for the coordinates of the crystal-pairs along which the annihilation photons are detected. A 72 Gbyte RAID disk system allows data acquisi-tion with a maximum storage rate of 40Mbytes/sec for list-mode data. The events can subsequently be converted into 32-bit data, wherein the coordi-nates of the lines joining the crystal pairs (and not the crystals themselves) are stored. Alternatively, the events can be histogrammed into sinogram data. Presently one uses 288 angular projections and 256 radial elements on each sinogram plane. Flexible axial compression modes, characterized by span (axial mashing of neighboring LORs) and ring difference (RD) [17] are available to further reduce the data. With no data compression, one obtains 10816 sinogram planes, corresponding to ~1.5GB of data in sinogram-mode. However routinely, the data are stored in compressed format (using a so-called span 3 and a maximum R D of 67), resulting in 6367 sinograms with a total size of - 0 . 9 G B . Transmission measurements are done with a Cs-137 point source, which is 1. Introduction to PET Imaging 37 driven by two servo motors and can sample the entire F O V in 2 min. Direct normalization is currently being commonly employed for the HRRT, with the normalization correction factors obtained from a long scan using a rotating rod source. The coincidence timing window can be set as low as 2 ns and can be incremented in steps of 2 ns. Currently, a coincidence time window of 6 ns is used. 1.8 The Aim of This Work and Contributions The work presented in this disseration has resulted in a number of publica-tions [40,41] and presentations [42-46]. As already mentioned in Sees. (1.5) and (1.7.1), our central aim has been to propose, implement and verify prac-tical image reconstruction algorithms compatible with quantitative imaging with the HRRT. To address some of the challenging properties of the H R R T (namely, the very large number of LORs as well as the presence of gaps be-tween the detector heads), we focused our attention to statistical list-mode reconstruction algorithms. Chapter 2 provides a general summary of analytic and statistical recon-struction algorithms, while chapter 3 presents an overview of motivations for list-mode image reconstruction. Various accelerated list-mode reconstruction algorithms were proposed, with particular emphasis on a novel convergent reconstruction method. A practical random correction technique was also proposed for use in list-mode reconstruction from the HRRT data. Chapters 4 and 5 represent the experimental core of this thesis work. The former elaborates upon the details of implementation and verification of statistical list-mode reconstruction on the HRRT. Furthermore, we extended the list-mode approach to dynamic (4D) image reconstruction, and demon-strated its quantitative accuracy and robustness compared to a number of conventional techniques. In addition, we proposed a modification of conven-1. Introduction to PET Imaging 38 tional data non-negativity constraints to remove the resulting overestimation biases, which were elaborately studies and quantified in this work. Chapter 5 is devoted to the proposal and study of novel motion compen-sation techniques, with validation on the HRRT. We developed an accurate scheme, applicable to both the histogram-mode and list-mode image recon-struction methods, which incorporated patient motion into the reconstruction task itself (unlike the existing purely event-driven approach). Subsequently, clear improvements in image quality (particularly in terms of axial unifor-mity) were demonstrated. The practicality of the technique was derived from the observation that an otherwise time-consuming motion averaging step was only required to be performed in the image space (and not in the projection space). A more detailed summary of each chapter, as well as concluding remarks and opinions have been presented in chapter 6. 2. 3D Image Reconstruction 39 2. 3D Image Reconstruction This chapter is intended to present an overview of 3D image reconstruction techniques applicable to P E T imaging. Both analytic and statistical recon-struction algorithms are discussed, with particular emphasis and elaboration on the latter, since statistical list-mode reconstruction is the central theme of this dissertation (for more discussion see chapter 3). 2.1 Introduction As discussed in the previous chapter, an event is recorded in the P E T mea-surement process if two photons are measured in coincidence. Denoting the coordinates for the measured photons using (xi,y\,Zi) and (£2 ,2 /2,^2), a line-of-response (LOR) may be traced in-between the two coordinates, as shown in Fig. (2.1). The image inversion or reconstruction task in positron emission tomogra-phy is aimed at finding the number of annihilation events / which occurred at a position x in the image space given the number of events in the measured L O R s 1 . Reconstruction techniques can be classified into two main categories: analytic and statistical. 1 With knowledge of scan duration, one may then calculate the activity at the given position. 2. 3D Image Reconstruction 40 Fig. 2.1: A PET event consists of detecting two coincident gamma rays. 2.2 Analytic Image Reconstruction In analytic reconstruction, one aims to reconstruct / from its projection line integral p(u,s) = / f(s + lu)dl (2.1) Joo where the pair (u, s) parametrizes the straight line along the unit vector u and passing through s, a point in 3D space. To avoid redundancy, the vector s is restricted to a 2D plane u x = {se R3\ s.u = 0} (2.2) since otherwise there exist (infinitely) many points, which would define the same parallel projection along a projection direction u. Therefore, for a given projection direction u, the vector s is restricted to a 2D plane perpendicular to u. This is depicted in Fig. (2.2a) for a spherical scanner. The integral transform mapping the function / (x) onto its line integrals is called the three-dimensional X-ray transform. 2. 3D Image Reconstruction 41 (a) Spherical scanner (b) Cylindrical scanner Fig. 2.2: (a) A spherical PET scanner, with an LOR parametrized by (u, s) passing through it. (b) A cylindrical PET scanner, with two LORs (u ,si) and (u, S2) passing through it, one measured and one unmeasured. Therefore the measured data are truncated since for certain directions, e.g. along u shown, the line integrals are not measured for all values of s. 2. 3D Image Reconstruction 42 2.2.1 Reconstruction from Non-Truncated, 2D Parallel Projections Introducing S2 as the unit sphere defining all possible directions for u, the projection function p will in general be defined on the tangent bundle to S2: T(S2) = {(u,s)| u € S2,s e VL-1-} (2.3) In practice, however, 2D parallel projections are not completely measured in all directions. Rather, projections are measured along only a subset Q C S2 of the unit sphere (called the aperture of the scanner). Furthermore, in 3D analytic P E T reconstruction, the data are often truncated. In other words, in most applications, for a given parallel projection u, contributions from all coordinates s in the image are not necessarily measured, and only a sub-region of the projection plane u - 1 (Eq. 2.2) is measured. A n example of this is encountered in the cylindrical geometry as shown in Fig. (2.2b). As it turns out [47], the inversion formula for such truncated geometries are all based on results obtained for the simpler case of a complete/non-truncated geometry r(n) = { ( u ) s ) | u e n ) s e u i } (2.4) wherein for any projection direction u e f i , measurements are made for all s in the projection plane t r 1 . In what follows, we shall first present an overview of analytic inversion techniques for the non-truncated case. In Sec. (2.2.8), methods of handling truncated projection data will be discussed. 2.2.2 Central Section Theorem The central section theorem is a fundamental theorem used in analytical re-construction in nuclear medicine imaging. We first define a rotated2 Carte-sian system (x',y',z') such that for a given projection direction u, the axis 2 This can, for instance, be achieved by rotating the coordinate frame through the az-imuthal angle <f>, followed by a rotation through the copolar angle 6. It can be shown [48] 2. 3D Image Reconstruction 43 x' lies along u, and subsequently, the coordinate axes y' and z' span the 2D plane u±. The 2D Fourier transform of Eq. (2.1) with respect to the variable s G u 1 is then given by P(u,v)=/7 f°° f(s + lu)exp(~2niv.s)dlds (2.6) JJu-^- J—oo where v = ( v » * v ) ( 2- 7) On the other hand, the 3D Fourier transform of the unknown distribution / (x) is given by F(V) = [f r f{s + lu)exp(-2mv.s)exp{-2mvxd)dlds (2.8) JJu1- J—oo where we have decomposed the 3D vector V into vx> and v. When V lies along the central plane (passing through the origin, and equivalent to the projection plane u - 1 defined earlier) which occurs when vx>=0 and V=v, Eqs. (2.6) and (2.8) become equivalent; in other words: F(v) = P(u,v) (2.9) This is the central slice theorem; namely: for a given projection direction ti, the central plane of the 3D Fourier transform F(V) of the 3D true activity / (x) is equal to the 2D Fourier transform of the 2D projection data along the same direction. 2.2.3 Orlov's Sufficiency Condition Orlov's sufficiency condition [49] states that any great circle on the unit sphere S2 must have a non-empty intersection with the set Q, of directions that this change in the coordinate frame is obtained mathematically via the transforma-tion: x' \ I cos</>cos0 sin^cos^ sin# \ I x v\ — \ -sin<£ cos<£ 0 ) [ y I (2-5) z' I \ —cos^ sinf? — sin</>sin0 cosf? J \ z 2. 3D Image Reconstruction 44 Fig. 2.3: Each point on the sphere S2 represents one 2D parallel projection. The set of measured projections Cl is in-between the two parallels shown. The set Q is intersected by the great circle normal to any frequency v. u along which complete measurements are made. To prove that Orlov's condition allows for reconstruction of the unknown activity / (x) from parallel projections, we shall work in the Fourier domain while sampling the space with v, and let us assume that Orlov's condition is satisfied. A n example of this is shown for the set 0 subtended between two parallels, as shown in Fig. (2.3). Then, for any vector v, it is possible to find at least one u G f2 which is intersected by the great circle orthogonal to the vector v, i.e., there is at least one projection direction u. along which measurements p(u, s) (see Eq. 2.1) are made, and therefore from which the 2D Fourier component P(u, v) can be recovered. Subsequently, one will arrive at the value of F(v) using the central section theorem (2.9), and will completely recover .F(v) by repeating this procedure for all frequencies v. The image activity distribution can finally be obtained by an inverse 3D Fourier transform of F(v) . On the other hand, if Orlov's condition is not satisfied, there exists at least one frequency v such that the great circle orthogonal to it does not intersect fl, 2. 3D Image Reconstruction 45 hence F ( v ) cannot be recovered for that particular frequency. 2.2.4 Image Reconstruction from Redundant Data In Fig. (2.3), for a given frequency v , there exists more than one projection direction u G Q (such that u .v = 0) which may subsequently be used to determine F(v). This demonstrated the non-uniqueness inherent in inverting the 3D X-ray transforms, whereas in the 2D case, the inversion is not over-determined3. The over-determination of the inverse 3D X-ray transform can also be interestingly observed from a simple dimensional analysis [50]: the unknown function /(x) depends on three scalar variables, whereas the data p(u, s) depend on the four scalar variables required to parametrize a straight line in R3. On the contrary, in the 2D case, a point x G R2 and a straight line are both parametrized with two scalar variables! The simplest approach to the issue of redundancy is to select only one of the estimates. In the absence of noise (i.e. consistent data), all P(u , v) values calculated along any projection directions u perpendicular to v are equal to F(v). Nevertheless, in the practical case in which noise is present, collective use of all the estimates can be used to enhance the signal-to-noise ratio, which has been the motivation in switching from 2D P E T to 3D P E T (Sec. 1.3). In what follows, we describe three image reconstruction techniques that make use of the entire 3D data set: (i) direct Fourier reconstruction, (ii) filtered-backprojection, and (iii) backprojection followed by filtering. 2.2.5 Direct Fourier Reconstruction Assuming Orlov's condition to be satisfied, for any given frequency v , one may obtain an estimate for the value of F(v) by simply averaging values of 3 This can be more clearly seen by noting that the set f2 in the 2D case is simply an equilateral circle on the unit sphere. 2. 3D Image Reconstruction 46 P(u, v) obtained along all projection directions u such that u.v = 0; in other words: ,-,/ x f0 P(u, vWu.vWu . . F v = f I,' \ J 2 - 1 0 /n<J(u.v)du The Dirac delta function in the integrand selects the projection directions which are perpendicular to the sampled frequency v: the 2D integral above is then restricted to a one-dimensional integral over the intersection of such projection directions with the set Q, as shown in Fig. (2.3). Using this technique, and sampling the frequency domain, one may subsequently take the 3D inverse Fourier transform of F(y) to recover the image. A more generalized approach to this problem is by weighting (not neces-sarily equally, unlike above) the P(u, v) values in order to obtain an estimate of ^(v). One therefore introduces a weighting function H(u, v) as such F(y)= [ H(u,v)P(u,v)5(u.v)du (2.11) To better understand the form of the weighting function H(u, v), we note that assuming an object distribution /(x) = <5(x) (i.e. a central point source), then the 3D Fourier transform satisfies .F(v) = 1; and similarly, by the central section theorem P(u, v) = 1. Substituting these results into Eq. (2.11), we arrive at the normalization condition: f iy(u,vWu.v)du = l (2.12) Weighting functions H(u, v) satisfying the above condition may be easily constructed: to see this we note that for an arbitrary function H(u, v) defined on T(Q), the function will satisfy the normalization condition (2.12), provided G is such that the 2. 3D Image Reconstruction 47 denominator is defined and does not vanish4. This observation explicitly demonstrates the redundancy inherent in the 3D X-ray transform, as different filters may be used for exact inversion of the transform5. 2.2.6 Filtered Back-Projection (FBP) The 3D inverse Fourier transform of Equation (2.11), which recovers the im-age activity distribution, can be shown [52] to be mathematically equivalent to a technique commonly referred to as filtered-backprojection (FBP); it pro-ceeds in two steps: Step 1: The filtering step pF(u,s)=[ h(u,s-s')p(u,s')d&' (2.16) Ju1-is a convolution between the 2D parallel projection data and the 2D in-verse Fourier transform of the weighting function H(u, v), which can alter-natively be performed using a frequency domain multiplication of P(u, v) and H(u, v): PF(u, v) = H(u, v)P(u, v) (2.17) 4 Another method by which a filter satisfying the normalization condition can be recon-structed is via additive normalization of an arbitrary function G where w is an arbitrary integrable function of u such that the denominator does not vanish. It can be easily shown that this construction also satisfies the normalization condition (2.12). As elaborated in Appendix A, this latter formulation is not a simple mathematical curiosity. Rather, it can be used to design filters suitable for the task of inverting truncated projections. 5 The simplest and earliest weighting function, referred to as Colsher's filter [51], is one which yields Eq. (2.10) (i.e. at any frequency v, all projections P(u, v) are weighted equally). This implies, in relation to Eq. (2.11), that Colsher's filter has the form g(u,v)= 1 (2.15) Jn S(u.v)du which can also be obtained by simply setting G(u,v) = 1 in Eq. (2.13). 2. 3D Image Reconstruction 48 x A Fig. 2.4: The backprojection of a ID parallel projection onto a 2D image. . since, by the convolution theorem, a multiplication in Fourier space is equiv-alent to a convolution in real space. In fact, due to presence of fast Fourier transforms (e.g. see [53]), the filtering step is commonly performed via mul-tiplication in the Fourier space, followed by an inverse Fourier transform. Step 2: The filtered projections pF(u, s) are then backprojected to obtain the reconstructed image /(x) This is shown in Fig. (2.4). For a given position x in the image domain, the backprojection step is an integral over all directions u which contribute to that position in the image. Using the aforementioned steps, /(x) may thus be reconstructed for all x in the image-domain. (2.18) 2. 3D Image Reconstruction, 49 2.2.7 Backprojection followed by Filtering (BPF) The weighting function, or filter H(u, v) is referred to as factorisable if it can be written as the product H(u, v) = H'(v)w(u) (2.19) Substituting this into the normalization condition (2.12), one arrives at /nw(u)<J(u.v)du from which it follows that if(u, v) is uniquely determined by w(u): ff(u,v)= " f f i (2.21) Incidentally, this is what one would obtain by using G(u, v) = iy(u) (i.e. a function Cr(u, v) independent of the frequency v) in Eq. (2.13). In the case of using a factorisable filter, it can be shown [52] that the filtering and back-projection steps in the F B P approach may be commuted, so that the back-projection step is performed first. Thus we have: Step 1: Backprojection 6n(x) = / p(u,x- (x.u)u)w(u)du (2.22) Step 2: 3D filtering in image-space / ( x ) = / /i'(x-x')fcn(x')dx' (2.23) The aforementioned B P F approach is therefore more limited than the direct Fourier and F B P reconstruction techniques, as it is only applicable when using factorisable filters. Nevertheless, in list-mode reconstruction (see chapter 4) in which measured events are not histogrammed, and are rather recorded and processed individually as they are measured, one does not have 2. 3D Image Reconstruction 50 access to projection data p(u, s), and it is therefore not possible to filter the projections. Alternatively, one may backproject separately the individual list-mode events, after which filtering may be performed. In analytic list-mode image reconstruction, therefore, the B P F approach is the only practical one. Comparison of Techniques and Comment on Discretization Issues Amongst the three aforementioned methods of inverting the 3D X-ray trans-form, the first method (direct Fourier reconstruction), involves the use of 2D (fast) Fourier transforms of the projection data which upon integration using Eq. (2.11) can evaluate F(v), thus yielding the reconstructed image by a final 3D inverse (fast) Fourier transform. The two other techniques (FBP and B P F ) , however, involve the use of computationally expensive 3D back-projection operations. The first approach is therefore expected to be more efficient. Nevertheless, the task of sampling F(v), at appropriate frequencies prior to taking the 3D inverse Fourier transform, may likely involve inter-polation of the available data [47], and can therefore be rather complex and time-consuming. At. the end of this section, we make one additional important note. Imple-menting backprojection operations as employed in F B P and B P F reconstruc-tion techniques requires6 some kind of discretization in the image function /(x). Nevertheless, these techniques are derived from the continuous line integral model, which therefore does not take into consideration discretized objects. In Sees. (2.4.1) and (2.4.2), we shall discuss how, by contrast, sta-tistical reconstruction techniques intrinsically incorporate discrete images in their formulations, and thus, can potentially regress discretization artifacts. 6 An exception to this is the 'natural pixel reconstruction' approach [54] in which the continuity of the object is intrinsically embedded in the reconstruction task. 2. 3D Image Reconstruction 51 2.2.8 Reconstruction from Truncated, 2D Parallel Projections In practice, measured 3D P E T data are often truncated, as discussed at the beginning of Sec. (2.2.1). Thus, one needs to consider the reconstruction of a function from the measurement of its 3D X-ray transform within the set r t r u n c ( 0 ) = { (u , s ) | i i e f i , s eu4} (2.24) where for a given direction u, measurements are only made on a subset u x of the projection plane u x (Eq. 2.2). Three approaches to the problem of reconstructing truncated, 2D parallel projections have been considered in the literature, and are summarized in Appendix A . The first two, unlike the third, are based on the assumption that the set of measured line integrals 7" t r u n c (Q) contains a subset To from which /(x) can be reconstructed (i.e. it satisfies Orlov's condition). Of course, in the absence of noise, the reconstruction from To is exact; nevertheless, due to introduction of noise into the data in practice, the challenge remains to utilize the redundant data to improve the signal-to-noise ratio. 2.3 Rebinning Algorithms The aforementioned analytic 3D reconstruction techniques are exact, but are typically more than an order of magnitude slower than the slice-by-slice re-construction of data acquired in the 2D mode (see Sec. 1.3). In this regard, various approximate or exact techniques, commonly referred to as rebinning algorithms, have been proposed which aim to sort (rebin) the 3D-acquired data into a stack of ordinary 2D data sets. These rebinned data are ge-ometrically equivalent to data collected in the conventional 2D mode, and therefore can be reconstructed using 2D reconstruction algorithms. This is summarized in Fig. (2.5). In this section, we shall present an overview of various existing rebinning algorithms. 2. 3D Image Reconstruction 52 Fig. 2.5: The rebinning procedure is summarized (courtesy of Dr. Universiteit Brussel). M . Defrise, Vrije 2. 3D Image Reconstruction 53 In this part, we choose to parametrize an L O R using the coordinates (s, <f),z,5 — tan#) where s and 4> are the radial and azimuthal angular co-ordinates, z is the axial coordinate of the point midway between the two detectors, and 6 is the copolar angle between the L O R and the transaxial planes (see Sec. 1.4). These parameters were illustrated in Fig. (1.9). Rebin-ning algorithms seek a mapping from the measured line integrals p(s, qb, z, 8) along the LORs, onto a rebinned, 2D data set consisting of pTe^(s,(j),z), wherein S is omitted since 9, and therefore S, vanish along the transaxial planes. 2.3.1 The SSRB and MSRB Algorithms The first rebinning algorithm introduced in 1987 by Daube-Witherspoon and Muehllehner [55] is very straight-forward. It makes the following approxima-tion: Preb(s> <!>> z) ~ P(s> z>S) (2-25) Let us consider the common case of having a P E T scanner with data trun-cated in the axial direction: the 5 values are then restricted to within an interval [0,5m a x]. Noting that the left-hand side of the aforementioned ap-proximation is independent of 5, one can improve the signal-to-no ise ratio by averaging all available estimates of Pj-Q^is, 4>, z); i.e. pveh(s,(j),z) ~ - — / p(s,<f>,z,5)d5 (2.26) This is commonly referred to as the Single-Slice ReBinning (SSRB) approx-imation. The algorithm is very simple to implement: it can be performed one-by-one for the acquired sinogram bins, and simultaneous consideration of the entire projection set is not required. However, the algorithm is only exact for radiation originating from the center of the L O R along which they are detected (see Fig. 2.6a), and therefore can result in significant axial blur-2. 3D Image Reconstruction 54 (a) (b) Fig. 2.6: (a) Single-slice rebinning (SSRB): Each oblique (6 > 0) event is assigned to the direct (9 = 0) sinogram centrally located between the two detec-tors, (b) Multiple-slice rebinning (MSRB): Each oblique event is frac-tionally allocated to direct sinograms that it traverses. ring, especially for regions away from the transaxial center of the scanner, in which case use of axial deconvolution has been suggested [56]. Alternatively, the Multi-Slice ReBinning (MSRB) algorithm [57] rebins the oblique (8 ^ 0) LORs into multiple direct (6 = 0) LORs: this is done by allocating a fraction of the counts received along each L O R to each of the LORs (with same transaxial coordinates s and 4>) whose z-values are traversed by the LOR. This is depicted in Fig. (2.6b). This algorithm can be more accurate than the SSRB algorithm for points distant from the transaxial center of the FOV; nevertheless, it surfers from a systematic blurring of the rebinned data (which can to some extent be compensated by use of axial filtering [57], at the expense of increased noise). 2.3.2 The FOREX Algorithm A series of rebinning algorithms based on Fourier techniques were proposed in the late '90s, which we summarize below. Defining the 2D Fourier transform 2. 3D Image Reconstruction 55 of a given sinogram corresponding to a fixed pair (z, 8) as /OO f2lT / e-iswe-ik<t>p(s,((>,z,5)dsd(f) (2.27) -oo J0 and the 3D Fourier transform of the data as /oo e-lzWzP(w,k,z,S)dz (2.28) -oo Defrise et al. were able in 1996 [58] to derive an exact rebinning relation 3> r e b(u;, k, wz) = elka7(wx, k, wz, 5) (2.29) where a = sin _ 1 o; (2.30) X = Vl - a2 (2.31) and a = ^ (2.32) w The reader is referred to Ref. [59] for a more compact and elegant deriva-tion of Eq. (2.29). This formulation can be directly used to rebin the 3D data into 2D data sets, referred to as the FOurier REbinning eXact (FOREX) al-gorithm. Details of the algorithm and its implementation are discussed in Refs. [58,59]. The difficulty with this algorithm is that it deals with 3D Fourier transforms of the projection data, and therefore cannot be directly applied to axially truncated data; therefore, the truncated data must first be estimated, which can be time-consuming. On the other hand, by considering Taylor expansions of Eq. (2.29) around a — 0, one can arrive at a class of approximate rebinning algorithms of in-creasing accuracy. The algorithms incorporating a0 and a1 terms are dis-cussed below. 2. 3D Image Reconstruction 56 By setting a = 0 in Eq. (2.29), we arrive at the zeroth order approxima-tion y r e b (w , fc ,w z ) ~ 7(w, k,wz,5) (2.33) for which a 3D inverse Fourier transform can be performed explicitly to arrive at: Preb(s> 0> z) - P(s' 0> 2 > 5 ) ( 2 - 3 4 ) which incidentally reproduces the SSRB approximation (2.25). 2.3.3 The FORE Algorithm Keeping Taylor expansion terms which are linear in a (i.e. o ~ a and x — 1)> one obtains yreh(w, k, wz) ~ e i f c i ^ ?(iu, k, wz, 8) (2.35) It is possible to explicitly calculate the inverse ID Fourier transform of (2.35) with respect to wz: /OO £ eizw*eik^'P(w,k,wz,S)dwz -OO /OO e*»*(*+^)<p<Wik,wz,5)dwz (2.36) -oo from which one arrives at k 8 Pveh(w,k,z)~P(w,k,z+~,8) (2.37) Averaging the right-hand side over all measured 8 values, we arrive at the FOurier REbinning (FORE) algorithm: 1 r&max hS Prph(w,k,z)~-— / P(w,k,z+—1S)d5 (2.38) r e D 8max Jo w Two key properties of the F O R E technique, compared to the F O R E X algo-rithm, can be summarized as follows: 2. 3D Image Reconstruction 57 1) F O R E does not require estimation of axially truncated data: this is be-cause it does not require taking an axial Fourier transform with respect to z and therefore the data need no longer be known for all values of z. 2) Implementation involves a ID interpolation in z, but does not require an interpolation in the frequency variable w (unlike F O R E X ) . A nice geometric interpretation of the F O R E algorithm is possible [58]: making the substitution z + ^ —• z in Eq. (2.37), for a given sinogram in the frequency domain (w, k) corresponding to fixed (z,8), we have k8 P(w,k,z,8)~ Preh{w,k,z- —) (2.39) yj This can be interpreted by saying that the value of P at the frequency (w, k) receives contribution mainly from sources located at a distance k8/w be-low the axial center of the L O R (i.e. z). A n example of this is depicted in Fig. (2.7). This approximate property can be viewed as a "virtual time-of-flight principle" since it provides information about the location of a source along the LORs, which is only possible in time-of-flight tomography. How-ever, it must be noted that this technique requires the use of all measured LORs (s ,0) in the sinogram (z,8) in order to take 2D Fourier transforms (unlike true time-of-flight tomography which determines the location of a source on an individual L O R basis). Furthermore, it must still be kept in mind that F O R E is an approximate rebinning method, expected to result in image distortions when the maxi-mum angle between the LORs and the transaxial slices, the angular aperture, exceeds a certain threshold. This is because the technique is based on first-order Taylor expansion of the exact relation (2.29) about a = 8wz/w which increases with increasing acceptance angles. The angular aperture above which the F O R E approximation exhibits artifacts is observed to be around 25° [60] at signal-to-noise levels and spatial resolutions typical of clinical 2. 3D Image Reconstruction 58 Fig. 2.7: Fourier rebinning (FORE): In the frequency domain, data from an oblique sinogram are allocated to a direct sinogram axially located k5/w below the central z value, as described by Eq. (2.39). studies. 2.3.4 The FOREJ Algorithm In 1999, Defrise and Liu [61] proposed an original approach to rebinning making use of the fact that the 3D X-ray transform of a function is a solution to the second-order partial differential equation first studied by John in 1938 [62] (thus the ' J ' in 'FOREJ ' ) . For sinogram data p(s,(f>,z,5) obtained in multi-ring P E T scanners, John's equation can be written in the form d2p(s, (p, z, 5) d2p(s, cf), z, 5) = _sSd2p(s, (j), z, S) (2.40) dzdcf) ' 88 ds d2z Using this relation, the authors were able to arrive at the F O R E J exact rebinning relation 1 fs Pveh{w,k,z) = / umax J U + W lax k5 {P(w,k,z+ —,5) w 5)d*P(w,k,z + %,5)_}d5 dwdz2 (2.41) 2. 3D Image Reconstruction 59 The first term of the algorithm reproduces the F O R E approximation (2.38), while the addition of the second term is what makes the algorithm exact. It can thus be seen that the F O R E J algorithm combines the advan-tages of the F O R E X and F O R E algorithms in that it is exact, and yet, it can be applied directly to axially truncated data (since it does not require performance of a Fourier transform in the axial z direction). However, it must be kept in mind that the second term in the algorithm involves a sec-ond derivative with respect to z, which can strongly amplify noise artifacts in the reconstruction task. Analytic image reconstruction techniques have a number of attractive prop-erties: they (i) are able to perform the reconstruction task in few steps and efficiently, (ii) exhibit linearity, and (iii) do not require parameter specifica-tion, except for the degree of smoothing which can be optimized easily for a given desired statistical level. However, (i) such techniques suffer from the approximations implicit in the line integral model on which the reconstruction formulas are based (see Sec. 2.2). For instance, one may observe that the line integral formula-tion (2.1) makes the fundamental assumption that the annihilation events occur along the L O R defined by the measurement, thus ignoring positron range and photon non-collinearity. (ii) Furthermore, analytic techniques are based on the assumption that the data are consistent: i.e. sum of all projection line integrals in any direction is constant: Nevertheless, this assumption is not true in practice since presence of statisti-cal noise can easily render the data inconsistent (this also being the case when 2.4 Statistical Image Reconstruction for allu (2.42) 2. 3D Image Reconstruction 60 noise artifacts are introduced in data correction steps, such as compensating for attenuation). (iv) Moreover, analytic techniques are applicable on an assumption of shift-invariance, whereas imaging systems commonly exhibit shift-variant properties. (v) Finally, analytic reconstruction techniques require careful consider-ation of data truncation, such as axial trucation and presence of gaps (see footnote 14 of chapter 1 and Sec. 2.2.8). On the contrary, this is not an issue in statistical techniques, due to the intrinsic modeling of the relation between the image and projection space. In this regard, statistical image reconstruction (SIR) techniques aim to incorporate a more complex modeling of the measurement process, so as to improve image quality in the reconstruction tasks. In general, SIR methods require five components [63]: 1) A finite parametrization of the imaged object. 2) A system matrix that models and relates the elements in the unknown image to the expected counts along each LOR. 3) A statistical model for how the L O R measurements vary around their expectation. 4) A n objective function to be maximized in order to find the image estimate. 5) A n algorithm for maximizing the objective function. Below, we present an overview of these components as found in the literature. 2.4.1 Object Parametrization As noted in Sec. (2.2.7), backprojection operations employed in analytic re-construction techniques require, in order to be implemented, a discretization of the image space. However, these techniques are derived from the con-2. 3D Image Reconstruction 61 tinuous line integral model, which does not take into consideration presence of voxels and thus cannot attribute differing values to voxels that are not entirely within the tube connecting a particular detector-pair: this can ul-timately produce severe artifacts. However, statistical reconstruction tech-niques, as originally observed by Shepp and Vardi [64], take as their starting position discrete models such as Eqs. (2.45) and (2.46), and can therefore use the discrete system matrix formulation designed to model relation of voxels to LORs to regress such artifacts, as we explain next. In order to finitely parametrize the object, most typically, the image domain is divided into rectangular voxels. Use of smoother bases (e.g. blobs) have also been suggested [65,66]. Blobs are spherically symmetric functions which handle interpolation issues more effectively than voxels. Nevertheless, voxels have the practical advantage of containing minimal support (i.e. no overlap), resulting in maximal sparseness in the system models, which in turn, relaxes computation times. Throughout this work, the voxel approach has been chosen. For the continuous distribution function / (x) , one may introduce the parametrization /(x) = ^ A A ( x ) (2.43) 3=1 where the image has now been divided into J voxels, Xj is the mean activity in the j t h voxel, and bj represents the jth voxel. 2.4.2 The System Matrix First, let us consider a model for coincidence detection. Assuming a radio-tracer distribution given by function / (x) , the expectation of the measured events along a certain L O R can be written as: fk = j P i(x)/(x)dx (2.44) 2. 3D Image Reconstruction 62 where j5j(x) represents the probability distribution of a positron generated at a position x resulting in an annihilation detected along the L O R i. Note that this model does not assume a line integral formulation, and therefore can potentially avoid some of the pitfalls of analytic image reconstruction. Incorporating object parametrization (2.43) into Eq. (2.44), one arrives at j ™i = X > A ' ( 2 - 4 5 ) 3=1 where Pn = J Pi(x)bj(x)dx. (2.46) Therefore, ptj is the probability of detecting a photon pair along L O R i produced from a positron generated at voxel j. The matrix P—(pij)ixJ, often referred to as the system matrix, has the ability to model a wide variety of effects. A variety of physical processes which are not considered in the line integral model, can instead be incorporated into the system matrix, as we describe below. One may decompose the system matrix P—(pij)ixJ as such [67,70]: P = WGB (2.47) The matrix B=(bij)jxj is introduced to account for voxel-to-voxel blurring, or spatial resolution, effects. It is therefore best suited for modeling of positron range. The operator G=(gij)ixj contains the elements gij used to calculate the geometric probability that a voxel j is detected along an L O R i, and as described above, can assign partial values to voxels that are not entirely within a given LOR. Furthermore, this operator can be used to model photon non-collinearity; that is, the uncertainties in the angular separation of the photon pair can be incorporated into the probability that an event generated in a voxel j is detected along an L O R i. 2. 3D Image Reconstruction 63 The matrix W=(wij)ixi, as elaborated by Ref. [68], allows modeling of L O R interactions: (i) In the original algorithms, the measured data n* were first corrected for attenuation and normalization effects prior to use in the reconstruction task. However, it was later proposed by Hebert and Leahy [69] that these correc-tions could also be included as weighting factors in the system matrix. This can be accomplished by noting that attenuation and normalization factors can be inserted in the diagonal elements of the matrix W=(wij)ixj, thereby allowing a weight to be assigned to each L O R to account for these effects. (ii) It is also possible [68] to incorporate into the matrix W the proba-bility that a photon incident on a particular crystal is detected in another crystal. In other words, this formulation can be used as a natural and accu-rate framework to model presence of detector blurring (Sec. 1.6.7), though this can highly complicate the calculation (as it renders the system matrix highly non-sparse). In Appendix B, we describe an ad hoc method in which detector blurring (including the parallax effect) are modeled in the image domain, so as to yield practical and feasible system matrices. As already mentioned, the photon non-collinearity effect should be mod-eled into the matrix P, however this leads to an increase in the number of sinogram elements to which each voxel contributes, and hence renders the system matrix highly non-sparse. As an approximation, one can assume that this effect is depth-independent and include it in the matrix W [68], or even more simply to include it as a finite resolution effect in the spatial matrix B [70]. 2.4.3 Statistical Modeling and Objective Functions The statistical model describes the distribution of each measurement about its mean, and therefore provides a measure of similarity between the actual 2. 3D Image Reconstruction 64 measurements rij and the expected counts n$, which can be calculated us-ing Eq. (2.45) given a current estimate for the voxel intensities. We first describe the formulation for an idealized P E T system in which effects of ran-dom coincidences and scattered events are neglected. This is evident in the original derivation of the M L - E M algorithm by Shepp and Vardi, wherein they emphasize: "We shall simply be forced to ignore these problems and to assume that the only source of difficulty is in the statistical fluctuations in the counting statistics" [64]. We start by noting that the counts emitted in a voxel j such that they are detected along an L O R i follow a Poisson distribution with mean PijXj. Since a variable obtained by summing a number of Poisson variables is also Poisson-distributed, it then follows that the distribution of total counts along any L O R i (which is the quantity we measure) must also be Poisson-distributed with mean j ni = Y,PiJXi (2-48) j=i Defining X=[Xi...Xj]T and n=[n\...nAT as vectors of image intensity and measured L O R counts, respectively, we note that the task is to reconstruct the former from the latter. One can define the Poisson likelihood function [64,71,72]: L{n | A) = II e _ " , (2-49) i=l ni-—* where n, is given by Eq. (2.48). The task is then to find a vector A producing ni values such that they maximize the Poisson likelihood function L(n | A). Alternatively, one can maximize the log likelihood function l(X) — In L(n \ X) which can be written in the form: i Z(A) = E { - n i + n i ln(n i )} (2.50) i=i where the term ln(nj!) has been dropped since it has no dependence on 2. 3D Image Reconstruction 65 Xj, and therefore has no effect on the maximization task. This approach to image reconstruction is often referred to as maximum likelihood (ML) reconstruction. The Ordinary Poisson Objective Function Turning our attention to corrections for randoms and scattered events in the reconstruction task, we note that as mentioned in Sec. (1.6.3), for P E T scanners with the delayed-coincidence measurement capability, distribution of random counts can be directly measured along with the prompts. In this regard, the measured delayed-coincidences are often subtracted from the measured prompts to obtain an estimate of the number of true coincidence events. A problem with this delayed-coincidence subtraction technique is that the pre-corrected data do not follow Poisson statistics. This is because sub-traction of delayed coincidences from measured prompts compensates for the coincidence events in terms of the mean but increases the variance. In this regard, the objective function (2.50) applied to data pre-corrected for randoms is not accurate. Defining f; and Si as the expected randoms and scattered events contributions along any L O R i, one notes that since the prompt events (not pre-corrected for randoms) are Poisson by nature, one can consider the log-likelihood function [73] ifi) = E + Si) + npi l n fa + ^ + s^} . (2.51) i=i where is the number of measured prompt events along an L O R i. To arrive at this objective function, we have noted that the expected number of prompt counts along an L O R i is given by fii + fi + S j , where Hi is the expected number of true coincidences along the L O R given by Eq. (2.48). Eq. (2.51) is often referred to as the ordinary Poisson (OP) log-likelihood function. 2. 3D Image Reconstruction 66 It must be emphasized here that fj in Eq. (2.51) is an expected value which can not be appropriately replaced by the measured delayed coinci-dences, rather estimates of mean random counts along the LORs must be used. To address this issue, two approaches are possible: i) using singles measurements at the detectors [74] to calculate the expected randoms con-tribution, or ii) variance reduction (smoothing) for the measured delayed events [75-77]. Advantages and limitations of these approaches will be dis-cussed in Sec. (3.5.1). It is the case in many P E T scanners that the prompt data are pre-corrected for the defection of randoms by real-time subtraction of the delayed coinci-dences, intended to minimize data transfer and processing times. Let us refer to the output data as n\. In this case, the OP log-likelihood can be written in the form: Here, the term (n- + fj) is used as a measure of the prompts data (which, for these scanners, are only output after subtraction of delayed coincidences). However, the problem with this approach can be seen as follows. Let us refer to the prompts and delayed coincidence counts as nf and nf, which are Poisson variables. The output data are given by We now recall that for a Poisson variable X, the mean is equal to the variance (i.e. E(X)=Vax(X)). Furthermore, we note that Randoms Pre-corrected PET Kx) = {~-(ni + n + Si) + (n- + n) \n(ni + n + si)}. (2.52) ni = n\ (2.53) E(nf) = ni + fi + ^ = Var(n?) (2.54) 2. 3D Image Reconstruction 67 and E(nf) = ^ = Var(nf) (2.55) It then follows that E « + fi) = E(nf) - E(ra?) + f4 = fk + f4 + s4 (2.56) and V a r « + n ) = Var(nf) + Var(nf) + 0 = nt + 2f{ + s{ (2.57) where we have noted that E(fj)=fj and Var(fj)=0 since f; is a constant random variable. It is therefore clear that the term (n- + f,) matches the prompts counts nf in mean, but is broader in variance, and therefore does not follow Poisson statistics. Due to the non-Poisson nature of the output data, other general ap-proaches have also been employed: (i) The algebraic reconstruction technique (ART) and its variations: Pro-posed first in 1970 by Gordon et al. [78] for 3D electron microscopy and x-ray photography, Herman and Meyer [79] were first to apply this technique to the P E T problem. Mathematically, A R T aims at solving the set of linear equations determined by the projection data (if consistent). If the system of equations is not consistent (the practical case), then, by introducing appro-priate relaxation parameters, the algorithm approximates a weighted least squares solution [80]. Several variations of the A R T algorithm, such as the multiplicative A R T (MART) , have also been proposed [78,81] (see also [115] for other variations. (ii) Gaussian approximation of the data: these techniques results in (weighted) least squares penalty terms [63]. For scans that are not pre-corrected for randoms, the aforementioned methods are suboptimal because they do not fully accommodate the Pois-son nature of the data. The A R T algorithm, and its variations, do not at 2. 3D Image Reconstruction 68 all incorporate any modeling of the statistical distribution of the data: an exception to this is the R A M L A algorithm, which is derived from the A R T approach to reconstruction, but can be shown to converge to the maximum of the Poisson likelihood (2.49). However, to guarantee convergence, a re-laxation parameter is needed which must be controlled and changed with every iteration to satisfy certain criteria. Similarly, especially in high resolu-tion P E T in which individual LORs often receive sufficiently low counts, the Gaussian approximation to the Poisson distribution is not accurate. (iii) For scans that are pre-corrected for randoms, another approach is the saddle-point (SD) approximation technique (obtained by a second-order Taylor series expansion in the Poisson-modeling). The reader is referred to Refs. [82,83] for more detail. According to [84], to date, the SD approach has been able to out-perform other approaches to reconstruction of randoms-precorrected data, but is relatively more difficult to implement. (iv) Another approach, easy to implement, which still makes use of Poisson-modeling has been proposed by Yuvaz and Fessler [85]. One may notice that the variable (n\ + 2fj) exhibits a mean that matches its variance; i.e. E(n'i + 2n) = E(n?) - E(nf) + 2f4 = fk + 2fj + S j .(2.58) and V a r « + 2fi) = V a r « ) + Var(nf) +0 = ^ + 2^ + ^ (2.59) where we recall again that fj is a constant random variable, and therefore has zero variance. One may then attempt to maximize the so-called shifted Poisson (SP) log-likelihood function defined about the variable (n- + 2fj): i K%) = E {-fa + 2fi + *0 + « + 2 f*) l n f a + 2 f * + ( 2 - 6 ° ) t=i Nevertheless, it must be noted that while the term (n- + 2fj) matches the true Poisson variable (nf + nf) in mean and variance, it is still not exactly a true 2. 3D Image Reconstruction 69 Poisson variable, and this approach, while superior to the OP approach for the aforementioned scanners, is still an approximation [85]. In this regard, designing P E T scanners with the capability to separately store prompt and delayed events can result in more accurate reconstructions of the true image activity. The H R R T is an instance of modern P E T scanners capable of such measurements. In this case, one may use the OP approach, which does not make any approximations in the statistical modeling. . 2.4.4 Reconstruction Algorithms Let us start by the Poisson log-likelihood function (2.50). Unfortunately, there is no closed-form expression for the estimate A that maximizes the like-lihood [86], which therefore necessitates iterative algorithms. There exist a number of algorithms that maximize the Poisson likelihood, such as the row-action maximum likelihood algorithm ( R A M L A ) [87] and the paraboloidal surrogates (PS) methods [88-90]. To date, however, the most popular algo-rithm to maximize Poisson likelihoods remains to be the expectation maxi-mization (EM) algorithm, as we further elaborate. The E M algorithm as presented by Dempster, Laird and Rubin [91] is a general approach to iterative optimization of likelihood functions when the data can be formulated in a complete/incomplete framework. Such a framework is applicable when the problem has a more natural formulation in terms of a set of unobserved data. At each iteration, the E M approach requires two steps: an expectation step (E'-step) followed by a maximization step (M-step). Often, these two steps are combined into one. The E M algorithm was first applied to Poisson likelihood maximization in emission tomography by Shepp and Vardi [64], followed by Lange and Carson [71]. In [92], Parra and Barrett were able to demonstrate applicability of the E M algorithm to list-mode likelihood maximization; list-mode E M 2. 3D Image Reconstruction 70 reconstruction algorithms shall be discussed in chapter 4. The combination of the E- and M-steps in the case of the objective function (2.50) yields the iterative algorithm: \m I n4 nf (2.61) where r^ = E%Ar (2-62) Here A™ denotes the image intensity in voxel j (j=l...J) at the mth iteration. The sensitivity correction factor Sj=Yli=iPij is a summation over all possible measurable LORs (i=l...I) and calculates the probability of an emission from voxel j being detected anywhere. The algorithm is iterative in that it starts from an initial estimate of the image vector A m , performs a forward projection step along all LORs to calculate the expected counts n™ based on the current estimate for image in-tensity, takes the ratio between the measurement and the expected counts for all the LORs, followed by a back-projection summation of these ratios onto the given voxel j , subsequently divided by the sensitivity correction fac-tor Sj, and followed by a final multiplication with the old estimate A™ from which a new estimate A™+1 is calculated7. It can be shown [71,93,94] that the estimates A™ monotonically converge to the.maximum of the Poisson likelihood function with increasing iterations. In other words, Z ( A m + 1 ) > Z(Am) unless A m = A m + 1 in which case we have converged to a solution that maximizes the Poisson likelihood function. As described in Sec. (2.4.3), the original derivations did not directly incor-porate presence of randoms and scattered events into the likelihood functions. 7 Notice that if A™ is such that the expected counts nf1 are equal to the measurements nj for all i, one then arrives at A™ + 1 = A™ so that the new update is the same as the old, as it completely matches the data. However, this is very unlikely to take place due to the presence of Poisson noise in the data, which renders it inconsistent. 2. 3D Image Reconstruction 71 For instance, using the delayed-coincidence subtraction technique, data were (and are) often simply pre-corrected for randoms. However, the subtraction technique faces two important issues: First, data pre-corrected for randoms are no longer Poisson and therefore the modeling is not accurate, as elabo-rated in Sec. (2.4.3). This also means that the algorithm (2.61) is not truly E M when applied to data pre-corrected for randoms, and therefore, conver-gence is not guaranteed. Second, the subtraction of delayed-events from the prompts (i.e. n\ = nf — nf) can result in negative sinogram bins, which can be a source of potential problems. This can occur since the delayed-coincidences are random variables which are themselves separately measured and therefore are not exactly equal to the number of random events measured along with true coincidences8. Consequently, a particular instance of the delayed-coincidence measurement can yield values that are larger than the actual randoms, and can also result in negative histogram bins upon subtraction from the prompts. This issue has typically been dealt with by setting negative sinogram values to zero, due to the nonnegative nature of Poisson random variables and since negative sinogram values can cause reconstruction algorithms to diverge [84]. This zero-thresholding can however result in positive bias in the final reconstructed image, especially in low statistics studies. For scanners that allow separate storage of prompts and delayed-coincidence events, the aforementioned issues can altogether be avoided via directly in-corporating the presence of randoms (and scattered events) directly into the OP objective functions as described by Eq. (2.51). Defining n™ = Zl/=i Pij^j1 as before, application of the E M algorithm yields [73] \m I 2^i=lPij i=l V nr + n + Si (2.63) 8 This is why one may not use interchangeably the terms "randoms" and "delayed-coincidences" : the two are different realizations (or instances) of the same Poisson random variable, and therefore are not necessarily equal. 2. 3D Image Reconstruction 72 which we shall refer to as the O P - E M algorithms. For scanners in which the data n\ are only output after real-time sub-traction of delayed-coincidences nf from the measured prompts n\ (i.e. n[ = n\ — nf), the data are no longer Poisson, and application of (2.61) (wherein Hi = n'j) is therefore not suitable. It is instead possible to use the shifted-Poisson (SP) approximation (as described in the previous section). Appli-cation of the E M algorithm to the SP objective function (2.60) yields the S P - E M algorithm: \m I L-,i=\ P%3 i—\ nf + 2fi (2.64) It must be noted that it is still possible that n'i + 2fi < 0, though it is less likely than n\ < 0. This means that the S P - E M algorithm has the added advantage that upon introduction of zero-thresholding, it is less likely to introduce positive bias in the S P - E M algorithm as compared to when the regular E M algorithm (2.61) is used (while this problem is non-existent in O P - E M reconstruction since n\ is always non-negative). In order to altogether eliminate any introduced positive bias for data from scanners that only output n-, Ahn and Fessler [84] have proposed likelihood approximations that allow negative sinogram values without requiring any zero-thresholding. The proposed techniques, however, can result in a very considerable computational demand. For instance, the modified M L - E M algorithm has sensitivity correction factors, normally computed only once, that depend on the current image estimate at every iteration! Alternatively, as elaborated in Sec. (4.2), we have implemented another delayed-subtraction approach based on relaxing the sinogram non-negativity condition. 2. 3D Image Reconstruction 73 2.5 Problems with Expectation Maximization Algorithms Expectation maximization algorithms suffer from two main drawbacks: First, —* —* the problem of determining A such that it maximizes 1(A) is generally very ill-conditioned and results in image intensities which are usually extremely noisy and exhibit edge artifacts [95]. For a nice theoretical treatment of noise properties of the E M algorithm, Ref. [96] should be consulted. Second, a very large number of iterations may be needed before they converge. Below, we elaborate upon techniques used to address these two issues. 2.5.1 Importance of Convergent Reconstruction Prior to discussing techniques to address (i) noise artifact and (ii) slow con-vergence issues encountered with regular E M reconstruction, we focus our attention on the issue of convergence itself.Monotonic convergence, in the strict statistical sense, was defined Sec. (2.4.4): the Poisson-likelihood func-tion must satisfy l(Am+1) > Z(Am) unless Am = Am+l in which case one has converged to the solution that maximizes the Poisson likelihood function. In a more practical sense, convergence can be defined as a constant improve-ment (with iterations) in figures of merit, such as resolution and contrast, which are more relevant to the clinical task. In our experimental studies on the HRRT, we have taken the latter approach in order to quantify convergent reconstruction techniques (see Sec. 4.3). One can argue that convergence properties can be very relevant to clinical medical imaging, since algorithm divergence could have unfortunate conse-quences [97]. For instance, the theoretical analysis of Qi and Huesman [98] in lesion detectability (using statistical reconstruction vs. FBP) applies only to convergent algorithms, whereas no such justification has been provided for any non-convergent algorithms. In this regard, in the course of completing this dissertation, we have paid particular attention to the study of conver-2. 3D Image Reconstruction 74 gent reconstruction algorithms (applicable to the HRRT) , as elaborated in Sec. (3.4). In the next two sections, we provide an overview of existing liter-ature, and the various challenges, in addressing the aforementioned existing issues in regular E M reconstruction, with emphasis on limitations/strengths of the various proposed- techniques in convergent reconstruction. 2.5.2 Tackling Noise Artifacts: Regularization Several regularization techniques have been proposed to remedy the prob-lem of noise artifacts of the E M algorithm. A l l techniques, categorized into three main headings below, effectively alleviate the noise problem with some degradation in resolution, with some techniques even resulting in non-uniform resolution as we described below: 1) Early termination: this approach uses the E M method alone and termi-nates the iteration at a stage prior to deterioration of the image quality using proposed criteria. Veklerov and Llacer [99], for instance, designed a stopping rule based on statistical hypothesis testing that determined the degree of contradiction between the task of likelihood maximization and the model assumption of Poisson-distributed data. Later, they introduced the concept of "feasibility" which indicated the physical consistency between the recon-structed images and the initial data [100]. A serious problem with stopping rules is that since it is known that different parts of an image converge at different rates9, one is therefore likely to obtain object-dependent (and hence non-uniform) resolution and noise characteristics. 2) Filtering techniques: Initially developed by Snyder and his colleagues as 9 There is existing theoretical literature on dependence of image quality on position within object (object-dependency) for the final image using the regularization techniques discussed in the next two parts (filtering and Bayesian techniques). Nevertheless, to the best of our knowledge, the object-dependency of resolution as a function of iteration has not been treated theoretically in the past. Empirically, of course, this has been demonstrated, e.g. see Ref. (.[101]). 2. 3D Image Reconstruction 75 the method of sieves [95,102], these techniques are used to obtain a blurred version of the (unregularized) image intensity. This is often implemented in the form of post-smoothing the final reconstructed image using Gaussian kernels. The real practical problem with this approach is that the (unregu-larized) algorithm has to be performed until convergence is reached prior to application of post-filtering. However, hundreds of iterations may be needed in order to reach convergence. One alternate approach is to use one of the accelerated reconstruction techniques, most of which exhibit uncertain con-vergence properties, prior to post-filtering. A n alternate approach to post-smoothing is to perform the filtering in-between the iterations [70,103,104]. However, it can be shown (theoretically and experimentally) that use of spatially invariant inter-filtering (e.g. convo-lution of images obtained after every iteration with Gaussian kernels of fixed width) in many iterative algorithms, and in particular in E M reconstruction, results in non-uniform resolution that would depend on the radioactivity dis-tribution in the image: more sophisticated filters are therefore required to allow for shift-invariant and object-independent image characteristics [105]. 3) Bayesian methods: Also referred to as penalized likelihood (PL) or maxi-mum a posteriori probability (MAP) methods, these techniques are originally derived from a simple application of Bayes' rule to image reconstruction. The probability function of the data (or measurement) vector n conditioned on the image vector A can be formulated using Bayes' equation: Here L(A | n) is the likelihood function, given by Eq. (2.49) for Poisson-—* distributed data, whereas L(n | A) is the a posteriori probability distribution of the image vector. In the maximum likelihood (ML) approach, the desired image vector is defined as the maximizer of L(X | n), whereas in the M A P approach, one maximizes L(n | A). In the Bayesian framework, specification 2. 3D Image Reconstruction 76 of the prior L(X) allows some a priori information to be incorporated into the image reconstruction task. Geman and McClure were first to apply this technique to regularized image reconstruction in nuclear medicine [106]. They used Gibbs priors [107] to ensure that values at nearby voxels did not differ too much from each other. To see how this works, we note that by taking the log of both sides of Eq. (2.65), and omitting L(n), since it is a constant for a given data set, M A P reconstruction requires finding the image vector A which maximizes B(n | A) = In L(A | ft) + In L(A) (2.66) One can think of the last term as a penalty term which can be defined so as to decrease when nearby voxels differ too much from each other. Examples of this approach are illustrated in Refs. [108,109]. Furthermore, it is possi-ble to extend the definition of the prior such that separations are penalized only if the neighboring voxels are thought to be in the same region, without unduly penalizing the larger separations which one foresees occurring at the boundary between two different regions of the image (e.g. see [110, 111]) In contrast to the use of the E M algorithm with early termination, the Bayesian approach to regularized image reconstruction has the property that the objective function (through incorporation of the penalty term) is solely responsible for the final image quality, while the particular algorithm used to maximize the objective function has no effect on the obtained image qual-i ty 1 0 (and thus can be varied to yield faster reconstruction techniques). Fur-thermore, there is evidence that M A P (or equivalently penalized likelihood) methods are able to outperform the method of sieves [112]. However, again similar to the inter-filtering approach, it can be shown that non-sophisticated penalty terms can lead to non-uniform and object-1 0 In other words, one is able to separate the objective function from the algorithm used to maximize it: see the opinion page of Dr. J. Fessler for further details: http://www.eecs.umich.edu/ fessler/misc/opinion,1998.html 2. 3D Image Reconstruction 77 dependent resolution properties in the image reconstruction task, even for space invariant tomographs (i.e. tomographs that exhibit uniform resolution across the field-of-view) [113]. 2.5.3 Accelerated EM Reconstruction Algorithms One major issue with E M algorithms is that many iterations may be required for the reconstructed images to converge such that the likelihood objective function is maximized. In fact, soon after the introduction of the E M al-gorithm for P E T reconstruction, attempts to accelerate the algorithm were proposed so as to render E M reconstructions practically feasible. In 1987, Kaufman [72] treated some of the numerical aspects of the E M algorithm and proposed ways of speedings up the numerical algorithm using some of the traditional techniques in numerical analysis. It was the introduction by Hudson et al. [114] of the ordered subset ex-pectation maximization (OSEM) algorithm for histogram-mode emission to-mography which generated considerable interest in accelerated reconstruction techniques. In the O S E M approach, the data are divided into LOR-based subsets. Using Si to denote the Ith subset (1=1... L), and m (m=l,2,...) as the iteration number which is only completed after a thorough loop through all the L subsets in the data, the algorithm can be written as: \ m,l—1 * j ieSt Z v b = i PibAb where *1 = £ P « (2-68) ieSi which can be similarly applied to OP- and S P - E M algorithms. Here, we note that the back-projection summation is performed over only a subset of the data at a time, allowing the image to be updated L times for every iteration through the data. O S E M algorithms are typically able to achieve an order 2. 3D Image Reconstruction 78 of magnitude speed-up of the E M algorithm. The only (easy-to-handle) in-creased requirement of this algorithm is that the sensitivity correction factors slj must be stored for all the subsets. This is in comparison to non-subsetized E M reconstruction wherein only the overall sensitivity factor Sj = Yl\ Pij is utilized. Nevertheless, the O S E M algorithm is not a convergent algorithm in gen-eral. Hudson et al. [114] were able to prove convergence only for a very special case, referred to as the "subset balance" condition, in which case values of slj were independent of the particular subset I. Alternately, since YA=I S\ — sj> this condition can be written as: s *~ L (2.69) However, this assumption is very restrictive, and in practice, one often ob-serves limit cycles, which are non-convergent and oscillatory variations in image likelihood as well as figures of merit (e.g. contrast, resolution) with further subsets and iterations into the data, as we also report in chapter 4. Alternatively, Byrne [115,116] has suggested a rescaled block n-iterative E M maximum likelihood (RBI-EMML) algorithm: \m,l A0 ~ ol 1 pls m,l—l + J 1 X m,l—l nj 2^6=1 PibAb test where p = maxj (2.70) (2.71) In the particular case when the subset balance condition (2.69) is satisfied, slj/sj, and therefore pl, equal l/L. Subsequently, Eq. (2.70) would reduce to the O S E M algorithm (2.67) since slj/(plSj) equals unity in this case. The R B I - E M M L algorithm, similar to the O S E M algorithm, is fairly straight-forward to implement, and simply requires an additional calculation of pl val-ues (which are independent of the emission data) for all the subsets. 1 1 The terms block and subset are interchangeably used in the literature 2. 3D Image Reconstruction 79 However, the algorithm is convergent only in the feasible case: i.e. when there is a non-negative vector A such that Y/j=\PijXj = rii for all measured LORs. Nevertheless in typical cases, due to presence of Poisson noise, we do not expect there to be such a non-negative vector (and this is why maximum likelihood approaches are attractive in the first place), and one therefore arrives at limit cycles. "Feedback" techniques have been suggested [117] to extract information from the limit cycles to construct suitable approximate solutions. It must also be noted that the R B I - E M M L algorithm bears some simi-larity to reconstructions with a relaxation parameter as used by Lewitt and Muehllehner [118] and Reader et al. [119] to accelerate histogram-mode and list-mode E M reconstructions, respectively. In subsetized histogram-mode, this can be written as A " 1 ' ' " 1 77-xf = (i - M) A - ' - 1 + li^j- £ P i W J V,-! (2-72) *j ieSi 2^6=i PibAb Here, unlike the term slj/plSj in Eq. (2.70), the relaxation parameter fj, needs to be determined on an ad hoc basis, and is independent of the particular voxel j and subset I. We also note that this algorithm is not guaranteed to be convergent, even in the feasible case. There has been interest by other research groups in deriving provably convergent versions of the fast OS methods. In [87], an alternate algorithm termed row-action maximum likelihood algorithm ( R A M L A ) was proposed along with a convergence proof. The authors have also introduced the block sequential regularized E M (BSREM) algorithm, which extends the R A M L A approach to the case of maximum a posteriori (MAP) reconstruction [174]. Two types of globally convergent relaxed ordered subsets algorithms were also presented in [90]: one by modifying the B S R E M algorithm to yield an algorithm convergent under more realistic assumptions, and the other by re-laxing the OS-SPS method (first introduced in [120]). One problem with 2. 3D Image Reconstruction 80 these formulations is that they are controlled by a relaxation schedule to ensure convergence and that it can be difficult to determine these sched-ules such that they lead to fast algorithms while simultaneously satisfying theoretical criteria to ensure convergence. In [121,122], Hsiao et al. derived a new convergent complete data ordered subsets algorithm for histogram-mode E M reconstruction (C-OSEM). They have shown that the proposed algorithm, which does not involve the use of relaxation parameters, monotonically decreases the complete data objective function, and furthermore demonstrated that while increase in log likelihood with the iterations is not guaranteed to be monotonic (though the authors have always seen this to be the case), nevertheless, the solution does converge to the maximum of the log-likelihood objective function. Details of this algo-rithm are presented in Sec. (3.4.2), wherein we have extended the approach to list-mode reconstruction of the emission data. 2.5.4 Summary Numerous techniques have in the past, and in recent years, been proposed to solve the 3D image reconstruction problem in P E T We have provided an overview of these algorithms under two main headings: analytic and sta-tistical image reconstruction. Analytic techniques have been presented in terms of direct 3D inversion techniques (such as filtered-backprojection), in cases when the measured data are/not truncated, as well as rebinning tech-niques, which aim to sort the 3D data into stacks of 2D sinograms, to be reconstructed subsequently using fast 2D reconstruction algorithms. Motivations for switching from the analytic approach to statistical image reconstruction (SIR) were also explained in this chapter, and various SIR techniques were elaborated, with particular emphasis on maximum likeli-hood expectation maximization (ML-EM) algorithms. Two major problems 2. 3D Image Reconstruction 81 with these algorithms (noise amplification and slow convergence) were also discussed and a brief overview of suggested solutions to these issues was pre-sented. The next chapter discusses concepts of list-mode acquisition as well as list-mode reconstruction of P E T data. 3. List-mode Image Reconstruction 82 3. List-mode Image Reconstruction In this chapter, we introduce and investigate list-mode image reconstruc-tion algorithms as a class of imaging techniques suitable for high resolution tomographs. 3.1 Introduction In positron emission tomography (PET) systems, there is a continuous ef-fort to increase sensitivity and to improve spatial resolution. This, in turn, brings about the need for different approaches to data collection and image reconstruction in order to make use of the increasingly high sampling ca-pabilities of modern P E T systems. To better see this, one may make the following interesting observation [123]: Moore's Law states that the number of transistors in an integrated circuit doubles every 18 months. By com-parison, the number of individual crystals in P E T tomographs has doubled approximately every two years for the past 25 years (Nutt's Law). The cor-responding number of LORs, which is measured by the square of the number of detector elements, has therefore grown faster than the available computing power. These are depicted in Fig. (3.1). A n important general conclusion can be drawn from this observation: assuming a continuation of the aforementioned trend in modern P E T scan-ners, novel image reconstruction techniques must be considered in order to efficiently reconstruct images from the full information available from these scanners. One important step in such a direction has been the introduction 3. List-mode Image Reconstruction 83 Transistor Court 1.000 10.0HO0O, 100 600 vm w 10 i Ui« Hwt*i of Transistors Ptt < pouwtwyis (tenths Pentium* El Pfocewcf Pwtttsam Pro Pfoceiwr % Pentium Processor f366m Proc«MOt Moore's Law 71 581 1 I 1 1 1 1 I 1 'SI ' « rater of Detects r tinner*-, pier PTT Ton osraiih Double 1 * MR" Nutt's Lfw UM MR MM Fig. 3.1: Moore's Law vs. Nutt's Law, as described in text. The version of the HRRT depicted is the single layer (~60K crystals), whereas most existing HRRT scanners are dual layer (~120K crystals) (courtesy of Dr. R. Nutt). of the list-mode acquisition capability in a number of modern P E T scanners. 3.2 List-mode Acquisition Conventional tomographs inherently histogram the collected data into sino-gram bins as the data are acquired. This is often referred to as histogram-mode acquisition. A limitation of this imaging modality is that in dynamic imaging, the number of frames, and durations of the individual frames, have to be specified prior to performing the scan. Furthermore, considering the increasing large number of LORs in modern scanners, it is quite likely in dynamic P E T studies to have the total number of LORs exceed the actual number of events measured per frame. Subsequently, storing the data in histogram-mode can be expensive. The main motivation for the introduc-tion of list-mode acquisition capability into a number of modern scanners, in-cluding the high resolution research tomographs (HRRT), has therefore been to enable data-acquisition while allowing the data to be later histogrammed into the desired dynamic frames, as well as potentially reducing the size of 3. List-mode Image Reconstruction 84 data-sets. List-mode acquisition is achieved by storing information regarding the acquired events as they are detected one-by-one in the form of a list. The recorded information, or attributes of the list-mode events, typically consist of coordinates of the two detectors along which the photons are detected in coincidence, along with the time-of-detection1. It is also possible to store further attributes for each event in the list-mode data, such as information about the time-of-flight, depth-of-interaction, and energy. Once the list-mode data are acquired, they can be binned and sub-divided into multiple frames of desired durations, and therefore greater utility is gained by removing the need to specify the frames in advance prior to the scan 2. 3.3 List-mode Reconstruction A n important possibility with list-mode acquisition is that it enables the use of list-mode algorithms in the inversion of the acquired data into the imaged object. List-mode reconstruction is referred to those class of image reconstruction algorithm that do not bin, or histogram, the acquired list-mode events into sinogram bins, and rather incorporate the processing of the individual list-mode events directly into the reconstruction task. Thus, we emphasize that list-mode acquisition is not identical to list-mode reconstruc-tion; rather, the former is a pre-requisite for the latter. In fact nowadays, list-mode acquired data are often subsequently histogrammed prior to image 1 In the case of the HRRT, actually, a more simplifying approach has been taken: time tags are inserted every 1 ms in-between the recorded events, thus removing the need to incorporate the time-of-detection information inside each event. In other words, since the detected events are sorted by time-of-arrival, introduction of time tags is a very practical method to collectively label all events recorded within every 1 ms. 2 This can be very important in research, wherein the practical timing resolution of the scanner, given the particular patient, can be determined by fitting models to dynamic data obtained using a variety of framing sequences, thus allowing optimization of frame selection from the particular data set. 3. List-mode Image Reconstruction 85 reconstruction. In this work, we have thus concentrated on taking the list-mode approach one step further: from mere list-mode acquisition to list-mode reconstruction. 3.3.1 Benefits of List-mode Image Reconstruction A large portion of this dissertation is focused on investigating benefits and applicabilities of list-mode reconstruction techniques for scanners with list-mode acquisition capability. The high resolution research tomographs (HRRT) (see Sec. 1.7) is an instance of such scanners, and experimental investigations reported in this work have all been performed on this scanner. Several bene-fits can potentially be gained by performing list-mode image reconstruction, instead of the conventional histogram-mode approach: 1) Faster reconstruction: In 3D P E T studies in which a low number of counts are acquired (most notably in dynamic P E T scanning), list-mode recon-struction can in principle be performed more quickly and efficiently than histogram-mode reconstruction. This is because when the reconstruction of low-count frames is required, the number of events acquired may be in-fact (much) less than the number of lines of response (LORs) in a full sinogram set, especially with high resolution scanners in which the number of LORs is typically very large (e.g. ~4.5B in the HRRT) . Use of list-mode recon-struction techniques implicitly ignores LORs along which counts were not recorded [124], and can therefore considerably improve the reconstruction speed especially for low-statistics frames. 2) Preservation of maximum sampling frequency: When histogram-mode re-construction methods are used, the data are often compressed in the axial or radial directions in order to accelerate the reconstruction tasks. This is 3. List-mode Image Reconstruction 86 sometimes referred to as data mashing by which it is meant that certain 'nearby' LORs are histogrammed into the same sinogram bin in order to re-duce the size of the sinogram data. For instance, in the case of the HRRT, with no data compression, the sinogram size is 1.5G bytes. Application of axial/radial compression schemes, however, has been shown to adversely affect axial/transaxial resolution of images reconstructed using 3D-OSEM, especially as one moves away from the center of the field-of-view (FOV) [35]. In the case of list-mode reconstruction, since events are considered one-by-one, sinogram data compression is in principle not needed, thus resulting in preservation of maximum sampling frequency at no extra cost in terms of time and data size. 3) Time-of-flight Positron Emission Tomography: It has been known since the early 1980s that P E T scanners capable of encoding time-of-flight (TOF) information would potentially reduce the statistical noise variance in P E T reconstruction [125,126]. T O F P E T is based on the observation that by measuring the difference of the arrival times of the 511 keV photons, a P E T camera could restrict the position of the positron emission to a subsection of the line segment joining the detector pair. This is depicted in figure (3.2). T O F P E T , especially in whole body scanning and for scans with high random fractions, is expected to considerably improve image noise behavior compared to conventional schemes in which T O F information is not incorporated [127]. Nevertheless, technological difficulties, namely slow electronics and the need for fast scintillators which were at the same time effective absorbing of anni-hilation photons, had limited development of T O F P E T until recently. With the continuous improvements in the technology of P E T imaging (e.g. faster electronics), and especially since the discovery of the scintillator L S O 3 , time-3 P E T scanners based on LSO have the potential of achieving significantly better coin-cidence timing resolution than the 6-12 ns FWHM typically achieved with BGO. 3. List-mode Image Reconstruction 87 Fig. 3.2: With conventional reconstruction (left) voxels along the LOR are incre-mented regardless of position along the LOR. With TOF reconstruction (right), each voxel is incremented by the probability (as measured by the TOF measurement) that the source originated at that voxel. of-flight P E T is now being actively reconsidered [128,129]. The important observation is that with the added attribute of time-of-flight measured for the acquired events, the one-to-one mapping from the LORs to the sinogram bins will no longer be the case; rather, many more sinogram bins (depending on the precision of the T O F modality) may be required to take into account the measured T O F information along each LOR. On the contrary, with list-mode reconstruction, one does not require the use of sinogram bins and instead processes the events one-by-one (and reads the T O F information meanwhile). Subsequently, the data generated from such scanners are most efficiently reconstructed using the list-mode approach if the T O F information is to be made full use of. 4) Accuracy and convenience in motion compensation: A n additional advan-tage of list-mode reconstruction is the increased accuracy and convenience with which motion correction can be implemented into the reconstruction procedure, as we shall elaborate in Sec. (5.2.2). This is the case since in histogram-mode reconstruction: (i) a motion-corrected L O R will not typ-ically correspond exactly to the center of a sinogram bin, and therefore 3. List-mode Image Reconstruction 88 an interpolation needs to be performed upon histogramming the event into motion-corrected sinograms. On the contrary, motion-corrected list-mode event coordinates can be maintained as continuous variables in list-mode reconstruction, thus potentially preserving a higher degree of accuracy in the reconstruction task resulting directly from better sampling of the mea-surement. Furthermore, (ii) the regularly employed sinogram-space used in histogram-mode reconstruction needs to be extended, in order to account for all measured events (including those that, upon being motion-corrected, do not correspond to actual detector pairs). On the contrary, list-mode process-ing of such events is convenient. 3.3.2 Analytic List-mode Image Reconstruction Of the analytic imaging techniques discussed in chapter (2), one is particu-larly suitable for list-mode reconstruction: the B P F algorithm (Sec. 2.2.7). This algorithm allows backprojection of list-mode events, which may sub-sequently be filtered in image-space to yield reconstructed images. Inter-estingly, prior to the popularity of the list-mode acquisition scheme, this approach was used in such scanners as the HIDAC (II) animal imaging sys-tem [130] and the P E T R R A whole-body scanner [131] as a data storage scheme [132]: that is, the individual data, as they were being measured, were backprojected onto the image space. In other words, instead of storing the data in 4D projection data format, they were stored as 3D backprojected images4. Image-based scatter and random correction steps were subsequently implemented to address data acquired and stored using this modality [24]. Other than the general shortcomings of the analytic approach compared to statistical reconstruction (as reviewed in Sec. 2.4), for a scanner with gaps in-between the detectors (particularly the HRRT) , gap-filling techniques (see 4 This does require a pre-specification of framing sequences, so that images in each frame are backprojected onto the image voxels for that frame. 3. List-mode Image Reconstruction 89 chapter 1, footnote 14) all require operating in the projection-space, thus making application of the purely list-mode B P F technique (without gap-filling) inaccurate. Therefore, in what follows, we shall restrict our attention to statistical list-mode image reconstruction algorithms. 3.3.3 Statistical List-mode Image Reconstruction In this work, we have particularly concentrated on implementation and study of EM-based algorithms for reconstruction of P E T data in list-mode. The list-mode expectation maximization (LMEM) reconstruction algorithm has been formulated from first principles [92]: denoting \™ as the image intensity in voxel j (j=l...J) at the mth iteration, and Pij as the probability of an emission from voxel j being detected along L O R i (often referred to as the "system matrix"), the L M E M algorithm is given by \m N 1 A m + i = _J_ y p ——t (3.1) 2^i=\ Pi] fc=l 2^6=1 PikbAb where refers to the L O R along which the fcth list-mode event is detected and ./V is the number of measured events. The sensitivity correction fac-tor Sj=Y^l=iPij is a summation over all possible measurable LORs (i=l...I) and calculates the probability of an emission from voxel j being detected anywhere. Comparing this algorithm with the histogram-mode version (Eq. 2.61), one notes that the two are different in that in the list-mode algorithm, the backprojection summation over the LORs is instead performed over the mea-sured events, while replacing n; (the number of events measured along L O R i) with the numeral 1. Running through the entire data, the two algorithms are thus exactly equivalent in mathematical terms. Differences arise in: (i) calculation times, wherein the histogram-mode algorithm passes through all the LORs, while the list-mode algorithm runs through the events (see the first 3. List-mode Image Reconstruction 90 point in Sec. 3.3.1); (ii) in accelerated schemes, the data subsets in histogram-mode and list-mode reconstruction are fundamentally different (see Sec. 3.4) Incorporation of Attenuation and Normalization Effects Following discussion in Sec. (2.4.2), we shall directly include the attenuation and normalization effects as weighting factors in the E M algorithm. We shall therefore decompose the system matrix P=(pij)ixj into P=WG where G=(gij)ixj is the geometric probability of an event generated at voxel j to be detected at L O R i, and the diagonal5 matrix W={wi)ixi allows a weight to be assigned to each LOR, to account for variations due to attenuation and normalization. Possible schemes incorporating the parallax effect have been discussed in Appendix B. Such matrix factorization results in the emergence of a cancellation in the algorithm, thus giving: \m N i The cancellation of everywhere, except in the sensitivity factor, which is only calculated once, considerably eases the direct inclusion of normal-ization and/or attenuation correction. In the case of histogram-mode E M , it has been observed [133] that appropriate modeling of normalization and attenuation as such, results in better image quality when compared to the unweighted case, in which the emission scan is simply pre-corrected by these factors. 5 This matrix will not be diagonal if one also includes effects of crystal penetration and inter-crystal scattering. Including such effects results in a high reduction in the sparseness of the matrix and increases the computational demand [68]. 3. List-mode Image Reconstruction 91 3.4 Accelerated List-mode Reconstruction Algorithms In Sec. (2.5.3), we provided an overview of existing accelerated histogram-mode reconstruction techniques. In the ordered subset E M (OSEM) ap-proach, the data are divided into LOR-based subsets, and the image estimate is updated every time the algorithm passes through a data subset. 3.4.1 Ordinary Subsetized List-mode EM Algorithm The histogram-mode data-subset approach can be conceptually applied to list-mode reconstruction as well [70,124]. One may consider event-based (instead of LOR-based) subsets, obtained by sub-dividing the list-mode data into segments that span a fraction of the total duration of the data 6. Dividing the data space into L subsets, we use Si to denote the Zth list-mode subset (1=1...L). We shall maintain the use of m (m=l,2,...) as the iteration number which is only completed after a thorough loop through all the L subsets in the data. We also use Aj 1 ' ' to denote the image estimate at the mth iteration and Ith subset. The subsetized list-mode expectation maximization algorithm (which we shall refer to as the S - L M E M algorithm) is then given by: ym,l—l Y X ^ = wa- ^ 9 i k i r J a A " 1 ' * " 1 ^ One must additionally note here that list-mode subsets exhibit a fundamental difference compared to sinogram-based subsets. This is because each list-6 For dynamic list-mode image reconstruction, each single frame being reconstructed should in principle be static. With a continually changing object, issues of scanner sensi-tivity and time resolution limit how short-in-duration a single frame may be in order to obtain sufficient counts. Consequently in practice, one is likely to encounter a non-static object for a given frame. To deal with this, therefore, it is best (as we have done in our implementation), to have each list-mode subset contain portions of data obtained at var-ious intervals throughout the frame, such that all list-mode subsets represent nearly the same object. 3. List-mode Image Reconstruction 92 mode subset can be thought of as a lower-statistics scan in its own right. This observation may point to another advantage of list-mode reconstruction. It has been shown [134] that the way in which sinogram-data subsets are chosen and ordered has an effect on the resulting reconstructions. The requirement of minimum variation in-between the data subsets is inherently fulfilled with list-mode subsets. The aforementioned subsetized algorithm, however, is not a convergent al-gorithm, and instead results in limit cycles: oscillatory alternations in image likelihood as well as figures of merit (e.g. contrast, resolution) with further subsets and iterations into the data. Starting from first principles and using the complete data approach as in [121,122], Khurd and Gindi [135] have been able to derive a convergent list-mode E M reconstruction algorithm. The au-thors have subsequently tested the convergence and speed-up achieved by the algorithm using simulated S P E C T data. In what follows, (i) we show in detail a derivation of the same algorithm using an approach based on re-visiting the histogram-mode technique, (ii) We go on to present an intuitive picture of how the algorithm proceeds using additive updates in image space, (iii) Furthermore, we propose a hybrid algorithm employing both the regular and convergent subsetized list-mode algorithms, (iv) Results of implementing the regular, convergent and hybrid algorithms applied to experimental P E T data are subsequently presented in Sec. 4.4. We first discuss issues of convergence as observed and tackled in histogram-mode reconstruction. 3.4.2 Convergent OSEM Reconstruction In the histogram-mode approach, Hudson et al. [114] were able to prove convergence of the O S E M algorithm only for an impractical special case, in which the subsets chosen corresponded to a restrictive "subset balance" 3. List-mode Image Reconstruction 93 condition in the matrix. In practice, while the algorithm is seen to perform considerably faster than the regular E M algorithm, it is often seen not to converge to a fixed point and instead results in limit cycles. In other words, the O S E M algorithm in itself does not maximize likelihood. Subsequently, there has been interest in deriving provably convergent ver-sions of the fast OS methods, as reviewed in Sec. (2.5.3). In [121,122], Hsiao et al. derived a new convergent complete data ordered subsets algorithm for histogram-mode E M reconstruction (C-OSEM). They have shown that the proposed algorithm monotonically decreases the complete data objective function, and furthermore demonstrated that the solution converges to the maximum of the log-likelihood objective function. Below we elaborate upon the algorithm. This algorithm has the advantage that it does not involve a relaxation schedule (as discussed in Sec. 2.5.3), converges to the fixed point of the maximum likelihood objective function, and can be extended to list-mode reconstruction, as we demonstrate shortly. To see how the C - O S E M algorithm works, we begin by defining dj as the complete data, as used in statistical derivations of M L - E M , representing the number of counts detected along an L O R i (i=l...I) that have originated from voxel j (j=l...J). However, one only measures and has knowledge of the incomplete data no total number of counts detected along a given L O R i (regardless of the voxel(s) from which the events have originated); i.e. ni = Y.Cij (3.4) j Dividing the data space into L LOR-based subsets, Si is used to denote the /th histogram-mode subset (1=1...L). We shall use ra (771=1,2,...) as the iteration number which is only completed after a thorough loop through all the L subsets in the data. We also use A™'' to denote the image estimate at the rath iteration and Zth subset, while as before is the probability of an emission from voxel j being detected along L O R i). 3. List-mode Image Reconstruction 94 The C-OSEM can then be written in the form [121,122]: ^ = ^ T # V ^ ' V Z G 5 ' ( 3- 5 ) 2^6=i 9ibAb m i — 1 \m,l 1 A3 ~ W L <m—l,s ij s=lieSs s=l+li€St (3.6) where C™'' is the current estimate for the complete data Ctj. Here one sets m 3 \ ™, 0 = \ m l ' L at the beginning of each iteration, while using some initializa-tion for Cff values. In place of the update Eq. (3.6), the ordinary O S E M algorithm performs the following \fl = — — (3.7) ^i€St Wi9ij i€st Thus, we see that the C-OSEM algorithm is different from the ordinary O S E M in that calculation of image updates at every subset (numerator of Eq. 3.6) is not limited to values for LORs in that subset only. Meanwhile, at any subset I, values of Cij are updated only for LORs i G Sf this explains why the update image at each subset can be computed nearly as fast as that of regular O S E M . 3.4.3 Convergent Subsetized List-mode EM Algorithm Similar issues, as in the O S E M algorithm, are present in subsetized list-mode reconstruction, and the proposed S - L M E M algorithm results in non-converging (e.g. limit cycles) behavior, as reported in Sec. (4.4). Using trans-formations as in [124], we are able to extend the histogram-mode formu-lation presented in Sec. (3.4.2) into list-mode reconstruction. By defining list-mode subsets as event-based subsets, as compared to LOR-based sub-sets in histogram-mode reconstruction, and replacing the summations over 3. List-mode Image Reconstruction 95 the LORs by summations over the events, while replacing n$ in Eq. (3.5) by the numeral 1, we arrive at the following list-mode reconstruction update equations: ~X^'1 = T1' wa- ^ 9ikWJ a A ™ ' ' " 1 ^ 2^i=l ^ihli] k&Si 2^ 6=1 9ikbAb \f = J2~A7's+ E ~x7~l,s (3-9) s=l s=l+l where A™'' is an intermediate image vector produced by the first update Eq. (3.8), subsequently used by Eq. (3.9) to arrive at the overall image esti-mate X™'1. Here, we shall set X™'°=X™l~1,L at the beginning of each iteration. One may readily note here that the first step of the convergent approach is similar to the regular subsetized list-mode (S-LMEM) algorithm (Eq. 3.3). That is, if step two of the above approach is replaced by Xj"l=XY's, one arrives at the S - L M E M algorithm. This is not the case in the histogram-mode approach, as used by others, since in the O S E M algorithm, at each subset of the data, the sensitivity correction factors depend on the particular subset (given by Y^i^Siwi9ij)' whereas this term does not appear in the update Eq. (3.6). On the other hand, since the list-mode data subsets are event-based and not LOR-based, the sensitivity correction factors are always given by Ei=iwi9ij-As shown in Eq. (3.9), the algorithm takes the form of additive updates in image-space, in that upon arriving at any subset I, the intermediate image updates which have been previously calculated for other subsets {Vs|s ^  1} are added to the update X™'1 calculated for the current subset I. We shall refer to this approach as the convergent subsetized list-mode E M (CS-LMEM) algorithm. We also note that it is easy to show that: xm,l = xm,l-l + ~xm,l _ yrn-1,1 ( 3 1 Q ) From this observation, it follows that by keeping track of the values of A™'' - 1 3. List-mode Image Reconstruction 96 and the values of X™'1 for all subsets Si,l=l...L, values of A™'1'1 can be recur-sively updated according to the above relation. This makes the calculation of image updates using the C S - L M E M algorithm nearly as fast as the regular S - L M E M algorithm. Nevertheless, as we demonstrate in Sec. (4.4), improvements in image quality achieved by the C S - L M E M algorithm are slower than those obtained using S - L M E M . This has encouraged us to propose and investigate a hybrid algorithm, as we discuss below. 3.4.4 Hybrid S/CS List-mode EM Algorithm We have found it very useful to investigate the possibility of combining the advantages of the S - L M E M and C S - L M E M algorithms into a hybrid algo-rithm. Namely, one typically notices, as also shown in Sec. (4.4), that the regular S - L M E M algorithm, in the first few subsets, is able to produce im-ages of higher quality (e.g. contrast, resolution) relative to the C S - L M E M algorithm, whereas the latter is able to exhibit convergent resolution and contrast behavior as the iterations proceed. The hybrid approach we have taken uses S - L M E M for the entire or part of the first iteration, followed by C S - L M E M in the rest of the calculation. 3.5 Random Correction in List-mode Reconstruction Various correction techniques for the contribution of random coincidences to the measurement process were discussed in Sec. (1.6.3). The HRRT is an instance of modern P E T scanners with the delayed-coincidence measurement capability, and the distribution of random counts can therefore be directly . measured and recorded along with the prompts events. In most histogram-mode reconstruction tasks, the prompt data are pre-corrected for the detection of randoms by subtraction of the delayed coin-3. List-mode Image Reconstruction 97 cidences. However, as discussed in Sec. (2.4.4), the subtraction can be a source of potential problems: First, the randoms-precorrected data do not follow Poisson statistics. Second, it can result in negative histogram bins, which can be a source of potential problems. The first issue has been tackled by means of practical approximations, such as the shifted Poisson model, as discussed in Sec. (2.4.3), while the second issue has typically been dealt with by zeroing negative sinogram values. This latter can in turn result in positive bias in the final reconstructed image [84], as also reported and discussed in this chapter. The ability of the HRRT to separately store prompt and delayed events can therefore allow a more accurate reconstruction of the true image activity. Here, we summarize the case for histogram-mode reconstruction, which is eas-ily extendable to the list-mode case. As also noted at the end of Sec. (2.4.3), the ordinary Poisson (OP) E M algorithm is well-suited for reconstruction of the H R R T data, since it incorporate accurate modeling of the Poisson statistics of the measured prompts and delayed events. As elaborated in Sees. (2.4.3) and (2.4.4), defining fi and s; as the expected randoms and scattered events contributions along any L O R i, one notes that since the prompt events are Poisson by nature, one can consider the log-likelihood function where n\ is the number of measured prompt events along an L O R i, and nf is the expected number of prompts along the LOR, given by 3.5.2 Ordinary Poisson EM Reconstruction l(\) = Y,[-nPi+<Hn\)\ (3.H) (3.12) 3. List-mode Image Reconstruction 98 Application of expectation maximization to the likelihood functions then yields the following O P - E M algorithm [73]: This algorithm poses computational difficulties which need to be ad-dressed. First, we note that a cancellation of Wi in the forward- and back-projection steps is no longer valid, requiring constant look-up of attenuation and normalization factors for LORs along which the events are being read. Furthermore, it must be noted that fj in Eq. (3.13) is an expected value which can not be appropriately replaced by the measured delayed coinci-dences, rather estimates of mean random counts along the LORs must be used. To address the latter, two approaches are possible: i) using singles mea-surements at the detectors [74] to calculate the expected randoms contribu-tion, or ii) variance reduction (smoothing) for the measured delayed events [75-77]. The first approach has the advantage of utilizing high statistics sin-gles measurements. At the same time, it increases the 'bandwidth' of the scanner, since delayed time windows need not be imposed for delayed coin-cidence measurements, and less saturation of counts would occur especially for studies involving high count rates. There are certain difficulties that render the application of either of the aforementioned approaches difficult. Both approaches do require constant look-up of calculated sinograms for the expected random events along the LORs, and as such will introduce additional time-costs to the reconstruction task. As for the first approach, in the HRRT, the singles rates can only be measured for the crystal block and not the individual crystals. This can result in a relatively coarse estimation of random rates at the individual crystals. This effect is further amplified by noting that while singles are only available for the crystal blocks, each crystal has a different efficiency due to intrinsic as Eb=i Wigib\f + ri + S i (3.13) 3. List-mode Image Reconstruction 99 well as object-dependent factors. This is because the contributions of singles events along the various lines passing through a particular crystal depend on the size and shape of the patient. Therefore, one cannot correctly use the trues normalization to compensate for variations in singles rate. In P E T brain imaging, compared to other P E T modalities, in which a less, though still existent, degree of object variation across the studies is en-countered, an approximate technique has been suggested [74]. It consists of scanning a reference phantom (e.g. a homogeneous cylindrical phantom) and measuring the singles as well as delayed-coincidence rates. Next, compar-ing the calculated expected delayed-coincidence rates (from the singles) and comparing with the measured delayed-coincidences, calibration factors can be obtained. These factors are then applied to subsequent patient scans in which only the singles rates are measured. This is clearly an approximate technique and remains to be tested and improved further. The above con-siderations therefore render the application of the singles approach difficult, conceptually and computation-wise, and the possibilities and advantages of this technique remain to be fully explored and addressed7. Second approach (random smoothing) is currently quite difficult to imple-ment on the HRRT. Due to the non-cylindrical shape of the H R R T as well as the depth-of-interaction measurement capability, the relation between the de-tector pairs and allocated sinogram bins for the LORs is not 1-to-l, i.e. some projection bins correspond to more than one physical LORs. This problem renders application of variation reduction (smoothing) difficult. In addition, this approach is computationally intensive. The H R R T contains 936 (8x8 7 One may note here that current scanners actually measure hits and not just singles (a hit is a detected photon, independent of whether the other photon is detected or not). For all existing scanners, the vast majority of hits are singles. However, as scanners become more and more sensitive to radioactivity (both in terms of coverage of FOV as well as detector sensitivities) the proportion of singles to hits would become less and less, and corrections (hardware or software) need to be implemented. 3. List-mode Image Reconstruction 100 dual-crystal) blocks, corresponding to nearly 120k crystals. With no data compression (i.e. mashing), this corresponds to about 3GB of (floating) ran-dom sinograms which need to be processed for each frame if this approach is to be taken. Alternatively, it is possible to take a non-analytic approach to improve variance on randoms measurement; namely, one may acquire them with a noticeably wider time window, but this technique will correspond to a potentially considerable increase in resource allocation and 'bandwidth' con-sumption, and will result in saturation effects for higher count-rate studies. 3.5.2 The Delayed Events Subtraction Technique Due to the aforementioned issues as well as the computational difficulties they pose, we have alternatively considered the delayed events subtraction method, adapted to list-mode reconstruction as elaborated in Sec. (4.2), as a practical random correction technique in the implementation of a feasible list-mode reconstruction algorithm applicable to the expected workload of the HRRT. Our approach will involve passing only through the list-mode data and does not require access to histogrammed data nor processing of them. Furthermore, as shown in Sec. (4.4), the algorithm does not exhibit the intrinsic bias observed in O S E M when zero-thresholding is applied to negative sinogram bins. Due to its practical implementation, using ~16 processors we have been able to achieve a processing rate of —l.OGB/hour, approaching our feasibility criteria as discussed at the beginning of chapter 5. 3.6 Scatter Correction in List-mode Reconstruction In this dissertation, along with the development of statistical list-mode re-construction algorithms, we have incorporated, out of the ten complicating factors in P E T imaging as introduced in Sec. (1.6), correction techniques for nine. Scatter correction remains to be implemented in the list-mode imaging 3. List-mode Image Reconstruction 101 method for the HRRT. This requires much dedicated research, to be built upon previous work on the subject. Of the various scatter correction techniques (summarized in Sec. 1.6.2), ones employing the Klein-Nishina formula are most attractive due to treat-ment of the scatter using basic physical principles. A number of such schemes are currently under investigation by our group. One possible approach has already been developed for the HRRT, namely the Watson scatter correction technique [28]. The original implementation of the algorithm was purely image-based (i.e. would compute the scatter contribution using only the re-constructed image). More recent versions [136] require the histogrammed data, in order to scale the calculated scattering components to compensate for external scatter. This current scatter correction algorithm takes under one minute to compute and is under constant modification and speed-up [137]. Another possibility is to use the Watson technique to estimate S j , the scattered events contributions along any L O R i, and to analytically include this term in the denominator of the E M algorithm, as shown in Eq. (3.13). Finally, we would also be interested in including the effect of scatter directly into the system matrix of the HRRT. Thus, one would be computing the probability that an annihilation generated in a voxel j would be detected along an L O R i, including the effect of scatter, which only requires access to ' the attenuation data (is independent of the emission image) [138,139]. This method, in our opinion, is potentially the most accurate scatter correction technique (amidst the most computationally intense), since it is not based on an initial estimate of the true activity distribution. This is in contrast to the other aforementioned techniques which require an initial estimate for the image, somehow calculated without true knowledge of the scatter distribu-tion. 3. List-mode Image Reconstruction 102 3.7 Summary In this chapter, we have introduced the concept of list-mode acquisition, and elaborated upon the list-mode image reconstruction approach, as an appro-priate consideration in current state-of-the-art and future (e.g. time-of-flight) P E T scanners. With particular emphasis on the statistical techniques, we have provided an outline of various methods for accelerated list-mode recon-struction. In particular, we have provided a framework, based on previous work [121,122,135], for a convergent list-mode E M reconstruction technique. Experimental studies of the various approaches, along with implementation issues (such as interpolation techniques in geometric projection), are pre-sented in the next chapter. 4. Reconstruction Implementation and Comparison 103 4. Reconstruction Implementation and Comparison Objective of this Work This chapter represents the experimental core of this thesis work in statistical list-mode image reconstruction. The results, along with the algorithms pro-posed in the previous chapter, have been published [40] and presented [45]. In this work, we have been interested in implementing practical list-mode recon-struction algorithms capable of coping with the considerable size of list-mode data (~200-300GB/week) expected to be generated from the H R R T once in fully operational mode. The issue of computation efficiency is therefore a very important one. Accelerated variations of statistical list-mode reconstruction have there-fore been investigated, with the aim of reaching and exceeding processing rates of 1.0-1.5GB/hour, to enable round-the-clock reconstruction of the data expected to be generated by the HRRT. The list-mode expectation maximization (LMEM) reconstruction algorithm was already introduced in Sec. (3.3.3). Several accelerated versions of the list-mode algorithm (i.e. sub-setized, convergent-subsetized and hybrid) were also elaborated in Sec. (3.4). In Sec. (4.1), we introduce the various projection techniques investigated (for accuracy and feasibility) in this work, followed by experimental methods and results presented in sections (4.3) and (4.4). 4. Reconstruction Implementation and Comparison 104 4.1 List-mode Geometric Back- and Forward-Projection Techniques In general, forward- and back-projection schemes can be performed using two main approaches: voxel-driven and LOR-driven. It has been suggested [48,140,141] that best results may be obtained when the back- and forward-projection operations are output driven: i.e. if back-projecting, the process should be voxel-driven and if forward-projecting, the process should be LOR-driven. A hand-waiving argument has been put forth [141]: if a projection is output driven, the operation would be a "many to one" operation rather than a "one to many" value operation. For instance, when back-projecting, an LOR-driven approach measures how each L O R contributes to all the voxels ("one to many"), whereas in the voxel-driven approach, one would be mea-suring the contributions of all the LORs to a given voxel at a time ("many to one"). However, this is not a concrete argument and we have in fact verified it not to be the case in our imaging task, as elaborated shortly. Due to the intrinsically LOR-based nature of list-mode reconstruction, only the LOR-driven projection operations may be utilized. This is one lim-itation of list-mode reconstruction, and is in a sense acquired due to the fact that one does not need to access the entire projection-space in list-mode re-construction (which is one important potential advantage of the technique in the first place, especially for low-statistic frames). In our reconstruc-tions, as shown later, we have not observed a degradation in image quality when switching from voxel-driven back-projection (used in histogram-mode) to LOR-driven back-projection. 4.1.1 Projection Techniques Three LOR-driven projection techniques were explored for use in list-mode reconstruction: 4. Reconstruction Implementation and Comparison 105 1) The Siddon method: Following work of Siddon [142], this technique is based on calculating the path length of intersection of a given L O R along each voxel, as depicted in Fig. (4.1a). 2) Trilinear interpolation: To understand this technique, let us consider an image grid with voxels of unit length in all directions. Trilinear inter-polation works by stepping through a given L O R with increments of unit length, as shown in Fig. (4.1b) for the 2D case. For back-projection then, one distributes the L O R value between the nearest voxels (four in 2D, eight in 3D). In forward projection similarly, the L O R value is obtained from the nearest eight (four) voxels in 3D (2D). 3) Bilinear interpolation: In bilinear interpolation, the length of incre-ments on a given L O R is chosen so as to ensure that the sampled points lie along the centers of the voxels in one direction, in order to eliminate interpo-lation in that direction. In our case, the dimension along which interpolation is eliminated is the transaxial (X or Y) direction along which the given L O R increases faster; e.g. Y direction for the L O R shown in Fig. (4.1c). This method is potentially less accurate (which we have not observed to be the shown later) than the trilinear counter-part since it makes the lengths of the increments LOR-dependent. However, the technique is faster since: (i) interpolation along one direction is eliminated; i.e. for each point on the LOR, an interpolation is performed over only four (two) nearest voxels in 3D (2D), and (ii) for oblique LORs, fewer samples per L O R are considered. Fig. (4.2) shows images obtained by back-projection of constant values along a projection direction at an oblique transaxial angle </>=30. Results are shown for the three aforementioned projection techniques. Horizontal profiles across these images can be used to compare the back-projection artifacts, as shown in Fig. (4.3a). We have observed that the artifacts vary according to the projection angle, yet typically, the artifacts are seen to be stronger in the Siddon case as compared to the bilinear and trilinear interpolation 4. Reconstruction Implementation and Comparison 106 / / / / \ r{ 7 / f / / (a) (b) (c) Fig. 4.1: Projection algorithms employing (a) the Siddon method, (b) trilinear, and (c) bilinear interpolation techniques are drawn, as elaborated in text. i r (a) Siddon A (b) Trilinear (c) Bilinear Fig. 4.2: Results of back-projecting constant values along a projection direction onto a 64x64 image at axial (9=0 and transaxial 0=30°. techniques1. We shall also demonstrate in Sec. (4.4) that: (i) reconstructed image quality is manifestly superior when performing trilinear and bilinear interpo-lations, compared to the Siddon technique, while the two former techniques perform comparably; and (ii) output-driven histogram-mode reconstruction yields nearly similar image qualities compared to LOR-driven list-mode re-construction. 1 The three methods are seen in Fig. (4.3b) to perform nearly similarly in the forward-projection operation. 4. Reconstruction Implementation and Comparison 107 1.5, .« 1 0.5 (a) 'A th yVy : 1 : : 10 20 30 40 Voxel 50 60 30 40 Voxel (b) Fig. 4.3: (a) Plots of horizontal profiles across the back-projection images depicted in Fig.( 4.2). (b) Projection vector obtained with forward-projection of a uniform circle along 9=0 and 0 = 1 5 ° . 4.2 Subtraction of Delayed Events and Non-negativity Constraints As elaborated and argued for in Sec. (3.5.2), for the HRRT, we have con-sidered the delayed events subtraction method as a practical technique to compensate for presence of randoms in the data. In this work, as explained later, we have imposed an image non-negativity constraint in the list-mode reconstruction algorithms. To better understand the constraint, we have performed studies of whether its incorporation into the reconstruction tasks can improve the overestimation bias that is likely to arise from the sinogram non-negativity constraint often used in histogram-mode reconstructions, as we describe below. We first consider the histogram-mode delayed coincidence subtraction 4. Reconstruction Implementation and Comparison 108 technique: with nf~ues = n\ — n\ from the measured prompts. Due to the Poisson nature of measured prompts and delayed coincidences, it is possible for any bin i to record more randoms than prompts. The constraint nj>0 has normally been imposed on the data in previous work [133,143], such that the value of ri; is set to zero for any L O R along which a negative value is obtained (i.e. the sinogram non-negativity constraint). Nevertheless, in low statistics scans, especially those with large random fractions, it is commonly observed that a noticeable number of sinogram bins exhibit negative trues counts after random correction. Imposing the sinogram non-negativity constraint would in turn introduce an overestimation bias in the reconstructed images. In this regard, we have instead considered using a weaker condition, namely the image non-negativity constraint, such that if upon processing a data subset, the correction to an image voxel is calculated to be negative (due to presence of considerable contribution from negative sinogram bins), the image voxel is not updated for that particular subset2. From Eq. (4.1), one makes the intuitive observation that by replacing the summation over the LORs with a summation over the number N of measured data (i.e. effectively reading.the sinogram counts one by one), and setting ni to 1, if the histogrammed event was a prompt event, and to -1, if it was a delayed-coincidence event, one arrives at the list-mode reconstruction algorithm: 2 Due to the multiplicative nature of the iterative E M algorithm, if image voxels were allowed to have negative values, then the subsequent updates would be adversely affected! (4.2) 4. Reconstruction Implementation and Comparison 109 where 1 A; is a prompt event — 1 A; is a delayed coincidence event (4.3) In this study, we implemented the statistical list-mode reconstruction scheme described by Eqs. (4.2) and (4.3) for the HRRT. We note that unlike truly EM algorithms such as Eq. (3.13) for which there is never any "divide by zero" error, it is possible in the above delayed events subtraction technique that the denominator of Eq. (4.2) would be zero (i.e. forward projection of the current image estimate along the L O R coordinates of a measured event is zero). In such cases, our algorithm neglects the particular events for which this occurs. The algorithm was accelerated using the three schemes presented in Sec. 3.4, namely, the S - L M E M , C S - L M E M and hybrid S / C S - L M E M variations. The image non-negativity constraint, described earlier, was also imposed. We also investigated the parallelization of the list-mode reconstruction code on a Linux cluster. The algorithms were tested with point source measurements as well as data sets spanning a wide range of count rates in order to investigate reso-lution, bias, contrast and noise properties for different data acquisition con-ditions. Convergence properties of the various schemes were also compared in contrast as well as resolution studies. Phantoms used and measurements performed: On the HRRT, the data can be axially spanned in various modes. In list-mode reconstruction, however, as discussed in Sec. (3.1), since the individual LORs are being read one-by-one, rate of processing the emission data in list-mode is independent of the axial span. One would thus be motivated to explore use of no spanning at all (i.e. span 0). However, performing the normalization measurement in span 0 4.3 Methods 4. Reconstruction Implementation and Comparison 110 is extremely time-consuming (as sufficient counts are needed in the LORs), has not yet been implemented on the HRRT, and remains to be studied. Therefore, we have instead investigated the effect of switching in-between axial spans of 3 and 9. In histogram-mode reconstruction, processing the data with an axial span of 3 is more expensive than a span of 9 (by nearly three times), whereas this is not the case in list-mode reconstruction. Two separate experiments were performed as we outline below. In order to allow direct comparison between the various algorithms, consistent axial spans (9 for first experiment and 3 for second experiment) were imposed, while a maximum ring difference of 67 was also used. In the second experi-ment, which involved resolution measurements, the effect of resolution degra-dation by switching from a span of 3 to a span of 9 was also investigated. In all the reconstructions, 16 subsets were used for the accelerated algorithms. The hybrid S / C S - L M E M algorithm consisted of having the first 8 subsets being iterated using the S - L M E M approach and subsequently switching to the C S - L M E M counterpart3. The following experiments and analyses were performed: Experiment 1 - Contrast Phantom This experiment was performed in order to study the robustness of the ran-dom correction technique as well as contrast vs. noise properties of the list-mode technique. A 20 cm long, 10 cm radius phantom was used. The phantom had three 5 cm diameter cylindrical inserts: one was solid plastic, one was filled with water (cold insert) and one was filled with a 1 8 F radioac-tivity concentration of 3.39 / iC i /ml (hot insert). The phantom itself was filled with a 1 8 F concentration of 0.622 / /Ci /ml ('background'), yielding a 3 This is because we have often observed that the image quality improves rapidly for the first half of the subsets in the first iteration of the S-LMEM algorithm, while it begins to exhibit oscillating behavior for the remaining subsets. 4. Reconstruction Implementation and Comparison 111 Index Total Activity(mCi) Trues Rate(kcps) Random Fraction(%) 1 2.7 814 70.9 2 1.8 591 46.8 . 3 1.3 425 32.0 4 0.95 208 16.2 5 0.19 69 7.73 Tab. 4.1: Table of Statistics for Frames used in the analysis hot insert to background ratio of 5.45. The total amount of radioactivity in the scanner field of view (FOV) at the beginning of the scanning procedure was 4.16 mCi. A series of sixteen 20 min long scans was acquired in list-mode, one hour apart. The measurement thus covered 8.7 radioisotope half-lives, yielding a final amount of radioactivity in the F O V of 0.00972 mCi. The subset of the scans listed in table (4.1) was used in the analysis. The listed scans were chosen for analysis so as to cover a random fraction (randoms/trues) range between 8% to 71%, which covers most clinically encountered situations. The following figures of merit were used in evaluating the reconstruction, and random correction methods: i) Quantitative Accuracy of the Random Correction: Reconstructed im-ages were compared with those obtained using the FORE+2D-FBP scheme, since the latter exhibits linearity with measured counts and can be used as a standard for quantitative accuracy. ROIs were selected on the hot and cold as well as the background regions of the reconstructed images for each of the studied frames. Five transaxial planes were selected for the calculation of mean and standard deviation of total counts in the ROIs. Tests of quan-titative accuracy were performed for the 3D-OSEM algorithm (with (a) no random correction, (b) the sinogram non-negativity constraint, and (c) the image non-negativity constraint) as well as the S - L M E M algorithm. 4. Reconstruction Implementation and Comparison 112 ii) Image Random Fractions: For each list-mode reconstructed frame from table (4.1), an image random fraction (RF = randoms/trues) was measured using: imageRF i j rc '3 3=1 3=1 £ A3T7 E (4.4) where A^ c and A " r c are reconstructed image intensities at a voxel j with/without random-correction being performed, respectively. As such, the image random fractions were calculated and compared for images reconstructed using the S - L M E M algorithm. Image R F values are not expected to be numerically equivalent to the random fraction in the acquired data: the true and the random events have a different spatial distribution and will be therefore dif-ferently affected by attenuation and sensitivity corrections. However, the ratio between the image R F and the acquired events R F must not vary as a function of acquisition condition if no count rate or number of counts depen-dent bias is introduced in the data by the random corrections. iii) Contrast vs. Noise Comparisons: Contrast vs. noise studies were per-formed for images reconstructed up to three iterations using the 3D-OSEM, S - L M E M , C S - L M E M and hybrid S / C S - L M E M algorithms. The contrast and the noise were estimated following approximately the N E M A N U 2001 protocol. The percent contrast QH for the hot cylinder was calculated by: = GAH,?A " 1 X 1 0 ° % ^ where CH and CB are the average counts in regions of interest (ROIs) placed on the reconstructed images of the hot insert and the background region, respectively, and A H / A B is the actual concentration ratio between the two re-gions (measured to be 5.45). The percentage noise (standard deviation/mean) was calculated by placing eight ROIs on different parts of the background image and averaging their values to yield a background mean value and its standard deviation. 4. Reconstruction Implementation and Comparison 113 Experiment 2 - Radioactive Paper Source This experiment was performed to study resolution and noise properties of the various interpolation techniques, as well as the ordinary, convergent and hybrid subsetized list-mode algorithms. Using the technique presented in Appendix C, which allows printing of radioactive patterns using a modified standard ink-jet printer, we imaged radioactive ( 1 8 F) point sources of size 0.7 mm placed at X=0,l,2,3,4,5 and 6 cm radially away from the center of the FOV. The sample also included a 1x7 cm rectangular area of uniform activity created for the purpose of monitoring noise behavior. For better visualization, a sample reconstructed image of the radioactive paper source is shown in Fig. (4.4). The middle row of point sources (which were printed over a background were not utilized for analysis in this work. The overall F W H M for any given point was measured by calculation of the root mean squared value of the measured point widths in the transaxial (X, Y) and axial (Z) directions. The percentage noise (standard deviation/mean) for a given reconstructed image was calculated in two ways: 1) Voxel noise: in which percentage variation of the individual voxels along the entire rectangle was measured. 2) ROI noise: in which the activity rectangle was sub-divided into eight small rectangular ROIs, and the percentage variation of the sum of counts in the ROIs was measured. i) Comparison of Projection Techniques: As elaborated in Sec. (4.1.1), three LOR-driven projection methods were considered and implemented for performance of back- and forward-projection on the HRRT: (a) the Sid-don method as well as (b) trilinear and (c) bilinear interpolation techniques. The data were subsequently reconstructed using the aforementioned pro-jection methods (three iterations of the S - L M E M algorithm were applied). Histogram-mode E M reconstruction was also applied to the data, wherein 4. Reconstruction Implementation and Comparison 114 tit -2 -4 -6 -6 -4 -2 0 2 4 6 X-direction (cm) Fig. 4.4: A sample reconstruction of the radioactive paper source (3 iterations of the S-LMEM algorithm). The lower row of points sources as well as the rectangular box were utilized for analysis of resolution and noise properties. projection algorithms used bilinear interpolation and were output-driven (i.e. voxel-driven back-projection and LOR-driven forward-projection) as dis-cussed in Sec. (4.1.1). Subsequently, the resolution (FWHM) and noise prop-erties of the reconstructed images were compared. ii) Convergent List-mode Reconstruction: In order to study convergent list-mode reconstruction algorithms, we performed the following compar-isons: a) plots of measured F W H M vs. iteration (for two selected point sources 1 and 5 cm away from the center of the FOV) were calculated and shown for three iterations of the 3D-OSEM, S - L M E M , C S - L M E M and hybrid S/CS-L M E M reconstruction schemes. b) Plots of measured F W H M vs. radial position were also depicted for these four schemes to compare the final reconstructed F W H M values across the FOV. Procedures (a) and (b) was performed for cases when i) Siddon and ii) bilin-4. Reconstruction Implementation and Comparison 115 ear interpolation techniques were used. c) Plots of noise vs. iteration were also calculated for the 3D-0SEM, S-L M E M , C S - L M E M and hybrid S / C S - L M E M reconstruction schemes. iii) Effect of Axial Spanning: We have also studied effects of degradation in resolution as one switches from (axial) span 3 reconstruction to span 9. The latter is expected to result in poorer F W H M values along the axial (Z) direction and is expected to degrade further with increasing distances from the center of the FOV. Values of overall F W H M vs. iteration were plotted for two point sources at X=3 cm and 5 cm from the center of the FOV: The measured degradation in F W H M (along Z direction) vs. radial position were also depicted for the S - L M E M , C S - L M E M and hybrid S / C S - L M E M reconstruction schemes. Code parallelization: The code parallelization was implemented on the Western Canada Research Gr id 4 , University of British Columbia. It con-sists of 345 x I B M eServers, using dual 3.0 GHz processors (2GB R A M ) . The message passing interface (MPI) software was utilized to parallelize the list-mode reconstruction code. The algorithm essentially consists of having several slave nodes to perform the actual forward and backward projections, results of which are passed to the master node for updating the current image estimate after every subset. 4.4 Results and Discussion Experiment 1 - Contrast Phantom Quantitative Accuracy of the Random Correction: Figs. (4.5) show plots of ra-tios between ROI counts in images reconstructed using 3D-OSEM and those obtained using FORE+2D-FBP. In each figure, three plots are shown corre-sponding to ROIs in the hot, background and cold regions. Figs. (4.5a,b,c) 4 http://www.westgrid.ca/ 4. Reconstruction Implementation and Comparison 116 (c) Fig. 4.5: Calculated ratios between total counts for images reconstructed using 3D-0SEM and those obtained using F0RE-2D-FBP, for the hot (top), background (middle) and cold (bottom) regions of the phantom, when (a) no random correction, (b) the sinogram non-negativity constraint and (c) the image non-negativity constraint have been imposed. For better visibility, the plots have been shifted by ten units with respect to one another. The case of reading 10M counts is shown, and qualitatively similar results have been observed in the 40M and 80M cases. The error bars indicate statistical variation for the five transaxial planes selected. 4. Reconstruction Implementation and Comparison 117 Frame 1 No Random Correction Sinogram Posit. Constr. Hot 3.8% 1.8% Background 21.7% 17.1% Cold 74.0% 65.2% Frame 5 No Random Correction Sinogram Posit. Constr. Hot 0.9% 0.4% Background 1.8% 1.9% Cold 10.4% 10.0% Tab. 4.2: Image Bias for 3D-OSEM reconstruction of Frames 1 and 5 in table (4.1) (10M counts) correspond to cases where 3D-OSEM was performed: (a) without any ran-dom correction, (b) with the sinogram non-negativity constraint and (c) with the image non-negativity constraint. The data reconstructed contained 10M counts. The reason this comparison was performed for the 3D-OSEM al-gorithm was because a similar comparison is not possible in list-mode re-construction, for which the sinogram non-negativity constraint cannot be imposed. In Fig. (4.5c), the values are seen to be consistent (within statistical error) for a wide range of event random fractions (8%-71%). This is seen not to be the case in the former two schemes, especially for the background and cold regions, which contain less trues and therefore are more sensitive to insufficient random correction. One is therefore clearly able to verify that, as predicted, imposing the sinogram non-negativity constraint can introduce an overestimation bias in the reconstructed images, especially for low-statistic scans. Table (4.2) shows percentage increase in ROI counts density for 3D-OSEM reconstructed images of frames 1 and 5 (71% and 8% random fractions) when reconstructed using schemes (a) and (b), when measured in comparison to 4. Reconstruction Implementation and Comparison 118 B a c k g r o u n d ~-i + . ._{ C p l d j j 1 2 3 4 5 Frame Fig. 4.6: Calculated ratios between total counts for images reconstructed using S-LMEM and those obtained using F0RE-2D-FBP, for the hot (top), background (middle) and cold (bottom) regions of the phantom. For better visibility, the plots have been shifted by ten units with respect to one another. The case of 10M counts is shown, and qualitatively similar results have been observed in the 40M and 80M cases. The error bars indicate statistical variation for the five transaxial planes selected. the case of reconstructing with the image non-negativity constraint only. Schemes (a) and (b) are seen to exhibit close values for overestimation bias percentages in each of the hot, background and cold regions, which can be explained by the low-statistic nature of the scans (10M counts) resulting in a notable fraction of histogram bins to measure more randoms than prompts, which are subsequently neglected in scheme (b). Similar qualitative patterns have also been observed in the cases of having 40M and 80M counts, with the difference that bias is seen to become less significant in scheme (b) (e.g. 54% for 40M counts and 35% for 80M counts, in cold region). Testing quantitative accuracy for the list-mode algorithm, Fig. (4.6) shows similar plots of ratios between images reconstructed using S - L M E M and FORE+2D-FBP. The plots are also seen to be consistent (within statistical error) for the hot, cold and background regions. Consequently, the S - L M E M 40 36 o 1 O O 4. Reconstruction Implementation and Comparison 119 Index Events R F Image R F Image RF/Events RF(%) 1 70.9 45.8 65.7 2 46.8 31.0 66.7 3 32.0 21.2 66.8 4 16.2 10.7 65.7 5 7.73 5.19 67.2 Tab. 4.3: Table of random fractions algorithm is seen to preserve reconstructed counts in various parts in the image for a wide range of random fractions. Ratios of Random Fractions: Image random fractions, calculated as de-scribed in previous section, along with random fraction of the acquired events are shown in table (4.3). The two values are not expected to be equal, as explained in the methods section, but for an accurate list-mode random cor-rection technique, they are expected to have same ratios independent of par-ticular frame. As can be seen from the table, there is only a very small change in the ratios over a wide range of random fractions. Contrast vs. Noise Comparison: Monitoring progression of image quality with iteration, Fig. (4.7) shows contrast vs. noise plots for 3 iterations (16 subsets - with results after every 4 subsets shown) of S - L M E M as well as 3D-OSEM on frame 4 (with random fraction of 16%). We have verified, for the wide range of random fractions and total counts considered, that the S - L M E M algorithm performs at least as effectively (in terms of contrast vs. noise) as 3D-OSEM. We have also seen, as shown in Fig. (4.7), that both algorithms exhibit limit cycles, commonly reported in the literature to arise from subsetization of the measured data. Figs. (4.8a,b) show plots of contrast vs. noise for images reconstructed using three iterations of the S - L M E M , C S - L M E M and hybrid S / C S - L M E M algorithms. One is able to clearly observe limit cycles in the ordinary S-4. Reconstruction Implementation and Comparison 120 1 7 1 8 1 9 2 0 2 1 2 2 2 3 1 7 1 8 1 9 2 D 2 1 2 2 2 3 N o I s e ( % ) N o i s e ( % ) (a) (a) Fig. 4.7: Contrast vs. noise plots for images reconstructed with S-LMEM and OSEM. Three iterations are performed with 16 subsets, with results shown every 4 subsets. Results are shown for (a) 20M and (b) 40M total counts. Presence of limit cycles is noticeable in both algorithms. L M E M approach, especially with lower statistics. The C S - L M E M approach, on the other hand, is able to eliminate the observed limit cycles. Using the hybrid technique, in order to combine the advantages of the S - L M E M and C S - L M E M algorithms (as explained in Sec. 3.4.4), the plots show that one is able to attain higher contrast values in fewer iterations, and yet maintain the non-cyclical behavior as the reconstruction proceeds. Experiment 2 - Radioactive Paper Source i) Comparison of Projection Techniques: Fig. (4.9) shows measured resolution values for the various points across the FOV, upon application of three iterations of the S - L M E M algorithm to the data. For comparison, results of application of histogram-mode E M re-construction are also shown, wherein voxel-driven back-projection and LOR-driven forward-projection (referred to as output-driven projection) were used, 4. Reconstruction Implementation and Comparison 121 60 55 h fr« 50 •40 h 35: A S - L M E M = i C S - L M E M • ' : Hybr id 14 16 : 8 20 22 24 26 Noise(%) (a) 60. 55 £9 50 45 4 o ; 35: A S . - L . M E M = I G S - L M E M • Hybr id 14 16 18 20 22 24 26 NoFse(%) (b) Fig. 4.8: Contrast vs. noise plots for images reconstructed using the S-LMEM (dot-ted), CS-LMEM (- -) and hybrid S/CS-LMEM (solid) algorithms with scan durations containing (a) 10M and (b) 5M total counts. Three iter-ations are shown. 2 6 i _ ^ , , , , , ^ 0 1 2 3 4 5 6 Radial Position (cm) Fig. 4.9: Plots of measured resolution vs. radial position for the S-LMEM algo-rithm implemented using the Siddon algorithm as well as bilinear and trilinear interpolation techniques. For comparison, results of output-driven (as described in text) histogram-mode OSEM reconstruction are also shown. Three iterations were performed. 4. Reconstruction Implementation and Comparison 122 Iteration Iteration (a) Voxel noise (b) ROI noise Fig. 4.10: Plots of noise vs. iteration for the reconstruction schemes described in the caption of Fig. (4.9). as recommended by previous investigators (see discussion in Sec. 4.1.1). As a side note: one is also able to observe space-variance of the point spread function, manifesting itself as a degradation in resolution as one moves away from the center of the F O V (seen in all the reconstruction tasks). This effect occurs due to a higher probability of inter-crystal penetration with higher angles of radiation incident on crystal fronts. Depth-of-Interaction (DOI) encoding is known to improve this problem, but has not reached com-plete space-invariance. In Appendix B, we present an approach to model the space-variance and anisotropicity of the point-spread function into the system matrix of the E M algorithm for the HRRT. Similarly, noise vs. iteration plots are shown in Fig. (4.10) for the afore-mentioned reconstruction algorithms, wherein (a) voxel-noise and (b) ROI-noise were considered, as described in the methods section. Three main observations can be made with respect to these figures: 1) Clearly, the Siddon technique performs noticeably poorly compared to the other interpolation methods (especially in terms of resolution). 4. Reconstruction Implementation and Comparison 123 2) Trilinear and bilinear interpolation techniques perform nearly similarly. 3) Histogram-mode reconstruction with output-driven projections does not perform better than list-mode reconstruction with LOR-driven projections. The current implementation of the trilinear interpolation technique is >3-4 times slower than the Siddon method, while Bilinear interpolation is comparable to the latter (only around 20% slower). This has therefore given us sufficient motivation to perform our reconstructions using projection al-gorithms which employ bilinear interpolation. ii) Convergent List-mode Reconstruction: Figs. (4.11a,b) show plots of reconstructed F W H M width vs. iteration for point sources located at X = l cm and 5 cm from center of FOV, with the data reconstructed using the 3D-O S E M , S - L M E M , C S - L M E M and hybrid S / C S - L M E M algorithms. Bilinear interpolation was used in the projection algorithms. The values of F W H M resolution are seen to change in a cyclical manner for the 3D-OSEM and S - L M E M algorithms. In Fig. (4.11a), for instance, the F W H M width reconstructed using the S - L M E M approach is seen to os-cillate between a low of 3.17 and a high of 3.23mm. Nevertheless, one clearly observes that in the C S - L M E M approach, due to its converging behavior, the F W H M widths improve with further iterations in a systematic and pre-dictable manner. One is also able to observe that the hybrid approach results in a faster de-crease in reconstructed F W H M width with less iterations while maintaining the non-cyclical behavior. For comparison purposes, similar plots are shown when using the Siddon method, as seen in Figs. (4.11c,d). Clearly, bilinear interpolation is seen to result in superior image qualities. Fig. (4.12) shows plots of measured F W H M values after four iterations for all the seven points located at X=0,l,2,3,4,5 and 6 cm from the center of FOV. We note from the plots that the histogram-mode and list-mode algorithms are able to achieve nearly similar F W H M values for a given point. Results 4. Reconstruction Implementation and Comparison 124 2.61 1 1 1 ' 1 1 2.61 1 ' ' ' 1 1 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 Iteration Iteration (c) Siddon — X = 1 cm (d) Siddon — X = 5 cm Fig. 4.11: Plots of reconstructed F W H M width vs. Iteration are shown for the 3D-OSEM, S-LMEM, and the CS-LMEM and hybrid algorithms for the . point source located at (a) 1 cm and (b) 5 cm from center of the FOV. (c) Final F W H M values are shown for all the reconstructed points sources using the aforementioned four schemes (after 3 iterations). 4. Reconstruction Implementation and Comparison 125 - • - 3D -0SEM 2 6 i — , , , , , , , — i 2 6 i — , , , , , , ,_ 0 1 2 3 4 5 6 0 1 2 3 4 5 6 Radial Position (cm) Radial Position (cm) (a) Bilinear (b) Siddon Fig. 4.12: Plots of reconstructed F W H M width vs. Iteration are shown for the 3D-OSEM, S-LMEM, and the CS-LMEM and hybrid algorithms for the point source located at (a) 1 cm and (b) 5 cm from center of the FOV. (c) Final F W H M values are shown for all the reconstructed points sources using the aforementioned four schemes (after 3 iterations). le n IU - • - 3D-0SEM S-LMEM - • - CS-LMEM Hybrid 11 \ 11 \ il il 0) %5 Z a B c 0 o £ 3 0.5 1.5 , Iteration 2.5 (a) Voxel noise -•- 3D-OSEM -•- S-LMEM -•- CS-LMEM -*- Hybrid 0.5 1.5 Iteration 2.5 (a) ROI noise Fig. 4.13: Plots of reconstructed F W H M width vs. Iteration are shown for the 3D-OSEM, S-LMEM, and the CS-LMEM and hybrid algorithms for the point source located at (a) 1 cm and (b) 5 cm from center of the FOV. (c) Final F W H M values are shown for all the reconstructed points sources using the aforementioned four schemes (after 3 iterations). 4. Reconstruction Implementation and Comparison 126 3.7 3.6 ? £ 3 . 5 § 3 . 4 TD £ I 3.31 3.2| 3.1 - • - X=5cm span 3 :1\ H H X=5cm span 9 !U - • • X=3cm span 3 - • • X=3cmspan9 0.5 1 1.5 Iteration 2.5 (a) 0.3 o jjj 0.25| b N 0.2| 0.15| § 0, | 0.05 IL < 0 -0.05' •*• S-LMEM •• • CS-LMEM •a- Hybrid J (b) 1 2 3 4 5 Radial Position (cm) Fig. 4.14: (a) Overall F W H M vs. iteration for images reconstructed using the hy-brid S / C S - L M E M algorithm with axial spans of 3 and 9: point sources located at 3 cm and 5 cm from center of F O V are shown, (b) Degra-dation in F W H M (along Z-direction) vs. radial position as one switches from axial span 3 to 9: all the point sources reconstructed using the S-L M E M , C S - L M E M and hybrid schemes (after 3 iterations) are shown. when applying the (a) bilinear and (b) Siddon methods are shown. The bilinear interpolation method is seen to outperform the Siddon technique for all the point sources. Furthermore, we note that the convergent and hybrid techniques perform at least as well as (if not better than) the regularly subsetized list-mode and histogram-mode E M algorithms. Noise vs. iteration plots are shown in Fig. (4.13) for the aforementioned reconstruction algorithms (using the bilinear technique), wherein (a) voxel-noise and (b) ROI-noise were considered. The convergent C S - L M E M algo-rithm converges very slowly, compared to the other algorithms, and after the first iteration exhibits poorer F W H M values. On the other hand, the hybrid algorithm is seen to be suitable for reconstructions where early termination is used (for time-cost considerations) as it reaches noise levels comparable to (and possibly, voxel-wise, better than) other non-convergent algorithms. 4. Reconstruction Implementation and Comparison 127 iii) Effect of Axial Spanning: Plots of F W H M vs. iteration for point sources at X=3 cm and 5 cm from the center of the F O V are shown in Fig. (4.14a). The image has been reconstructed using the hybrid algorithm with axial spans of 3 and 9. It is seen, as expected, that the effect of axial spanning is more significant for the point source more distant from the center of the FOV. To see this more clearly, Fig. (4.14b) shows plots of final F W H M values (along the Z-direction) for the various point sources reconstructed using the S - L M E M , C S - L M E M and hybrid algorithms. The resolution along the Z-direction is seen to degrade further as one moves away from the center of the F O V for the various schemes. We have also checked the resolution along the transaxial direction and have observed no change to occur upon switching in-between the axial spans, as expected, since this is only expected to affect the resolution in the Z-direction. Parallelization of the Reconstruction Task: Parallelization was success-fully implemented using MPI for the list-mode algorithms. It was found that having the slave nodes indirectly accessing list-mode data did improve the efficiency compared to the case of the master node distributing the ac-tual workload to all the processors. This required multiple read access to a single file as supported by the cluster. With this data on the cluster, us-ing 16 subsets and 16 processors, we are able to reach a processing rate of 0.5-l.OGB/hour, depending on the number of counts per subset5. 4.5 Dynamic PET List-mode Image Reconstruction In this section, we present an investigation into the extension of the aforemen-tioned list-mode image reconstruction techniques developed for the H R R T to dynamic P E T imaging (see discussion in Sec. 1.5). Our dynamic scheme consists of independently reconstructing the events obtained withing each in-5 With too few counts per subset (i.e. too many subsets in the data), the task is not as efficiently parallelized. 4. Reconstruction Implementation and Comparison 128 dividual timing sequence (i.e. dynamic frame), with quantitative corrections applied to each generated image. The extension from 3D imaging to dynamic (or 4D) imaging necessarily requires corrections for the following factors: i) Detector deadtime (see also Sec. 1.6.5): Currently, the deadtime correc-tion scheme used on the H R R T involves a global scaling of the reconstructed images within each frame as determined by the average singles rates within each frame6. The scheme involves application of the following deadtime mul-tiplicative factor to the reconstructed images: deadtime factor = exp(aS') (4.6) where S indicated the measured singles rates per block (the H R R T measures the singles rates only at the block level, see Sec. 3.5.1 for implications), and a is a factor determined experimentally to be 7 .7004xl0 - 6 when LLD=350 ke-Vand 8.94xl0" 6 when LLD=400 keV(LLD is the Lower Level energy Dis-criminator, events with energies below which are rejected by the scanner7). ii) Decay of radioactivity: As mentioned in Sec. (1.6.6), decay correction is relatively the most trivial of all corrections to be applied to reconstructed images. It requires a global scaling of the final reconstructed image by a factor determined by the start and end times within which the events are acquired8. 6 It has been determined by the manufacturing company that singles rates are not noticeably affected by deadtime effects, compared to the deadtime saturation effect en-countered by the coincidence events. Thus, the major bottleneck in the processing can be attributed to the coincidence detection system which saturates the acquired coincidences globally, as mentioned in Sec. (1.6.5). 7 Most HRRT scanners, including the Vancouver HRRT, are currently setting the LLD to 400 eV. 8 For a given dynamic frame (counts within which are reconstructed into a single im-age) in ordinary S-LMEM reconstruction, all-but-the-last subsets in the frame merely contribute to improved image estimates, and it is only the counts in the very last subset which determine (quantitatively) the final image. This implies that we need only im-plement deadtime and decay corrections for the counts occurring within the durations spanned by the very last subset. In other words, scaling the previous image estimates for 4. Reconstruction Implementation and Comparison 129 4.6 Methods and Results In what follows, we study the quantitative accuracy of our dynamic list-mode image reconstruction technique, and compare it to the ones readily available for the HRRT: namely (i) FORE+2D-FBP, and (ii) 3D-OSEM. Contrast Phantom: We used a phantom similar to the one described in experiment 1 of Sec. (4.3). The phantom was this time filled with a n C concentration of 0.311 /xCi/ml ('background'), with a 1.61 / /C i /ml activity in the hot insert. The 'cold insert' was filled with only water. A n initial trues rate of 1.6 Mcps were observed. Eighteen dynamic frames (each 5 minutes in duration) were considered and 16 subsets were used in each reconstruc-tion. The measurement thus covered 4.4 radioisotope half-lives. The random fraction was 25% in the first frame and 5% in the last frame. The following comparisons were performed: i) Time activity curve (TAC) comparisons: Plots of mean reconstructed voxel intensity (for the three hot, cold and background regions) were obtained for all the reconstructed frames. For a quantitatively accurate reconstruction algorithm, the T A C curves are expected to be constant. ii) Axial profile comparisons: Mean reconstructed voxel intensities within each axial plane (for the hot, cold and background regions) were plotted as a function of the axial plane. The corresponding curves for the various frames were overlayed for a visual comparison. iii) Contrast recovery comparisons: The percent contrast for the hot cylin-der was calculated using Eq. (4.5). The plots were depicted as a function of the number of acquired counts per frame. deadtime and decay effects has no impact, since A ' + 1 is not affected by scaling of A* in the E M algorithm. In convergent C S - L M E M reconstruction, however, images reconstructed from each subset contribute (additively) to the final image, and therefore correction factors need to be applied to all the intermediate images. 4. Reconstruction Implementation and Comparison 130 10 10 counts per frame (million) (a) List-mode Background 0 o » o o 6 o o o o o o « o o o < 10 counts per frame (million) (b) FORE+2D-FBP £ 0.061 Hot.-., . Background . >. O o O': '0:o" 9 • . • • ' • • « • • • ' • 0 ' C o l d - ; . ^ v • • • • * * 10 10 counts per frame (million) (c) O S E M Fig. 4.15: Time activity curves (TACs) for 18 frames each 300 s in duration. First frame has a random fraction of 25%, last frame 5%. One iteration and 16 subsets are used in list-mode and OSEM reconstructions. 4. Reconstruction Implementation and Comparison 131 ROI total counts per Z-plj ROI total counts per Z-plane &2\ I 20 40 60 80 100 120 140 160 180 200 Axial Plane 20 40 60 80 100 120 140 160 180 200 Axial Plane (a) List-mode (b) FORE+2D-FBP ROI total counts per Z-plane 20 40 60 80 100 120 140 160 180 200 Axial Plane (c) O S E M Fig. 4.16: Axial uniformity comparisons: profiles were drawn through the hot (top), cold (bottom) and background (middle) regions in each of the reconstructed images. Results: Fig. (4.15) shows images of time activity curves obtained for the three reconstruction schemes: (a) S - L M E M (b) FORE+2D-FBP and (c) 3D-OSEM. The list-mode algorithm is seen to yield a relatively flat T A C , similar to the FORE+2D-FBP case. However, the histogram-mode scheme yields time activity curves which (especially for the cold and background regions) increases with higher count-rate frames. This is because the cur-rent implementation of dynamic histogram-mode reconstruction imposes a sinogram non-negativity constraint, which as we have shown in previous sec-4. Reconstruction Implementation and Comparison 132 tions, results in a positive bias with frames with higher activities (thus higher random fractions). Axial profiles of the mean ROI activity have been depicted in Fig. (4.16) for the hot (top), cold (bottom) and background (middle) regions. The profiles for the 18 frames have been drawn overlaying one another. The list-mode algorithm is clearly seen to outperform the FORE-f-2D-FBP algo-rithm in terms of axial uniformity. We note here that the power of statistical Poisson modeling of the measurement process clearly exhibits itself on the HRRT, wherein, due to the enormous number of LORs, typical dynamic frames, rarely exhibit counts per L O R to be more than 1 or 2, thus ren-dering the E M algorithm (as compared to Gaussian techniques, and simple analytic methods) a very powerful solution to the task of image reconstruc-tion. This exhibits itself particularly in terms of the reconstructed noise, axial uniformity, etc. as we observe here. Furthermore, due to the afore-mentioned zero-thresholding bias, the O S E M algorithm (distributed by the manufacturing company) does not yield overlapping axial profiles for the various frames, and performs poorly. Calculated percentage contrasts are depicted in Fig. (4.17) for the various frames. Best uniformity is observed for the list-mode technique, while the O S E M approach (with zero-thresholding of negative sinogram bins) performs poorly, as we also observed in Fig. (4.5b) and commented in Sec. (4.4). 4.7 Conclusion Practical statistical list-mode reconstruction algorithms were developed for the high resolution research tomograph (HRRT). A n image non-negativity constraint was employed in the algorithm. This constraint was shown, us-ing comparisons in histogram-mode 3D-OSEM reconstruction, to effectively remove the overestimation bias typically encountered when a sinogram non-4. Reconstruction Implementation and Comparison 133 Comparison of Percentage Contrast Comparison of Percentage Contrast : Hot contrast Cold Contrast • . : Hot contrast. • Cold Contrast 10 1 0 ' counts per frame (million) (a) List-mode 10 10 10 counts per frame (million) (b) FORE+2D-FBP Comparison of Percentage Contrast 3 f 60 .•I Hot contrast Cold Contrast counts'per frame (million) (c) O S E M Fig. 4.17: Plots of percentage contrast for the various dynamic frames. 4. Reconstruction Implementation and Comparison 134 negativity constraint is used. Results indicated that images were comparable in terms of statistical properties, contrast vs. noise as well as resolution with images generated by 3D-OSEM. Furthermore, robustness of the list-mode technique was demonstrated in dynamic P E T image reconstruction, upon the inclusion of appropriate deadtime and decay correction factors (demon-strating the quantitative accuracy of the technique). Three LOR-driven projection techniques were considered: (a) the Sid-don method as well as (b) trilinear and (c) bilinear interpolation methods. We were able to demonstrate that the Siddon approach performs poorly compared to the other two algorithms (especially in terms of F W H M ) . Our method of choice was therefore the bilinear approach as it was considerably faster than the trilinear interpolation technique and produced comparable image qualities. Using regular subsetized list-mode reconstruction, we were able to ob-serve limit cycles: oscillatory alternations in image quality parameters (such as contrast and resolution) with, further subsets and iterations into the data, similar to what is commonly encountered in ordered subset histogram-mode reconstruction. To address this issue, we implemented a convergent list-mode E M reconstruction algorithm, based on previous work [121,122,135], and in-vestigated its properties using experimental P E T data. It was demonstrated that the algorithm is robust and does not result in limit cycles. A proposed hybrid algorithm was shown to combine the advantages of the ordinary and the convergent list-mode algorithms (i.e. it was able to reach a higher image quality in fewer iterations while maintaining a convergent behavior), making it a good alternative to the ordinary subsetized list-mode E M algorithm. 5. Motion Compensation: Beyond the Event-Driven Approach 135 5 . Motion Compensation: Beyond the Event-Driven Approach This chapter represents the core of this thesis work in the proposal and vali-dation of accurate and practical motion compensation techniques applicable to high resolution 3D P E T . The proposed techniques along with the experi-mental results have been published in [41]. 5.1 Introduction Recent developments in 3D positron emission tomography (PET) systems have enabled the spatial resolution to reach the 2-3mm F W H M range. With such improvements in spatial resolution, small patient movements during P E T imaging become a significant source of resolution degradation. One method to correct for patient movement involves gating of detected events into multiple acquisition frames (MAF) , with the use of an external moni-toring system, followed by spatial registration and then summation of recon-structions from the acquired frames [145,146]. However, the major limitation of the M A F approach is that it is unable to correct for motion during a frame. Subsequently, in the presence of con-siderable movement, one may need to acquire many low-count frames, and subsequently spatially align them, to better correct for patient motion. Lack of an adequate number of acquired events in the individual frames can in turn adversely affect the quality of the final reconstructed image, and an increased number of frames will also lead to increased (histogram-mode) reconstruction times. 5. Motion Compensation: Beyond the Event-Driven Approach 136 Correction of individual lines-of-response (LORs) for motion has alter-natively been suggested to achieve optimal motion correction [147]. This is what we shall refer to as the event-driven approach, by which it is meant that in either of the histogram-mode or list-mode reconstruction techniques, motion correction is performed by transforming the LORs along which the events are measured to where they would have been measured had the object not moved. To this end, motion-tracking systems have been used for accurate real-time measurements of position and orientation of the patient (see, for instance, Ref. [148] for Polaris, a system based on opto-electronic position sensitive detectors). In the histogram-mode approach, this introduces an event-by-event rebinning technique, resulting in motion-compensated sino-grams, which are are subsequently reconstructed using any of the common reconstruction algorithms [149]. However, it can be argued that the purely event-driven approach can produce image artifacts as compared to a more comprehensive modeling of the image-data relation, as we also demonstrate experimentally in this work. This is because one can observe [150,151] that certain events that would have been detected in some LORs may have exited the P E T scanner (e.g. axially or through detector gaps) undetected due to object motion. Regular recon-struction methods do not employ knowledge of missing data, and therefore would assume simply that nothing was detected. One further notes that the converse also requires consideration: • upon transformation of certain LORs along which the events are detected to LORs along which the events would have been detected, had the object not moved, the calculated LORs may correspond to no actual detector pairs. In other words, motion can result in some LORs which correspond to undetectable regions to exhibit non-zero probabilities of detection, which also needs to be properly modeled. Taking these effects into consideration, it is therefore not sufficient to 5. Motion Compensation: Beyond the Event-Driven Approach 137 merely correct the events for motion, rather one must also modify the algo-rithms themselves. Sections 5.2 and 5.3 discuss incorporation of a comprehen-sive modeling of motion into the histogram-mode and list-mode E M recon-struction. Both sections also describe (subsection B) an alternative, image-space based method that has the potential of being considerably faster in presence of frequent motion especially in high resolution tomographs. Meth-ods and results for experiments on transaxial and axial motion are presented in sections 5.4 and 5.5. 5.2 Motion Correction in Histogram-mode EM Reconstruction Similar to the previous chapter, denoting \™ as the image intensity estimate in voxel j (j=l...J) at the mth iteration, and as the number of events detected along L O R i (i=l...I), the histogram-mode E M algorithm can be written as XT 1 2^i=l Pi] i=l 2^b=lPibAb where Pij is the probability of an emission from voxel j being detected along L O R i, and constructive summation is performed over those LORs for which Pij^O. In the weighted scheme, the system matrix P=(pi3)ixj can be written as P=WG where G=(gi3)ixj denotes the geometric probability of an event generated at voxel j to be detected at L O R i, and the diagonal matrix W=(wi)ixi allows a weight to be assigned to each LOR, to incorporate nor-malization and attenuation effects. One then arrives at: Xm 1 n-53i=i ^%9ii i=i 2^6=1 9ibAr wherein the sensitivity correction factor Sj=Y2i=iPij is a summation over all possible measurable LORs (i=l...I) and calculates the probability of an 5. Motion Compensation: Beyond the Event-Driven Approach 138 emission from voxel j being detected anywhere. Here we make an observation: For any L O R i not corresponding to actual detector pairs, Wi is zero and therefore may not be canceled in the numerator and denominator of the E M algorithm. However, this is not a point of concern since such an L O R would not receive any counts, i.e. rii=0, and therefore, the back-projection summation is performed only over those LORs that can correspond to actual detector pairs. This will not be the case in presence of motion, as we discuss later 5.2.2 Modification of the Histogram-mode EM Algorithm The event-driven histogram-mode motion correction E M scheme (as for in-stance elaborated in Ref. [152]) consists of reading each list-mode event, producing correction factors (CFs) for the particular event (such as normal-ization), transforming the L O R in which the event is detected into the L O R in which the event would have been detected had the object not moved, and finally histogramming the event at this calculated L O R along with the application of the CFs. The motion-corrected sinograms, which have also been pre-corrected for the CFs, are subsequently reconstructed using the unweighted E M algorithm given by Eq. (5.1). Once a motion-corrected sinogram is obtained, some LORs may be trans-formed into undetectable LORs (i.e. LORs for which no actual pair of detec-tors exist). On the other hand, some detectable LORs exhibit fewer counts than they would have if the object had not moved, since corresponding events have passed through undetectable LORs. These missing LORs can in prin-ciple be located: i) radially out of the field of view, ii) axially out of the scanner, iii) exceeding the maximum allowed ring difference in the scanner's acquisition mode, or iv) in regions corresponding to gaps in between the detectors. 5. Motion Compensation: Beyond the Event-Driven Approach 139 L O R Object rn JM-. Fig. 5.1: An event detected at an LOR i' generated in voxel j, which has been translated to voxel j' = M ( j ) at time of detection, would have been detected at LOR i = if the object had not moved. The first case may be ignored as it can be safely assumed that the object stays in the radial field of view of the scanner all the time, whereas the second and third cases are especially important in 3D P E T imaging in which small rotations in the object can result in many LORs to exit the F O V axially or exceed the maximum allowed ring difference, and vice versa. The last possibility is expected to be significant in non-cylindrical designs. In the octagonal design of the high resolution tomograph (HRRT) [35], for instance, gaps existing between the eight detector heads of the scanner occupy over 10% of the sinogram space, as shown for a typical sinogram in Fig. (1.14). One recently suggested method by Thielemans et al. [151] involves scaling of counts recorded in the motion corrected sinogram bins, where the scale factors are computed using averaging of L O R weighting factors. This can be thought of as a "motion pre-correction" technique, which requires consider-ation of noise enhancement due to small scale factors division. The authors have then suggested use of forward-projection of an initial estimate of the im-age, in combination with or replacing the calculated scale factors (by which 5. Motion Compensation: Beyond the Event-Driven Approach 140 the data are divided) when these factors are below a certain threshold. We now note that such consideration can alternatively be fully addressed by modifying the system matrix itself so as to take motion into account ("mo-tion weighting" of the system matrix as compared to "motion pre-correction" of sinograms). This, as we shall demonstrate, results in modification of the sensitivity correction factors, and can potentially result in less noise artifacts. This is because, compared to individual scaling of sinogram bins, only the summation of motion-compensation factors over all directions is used1. Fur-thermore, as we demonstrate in Sec. 5.2.3, calculation of such correction can be performed in the image-domain, compared to the LOR-domain, and can thus potentially result in considerable speed-up of the algorithm. In this regard, we first introduce an invertible operator & which models the motion of the object by transforming the L O R i along which an event would have been detected in absence of any motion, to the L O R i' along which the event is detected at time t. A n instance of this is shown in Fig. (5.1), where an event generated from a voxel j, currently located at a position j' due to some transformation M( j ) , has been detected along an L O R i', and therefore must be histogrammed along the motion-corrected L O R i=£j~1(i'), as is shown. We next introduce g\,^ as the motion-dependent geometric probability of detecting an event generated in voxel j along an L O R i! at a given time t. The superscript t indicates knowledge of object orientation and position with 1 This observation is conceptually very similar to relative noise reduction in "weighted" schemes {attenuation/normalization correction) as compared to pre-correction schemes. Instead in this work, we are concerned with motion correction. 5. Motion Compensation: Beyond the Event-Driven Approach 141 respect to the origin of the time axis' i 2 . Also, defining 1 if L O R i corresponds to a detector pair 0 otherwise (5.3) such that \i\5i = 0} is the set of all LORs that correspond to no actual detector pairs, we write where we use gjj to calculate the purely geometric overlap between a voxel j and an L O R i, and Si is used to incorporate whether/not the L O R i can be detected by the scanner. Next we note that according to the frame of reference of a given voxel j , from a purely geometric point of view, an L O R %' at time t is equivalent to a calculated L O R i=£jt~1(i') at time t=0, since all LORs at time t=0 have now been transformed by the motion operator £> t{}. We thus note that the geometric overlap between a voxel j and an L O R i' at time t, is equal to the overlap it would have had with the calculated L O R i had the object not moved (which is the case at t=0); that is " Denoting Ni and Ai as the attenuation and detector normalization factors for an L O R i, incorporation of these factors in presence of motion must now be addressed. In the unweighted scheme, these factors are applied onto the acquired L O R events that are histogrammed into appropriate motion-compensated sinogram bins. Alternatively, in weighted schemes, one or both of these factors are instead used inside the reconstruction algorithm itself. The normalization factor for an L O R i' along which an event is detected 2 It is often assumed that the patient does not move between the attenuation scan and the start of the emission scan. Nevertheless, the more general case of having the patient move between the two scans can be treated by motion-correcting with respect to the time at which the attenuation scan was performed. 9ii = 9iA (5.4) 9i'j=9ij w h e r e i = &t1(i') (5.5) 5. Motion Compensation: Beyond the Event-Driven Approach 142 Time=t, Af\Nt A r \ N v Af\Nr Fig. 5.2: An event detected at time ti along an LOR i', which would have been detected along an LOR i if the object had not moved, has the same atten-uation correction factor as measured for Ai at time t\. The normalization correction factor however is LOR-specific and changes with motion. is given by the value of JVj/ for the L O R itself, independent of any motion. However, this is not the case for attenuation correction, in which case the factor is given by the value of the attenuation factor at L O R i along which the event would have been detected had the object not moved. This is depicted in Fig. (5.2). It follows that the weighting factor w{, for any L O R z'=£(i) can be writ-ten as Ni, N-weighted AiNi< AN-weighted ( 5 . 6 ) Introducing P=(Pij)ixj as the motion-compensated system matrix, p^ must indicate the probability, for the course of the entire scan, that an event generated in object voxel j is finally binned into L O R i. It must therefore in-corporate time-weighted probability-of-detection (w^gf^) contributions from 5. Motion Compensation: Beyond the Event-Driven Approach 143 any L O R i! that could record events that would have been detected in L O R i had the object not moved; i.e. 1 rT Pij = fj0 wi'9\'jdt (5-7) It then follows that Eq. (5.7) combined with Eqs. (5.4) and (5.5) can be written as Pij ±g% jT'w^dt (5.8) In the AN-weighted scheme, for instance, upon replacing p^ in Eq. (5.1) with and dropping the superscript in for convenience, one obtains, after cancellations, the following AN-weighted reconstruction algorithm: A r ' - f E f e w ™ (5-9) b3 i-1 Zv6=l !JibAb where h=XJl9ijAi FN^dt (5.10) 1 i=i J o 5.2.2 A Note on Convenience and Accuracy It must be noted, in histogram-mode reconstruction,, that in order to make use of all measured events, the regularly employed sinogram-space has to be extended in order to allow histogramming of all motion-compensated LORs, including those that do not correspond to existing detector pairs. In other words, the back-projection summation in the numerator of Eq. (5.9) needs to be performed over all observed counts, and not merely those that corre-spond to detector pairs 3. List-mode reconstruction of the data, therefore, is somewhat more convenient, in that such sinogram extension is not re-quired since the individually measured events are motion-compensated and processed one-by-one. 3 Whenever n^O, it is necessarily the case that p^-^O and therefore cancellation of weighting components of p^ in the back- and forward-projection steps is valid 5. Motion Compensation: Beyond the Event-Driven Approach 144 Another observation may also be made: the L O R i (calculated from the measured L O R i') does not typically correspond exactly to the center of a sinogram bin and an interpolation needs to be performed upon histogram-ming the event into motion-corrected sinograms. However, as described in Sec. (5.3), list-mode event coordinates, upon being motion-corrected, can be maintained as continuous variables in list-mode reconstruction, therefore pointing to a potential advantage in terms of inherent accuracy in list-mode reconstruction. 5.2.3 An Alternate, Fast Method for Calculation of Sensitivity Factors For a given voxel j, the task of calculating the motion-weighted sensitivity correction factor Sj (Eq. 5.10) requires integrating over the entire duration of the scan to derive the properly motion-corrected normalization factors, which are subsequently back-projected along with the attenuation factors. This calculation can therefore be very time-consuming for frequent motion in high resolution scanners. In what follows we propose an alternative approach in the calculation of motion-weighted sensitivity factors, which can potentially result in very efficient calculation times for high resolution scanners, as we explain. The technique is applicable to the case where all measured LORs are histogrammed and considered in the reconstruction task, including those that do not correspond to actual detector pairs after being corrected for motion. We make the following very useful observation: at any given time t, cal-culation of the time-dependent geometric factor gj,j can be performed in another way: instead of mapping L O R i' into a motion-corrected L O R i (the approach taken in Eq. 5.5), one can map the object voxel j to the new voxel f it has moved to at time t; i.e. (5.11) 5. Motion Compensation: Beyond the Event-Driven Approach 145 where M t is an image-space based motion-tracking operator, as depicted in Fig. 5.1. Mathematically, we have the following identity: 9ij = 9i'j' (5-12) The above relation can lead to considerable speed increase in calculation of Sj for schemes in which data are pre-corrected for attenuation (i.e. unweighted and N-weighted schemes), as we show here. In the N-weighted scheme, for instance, we have Aj ^ - r Tli/Ai 9ij — s m bj j = i 2^b=i9ibAb where Sj is given by 1 ^ T 1 T* ^ = 7rT,9n ( N A ' d t =r[ T,9ijNA>dt (5.14) T i=i J o T J o » = i We next note that the summation over all i in the above equation involves consideration of all i'=L(i), and thus the summation index i can as easily be replaced by i'. Combining this observation with Eq. (5.12) gives s3 = ^ fTY.9i,rNA>dt (5.15) and therefore 1 fT Sj = — I Sjdt where Sj = Y^9VjNiA' = (5-16) 1 J o i' i while noting that s • is simply the standard sensitivity correction factor which needs to be calculated only once for all voxels. In other words, in the calculation of Sj for any voxel j, instead of having to perform time-averaging in the L O R domain, one can do so in the image domain, by evaluating sensitivity factors (which are calculated once for the object at t=0) at voxels j' over time. This corresponds to tracing the motion of voxel j with time, as measured by the motion-tracking system. 5. Motion Compensation: Beyond the Event-Driven Approach 146 Comparing Eqs. (5.9) and (5.13), we make the following observations: (I) The first proposed algorithm does allow for incorporation of attenuation into the system matrix, whereas the latter requires pre-correction of the emission data for attenuation. This can result in an increase in image noise (e.g. see Ref. [133]), though less increase is expected with better accuracy and statistics in the attenuation and emission measurements. On the other hand, the gain in computation efficiency, especially in full motion compensation, can be very significant, as we shall argue for. (II) The sensitivity correction factor in the first algorithm (Eq. 5.10) involves, for each L O R i, a time-integral over N^Si' for all LORs i' along which L O R i has come to be aligned due to motion at some point during the scan. In other words, the motion-tracking operator C t {} must be applied for all the times t considered to every L O R i. Following this time-averaging, the resulting factors are back-projected. In the second algorithm, on the other hand, the sensitivity correction factor (Eq. 5.16) is given by a time-integral over the standard factors Sj (which is computed once by back-projection of NiSi) for all voxels j' to which voxel j had moved at some point during the scan. Since with current, high-resolution scanners, the L O R domain is typically much larger than the image domain (for instance, in the HRRT, with span 3 and maximum ring difference of 67, one deals with ~470M LORs whereas one has 256x256x207~14M image voxels), the second approach implies a consid-erable potential speed-up in full motion compensation of motion-corrected sinograms. This speed advantage increases with the number of motion-monitoring intervals (i.e. with degree of "exact-ness" in monitoring of motion) used in the computation. Nevertheless, scanner, motion and task-dependent studies remain to be performed in order to investigate the trade-off between image noise (first observation) and computation speed (second observation). As we shall show, this technique will also be applicable to list-mode recon-5. Motion Compensation: Beyond the Event-Driven Approach 147 struction. 5.3 Motion Correction in List-mode EM Reconstruction One must observe that while motion-compensated histogramming involves interpolating the transformation C^ 1(z /) into an actual sinogram bin i, in the case of list-mode reconstruction one can work with L O R coordinates, as compared to rebinned L O R positions, thus potentially preserving a higher de-gree of accuracy in the reconstruction task. In other words, this improvement results directly from better sampling of the measurement. In this regard, the probability element g^ can be replaced by the operation Sj(i), which calcu-lates the geometric probability of an event generated at voxel j to occur at L O R \=LTl(i'), a continuous variable, holding the exact coordinates of L O R i' after being motion-corrected. The list-mode expectation maximization (LM-EM) reconstruction algo-rithm has been previously formulated by Parra and Barrett [92]. Investi-gation of the applicability of the algorithm and its accelerated versions are nowadays common in the literature (e.g. see Refs. [70,124,153]). Follow-ing the list-mode approach, we are able to show in Appendix D that full consideration of motion results in the following E M algorithm: \m N 1 A™+1 = Y S7-(ifc)—1= —— (5.17) where ik—^t1'(i'k) with i'k denoting the L O R along which the kth event is detected (k=l...N), and s*- is a time-dependent sensitivity correction factor: the probability at time t that an emission from voxel j is detected anywhere. The overall time-averaged sensitivity correction factor Sj =. ^ JQ s^dt can be calculated on an LOR-domain approach such that any L O R i' is trans-formed to the corresponding motion-corrected L O R i=£j71(i') for the calcu-5. Motion Compensation: Beyond the Event-Driven Approach 148 lation of time-dependent attenuation and geometric factors; i.e.: (5.18) which has also been suggested by Qi and Huesman [154] in order to maximize the log-likelihood function of list-mode data. Nevertheless, similar to the approach in Sec. (5.2.3), we notice that in the N-weighted scheme, again using Eq. (5.12), one can write which reproduces Eq. (5.15). Therefore, we propose the following N-weighted algorithm: The studies performed in this work involve phantom studies with manual repositioning of the source. Two separate studies were performed, one in-volving transaxial rotation, and the other, axial translation. The technique described in this work, however, is fully applicable to 3D motion, which is the case commonly encountered in practice. Upon the availability of a motion tracking system to us in the near future, the experimental studies will be extended to the general 3D case. The two experiments are elaborated below. Experiment 1 - Transaxial Rotation A n F-18 line source was inserted axially into a 70-cm long N E M A phantom (20 cm diameter), placed 4cm away from, and parallel to the central axis of (5.19) 5.4 Methods Phantoms used and measurements performed 5. Motion Compensation: Beyond the Event-Driven Approach 149 the cylinder. The measured true count rate was 124k/s with a random frac-tion of 13%. Three separate frames each of duration 4 minutes were acquired with the cylinder manually rotated each time by approximately 45 degrees around the z-axis. Experiment 2 - Axial Translation of an Extended Source Via the technique presented in Appendix C, which allows printing of ra-dioactive patterns using a modified standard ink-jet printer, we imaged an F-18 "W"-sign (~4x4cm, 4.81mCi) printed on regular paper. Scans were performed with the source positioned (i) nearly at center of the scanner (1 minute), and also shifted axially by (ii) 5cm (3 minutes) and (iii) 9cm (3 minutes) with respect to the first frame. In both studies, the separate motion frames were then combined into one single frame to study the proposed motion correction techniques. Detector normalization correction factors were obtained from a 12 hour scan using a rotating rod source. Motion Measurement and Compensation The center of the cylinder used in the first study was expected to have un-dergone small translations during the manual rotation procedure. In this regard, exact inter-frame rotations and translations of the phantom were determined by comparison of separate reconstructions for the three frames. The center of the line source reconstructed in each frame was found by means of fitting a Gaussian profile. The relative translational and rotational shifts in-between the three centers were then found by means of matching one with another. The motion coordinates (with respect to the first frame) for the second and third frames were found to be: A X = 3 , A y = 7 , A#=40° and 5. Motion Compensation: Beyond the Event-Driven Approach 150 AX=9,AY=3,A(9=85 0 , respectively, where the displacement units (i.e. voxel dimensions) are 1.2mm. We denote the L O R coordinate system using (r, 9, z, 5) where r and 9 contain the transaxial radial and angular coordinates, while z and S define the axial position of the center of the L O R and the ring difference between the end-points. In the aforementioned case of transaxial motion, the two elements of the L O R affected by such motion are r and 9. These two follow the relation r = Xcos(9) + Ysm(9) (5.21) for an L O R passing through an image voxel with coordinates (X, Y, Z). It follows that for an event detected at a given time for an object rotated (counterclockwise) by an angle A9 with respect to its position at i=0, the appropriate LOR-domain motion operator becomes LR{r, 9, z, 5) = (r, 9 + A9, z, S) (5.22) whereas for an object translated by ( A X , A y ) , we would have £ T ( r , 9, z, 8) = (r + AXcos((9) + Ay S i n (0 ) , 9, z, 6) (5.23) where the superscripts R and T represent the transaxial rotation and trans-lation operations. Similarly, the equivalent image-domain motion operators are MR{X,Y,Z) = ( cos (A0)X-s in (A0)y , , . sin(A69)X + cos(A0)y) 1 > and MT(X,Y) = (X + A X , y + A y ) (5.25) In the axial motion study, the motion operator for an axial shift by AZ is given by &A(r, 9, z, 5) = (r, 9,z + AZ, 5) (5.26) 5. Motion Compensation: Beyond the Event-Driven Approach 151 where the superscripts A represent the axial translation operations, while the equivalent image-domain motion operator is MA(X, Y, Z) = (X, Y,Z + AZ) (5.27) The aforementioned LOR-domain and image-domain operators were subse-quently used for incorporation into the proposed histogram-mode and list-mode reconstruction methods. Reconstruction Schemes Throughout the studies presented in this work, a span of 3 and maximum ring difference of 67 were used. The motion-compensated histogram-mode (Eq. 5.13) and list-mode (Eq. 5.20) algorithms proposed in this work were implemented on the HRRT. In chapter 4, details of implementation and vali-dation of the regular list-mode E M algorithm with random events correction for the H R R T were discussed, and the statistical properties were compared with histogram-mode (3D-OSEM) reconstruction. Four schemes were considered in each of the histogram-mode and list-mode tasks: (I) Data for the non-moving phantoms (single phantom position) were first reconstructed. In addition, data for the motion studies were re-constructed (II) without any motion correction, (III) using the conventional purely event-driven technique (i.e. without modification of the sensitivity cor-rection factors), and (IV) with the proposed motion correction algorithms. In histogram-mode reconstruction, two instances of scheme III were con-sidered: (a) one in which data allocated to the H R R T detector gaps, after motion corrected histogramming, were ignored and (b) when gap data were not dropped and were used in the reconstruction. The results were subse-quently compared using measurement of relative count loss and resolution (line-source study) as well as radial/axial profile comparisons. 5. Motion Compensation: Beyond the Event-Driven Approach 152 (a) (b) 10 120 130 140 160 160 170 Radial Profile Fig. 5.3: (Top) Sinograms of a moving line source (a) without and (b) with motion-compensated histogramming. Clearly, upon appropriate histogramming, some counts are histogrammed into bins corresponding to detector gaps (shown by arrows), signifying that they would not have been detected had the object not moved. (Middle) Images reconstructed using histogram-mode algorithm applied to (c) data with no motion and (d) data for mo-tion when reconstructed without motion correction. (Bottom) (e) Image resulting from the proposed reconstruction algorithm given by Eq. (5.13). Radial profiles through this image as well as the one resulting from no motion data are shown in (f). Three iterations were applied to the data. Direct plane #110 is shown (randomly selected). 5. Motion Compensation: Beyond the Event-Driven Approach 153 (a) (b) P I Radial Profile (c) (d) Fig. 5.4: Images reconstructed using list-mode algorithm applied to a) data with no motion, as well as data for motion when reconstructed b) without and c) with motion correction, (d) Radial profiles through imaged line source in cases a) and c) are shown. Three iterations were applied to the data. Plane 110 is shown (randomly selected). 5.5 Results and Discussion Histogram-mode Reconstruction Experiment 1: Fig. (5.3a) shows a typical sinogram acquired without any motion-correction applied to the line source transaxial motion data. The presence of three distinct sinogram patterns as well as the detector gaps are clearly observable. Fig. (5.3b) shows the resulting sinogram when motion-compensated his-togramming is performed. The figure clearly exhibits non-zero counts in 5. Motion Compensation: Beyond the Event-Driven Approach 154 certain histogram bins corresponding to detector gaps. Such events, not cor-responding to any existing detector pairs, are detected due to object motion, as predicted in Sec. 5.1. Reconstruction of the motion-compensated sinograms, while ignoring data allocated to detector gaps, using the conventional event-driven technique (i.e. scheme III while ignoring gap data) was seen to considerably underesti-mate the image intensity (by 18%), compared to the reference image (scheme I). On the other hand, for the reconstruction schemes III (conventional, this time not ignoring gap data) and IV (proposed), count losses of 2% and 1% were observed, respectively. Furthermore, scheme III resulted in a recon-structed image resolution of 4.5mm, while scheme IV showed a resolution of 4.4mm. In comparison, scheme I produced a resolution of 4.1mm. Thus, it was observed that ignoring gap data resulted in a reconstructed count bias (plus clearly, discarding of a portion of measured data corresponds to a loss in statistics). Furthermore, schemes III and IV did not result in noticeably different image qualities. However, as we shall demonstrate, in the axial motion study with an extended object, the two schemes result in significantly different final images. Figs. (5.3c,d,e) show final images reconstructed using schemes I, II and IV. Radial profiles through Figs. (5.3c&e) (i.e. no motion study vs. proposed algorithm) are also shown in Fig. (5.3f). Comparing the two profiles, a slight broadening of resolution, from 4.1mm in scheme I to 4.4mm in scheme IV, was measured which can likely be attributed to the nature of motion measurement we have used (since an accurate motion tracking system is not yet available to us). List-mode Reconstruction Experiment 1: Similar to the previous analysis, images reconstructed from 5. Motion Compensation: Beyond the Event-Driven Approach 155 the line source (transaxial motion study) using schemes (I), (II) and (IV) are shown in Figs. (5.4a,b,c). The total number of counts in images reconstructed using the three schemes were confirmed to be within 0.5%. Radial profiles were subsequently drawn through resulting images from schemes (I) and (IV), as shown in Fig. (5.4d). Scheme I resulted in an imaged resolution of 4.0mm, while scheme IV showed a slightly-broadened resolution of 4.3mm. This, we attribute to the nature of motion measurement we have used, as previously mentioned. However, it was again observed that the conventional scheme (III) did not result in a noticeable decline in image quality. That is, a count loss of 1.5% and resolution of 4.4mm were measured compared to the proposed scheme (IV), which resulted in a count loss of 0.5% and a resolution of 4.3mm. This, as in histogram-mode study, is likely due to the transaxial nature of motion in the line source, and as we demonstrate below, the difference is very significant in the next study. Experiment 2: For the extended source motion study, resulting final images reconstructed using the four schemes (I), (II), (III) and (IV) are shown in Figs. (5.5a,b,c and d). One can clearly observe a non-uniformity in the re-constructed radioactive "W"-sign, when using the conventional scheme (III), whereas this is not the case in the scheme (IV) proposed in this work. Fur-thermore, upon drawing profiles through images from schemes (III) and (IV) as shown in the figures, one clearly observes, in addition to the axial non-uniformity artifact visible in the conventional scheme, an overall quantitative loss, measured to be 17% at axial plane 90 (corresponding to the left edge of the sign). Combined with the non-uniformity artifact, the image voxels are underestimated by nearly 50% at axial plane 150 (the right edge of the sign). These artifacts can be explained by the fact that in the conventional scheme, the sensitivity factors are not corrected for in accordance with object 5. Motion Compensation: Beyond the Event-Driven Approach 156 motion, and therefore result in overall count underestimation as well as non-uniformity upon motion towards less sensitive regions in the FOV. We have thus shown in the new experimental data, that the nature of the inaccuracy is not simply a global scaling and can result in loss of uniformity, whereas the proposed algorithm, achieves better uniformity. On a last note, we point out that scatter correction is yet to be incorpo-rated into the proposed algorithms in this work. We are currently evaluating different schemes. One possible approach has already been developed for the HRRT, namely the Watson scatter correction technique [28]. The algorithm is image-based (i.e. computes the scatter contribution using the reconstructed image) and therefore has the practical advantage of being directly applicable to images generated using our proposed methods. 5.6 Conclusion In this chapter, we have argued for and experimentally demonstrated that the event-driven approach to motion correction, in which one merely trans-forms LORs along which events are detected to LORs along which they would have been detected, can result in image artifacts. The nature of the inac-curacy was shown not to be necessarily restricted to a global scaling, and could manifest itself, for instance, as a loss in uniformity. This inaccuracy was attributed to the existence of LORs corresponding to no actual pairs of detectors (e.g. LORs axially out of the FOV) and their "interaction" with the detectable LORs due to motion. To address the issue, a more comprehensive modeling of the image-data relation was considered. It was subsequently shown that appropriate system matrix modeling of the aforementioned effects into the histogram-mode as well as list-mode reconstructions introduced time-weighted sensitivity correc-tion factors. These factors needed to be implemented in addition to compen-5. Motion Compensation: Beyond the Event-Driven Approach 157 Fig. 5.5: Images reconstructed for a radioactive printed "W"-sign (coronal view) when (a) no motion is introduced; and with motion (three axial positions: 1 min - 3 min - 3 min) when (b) no motion compensation, (c) the purely event-driven approach (conventional motion correction) and (d) the pro-posed algorithm given by Eq. (5.20) are applied to the data. Plots of uniformity profiles drawn through images in (c) and (d) are shown in (e). Five iterations were applied to the data. 5. Motion Compensation: Beyond the Event-Driven Approach 158 sation of the measured events for motion, with the conventional event-driven approach only performing the latter. The resulting motion-compensated sensitivity factors, at first instance, appeared to be potentially very time-consuming to compute for the general case of frequent motion throughout the scan. However, closer inspection of the sensitivity factors revealed that use of attenuation pre-correction for the data would allow computation of time-averaging of sensitivity correction factors to be performed in the image-domain. This was shown to potentially reduce reconstruction time significantly when correcting for frequent motion especially in high resolution scanners. 6. Summary, Opinion and Conclusions 159 6. Summary, Opinion and Conclusions 6.1 Summary This section briefly summarized the main points from the chapters in this thesis. Introduction to PET Imaging (Chapter 1): In this chapter, we introduced the P E T imaging technique. Main applica-tions of P E T imaging, along with a summary of the main characteristics of modern P E T scanners were presented. Motivations for and challenges in switching from 2D P E T to 3D P E T imaging were also discussed. The impor-tance of dynamic P E T imaging was explained. The chapter also elaborated upon the causes of degradation in image quality and quantitative accuracy. Finally, details of the high resolution research tomograph (HRRT) were ex-plained and the challenges in imaging with this scanner were mentioned. 3D Image Reconstruction (Chapter 2): In this chapter, we elaborated upon the two main approaches to 3D image reconstruction: analytic and statistical. Analytic techniques (3D reconstruc-tion as well as rebinning algorithms followed by 2D reconstruction) were introduced. We focused our attention, however, on statistical image recon-struction techniques, which were seen to be particularly suitable for imaging using the HRRT. Advantages of these techniques, their important proper-ties, and the various approaches within statistical image reconstruction were 6. Summary, Opinion and Conclusions 160 discussed. We paid particular attention to the various classes of expectation maximization (EM) algorithms, and introduced the importance of conver-gent image reconstruction, and of dealing with the various issues in the E M reconstruction technique. List-mode Image Reconstruction (Chapter 3): In this chapter, we introduced the concept of list-mode acquisition, and the possibility of list-mode reconstruction. Various advantage for the list-mode reconstruction techniques were propounded. We argued that the statistical list-mode reconstruction technique is very much suitable for reconstruction using the state-of-the-art HRRT, which contains a very large number of pos-sible lines-of-response (LORs). Various accelerated list-mode reconstruction algorithms were proposed (experimentally tested in the next chapter), with particular emphasis on convergent accelerated image reconstruction. A prac-tical random correction technique was also proposed for use in list-mode reconstruction from the H R R T data. Reconstruction Implementation and Comparison (Chapter 4) '• In this chapter, implementation details necessary for the task of list-mode image reconstruction were depicted. Various geometric back- and forward-projection techniques were proposed, and amongst them, the bilinear inter-polation technique was selected as the method of choice, due to its relatively powerful combination of speed and accuracy. The method was compared to histogram-mode O S E M reconstruction, commonly used with a sinogram zero-thresholding criterion. We demonstrated that a milder non-negativity constraint can relax the overestimation bias encountered with O S E M recon-struction of low-count frames. We also extended the list-mode approach to dynamic (4D) approach, and demonstrated its robustness (uniformity in time-activity-curves, axial profiles, percentage contrast) compared to other 6. Summary, Opinion and Conclusions 161 techniques. Motion Compensation: Beyond the Event-Driven Approach (Chapter 5) : In this chapter, we reviewed existing motion compensation techniques, and argued for the implementation of techniques that move beyond the purely event-driven motion compensation techniques. We proposed efficient and accurate motion compensation techniques (both in list-mode and histogram-mode imaging), and demonstrated the relative improvements encountered in terms of image quality figures of merit. 6.2 Opinion and Suggested Future Directions Our main challenge in the course of this work was the presence of numerous areas of further investigation applicable to our technique (and the obvious necessity not to pursue completely all areas of interest!). In what follows, we summarize main areas of further research that could yield important im-provements to the list-mode image reconstruction technique proposed, im-plemented and studies in this thesis. Use of splines in interpolation tasks: We have already demonstrated how significant the effect of the interpola-tion method of choice in the forward and backprojection tasks can be on final reconstructed image qualities. In particular, we showed in Sec. (4.1.1) that switching from the Siddon method to bilinear interpolation improves the F W H M at the center of the F O V from 3.1 mm to 2.7 mm (by ~15%). One area of research of much potential is the use of more suitable interpola-tion techniques. Use of splines, along with their very useful properties, seems to be a promising area of research, with regenerated interest in the recent years [155,156]. 6. Summary, Opinion and Conclusions 162 Ordinary Poisson List-mode Image Reconstruction: We identify that the delayed events subtraction technique can be replaced by the ordinary Poisson technique for more accurate modeling of the Poisson process. The difficulties with this approach, as elaborated in Sec. (3.5.1), mainly involve the fact that in the H R R T the singles events are acquired at the block level, and not the crystal level, and thus estimation of the mean random rates at the crystal level is difficult. To counter this problem, a recently proposed approach1 involves the use of delayed coincidences (which unlike the singles data are acquired at the crystal level) to estimate the singles rates at the crystal level. This technique, however, will result in increasing computational demands (in the calculation of the smoothed delayed coinci-dences, and constant look-up of calculated mean random coincidence rates). Improved 4D Image Reconstruction: The current approach we have chosen to imaging of the dynamic list-mode data involves independent reconstruction of the dynamic frames as specified by the user. However, it is worth noting that an alternate technique would involve using of the image reconstructed from a frame as an initial estimates for the image to be reconstructed for the subsequent frames: one may note that since iterative reconstruction techniques are often not run to conver-gence (see discussion in Sec. 2.5.2), the starting image estimate can play an important role in determining the accuracy of the final image [157]. In this alternate approach some type of filtering of each reconstructed image should be performed prior to use as an initial estimate for the subse-quent frame, so as to avoid increasingly noisy images as further frames are reconstructed [158]. Nevertheless, this approach suffers from a systematic problem. The final image produced in a frame will be potentially biased by 1 Internal documentation of the manufacturing company. 6. Summary, Opinion and Conclusions 163 the initial estimate, and this, in our opinion, would produce problems when studying fast dynamics in which adjacent frames differ noticeably. The ultimate approach, in our opinion, to reconstructing dynamic list-mode data involves truly 4D spatio-temporal reconstruction. This technique would not involve framing of the data into dynamic frames (activities within which are assumed to be static), thus making maximal use of the temporal resolution of the scanner. Rather, it would involve the use of temporal basis function to represent the activity in each voxel, and estimating the coeffi-cients of the basis functions by the collective use of the data. A possible approach to this problem is presented by Nichols et al. [159]. Scatter Correction: Scatter correction remains to be implemented in the list-mode image recon-struction method for the HRRT. This requires much dedicated research, to be built upon previous work on the subject. We have already presented our views and suggested future directions in Sec. (3.6). Modeling of the Detector Blurring: In Appendix B , we described a method in which detector blurring (including the parallax effect) were modeled in the image domain, in order to avoid a highly non-sparse system matrix. However, computational issues are only temporary, and long-term research should not view them as permanent ob-stacles. In our view, a better approach to modeling of detector blurring would involve its effective modeling into the forward and back-projection operations. This method would involve extension of the strip-integral ap-proach [160], which currently accounts for the finite detector width rather than using overly idealized line integrals, to account for inter-crystal blurring as well. In this way, the inter-crystal penetration (which results in the paral-lax effect), and the asymmetry associated with it (which is highly dependent 6. Summary, Opinion and Conclusions 164 on the individual LOR's angle of incidence), can also be incorporated into the system matrix. 6.3 Conclusion The work presented in this dissertation has been a result of the need for 3D image reconstruction algorithms suitable for high resolution positron emission tomography (PET). This has been particularly the case for the high resolution research tomograph (HRRT): a 3D-only state-of-the-art dedicated brain tomograph, with a very large number of lines-of-response (LORs) which it is able to measure (~4.5xl0 9 ) , exceeding most modern P E T scanners by 2-3 orders of magnitude. We proposed, studied and implemented statistical list-mode E M image reconstruction algorithms. In particular, we considered convergent accelerated algorithms, and demonstrated their applicability, es-pecially when combined with the ordinary accelerated algorithms (i.e. hybrid ordinary-convergent) in terms of convergence properties and speed. 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Med. Imag., vol. 20, no. 4, pp. 280-288, 2001 A. Analytic Image Reconstruction from Truncated Projections 182 A. Analytic Image Reconstruction from Truncated Projections In this appendix, we shall present a brief overview of the three exising analytic reconstruction techniques used on truncated data. The Universal Aperture Approach The universal aperture of a region of space B (for a particular scanner ge-ometry) is the set of all projection directions along which projections of B are measured (i.e. not truncated): ^univ(^) = {u €E f i | Vs G B,p(u,s) is measured} (A. l ) With the assumption that the universal aperture of the support B of the unknown function satisfies Orlov's condition (so that it is sufficient for inver-sion of the data), then one may consider set of projections T 0 = T ( f i u n j v ( B ) ) and reconstruct an image estimate using one of the methods described in the previous sections (for non-truncated data). This simple solution however will only be acceptable for some applications where 0 u m v ( S ) is not much smaller than fi (e.g. see [161]), as it does not attempt to utilize the data measured along directions outside the universal aperture. A n extreme example of this approach is to merely use the data obtained in the 2D-PET modality. The 3DRP Scheme: Forward Projection of the Unmeasured Line Integrals This method, commonly referred to as the 3D reprojection (3DRP) algo-rithm, proposes to recover the truncated data by using a first estimate n 0 (x) A. Analytic Image Reconstruction from Truncated Projections 183 of the image, which has been reconstructed from the data subset To as dis-cussed above, to calculate line integrals along the truncated directions which are not meaured. Subsequently, combining the measured data with the calcu-lated projection line integrals, one obtains the non-truncated set T(Q) from which the solution may be reconstructed. A n example of this approach used with the cylindrical geometry is discussed by Kinahan and Rogers [162]. We have discussed already that due to the inherent redundancy in the 3D X-ray transform, different filters may be utilized to reconstruct the image, as long as the normalization condition (2.12) is satisfied. Let us consider the filtered-backprojection algorithm (Sec. 2.2.6). With truncated data, the filtering step (2.16) can not be appropriately performed, since for a given projection direction u, one does not have access to p(u,s) for all s G u- 1.! However, it is possible to design the convolution kernel such that it vanishes in specific regions of i r 1 , thereby allowing the calculation of the convolution (2.16) even when the projections are truncated. In other words, assume one is able to to design a convolution kernel g satisfying where is the subset of the projection plane for which measurements are made, then for any direction u, values of p(u, s) for s ^ are not required in the calculation (2.12). Ref. [47] should be consulted for a discussion of how such filters may be constructed. However, once the convolution kernel g is constructed, it will not necessarily satisfy the normalization condition (2.12): an exception to this is the true three-dimensional reconstruction' (TTR) filter proposed by Ra et al. [163]. Otherwise, one can use the additive normaliza-tion formulation (2.14) which we write here as Use of Suitable Filters for Inversion of Truncated Data (A-2) 7Y(u,v) = G(u ,v )+L(u ,v ) (A.3) A. Analytic Image Reconstruction from Truncated Projections 184 where G is the Fourier transform of the convolution kernel g (as discussed above) and L ( U ' V ) - SaG(u',v)5{&.v)d& ( A - 4 ) One may then define w(u) as the characteristic function of the universal aperture w(u) - J 1 i f fl G 0 u n i v ( 5 ) (A 5^  With this choise of w, the correction filter L will only be applied to completely measured projections, and G is already designed to work in the presence of truncated data. It must be noted [47] that G must still not be too different from an exact filter (i.e. the term fn G(u, v)5(u.v)du must be close to unity). Otherwise, the contribution of the correction filter L, which only deals with a subset of the data fiumv(B), would be large, and signal-to-noise ratio might be noticeably degraded1. 1 The T T R filter already satisfies the normalization condition, and therefore L = 0 in this case. The T T R algorithm does make a mild assumption: that for any given point x in image-space, one can find a plane passing through x along which all line integrals are measured (an assumption that is satisfied for most existing scanners; notable exceptions to this are scanners with partial rings, as well as scanners with gaps in-between the detectors, such as our very own HRRT!). B. Space-variant and Anisotropic Resolution Modeling 185 B. Space-variant and Anisotropic Resolution Modeling In this part, we have proposed and investigated a method of incorporating the effect of inter crystal penetration (resulting in the parallax effect as discussed in Sec. 1.6.7). We first presented this work in [43]. Introduction One issue common to P E T scanners is the space-variance of the point spread function (PSF): manifesting itself as resolution degradation as one moves away from the center of the field-of-view (FOV). This effect is mainly caused by the dependence of inter-crystal penetration on angle of incidence of radia-tion [164]. As radiation occurs in voxels increasingly distant from the center of the FOV, it is more likely for the radiation to reach crystal fronts at higher angles of incidence, and to subsequently penetrate and be recorded in nearby crystals, ultimately degrading image resolution for such voxels (commonly re-ferred to as the parallax effect). Measurement of depth-of-interaction (DOI) within the crystals is known to minimize this problem, but its implementation has not achieved complete spatial invariance for resolution (see for e.g. [35]). In order to reconstruct images with uniform resolution properties in such systems a number of ap-proaches are possible. The regularization approach (see Sec. 2.5.2), initially introduced to suppress noise artifacts in statistical image reconstruction, can be used to yield images with uniform resolution properties: appropriately space-variant smoothing filters or penalty terms have previously been de-B. Space-variant and Anisotropic Resolution Modeling 186 signed for use in inter-filtering [105], post-filtering or penalized-likelihood reconstruction for space-variant imaging systems [165,166]. In expectation maximization (EM) algorithms, the system matrix itself may be utilized to incorporate space-variance of the system: an attempt to do so is elaborated in this part. Furthermore, spatial resolution for various scanners is commonly reported in the literature to be anisotropic (i.e. the PSF at a given position in the F O V exhibits distinct values along the axial and the two transaxial directions). This should be critical in correct modeling of the parallax effect, which ex-hibits anisotropic resolution at positions away from the center of the FOV. Simultaneous modeling of these two factors can allow more accurate charac-terization of degradation of each of the axial and the two transaxial resolution widths as one moves away from the center of the FOV. Modeling of Space-Variance and Anisotropicity In what follows, we describe how modeling of image resolution may be in-corporated into the list-mode E M reconstruction technique. Since modeling is included into the system matrix, which is similarly utilized in histogram-mode E M reconstruction, extension of such space-variant and/or anisotropic modeling to histogram-mode E M algorithm is straightforward, as pursued in [167] with space-invariant and isotropic resolution kernels. Denoting A™ as the image intensity in voxel j (j=l...J) at the mth iter-ation, and p^ as the probability of an emission from voxel j being detected along L O R i, the list-mode expectation maximization (LM-EM) reconstruc-tion algorithm is given by [92] d J k=l 2^a=l PikaAa where ik refers to the L O R along which the kth list-mode event is detected, B. Space-variant and Anisotropic Resolution Modeling 187 N is the number of measured events, and the sensitivity correction factor Sj=2_]i=iPij is a summation over all possible measurable LORs (i=l...I) We next note that the system matrix or operator P=(Pij)ixj may be decomposed as [67,70]: P = W G B (B.2) where, as elaboreated in Sec. (2.4.2), B=(bi3)jxj, G=(gi3)ixj and W=(wi3)ixi model voxel-voxel, voxel-LOR and L O R - L O R interactions, respectively. These matrices shall be referred to as spatial blurring, projection and sensitivity op-erators. Sensitivity variations due to attenuation and normalization can be taken into account using the diagonal elements of the sensitivity matrix W . Ideally, inter-crystal scattering and the parallex effect should also be incorporated into W , an example of which has been shown in [68]. However, including these effects can result in a high reduction in the sparseness of the matrix and can significantly increase the computational demand, especially for high resolution tomographs. For a scanner capable of measuing DOI information and achieving nearly isotropic resolution, Qi and Huesman [168] have argued using symmetry argu-ments that crystal scattering can instead be performed using space-invariant spherically symmetric blurring functions in image space (i.e. in the spatial blurring matrix JB) 1 . As explained above, we shall therefore be using a diagonal matrix W=(WAIX Upon substituting Eq. (B.2) into Eq. (B. l ) , and upon cancellation of the di-agonal elements Wi) in the forward and back-projection steps, and using 1 It has been suggested by Reader et al. [29] that image scatter may also be modeled by this blurring matrix. However, as also noted by Qi and Huesman [168], since soft issue has a long attenuation length, object scatters do not have the local property compared to crystal scatters, and therefore, modeling crystal scatters in the spatial blurring matrix B seems more suitable than for object scatters. Moreover, image scatter is object-dependent, and therefore, the system matrix needs to be object-dependent to correctly account for this effect, which can only complicate the matter! B. Space-variant and Anisotropic Resolution Modeling 188 S=[si...Sj]T and Km=[\™---Xj]T to denote J-dimensional vectors of image sensitivity and image intensity (at iteration m), Eq. (B.l) can be written in the compact form [70]: A m A m + 1 — x B? S N y BP i f c — w - (B.3) where vectorial multiplication and division operations are performed on an element-by-element basis, and FP i f c and B P i / t denote geometric operators which perform forward- and back-projection along L O R ik along which the kth list-mode event is detected. We find it intuitively helpful to think of B as an operator that mod-els the PSF response of the system by effectively blurring image intensi-ties of all voxels into the neighboring voxels, which are then subsequently forward-projected. Upon application of the back-projection operation, the "de-blurring" operation BT is then invoked in order to position back-projected values from neighboring voxels back to the appropriate voxels. In previous works, the spatial-blurring component B of the system matrix has been represented as a set of space-invariant and isotropic convolution ker-nels p. In this work, we have explored additional incorporation of the parallex effect into B, and have subsequently considered using position-dependent and anisotropic kernel dimensions. Defining !$=(X, Y, Z) as the position vector, each space-variant kernel p{Pl) has been modeled as an anisotropic Gaussian, with the distinct widths (ox(R*), oy(lt), oz{Pt)) along the transaxial (X,Y) and axial (Z) directions dependent on position-inside-FOVof any voxel which is being blurred. Various functional forms may be considered for the modeling of spatial distribution of the three kernel widths. We have considered exponential and inverse-Gaussian forms of degradation of image resolution, as given by <7i(i?) = ^ ( O j e t e ^ f e ^ t e ) (B.4) B. Space-variant and Anisotropic Resolution Modeling 189 Tab. B.l: Characterization of Space-variance and Anisotropicity ^x(O) Lxx LXY Lxz ay(0) LyX LyY LyZ <rz{0) LzX LZY Lzz and (7,(7?) = a i ( 0 ) e V 2 ^ ; e V 2 ^ ; e V 2 ^ ; (B.5) where i=x, y or z. We therefore note that twelve parameters would be needed for this general parametrization of the blurring operator p{ht), as shown in table B . l . The horizontal extension of the table models XYZ-variant degradation of image resolution, and the vertical takes its anisotropicity into consideration. Measurement of Finite Resolution Kernels In principle, object-independent components of blurring can be collectively modeled by scanning point sources positioned at various transaxial and axial positions in the F O V and reconstructing the scan data without any resolu-tion modeling. The resulting spatial profiles can then be fit to yield the resolution kernel characteristic lengths. If it is desired to inherently include modeling of positron range, the point sources need to be inserted inside a medium of the same density as the regions in patients/animals in which emissions occur. Moreover, since magnitude of positron range depends on the positron energy, which varies widely among isotopes, such measurements will be isotope-specific. It must be noted that for the common case of 90°-transaxial-rotation symmetry (encountered, for instance, in circular or octagonal designs), it B. Space-variant and Anisotropic Resolution Modeling 190 Fig. B.l: Resolution relations implied by the common case of 90°-transaxial-rotation symmetry. must be the case that <7x(0) = <7y(0) (B.6) (i.e. (x/y)-resolution values at the center of the F O V are equal), and that the characteristic lengths satisfy Lxx = LyY LyX = L X Y LzX = L Z Y by symmetry. These identities have been illustrated in Fig. (B. l ) . It therefore follows that characteristic lengths obtained from the fitting of spatial distribution of resolution along one of the transaxial directions can be determined from the corresponding one in the other transaxial direction. Methods Tomograph: Data were taken on the high resolution research tomograph B. Space-variant and Anisotropic Resolution Modeling 191 (HRRT) [35]. This scanner has an octagonal design, with the detector heads consisting of a double 1 cm layer of L S O / L Y S O crystal layers for a total of 119,808 detector crystals (crystal size 2.1 x 2.1 x 10 mm 3). The total number of possible LORs is 4.486xl0 9. Phantom used and measurement performed: Four C - l l line sources, ori-ented axially, were simultaneously positioned in the horizontal (i.e. Y=0) direction of the transaxial plane by distances of X=0, 4, 8 and 12 cm from the center of the scanner. A scan duration of 7 minutes was considered re-sulting in 60.OM trues and 4.9M randoms. This study has been performed for the transaxial directions and is currently also being extended to the axial direction. Detector normalization correction factors were obtained from a 12 hour scan using a rotating rod source. Data analysis: Details of implementation of the list-mode E M algorithm on the HRRT are explained in chapter (4). In this work, in order to deter-mine variation of image resolution for the scanner, we have used the following technique: the simultaneously-scanned line-sources have been reconstructed by space-invariant and isotropic Gaussian kernels for a number of times with different a values ranging from 0 to 5 mm. Reconstructions were also per-formed using space-variant and anisotropic Gaussian kernels modeled with exponential as well as inverse-Gaussian forms of degradation of image reso-lution. Subsequently, transaxial widths (wx(X) and wy(X)) of the four lines sources in the source were measured and compared (further elaborated in next section) at X=0,4,8,12 cm. Results Space-invariant and isotropic resolution modeling: Fig. (B.2) illustrates mea-sured transaxial widths (wx(X) and wy(X)) of the line sources as recon-structed for a wide range of Gaussian kernel widths. The following observa-B. Space-variant and Anisotropic Resolution Modeling 192 oi i i i i i o' 1 1 1 1 1 0 1 2 3 4 5 0 1 2 3 4 5 Modeled o Modeled o Fig. B.2: Observed transaxial (x/y)-resolution values as functions of space-invariant, isotropic kernel a values. Points at which the curves are each minimized are indicated using circles. tions can be made: (i) Improvement in reconstructed line widths upon performing space-invariant and isotropic modeling is significant compared to the case of performing no modeling whatsoever, i.e. (ax=ay=0). (ii) For each imaged position, there exist unique intermediate modeling widths (referred to as ox and ay) that minimize the reconstructed line widths. These points are marked using circles on each plot in the figure and are interpreted as modeling widths that best model PSF at the particular X-position. (iii) ox and ay are not necessarily equal to one another for any imaged po-sitions (anisotropicity). (iv) Increasingly larger values of ax and ay are observed as one moves away from the center of the F O V (space-variance). Space-variant and anisotropic resolution modeling: The spatial distribu-tion of ox and ay (as obtained at X=0,4,8,12 cm) were then separately fit using exponential and inverse-Gaussian functions, from which values of Lxx B. Space-variant and Anisotropic Resolution Modeling 193 and Lyx (and therefore Lyy and Lxy as given by symmetry Eqns. (B.7a) and (B.7b)) were determined. The resulting estimated characteristic lengths were then incorporated into the space-variant and anisotropic system matrix, and the data were subsequently reconstructed with the new system matrix. For any reconstructed image, we have introduced an error measure: where i=(x,y), X=(0,4,8,12 cm) (and therefore N=8 summations are per-formed), Wi{X) indicates the line widths obtained using the particular model-ing under consideration and w™in{X) indicates the minimum width obtained using the procedure described in data analysis and illustrated in Fig. (B.2). The above metric therefore measures the percentage difference between the reconstructed line widths and the minimum reconstructed line width, and averages it over the two transaxial directions and the four positions across the FOV. Value of A would therefore approach zero when a particular type of modeling achieves minimum observed widths simultaneously for all the lines across the FOV. Fig. (B.3) plots A(a) as a function of the various space-invariant, isotropic a values. Note that the two straight lines in the figure are for space-variant kernels and are plotted for comparison (and are not related to space-invariant models). The error measure has a minimum value of A=0.914 obtained at <r=3.1 mm. By contrast, A values of 0.4169 and 0.5348 are obtained for exponential and inverse-Gaussian modeling of degradation of resolution across the FOV, as also shown in the figure. It is clearly seen that the latter error values are less than any A value obtained using regular modeling of a values. A = (wi(x)-wrn(x)) wmmfX) X 100% (B.8) B. Space-variant and Anisotropic Resolution Modeling 194 } i , , , 2 2.5 3 3.5 4 Space-invariant, isotropic a Fig. B.3: Variation of A as a function of modeled space-invariant, isotropic cr val-ues, as described in text. Values achieved by using exponential and inverse-Gaussian modeling of degradation of resolution are also shown. Conclusion In conclusion, we have shown that the introduction of a space variant and anisotropic point spread function (PSF), compared to a space-invariant and isotropic PSF, can improve the resolution across the FOV. However, we must recognize this approach as what it is: an ad hoc method intended to render the otherwise-highly-non-sparse system matrices feasible and practical. Fur-thermore, symmetric (though anisotropic) PSFs were employed in this work. However, blurring caused by the parallex effect is asymmetric and the de-gree of this assymetry is highly dependent on the individual LORs (since it increases with higher angles of incidence). For space-variant tomographs, therefore, the ideal approach consists of modeling inter-crystal scattering and the parallex effects into W, the operator which models L O R interactions in data-space. C. Printed Sources for PET 195 C. Printed Sources for PET Introduction In order to characterize the spatial resolution of a particular positron emission tomograph, one often images a point source and estimates the full width at half maximum (FWHM) of the point spread function (PSF). This allows a measurement of the response of the tomograph. In order to accurately assess the system PSF, the source dimensions must be much smaller compared to the PSF F W H M . The spatial resolution achievable with modern tomographs is of the order of few millimeters (~l-3 mm). It is thus becoming increasingly difficult to manufacture practical point or line sources that are sufficiently small so as not to influence the determination of the PSF F W H M . In this section, we describe a novel technique to print radioactive point sources on paper using high concentrations of F-18 and using a modified stan-dard ink-jet printer (HP DeskJet). This technique requires minimal human intervention, thus allowing to safely deal with relatively high concentrations of radioactivity. Printed point sources have been previously developed and successfully used in S P E C T applications where gamma emitters are used as radioisotopes [169-172]. Their feasibility for P E T imaging, although an-ticipated in principle, has never been practically demonstrated [171]. P E T imaging of point sources is complicated by the fact that the positron must annihilate to produce 511 keV gamma rays, as discussed in Sec. (1.1.2). Therefore, due to the finite positron range, it is not clear whether ordinary paper would provide enough material to enable sufficient number of annihi-C. Printed Sources for PET 196 lations in a reasonable length of time. Use of further attenuating material on top of the paper sources, while allowing improved rate of annihilations, might noticeably degrade the measured resolution. Resolution in P E T is typically measured using F-18 since this radioisotope emits the lowest energy positron amongst those commonly used in P E T (see table 1.2), and therefore positron range of F-18 (approximately 1 mm in water or tissue) renders it most suitable for resolution measurements. This also implied that positrons generated by F-18 require (on the average) the least amount of attenuating material to annihilate. Consequently, the proposed approach uses an F-18 ink solution to print the point sources. The results of testing the printed sources on the high resolution research tomograph (HRRT) are reported in what follows. Ref. [173] should be con-sulted for results of testing this technique performed by our group,on the Siemens/CTI E C A T 953B as well as the Concorde microPET R4. The reso-lution of these tomographs spans quite a wide range (approximately 1.8 mm for the microPET, 2.8 mm for the H R R T and 5.5 mm for the E C A T 953B) and therefore has offered a large range of source testing conditions. Manufacturing of the Printed Sources A n important consideration when manufacturing the radioactive sources was the requirement of no human intervention since relatively high amounts of radioactivity are required to produce a very high radioactivity concentration ink solution. This was achieved by removing the original inkjet cartridge from the Hewlett Packard Deskjet printer (model C2170A) and installing a modified cartridge. A n HP 51626A cartridge was disassembled to drain the ink, to remove all other components, and to insert an adapter which was custom machined to fit an extant round port in the bottom of the cartridge. A swagelok (Columbia Valve & Fitting) fitting coupled the adapter to a C. Printed Sources for PET 197 Fig. C.J: Left: Side View, cut lines on printer ink cartridge. Right: Side View, section and assembly of modified ink cartridge. 22 gauge needle allowing the use of standard luer fittings for fluid delivery. These modifications minimized the volume of liquid which needed to be in the cartridge to allow printing. A 3mL vial was mounted on the printer and connected through remote controlled solenoid valves to the printer cartridge and a supply of low pressure (1-2 psig) helium. F-18 was produced in a conventional niobium bodied water target by irradiating 180-H20. For these tests the vial was preloaded with 0.2mL of ink previously removed from the cartridge and the irradiated water was added directly to this vial with no pre-treatment. Specific activity was not measured. This solution was then transferred to the previously modified cartridge (Figure C. l ) using a low pressure helium system. The pattern of radioactivity to be printed was created as a drawing in AutoCAD drafting software or Microsoft Word where the diameter of the dots or thickness and length of the line were specified. The initial page to be printed started with a large block of solid ink approximately 4 cm by 16 cm to ensure a uniform flow of ink before the printing of the sources of interest. The source distributions C. Printed Sources for PET 198 imaged in the tomographs and in the phosphor imager were all printed on multi-purpose 201b paper (Econosource). Effect of the Radioactive Ink and Paper Choice on Point Source Size Two additional aspects of the printed source manufacturing were investi-gated: affect of the addition of radioactivity to the ink and impact of the choice of paper. To determine if the addition of ink would degrade the source size, ten 0.64 mm radioactive and ten non-radioactive sources were printed with the same printer, and subsequently the points were enlarged with a pho-tocopier to determine their diameter.- Furthermore, taking into account the outcome of this experiment, the influences of five kinds of paper on point size degradation were investigated using cold ink. The paper types studied were: HP Premium Photo paper, Econosource paper, Lexmark Premium InkJet paper and heavy bond paper. Results: The mean value of the size of the sources produced with the radioac-tive ink was 0.67 ± 0.03 mm, while for those produced with the cold ink was 0.68 ± 0.04 mm, demonstrating that the procedure of introducing radioac-tivity into the ink did not affect printing quality. Both the HP Premium Photo paper and the Econosource photocopy paper gave spots that had di-ameters within 5% of the intended diameter as determined by the photocopy enlargement method. These two types of paper also gave the most homoge-neously spherical points. The other papers tested gave spots that were on average 16% larger than the intended diameter (range 5-32% increase). The Econosource paper was therefore chosen for all the imaging experiments. C. Printed Sources for PET 199 Point Source Manufacturing Reproducibility Nine 2 mm diameter point sources were printed and their individual radioac-tivity was measured in a dose calibrator of known accuracy. The manufac-turing reproducibility in terms of radioactivity was assessed by evaluating the radioactivity mean and standard deviation between the sources. The same procedure was performed with sixteen 0.5 mm point sources. Typical radioactivity levels for the 0.5 mm and 2 mm diameter point sources were approximately 1 mCi and 9 mCi, respectively. 0.5 mm sources: The radioactivity mean value and standard deviation be-tween the 16 sources were 1.01 ± 0.08 mCi indicating a variability of approx-imately 8%. 2 mm sources: The radioactivity mean value and standard deviation between the nine sources were 9.27 ± 0.3 mCi, indicating a variability of approxi-mately 3%. Imaging of Point Sources Sixteen 0.5 mm diameter point sources were printed on a 4x4 grid with 3.5 cm separation between the points. This way the grid coordinates spanned approximately one quarter of the FOV: one edge of the grid was located at the centre of the F O V (X=Y=Z=0) while the other edge (diagonally away from the center of the FOV) was placed at X = Z = 10.5 cm, y=0 cm. The sources were scanned in air and sandwiched between two A l layers. Radial and axial profile F W H M s were obtained by averaging over the F W H M of the profiles of the sources located at different Z values but at the same X value. Data were reconstructed using one iterations and 16 subsets of the subsetized list-mode E M (S-LMEM) reconstruction algorithm (using the Siddon projection technique) as described in chapter (4). C. Printed Sources for PET 200 -3.5/ , 2 5-1 -r^ , , i j ' ' l l l l ? 2 4 5 3 l t DUtants &cm«at« {aa.} Fig. C.2: Axial (squares) and radial (diamonds) resolution measured for the HRRT. Data were acquired for 10 minutes. The sheet of paper with the printed sources was taped to a lateral styrofoam support (placed away from the printed sources). The F W H M values were estimated following the N E M A NU2-2001 protocol [144]. results: The F W H M of the axial and radial resolution as a function of radial position measured with paper alone on the H R R T is shown in Fig. (C.2). The resolution follows the expected pattern dictated by the parallax effect, as discussed in Appendix B, manifesting itself as a space-variant point spread function. It must also be noted that the obtained F W H M values are dependent on the particular reconstruction algorithm used, parameters in the algorithm (e.g. number of iterations) as well as the particular interpolation techniques employed in projection operations in the reconstruction task as we have elab-orately investigated in Sec. (4.4) and clearly shown, for instance in Fig. (4.11). C. Printed Sources for PET 201 4 5 0 0 i^—^i^^ .4000, j 3500- • i - i ,..:3000K < 2500 -~j 1 2000 o o 1500 •; > • - ! ; -lOOOi i . 500 I ••! 0 -4„-^Xl. 80 100 120 140 : : ; ::; 1#r:: Radial isn Fig. C.3: An example of a source profile imaged with (solid) and without (dashed) one layer of A l foil. The two profiles nearly coincide with one another. In addition, an example of a reconstructed source profile with and without one layer of A l foil is shown in Fig. (C.3). The F W H M values were seen to be consistent in both cases for the HRRT. This effect was more elaborately tested on the microPET scanner, as we report next. Effect of Additional Attenuating Medium The effect of the additional attenuating material on both the resolution eval-uation and the number of counts originating from the source region was extensively tested on the microPET scanner as reported in [173]. This scan-ner was selected since it has the best resolution amongst those used and was thus deemed to provide the most sensitive test to possible changes in the resolution evaluation due to the presence of the attenuating medium. The effect of the attenuating medium on the source profiles was investigated by comparing the F W H M and F W T M of the PSF with and without the addi-tional A l foils. The effect of additional attenuating medium on the number of acquired counts was investigated by comparing the number of counts in C. Printed Sources for PET 202 a region of interest placed around the source image obtained from the scans with paper alone after scanning with and without the A l foil. Results: No overall consistent difference in the F W H M values were ob-tained with and without the A l layers. In particular the results obtained in air and with a single A l layer agree within approximately 0.1 mm which was found to be the measurement reproducibility. For the F W T M the values with and without one A l layer coincided almost completely. The effect of the attenuating material on the number of detected events originating from the source position could not be estimated directly from number of total acquired counts; many annihilation in-fact occur in the scan-ner material such as the gantry when the source is placed in air. The effect was thus estimated by evaluating the count density in a region of interest placed around the source in the various scanning situations. A 3.4, 5.3, 6.1 and 6.7 fold increase in the count density was observed when adding 1,2,3 and 4 A l layers on each side respectively. Discussion It has been shown that it is possible to manufacture printed positron emit-ting point sources by developing an automated method to print radioactive point sources that does not require human handling of high amounts of ra-dioactivity. Ordinary paper was found to be of sufficient quality to provide highly reproducible and reliable printed sources as determined by an inde-pendent radioactivity measurement. We have also shown that ordinary paper thickness provides enough material for a significant fraction of the positrons to annihilate - however if counting statistics need to be maximized, a thin layer of attenuating material can be added without negatively affecting the resolution evaluation. We have described a novel technique to manufacture printed positron-C. Printed Sources for PET 203 emitting point sources which does not require human handling of radioactiv-ity. Ordinary paper was found to be of sufficient quality to provide highly reproducible and reliable printed sources as determined by an independent radioactivity measurement. We have also shown that ordinary paper thick-ness provides enough material for a significant fraction of the positrons to annihilate. However, if higher counting statistics are required in a particular task, a thin layer of attenuating material (e.g. A l foil) can be added with-out adversely affecting the imaged resolution1. Consequently, we have made use of this practical technique in the course of this work for characterization of resolution achieved via various reconstruction (chapter 4) and motion-compensation (chapter 5) techniques. We have also utilized this method for printing and study of extended sources (e.g. see Figs. 4.4 and 5.5) used to investigate noise and uniformity properties, as elaborated in the subsequent chapters. 1 For instance, it was found for the microPET scanner, that even a single 0.016 mm thick Al layer placed on each side of the paper increases the number of counts originating from the source position by a factor of 3.4. In future studies, it might be possible and desirable to laminate the paper sheet within clear plastic, which would have the added advantage of allowing visual determination of the source location while providing rigidity and attenuation. D. Derivation of the List-Mode EM Reconstruction Algorithm with Motion Compensation 204 D. Derivation of the List-Mode E M Reconstruction Algorithm with Motion Compensation In what follows, time-of-detection will be treated as a continuous random variable, and therefore probability density elements, denoted using lower case p, will be utilized in addition to regular probability elements, denoted here using upper case P. We define p(A\Dj,j) as the probability density that an event generated in voxel j leads to a measurement A in the detector, assuming the event will be detected (denoted by D3). The list-mode acquired measurement AERd, where d denotes the dimensionality of the measurement space, can contain such information about the event as, for instance, the L O R il along which it has been detected, time-of-detection t and energy. Parra and Barrett [92] have previously derived the list-mode E M recon-struction algorithm from first principles, which (written in our own notation1) is given by: Xrl = r £ « W ^ M ^ b w (D-D where Afc refers to the coordinates of the A;th list-mode event (k=l....N) and s • is defined to be the probability that an event generated from voxel j leads to a measurement anywhere (as we note later, this term must incorporate presence of motion, to correspond to its definition). 1 One needs to carefully note that Ref. [92] does assume, in the definition of the prob-ability density elements, that the emitted events will be detected (i.e. Dj) but does not explicitly write this out in the derivations. D. Derivation of the List-Mode EM Reconstruction Algorithm with Motion Compensation 205 We next invoke the relation: J A m p ( A , D j | j ) p(A\j) where we have dropped Dj in the numerator of the last term because an event which leads to a measurement A is necessarily detected. Furthermore, we note that p(Dj\j), the probability that an event generated from voxel j is detected (along any L O R and at any time), corresponds exactly to the overall sensitivity correction factor Sj\ i.e. P'DM)^ (D.3) This term, which has no time dependence, must take into account presence of motion throughout the scan, as we show below. Let us consider any measurement A={i',t} to indicate the L O R %' and time t at which the measured event is detected. We then have p{A\j) =p(i',t\j) = P(i%j)p(t\j) = P(i'\t,j)± (DA) where under the assumption that image intensity Xj at any voxel j does not change over duration T of the scan, one notes that p(t\j), the probability density that an emission at voxel j has occurred at time t, is a constant (i.e. p(t\j)=7p). The term P(i'\t,j) indicates the probability that an event generated in voxel j at time t will be detected at L O R i'. This corresponds exactly to the concept of a time-dependent system matrix. It then follows that the probability of detecting (along any L O R and at any time) an event emitted at voxel j is given by J° v T J o i In other words, the probability that a photon emitted at voxel j is detected should include a time-average of the time-dependent sensitivity correction D. Derivation of the List-Mode EM Reconstruction Algorithm with Motion Compensation 206 factors Sj for the course of the scan, i.e: = ^[s)dt where s) = £ P(i'\t, j) (D.6) Now, similar to the approach of Sec. 5.2.1, the attenuation and normalization-weighted probability elements at time t are given by: P(i%j) = AiNi,9j(i) where i = L71(i') (D.7) Combining Eqs. (D.2), (D.3), (D.4) and (D.7), one has p(A\Dj,j) = T _ J (D.8) which upon being substituted into Eq. (D.l) results in \m N -i ^ 1 = t £ w = J O T F (D'9) where Sj, given by Eq. (D.6), is the overall motion-compensated sensitivity correction factor. 

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