Citation |

- Permanent Link:
- https://ufdc.ufl.edu/UFE0003440/00001
## Material Information- Title:
- Count Rate and Spatial Resolution Performance of a 3-Dimensional Dedicated Positron Emission Tomography (PET) Scanner
- Creator:
- ABU-AITA, RAMI RIMON (
*Author, Primary*) - Copyright Date:
- 2008
## Subjects- Subjects / Keywords:
- Coincidence ( jstor )
Data acquisition ( jstor ) Data processing ( jstor ) Dead time ( jstor ) Imaging ( jstor ) Modulation transfer function ( jstor ) Pets ( jstor ) Photons ( jstor ) Pixels ( jstor ) Spatial resolution ( jstor )
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- Copyright Rami Rimon Abu-Aita. Permission granted to University of Florida to digitize and display this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Embargo Date:
- 4/30/2004
- Resource Identifier:
- 55893148 ( OCLC )
## UFDC Membership |

Downloads |

## This item has the following downloads:
abuaita_r ( .pdf )
abuaita_r_Page_089.txt abuaita_r_Page_061.txt abuaita_r_Page_013.txt abuaita_r_Page_113.txt abuaita_r_Page_079.txt abuaita_r_Page_064.txt abuaita_r_Page_112.txt abuaita_r_Page_057.txt abuaita_r_Page_017.txt abuaita_r_Page_052.txt abuaita_r_Page_117.txt abuaita_r_Page_014.txt abuaita_r_Page_009.txt abuaita_r_Page_056.txt abuaita_r_Page_093.txt abuaita_r_Page_082.txt abuaita_r_Page_034.txt abuaita_r_Page_067.txt abuaita_r_Page_092.txt abuaita_r_Page_058.txt abuaita_r_Page_030.txt abuaita_r_Page_023.txt abuaita_r_Page_078.txt abuaita_r_Page_038.txt abuaita_r_Page_001.txt abuaita_r_Page_097.txt abuaita_r_Page_036.txt abuaita_r_Page_007.txt abuaita_r_Page_047.txt abuaita_r_Page_020.txt abuaita_r_Page_002.txt abuaita_r_Page_026.txt abuaita_r_Page_119.txt abuaita_r_Page_068.txt abuaita_r_Page_037.txt abuaita_r_Page_012.txt abuaita_r_Page_102.txt abuaita_r_Page_050.txt abuaita_r_Page_115.txt abuaita_r_Page_040.txt abuaita_r_Page_108.txt abuaita_r_Page_024.txt abuaita_r_Page_008.txt abuaita_r_Page_084.txt abuaita_r_Page_072.txt abuaita_r_Page_062.txt abuaita_r_Page_054.txt abuaita_r_Page_032.txt abuaita_r_Page_033.txt abuaita_r_Page_099.txt abuaita_r_Page_081.txt abuaita_r_Page_106.txt abuaita_r_Page_116.txt abuaita_r_Page_063.txt abuaita_r_Page_086.txt abuaita_r_Page_045.txt abuaita_r_Page_071.txt abuaita_r_Page_031.txt abuaita_r_Page_077.txt abuaita_r_Page_006.txt abuaita_r_Page_035.txt abuaita_r_Page_022.txt abuaita_r_Page_103.txt abuaita_r_Page_048.txt abuaita_r_Page_090.txt abuaita_r_Page_046.txt abuaita_r_Page_015.txt abuaita_r_Page_101.txt abuaita_r_Page_055.txt abuaita_r_Page_080.txt abuaita_r_Page_051.txt abuaita_r_Page_083.txt abuaita_r_Page_059.txt abuaita_r_Page_088.txt abuaita_r_Page_118.txt abuaita_r_Page_107.txt abuaita_r_Page_053.txt abuaita_r_Page_042.txt abuaita_r_Page_110.txt abuaita_r_Page_070.txt abuaita_r_Page_096.txt abuaita_r_Page_094.txt abuaita_r_Page_019.txt abuaita_r_Page_029.txt abuaita_r_Page_044.txt abuaita_r_Page_098.txt abuaita_r_Page_039.txt abuaita_r_Page_011.txt abuaita_r_Page_075.txt abuaita_r_Page_027.txt abuaita_r_Page_120.txt abuaita_r_Page_114.txt abuaita_r_Page_069.txt abuaita_r_Page_043.txt abuaita_r_Page_025.txt abuaita_r_pdf.txt abuaita_r_Page_016.txt abuaita_r_Page_104.txt abuaita_r_Page_074.txt abuaita_r_Page_095.txt abuaita_r_Page_010.txt abuaita_r_Page_111.txt abuaita_r_Page_041.txt abuaita_r_Page_060.txt abuaita_r_Page_049.txt abuaita_r_Page_004.txt abuaita_r_Page_028.txt abuaita_r_Page_109.txt abuaita_r_Page_073.txt abuaita_r_Page_100.txt abuaita_r_Page_085.txt abuaita_r_Page_065.txt abuaita_r_Page_105.txt abuaita_r_Page_087.txt abuaita_r_Page_005.txt abuaita_r_Page_091.txt abuaita_r_Page_066.txt abuaita_r_Page_003.txt abuaita_r_Page_021.txt abuaita_r_Page_018.txt abuaita_r_Page_076.txt |

Full Text |

COUNT RATE AND SPATIAL RESOLUTION PERFORMANCE OF A 3-DIMENSIONAL DEDICATED POSITRON EMISSION TOMOGRAPHY (PET) SCANNER By RAMI RIMON ABU-AITA A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2004 Copyright 2003 by Rami Rimon Abu-Aita To my parents ACKNOWLEDGMENTS I would like to thank several people who contributed to the completion of this work. First, I would like to express my sincere thanks to Dr. David R. Gilland for his support, encouragement, guidance, and most of all his trust, patience and, sense of humor throughout the last two years of graduate work at the University of Florida. I would also like to thank Professor Wesley E. Bolch, Dr. David E. Hintenlang, and Dr. Manuel M. Arreola for their quality teaching and for taking the time to be on my supervisory committee. The committee provided invaluable comments that truly made this work complete. I also thank the faculty, staff, and students of the Nuclear and Radiological Engineering Department. I also thank Michael Nicole, Mike Frame, and the technologists of the Nuclear Medicine Department of Shands Radiology for their willingness to share time and facility. Without their assistance, this investigation would not have been possible. Moreover, I thank Ben Chase, Ed Sokolowski, and Renate Gill of Philips/ADAC Medical Systems for providing assistance and guidance that lead to the success of this investigation. I am very grateful to my parents, Rimon and Khitam Abu-Aita, and my brothers, Wadie, Giovanni, and Marco, for their love and support and for accepting a long-term separation that lasted about five and a half years. I truly thank my parents for giving me the opportunity to pursue higher education in the United States. I thank both of my uncles, Awni and Issa Abu-Aita, for their support and encouragement. Finally, I thank God for giving me the strength to finish my work. TABLE OF CONTENTS page A C K N O W L E D G M E N T S ................................................................................................. iv LIST OF TABLES ............................... ............. .............. viii LIST OF FIGURES ......... ......................... ...... ........ ............ ix A B S T R A C T .............................................. ..........................................x iii CHAPTER 1 IN TRODU CTION ................................................. ...... ................. 2 BA CK GROUND ........................................................ .......... .......... .... .... Count Loss and D ead Tim e .................................................. ............................ 5 The Effects of High Count Rate on Spatial Resolution ............................ ...............8 Random and Scatter Coincidence Events .......................................... ...............8 N oise E equivalent C ount R ate......................................................................... ... ... 10 3 M ATERIALS AND M ETHOD S ........................................ ......................... 12 C am era D description .................. ........................... ................... .. 12 P h antom D description ........................................ .............................. .................... 13 Allegro D ata A acquisition and Processing .............................................................. 14 A llegro D ata A acquisition Param eters ............................................... ............... 15 Allegro Data Processing Param eters ....................................... ............... 16 F ile S tru ctu re ..................................................................... .. 16 NEMA Data Processing and Analysis Using MATLAB ..........................................18 C o u n t R ate s ................................................................................................... 1 8 Spatial Resolution................... .......................... ........ 22 4 R E S U L T S .............................................................................2 4 C o u n t R ate ................................................................2 5 Spatial R solution ............... .................................... ...... ........ ........ ....... 1 5 D ISC U S SIO N ............................................................................... 37 6 CON CLU SION S .................................. .. .......... .. .............40 APPENDIX A N EM A 2001 (SECTION 4) ............. ................................... ...................................42 B HOW TO COPY SPECIFIC FILES TO A CD ................... ......................... 49 C A SHELL SCRIPT FOR SCALING FACTORS ............... .................. ............50 D MATLAB CODE FOR ORGANIZING ACQUISITIONS ........................................51 E MATLAB CODE FOR SCALING FACTORS .............. ............ .....................53 F MATLAB CODE FOR DATA PROCESSING AND ANALYSIS...........................54 G MATLAB CODE FOR CONTINUING PROCESSING AND ANALYSIS.............61 H MATLAB CODE FOR CURVE FITTING..................................... .....................64 I MATLAB CODE FOR PRODUCING MTF CURVES .........................................68 J FIN A L SU M PR O JE CTION S........................................................................ ...... 71 K FITTED FINAL SUM PROJECTIONS ..................... ............... 82 L MODULATION TRANSFER FUNCTIONS .................................... ...............93 R E F E R E N C E S ........................................ ....................................................... .... 10 4 BIOGRAPHICAL SKETCH ............................................................. ............... 106 LIST OF TABLES Table pge 1 Initial conditions of im age acquisitions ................................ ........ ................... 24 2 Processed coincidence count rates of all acquisitions....................................26 3 P aram eters of fitted curves ....................................................................................31 4 FWHM calculated from the fitted final sum projections .............. ...................33 LIST OF FIGURES Figure page 1 Pulse before and after shaping .............................................................................6 2 Possible coincidence events ............................................... ............................ 9 3 M odule A ssem bly .......................... ...... ..................................... .. .... .. 12 4 Phantom utilized in the experiment .......................... ........................... 13 5 Illustration of sinogram s shifting process. .................................... .................20 6 Example of determining random and scatter coincidences..............................20 7 Final sum projections of selected acquisitions................... ....... ..................25 8 Final sum projection of acquisition 1 ............................................ ............... 71 9 Final sum projection of acquisition 2.......................... ...... .... .............. 72 10 Final sum projection of acquisition 3 ............................................ ............... 72 11 Final sum projection of acquisition 4............... ....... .... ..... .. .. ............ 73 12 Final sum projection of acquisition 5 ............................................ ............... 73 13 Final sum projection of acquisition 6............................................ ............... 74 14 Final sum projection of acquisition 7............................................ ............... 74 15 Final sum projection of acquisition 8 ............. .... .. .... ..... .. .. ............ 75 16 Final sum projection of acquisition 9............................................ ............... 75 17 Final sum projection of acquisition 10............................ .. .. ...............76 18 Final sum projection of acquisition 1 1.............. ...... ................ .. ............. 76 19 Final sum projection of acquisition 12............... ........ ......... .... ........... 77 20 Final sum projection of acquisition 13 .............. ...... ................ .. ............. 77 21 Final sum projection of acquisition 14 ........................................ ...............78 22 Final sum projection of acquisition 15 ................................... ......... ............... 78 23 Final sum projection of acquisition 16 .................................... ........ ............... 79 24 Final sum projection of acquisition 17 .................................... ........ ............... 79 25 Final sum projection of acquisition 18 ................................... ......... ............... 80 26 Final sum projection of acquisition 19 .................................... ........ ............... 80 27 Final sum projection of acquisition 20 ............................................ ...............81 28 Final sum projection of acquisition 21 .................................................................. 81 29 Count rates versus time into experiment.......................................................27 30 C ount rates versus activity ............................................... ............................ 27 31 True count rate versus time into experiment................................. ...... ............ ...28 32 True count rate versus activity ........................................... .......................... 28 33 Noise Equivalent Count rate versus time into experiment............ ...............29 34 N oise Equivalent Count rate versus activity ................................. ..................... 29 35 Randoms count rate and its fitted equation versus activity ............... ...............30 36 Final sum projection of acquisition 1 and its fitted gaussian curve .........................82 37 Final sum projection of acquisition 2 and its fitted gaussian curve .........................82 38 Final sum projection of acquisition 3 and its fitted gaussian curve .........................83 39 Final sum projection of acquisition 4 and its fitted gaussian curve .........................83 40 Final sum projection of acquisition 5 and its fitted gaussian curve .........................84 41 Final sum projection of acquisition 6 and its fitted gaussian curve .........................84 42 Final sum projection of acquisition 7 and its fitted gaussian curve .........................85 43 Final sum projection of acquisition 8 and its fitted gaussian curve .........................85 44 Final sum projection of acquisition 9 and its fitted gaussian curve .........................86 45 Final sum projection of acquisition 10 and its fitted gaussian curve .......................86 46 Final sum projection of acquisition 11 and its fitted gaussian curve .......................87 47 Final sum projection of acquisition 12 and its fitted gaussian curve .......................87 48 Final sum projection of acquisition 13 and its fitted gaussian curve .......................88 49 Final sum projection of acquisition 14 and its fitted gaussian curve .......................88 50 Final sum projection of acquisition 15 and its fitted gaussian curve .......................89 51 Final sum projection of acquisition 16 and its fitted gaussian curve .......................89 52 Final sum projection of acquisition 17 and its fitted gaussian curve .......................90 53 Final sum projection of acquisition 18 and its fitted gaussian curve .......................90 54 Final sum projection of acquisition 19 and its fitted gaussian curve .......................91 55 Final sum projection of acquisition 20 and its fitted gaussian curve .......................91 56 Final sum projection of acquisition 21 and its fitted gaussian curve .......................92 57 Adjusted FWHM as a function of time into experiment......................................34 58 Adjusted FWHM as a function of activity .................................... ............... 34 59 Modulation transfer function of acquisitions 1, 5, 10, and 21 ..............................35 60 Modulation transfer function of acquisition 1 ................... ......................... 93 61 Modulation transfer function of acquisition 2 ................ ............................... 94 62 Modulation transfer function of acquisition 3 ................... ......................... 94 63 Modulation transfer function of acquisition 4 ................ ............................... 95 64 M odulation transfer function of acquisition 5..................................... ..................95 65 Modulation transfer function of acquisition 6 ................ ............................... 96 66 Modulation transfer function of acquisition 7 ................ ............................... 96 67 Modulation transfer function of acquisition 8..................................................... 97 68 Modulation transfer function of acquisition 9 ................ ............................... 97 69 Modulation transfer function of acquisition 10...................................................... 98 70 Modulation transfer function of acquisition 11 ............................................. 98 71 Modulation transfer function of acquisition 12...................................................... 99 72 Modulation transfer function of acquisition 13..................... .............................. 99 73 Modulation transfer function of acquisition 14....................... .... ........... 100 74 Modulation transfer function of acquisition 15....................... ... ........... 100 75 Modulation transfer function of acquisition 16...............................................101 76 Modulation transfer function of acquisition 17...............................................101 77 Modulation transfer function of acquisition 18............................................... 102 78 Modulation transfer function of acquisition 19............................................... 102 79 Modulation transfer function of acquisition 20..................................................103 80 Modulation transfer function of acquisition 21 ....................... .... ...........103 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science COUNT RATE AND SPATIAL RESOLUTION PERFORMANCE OF A 3-DIMENSIONAL DEDICATED POSITRON EMISSION TOMOGRAPHY (PET) SCANNER By Rami Rimon Abu-Aita May 2004 Chair: David Gilland Major Department: Nuclear and Radiological Engineering The count rate and spatial resolution performance of a state-of-the-art 3- dimensional dedicated positron emission tomograph at Shands Medical Plaza were assessed. A 70-cm-long polyethylene phantom and a line source were both utilized to better study the effect of activity from outside the field of view on the counting rate and spatial resolution of the scanner. A series of 21 tomographic images were acquired with a fluorine-18 radionuclide. The computed true, scatter, random, and noise equivalent count rate coincidences were plotted as functions of activity, as described by NEMA 2001 protocol. The spatial resolution was studied by calculating the full width at half maximum (FWHM) of the fitted final sum projections. The FWHM decreases over time as activity decays, and is constant for the last 5 acquisitions with activities ranging from 0.22 mCi for the 17th acquisition to 0.08 mCi for the 21st acquisition. Moreover, the modulation transfer function was computed to completely describe the spatial resolution at varying activities. It was concluded that both the count rate and the spatial resolution were affected by the increase in activity, especially at high counts. It was found that the peak NEC rate occurred at 31.04 kcps at an activity of 5.876 mCi. The scatter fraction of the system was found to be about 40% indicating the disadvantageous high scatter coincidence events due to the contribution of scatter from out of the field of view. Finally, the peak NEC rate is a useful measure of the amount of activity that should be employed in a clinical patient scan. Also the trends of the true, scatter, and random count rate curves show the behavior of the scanner studied. CHAPTER 1 INTRODUCTION There is increasingly wide PET application for clinical diagnosis, due to the improvement in the diagnostic accuracy of this imaging modality through the development of new data acquisition and processing systems and the introduction of new positron-emitting radiopharmaceuticals. Positron emission tomography (PET) with fluorine-18 fluoro-2-deoxy-glucose (18F-FDG), a glucose analog, is a unique biologic imaging tool that can be used to obtain functional information from the living human body. After administration of the 18F-FDG radiopharmaceutical to a patient, it takes part in physiologic processes. The unstable, neutron-deficient nuclide decays by emitting a positron, which will annihilate with a nearby electron to create two 511 keV photons that are emitted essentially back-to-back at an angle of 180 degrees. Detection of the two photons within a narrow time window is called a coincidence event, where the assumption is that the photons originate from a single annihilation occurring along the line-of-response defined by the direction of the photons. The radioactive decay is continuously taking place and can be detected from the emitted gamma rays, allowing the monitoring of the distribution of the tracer concentration. The widely used 18F-FDG has an advantage of becoming trapped in the tissue and reaching a near equilibrium state approximately 45 to 60 minutes after injection, therefore keeping the tracer concentration almost constant with time.1 Oncology imaging using PET with s1F-FDG has proven to be effective in evaluating the extent of disease for several types of cancer.2 FDG-PET is useful for defining the degree of malignancy and for differentiating recurrent tumors from necrosis after therapy in patients with brain tumors.3 Data acquisition is substantially different in PET than it is in planar nuclear medicine and single photon emission computed tomography (SPECT). In PET, annihilation coincidence detection (ACD) is used in lieu of absorptive collimation to determine the directionality of the detected photons. Because of this technique, the sensitivity, the rate at which the system detects counts per unit of activity, of PET is substantially higher than in single-photon nuclear imaging. ACD also avoids the degradation of spatial resolution with distance from the detector.4 There are two basic ways of collecting data in PET. These are referred to as 2-dimensional (2D) and 3- dimensional (3D) PET. Some PET scanners can acquire data in both 2D and 3D modes whereas others can only acquire data in 3D mode, like the dedicated PET scanner considered in this investigation. With 2D PET, thin rings (- 1 mm thick) of lead or tungsten, known as septa, are often placed between the detector rings to reduce the interplane scatter. These septa are more like the antiscatter grids used in radiography. They do not provide spatial definition but simply reduce the amount of interplane scatter incorporated into the data acquisition. On the other hand, the absence of septa in 3D PET increases the sensitivity by a factor of 4-6 but also increases the scatter fraction as well as the random coincidences from activity that is out of the field of view.5 The increased sensitivity to true events is partially offset by the higher sensitivity to scattered and random coincidences. Consequently, the overall advantage of 2D versus 3D PET whole- body imaging is unclear.6 In the last 11 years, whole-body "F-FDG studies have become the predominant type of PET study performed by most centers.7 Therefore, it is more relevant than before to measure the performance of the PET scanners under conditions that better represent whole-body studies. The phantom that was used for evaluating the performance of the scanner was a 19-cm-long phantom. The phantom's volume was more comparable to the head volume, whereas the body is much larger. Nowadays, the most significant change in testing the PET counting rate performance is the change from the 19-cm-long phantom to a 70-cm-long phantom, while keeping the diameter (20 cm) of both phantoms unchanged. The 70-cm phantom is a better approximation to the activity distribution in whole-body studies, because the effects of out-of-field activity (OFA) are included in the measurements. The 19-cm, on the other hand, continues to be used to test the performance of scanners used primarily for brain imaging, particularly for scanners that are dedicated brain imaging instruments. Most patient PET studies are not performed under conditions of low counting rate losses or negligible random rates. At higher activity levels, coincidence events are lost because of system dead time, whereas the rate of random coincidences rises.6-11 It is necessary to measure the counting rate performance, both dead-time losses and random events, as a function of activity to understand the scanner's behavior for a wide range of scanning conditions. Different studies have examined the axial and transverse spatial resolution within the field-of-view (FOV) measured with a point source.8 9, 12 These studies, however, do not take into account the effect of varying activities on the resolution. The increase in count rate increases the probability that pulses from multiple coincidences will be integrated together to cause spatial distortion and degrade the spatial resolution. The consequences of high count rate on spatial resolution are therefore investigated. The aim of this study was to assess the count rate capabilities of the 3D dedicated PET scanner and determine the effects of the high count rate on the system's spatial resolution. To quantify the changes in spatial resolution, the full width at half maximum (FWHM) of line source images and modulation transfer functions (MTF) were computed from images of a decaying line source in the 70 cm cylindrical phantom. The practicality of obtaining both count rate and spatial resolution data in a single acquisition was also determined. CHAPTER 2 BACKGROUND Count Loss and Dead Time An incident 511-keV annihilation photon that is absorbed in the scintillator generates a pulse of light that is converted into an electronic signal and amplified by the photomultiplier tubes (PMTs). The outputs from the photomultipliers are used to localize the incident photon and as a measure of the energy to reject photons that have scattered before reaching the detector. The overall count rate performance of the detectors is dependent on a number of factors such as pulse pile-up and system dead time or pulse resolving time Pile-up within the crystal occurs when two photons from different annihilations arrive so closely spaced in time that they cannot be distinguished as two separate photons. The light output of such an event is the sum of the two photon energies that will, in general, exceed the upper energy level discriminator (ULD), and therefore be discarded. Both photons are subsequently lost. This type of signal pile up is called post-pulse pile-up because a pulse is received while another is being integrated. A second source of count rate loss, known as pre-pulse pile-up, occurs when the positional and energy determination of a photon is still in process when a second photon arrives. Since the detection system is dead when the second photon arrives, the count will be lost. Such a process has the characteristics of a saturating or paralyzing system. A paralyzable system is one for which each event introduces a dead time whether or not that event actually was counted. Thus an event occurring during the dead time of a preceding event would not be counted but still would introduce its own dead time during which subsequent events could not be recorded. To reduce pulse pile-up at high count rates, scintillators with short decay times are essential. The decay constant for the Gadolinium Oxyorthosilicate (GSO) scintillation crystal considered in this study is 56 nanoseconds. Scintillation light of GSO is emitted faster than Sodium Iodide (NaI(Tl)). NaI(Tl) has long been used as a scintillation crystal in most nuclear medicine imaging systems. Nal has a large decay constant of 230 nanoseconds. To further reduce pile-up and improve the high-count rate performance of the system, the amplifier shapes the pulse therby reducing it to 100 nanoseconds [Fig. 1]. I 22M ns Before Pulse Shaping ter Pulse Shaping FIG. 1. Pulse before and after shaping Because of dead time losses, the observed counting rate R0 is less than the true counting rate Rt, where the latter is the counting rate that would be recorded if there was no dead time and is proportional to the source activity. At low activity, an increase in count rate is proportional to the increase in activity. However, at high activity levels, the proportionality is lost, as the system is unable to handle the increasing count rate. At sufficiently high activity levels, the count rate actually decreases with increasing activity as the system becomes paralyzed. The relationship between the observed and true counting rates and the paralyzable dead time is given as follows: Ro =Rte-Rt (1) Notice that the observed counting rate rises to a maximum value given by R a (2) 2.718T Then the observed counting rate actually decreases with a further increase in true counting rate. This is because additional events serve only to extend the already long dead time intervals without contributing to additional events in the observed counting rate. The equation for the degradation of count rate due to pre-pulse dead time can be extended to coincidence counts by noting that both annihilation photons must be detected. If only one of the paired photons is detected, a coincidence event will not be registered but instead will be lost. Therefore, the coincidence count rate, Rc, can be expressed as R, = fRte-Rt -t = fRte-2Rt (3) where f is the fraction of the true single event rate, Rt which will create coincidence events. Note that for the coincidence case, Rt refers to the rate at which photons interact with the scintillation crystal, and is assumed equal at both detectors. Similarly, the equation for the degradation of coincidence count rate due to post- pulse dead time is given as R, = fRte-2Rtre 2Rtn = fte2Rt(T+n) (4) where T, is the integration time and the sum ( + n ) is the effective dead time constant. The Effects of High Count Rate on Spatial Resolution As pulses arrive after the integration process has begun, the extraneous light may be integrated with the original pulse. The extra signal contains additional spatial information that, when integrated with the spatial information of the original signal, may misplace the gamma ray's origin in the patient. This misplacement will create spatial distortion and worsen spatial resolution. The inaccurate positional information is reduced by the post-pulse energy discrimination levels. An alternative to energy thresholds employed by the detection system, local centroid algorithm,13 and hence local centroid position, is used. The local centroid algorithm ensures that the PMTs that may receive light from a pile up event are excluded from the position calculation. Random and Scatter Coincidence Events In addition to detection of the true coincidence events, PET imaging with coincidence detection can result in two other undesirable types of events; scatter and random events. Scattered coincidences occur when one or both of the gamma rays undergo a Compton scatter interaction inside the body. This process changes their direction and reduces the energy of the photon. The change in direction results in misidentification of the gamma-ray origin as shown in Figure 2B. The fraction of gamma rays which get scattered depends on the scattering media and path length through the body. Therefore, the contribution from scattered events is more evident in abdominal imaging than in brain imaging. Although the gamma rays that are scattered have their energy reduced below 511 keV, the energy resolution of most PET systems is insufficient to use this as an effective means of scatter rejection. Thus, many of the scattered events are accepted and subsequently lead to falsely positioned data despite the scatter correction techniques.14, 15 FIG. 2. Possible coincidence events; a graphical representation of true (A), scatter (B), and random (C) events.9 Random, or accidental, coincidences arise when two photons from different annihilations are detected within the coincidence window and recorded as a coincidence. This situation may arise either when the partner photons are scattered out of the FOV [Fig. 2C], or when the two uncorrelated photons simply arrive more closely in time than the true coincidence. The random coincidence rate increases with the singles rate on the detectors because the probability that two uncorrelated photons will arrive within coincidence window increases. The random coincidence rate is proportional to the square of the activity and is, therefore, a particular problem for high count rate studies. The randoms rate (R) is strictly related to the singles counting rate (S and S2) of each detector and to the coincidence time window width (c, ) by the following relation1' 9 R = 2.T,.Si.S2 (5) Therefore, a quadratic increase in random events will be observed by increasing the radioactivity, whereas a decrease will result from reducing Tz Random coincidences contribute to the background in the image, which can lead to a loss of image contrast. Efforts have been made to increment the net trues counting rate without incrementing random coincidences, by implementing faster systems with narrower coincidence time windows. The coincidence time window for the GSO crystal is 8 nanoseconds. Noise Equivalent Count Rate The relative noise in an acquisition is the standard deviation C of the total counts divided by the mean value N of the total counts. Because counts in a nuclear medicine acquisition follow a Poisson distribution, the relative noise is simply 1/ fN. When random and scatter coincidence events are removed to determine the true count rate, the mean counts are reduced but the standard deviation remains the same assuming no change in randoms or scatters. The distribution is no longer Poisson and the relative noise is now ,N /(N-A), where A is a constant representing the random and scatter coincidence events that were removed from the total counts. The noise equivalent count (NEC) rate is a useful parameter indicating the noise component of PET data. It is a useful predictor of the PET scanner performance because it combines the effects of signal and noise in the system in a single metric. NEC rate provides a quantitative framework in which to analyze design options that may increase true, scattered, and random coincidences, by varying degrees, to assess their impact on imaging performance. The NEC rate represents the ratio between net trues and prompts. The NEC rate follows Poisson statistics and has a relative noise equal to the relative noise of the true events after the random and scatter coincidence events have been removed. Setting the relative noise equal to 1/ NEC, then the NEC rate is given by the following equation: NEC- (N (6 N 11 It is worth mentioning that the best imaging condition is reached when acquisitions are performed by placing an activity in the FOV corresponding to the peak NEC rate. CHAPTER 3 MATERIALS AND METHODS Camera Description The camera used was a Philips Allegro Positron Emission Tomography (PET) system. The system is equipped with 420 Photo-Multiplier Tubes (PMTs) and 28 modules. Full-ring detection geometry is defined by placing the modules side by side, coupled to the PMTs closely packed in a hexagonal array. Each module is populated with an array of 638 Gadolinium Oxyorthosilicate (GSO) crystals, totaling 17,864 crystals. Each crystal has dimensions of 4x6x20 mm. The modules are arranged around the patient port with the GSO crystals facing the patient. The 420 PMTs are mounted on the opposite side of the 28 modules which collect light from the scintillation crystals [Fig. 3]. By surrounding the patient with 28 position-sensitive segments, data can be acquired simultaneously along any parallel line and at any right angle. PMT (420 REQUIRED) MODULE 12' REQUIRED) -G SO CRYSTALS (638 PER MI:DULE) (17,864 TOTAL) FIG. 3. Module Assembly The axial field-of-view (FOV) of the camera is 18 cm and the transverse FOV is 57.6 cm. The system operates in 3D mode, and therefore no physical collimators are involved. The scanner houses a 137Cs source, 662 keV, for generating transmission images during patient scanning. The transmission data is used to correct the emission images for photon attenuation in the patient's body. Transmission scans are not used in this investigation because, according to the National Electrical Manufacturers Association16 (NEMA), attenuation corrections should not be employed when measuring the count rate capability of the scanner. NEMA is further discussed in the following sections. Phantom Description The phantom considered here is described in section 4 of the National Electrical Manufacturers Association (NEMA) NU 2-2001 document of performance standards for PET scanners.16 A list of the NEMA protocol is provided in Appendix A. The test phantom [Fig. 4(A), and 4(B)] is a solid circular cylinder, as shown below, Center of cylinder at center of transerse FOV A B FIG. 4. Phantom utilized in the experiment; (A) Phantom and line source, (B) Cross section of the cylindrical phantom showing line source inserted 4.5 cm off center. composed of polyethylene with an outside diameter of 20 cm and with an overall length of 70 cm. A 6.4 mm hole is drilled parallel to the central axis of the cylinder, at a radial distance of 4.5 cm. The cylinder consists of 4 segments that are assembled together during testing. The assembly of the completed phantom was checked to insure a tight fit between adjacent segments, as even very small gaps will allow narrow axial regions of scatter-free and attenuation-free radiation. The test phantom line source insert is a clear polyethylene coated plastic tube that is 80 cm in length, with an inside diameter of 3.2 mm and an outside diameter of 4.8 mm. The central 70 cm of this tube was filled with an sF-FDG initial activity of 19.26 millicuries (mCi) at 3:30 PM and threaded through the 6.4 mm hole in the test phantom. Notice that the phantom in this experiment is longer than the axial FOV of the scanner, which allows studying of the effects of out-of-field activity on the count rate. For consistency, the phantom was rotated such that the line source is at the lowest position (i.e., nearest to the patient's table), because the measured result will depend on the relative orientation of the line source and the table. Allegro Data Acquisition and Processing The phantom was centered in the 18 cm field of view of the scanner and the dynamic NEMA-Countloss protocol was used to acquire data at different activities. Each acquisition lasted for 20 minutes. A delay time of 20 minutes was used between acquisitions. The first acquisition began at 4:38 PM with a calculated initial activity of 12.536 mCi. A total of 21 tomographic acquisitions were taken every 40 minutes. Table I, in the results chapter, lists the initial conditions of the image acquisitions, including the times of the acquisitions, the initial activities at the beginning of each acquisition, and the time into the experiment. A complete description on how to acquire data using the Allegro PET scanner is given in the following sections. Allegro Data Acquisition Parameters The Allegro workstation consists of two monitors; one for acquisition, and another for processing. File Management was chosen from the acquisition monitor. Invoked within the File Management is the Acquisition menu that is used to setup the parameters for the acquisition. By choosing Set Up Acquisition, one can enter all information pertinent to that acquisition, such as study date, phantom weight, etc. The information required here are easy to understand and require no previous experience. Notice that the directory where the output raw data will reside does not need to be specified. After the acquisition is finished, the raw data will be transferred to the patient directory automatically and will be easily identified. However, the filename should be entered to be able to identify the correct file. For further clarification on the acquisition parameters and any other software setup information, one can refer to the Allegro user's manual.17 The dynamic DefaultNEMACountloss is selected as the acquisition protocol. Typically, acquisitions are performed using a preset protocol. In this experiment, however, it was necessary to change some of the acquisition parameters, which is done by selecting to edit the protocol. The editing process consists of specifying the acquisition time and adding more frames (or acquisitions) at the end until a small activity is left in the line source. The number of frames added depends on the starting activity and the acquisition time chosen. The durations of all the frames could be added to get a good prediction of what the last frame should be. After acquiring, a file containing the raw data is generated and saved in the patients' directory. This file is further processed as shown in the following section. Allegro Data Processing Parameters The raw data file, the output at the end of the last acquisition, is then processed on the processing monitor. To do so, the NEMACountloss reconstruction protocol is selected by highlighting the file, and choosing Petview, Reconstruct Sinogram, and Research Protocol, respectively. The reconstruction protocol is then edited to generate the Fourier rebinned18' 19 data to be use in the data analysis programs. The parameters of NEMACountloss reconstruction protocol are already set correctly as specified by the NEMA standards for testing the count rate performance of positron emission tomographs. Examples of these parameters are the background subtraction, attenuation correction, and decay correction, which are all turned off during the reconstruction. However, the user needs to save the Fourier rebinned (FORE) sinograms using the FT Output Sinogram option, found in the advanced parameters menu. A filename can also be specified for the rebinned data. Notice that in the advanced parameters, there is an option provided by the manufacturer to turn the Single Slice Rebinning8 (SSRB) on. The SSRB, however, was not used because it is believed that this rebinning algorithm is not functioning at all. Despite consultation with the software experts at Philips, this matter has not been resolved. The manufacturer, however, suggested using the FORE for this experiment. File Structure The reconstruction program operates on the raw data to produce an interpolated scan file. The structure of this interpolated file is as follows. It has a main header that is 512 bytes long. As a confirmation, the headers display acquisition parameters and patient information. The main header contains information such as the date of the experiment or patient scan, the patient ID, isotope used, etc. The main header is followed by a directory record, which is 512 bytes long. Every directory record is followed by 31 slices. If the file contains more than 31 slices, a new directory record is created after slice number 31. Sinograms and directory records are added to the file as needed until all data are stored. Each slice or sinogram within the file has a subheader. The size of the subheader is also 512 bytes followed by data. The data in the sinogram are always written as signed short (2 bytes) variables. The sinograms are treated as 256 by 192 arrays. The overall structure of the Fourier rebinned file is the same as the interpolated file. However, the interpolated file contains 45 tilted slices at each tilt angle. There are 7 tilt angles for each sinogram. Thus, there are 7 tilted slices at each position along the axis parallel to the axial field-of-view, including the zero tilted sinogram or direct sinogram. These oblique sinograms are collapsed into a single sinogram for each respective slice using the FORE algorithm to produce only direct sinograms suitable for NEMA processing. Therefore, reducing the total number of slices from 315 to 45 slices per acquisition. The Fourier rebinned file and the interpolated file are both output by the reconstruction protocol. To recap, in the interpolated file each event is binned into 4-dimensional projection coordinates; Transverse distance r, projection angle tilt or off-plane angle 0, and axial distance y'. FORE is then employed to convert these 4-dimensional sinograms into 2- dimensional sinograms by collapsing the oblique sinograms into a single sinogram for each respective slice while conserving the number of counts in the sinogram. The resulting sinogram is a 256 by 192 pixel matrix with a signed short format. The pixel's linear size is 2.25 mm. No corrections for dead time, attenuation, or random events are employed during the acquisitions. A total of 45 sinograms per acquisition are obtained in order to span the length of the line source in the 18 cm field of view. In addition, the Fourier rebinned file is then copied to a specified directory to be burned on a CD. Appendix B provides more information on how to copy and burn files to a CD. All pixels in a sinogram are multiplied by a scaling factor to correct the number of counts in each pixel. This is done when running the MATLABt codes, discussed in the following section. The scaling factors are obtained using a shell script provided by the software engineer at Philips Medical Systems [appendix C], which is a small program that can be run in a Unix environment. The output from this program is three columns. The first is frame number, running from 1 to 21. The second is a slice index, typically running from 1 to 45. The third value is the sinogram slice scale factor. To run this program in Unix, one should refer to the commands shown in appendix C. NEMA Data Processing and Analysis Using MATLAB Count Rates MATLAB codes, written by the author of this paper, were employed to perform the NEMA data analysis for processing the sinograms data obtained after the experiment. These codes are given in appendices D through I and will be briefly discussed here. Notice that comments are embedded within each code to make them easy to understand and user friendly. These MATLAB codes need to be run successively. Because the output from the first program [appendix F] will become the input to the second program [appendix G] and the output from the second program will become the input to the third program [appendix H], and so on. Appendix D shows the code for extracting the sinograms from the raw data file. The extracted sinograms of each acquisition are output MATLAB: The language of technical computing by MathWorks, Inc. 2002 to different acquisition files numbered 1 through 21. The main header of the original file, the subheader of each sinogram, and the directory records are all not included in the output acquisition files. Therefore, the output data are only the sinograms data needed for processing. This step of reorganizing the data such that no header information is embedded within the actual sinogram data is significant for it reduces the amount of code necessary in processing the sinograms. Appendix E has the scaling factors code. This is a simple piece of code written to organize the scaling factors in a format suitable for processing when used by the programs to follow. Again, the order in which these programs are run is important and follows the order shown in the appendices. Section 4 of the NEMA standards publication NU 2-2001 for performance measurements of positron emission tomographs summarizes the steps used to determine scatter fraction, count losses, and randoms measurements. The MATLAB codes in appendices F and G were both used for further processing of the data. In these codes, pixels that were 12 cm or further away from the center of the phantom were set to zero counts for each sinogram. For each projection angle within a sinogram, the location of the line source's center was determined as the pixel where the highest count value occurred. The count data within that projection angle were then shifted in the transverse direction so that the center of the line source aligned with the center of the sinogram in the transverse direction [Fig. 5]. After the shift, each sinogram was compressed into a one- dimensional profile, or sum projection, by summing the pixel values along the angular direction. A B FIG. 5. Illustration of sinograms shifting process; (A) Sinogram, (B) Shifted sinogram. In slight deviation from NEMA, these profiles were then summed over all sinograms for the particular acquisition to create a final sum projection to describe each acquisition. It was from these final sum projections, rather than the sum projections of the individual slices, that the count rates were determined. From these count rates, the activity where the peak rate occurs was found. Figure 6 illustrates how the random and scatter coincidence events were determined for each acquisition. The counts at the pixels 2 cm to either side of the center of the final d maximum pil unscattered SLij counts C Ri/ j S scaltered cawnis radial distance from maximum pixel FIG. 6. Example of determining random and scatter coincidences; counts versus radial distance from the center pixel.16 sum projection were averaged and then multiplied by the number of pixels equal to a distance of 4 cm. This product was added to counts outside the 4 cm strip to create the random plus scatter counts for acquisition j, C+, To separate the scatter and random components, the final acquisition j', where the random coincidence rate and the loss in count rate due to dead time are both presumed less than 1%, was considered. With the dead time equation and equations for coincidence count rates7' 1, it was verified that the dead time losses and the random count rates in the final acquisition would both be less than one percent. Letting the random coincidence counts be negligible, the scatter and random counts, C +s,. were divided by the total counts of acquisition j', CTor,; to calculate the scatter fraction SF: C SF = r (7) C TOTj This scatter fraction was constant for all acquisitions because the geometry and attenuation media remained constant. Variations in design cause PET scanners to have different sensitivities to scattered radiation. The scatter fraction is measured at a sufficiently low counting rate that random coincidences, dead-time effects, and pileup are negligible. The total event rate, RTOTj for each acquisition is the total counts for that acquisition divided by the acquisition time Tacq,j: COTj (8) RTOT --T acq,j The true event rate, free of scatter or random coincidences, is CTOTj Cr+sJ acq,j The scatter and random count rates can be found by using the following equations: SF Rs = Rtj (10) s, 1- SF R = R TOT, J (11) R R11 1 -SF The noise equivalent count rate was determined by the following equation: R2 R NEC = t (12) RTOT,j To determine if the random coincidence count rate increases with the square of the activity, a function was fitted to the plot of computed random count rates as a function of activity [appendix H]. The fitted function was of the following form: F(x; a)= ax2 (13) where x is the activity in millicuries (mCi), and [a] is a constant. Spatial Resolution The final sum projection of each acquisition was fitted to a Gaussian function plus a constant, equation 14, using the nonlinear least squares method in MATLAB [appendix H]. To further verify the fitted curves, the final sum projection of each acquisition was plotted on the software package Kaleida Graph 3.5 and a function was fit to the plot. An online curve-fitting tool was also used for the same purpose. The fitted function that was always considered was a gaussian plus a constant. The parameters of the fitted functions were all the same within three significant figures. The Kaleida Graph parameters, however, were slightly different. The FWHMs, calculated based on the fitted parameter of the standard deviation, are, therefore, only rough calculations due to errors that are likely to be associated with the fitting process. A trial version of this software package was used. http://zunzun.com: An interactive 2-D and 3-D data-modeling tool. The spatial resolution was then determined by calculating the full width at half maximum (FWHM) for each acquisition using equation 15. The standard deviation C in equation 15 is obtained from the fitted functions. The FWHM was adjusted for the finite diameter of the line source (3.2 mm) using equation 16. This equation was based on the practice of squaring components of resolution, adding the squared components together, and then taking the square root of this sum to determine the final resolution. F(x; a, b, xo,) = a+ be -5Ox-x)2 /T2 (14) FWHM = 2.35 y (15) FWHMa(mm) = [(FWHM(mm))2 (3.2)2]1/2 (16) To further explore the spatial resolution of an imaging system, the modulation transfer function (MTF) is preferred. By using MATLAB [appendix I], an MTF was calculated for each acquisition by computing the discrete Fourier transform of the final summed projection of each acquisition. The MTFs were each normalized to a value of unity at the zero frequency. This analysis on the spatial resolution is confined to only the resolution in the transverse direction because of the type of geometry and analysis followed. CHAPTER 4 RESULTS Table 1 lists the initial conditions of the image acquisitions, including the times of the acquisitions, the initial activities at the beginning of each acquisition, and the time into the experiment. TABLE 1. Initial conditions of image acquisitions Acquisition Start time 4:38 PM 5:18 PM 5:58 PM 6:38 PM 7:18 PM 7:58 PM 8:38 PM 9:18 PM 9:58 PM 10:38 PM 11:18 PM 11:58 PM 12:38 AM 1:18 AM 1:58 AM 2:38 AM 3:18 AM 3:58 AM 4:38 AM 5:18 AM 5:58 AM End time 4:58 PM 5:38 PM 6:18 PM 6:58 PM 7:38 PM 8:18 PM 8:58 PM 9:38 PM 10:18 PM 10:58 PM 11:38 PM 12:18AM 12:58 AM 1:38 AM 2:18 AM 2:58 AM 3:38 AM 4:18 AM 4:58 AM 5:38 AM 6:18 AM Time into experiment (min) 0 40 80 120 160 200 240 280 320 360 400 440 480 520 560 600 640 680 720 760 800 Initial activity (mCi) 12.536 9.738 7.564 5.876 4.564 3.545 2.754 2.139 1.662 1.291 1.003 0.779 0.605 0.470 0.365 0.284 0.220 0.171 0.133 0.103 0.080 Count Rate The final sum projections of acquisitions 1, 5, 10, 15, and 21 are shown in Figure 7. 4 x 10 2.5 2 o) o 1.5 rw I--0 0 0 acquisition 15 acquisition 1 0.5- acquisition 5 acquisition 21 \ acquisition 10 0 -20 -10 0 10 20 Distance along transverse axis FIG. 7. Final sum projections of selected acquisitions Figure 7 illustrates the changes in count rate, width of the peak, and background level due to random and scattered events. The background decreases over time indicating that the random and scatter coincidences together are decreasing. The final sum projections of all acquisitions are shown in Figures 8 through 28 [Appendix J]. Notice that the y-axis of Figures 8 through 28 is changed to reflect the changes in the count rates for each separate acquisition. The total paired events, and total, true, scatter, random, and NEC count rates are all shown in Table 2. The scatter fraction was calculated to be 0.403. This scatter fraction was computed from the last acquisition. TABLE 2. Processed coincidence count rates of all acquisitions Acquisition 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 Total Paired Count RateCount RateCount RateCount RateCount Rate Events [106 ] 343.88 294.66 246.74 203.00 164.77 132.65 105.81 83.89 66.32 52.03 40.83 31.92 24.95 19.49 15.17 11.88 9.27 7.22 5.65 4.37 3.44 Total [cps] 286564.15 245551.51 205616.61 169163.43 137305.24 110538.73 88176.77 69904.97 55268.48 43355.99 34022.45 26602.83 20795.78 16244.02 12638.47 9898.20 7724.92 6016.78 4705.65 3640.10 2866.02 Trues [cps] 83176.54 83854.61 79714.53 72462.72 63534.72 54306.58 45378.88 37285.31 30308.12 24302.21 19392.04 15351.38 12124.09 9545.14 7469.23 5868.92 4594.00 3584.41 2802.28 2166.92 1706.42 Randoms [cps] 147239.97 105091.52 72091.44 47785.34 30881.93 19572.93 12165.25 7450.51 4501.10 2648.76 1539.97 888.63 487.43 255.52 127.20 67.52 29.78 12.74 11.71 10.41 7.70 Scatter [cps] 56147.65 56605.37 53810.64 48915.37 42888.59 36659.22 30632.65 25169.15 20459.25 16405.01 13090.44 10362.82 8184.27 6443.37 5042.04 3961.77 3101.14 2419.63 1891.66 1462.76 1151.91 NEC [cps] 24142.36 28635.93 30904.15 31040.07 29399.17 26680.29 23353.58 19886.92 16620.37 13622.05 11053.03 8858.64 7068.43 5608.81 4414.25 3479.85 2732.04 2135.36 1668.80 1289.95 1016.00 Figures 29 through 34 illustrate the counting rate performance of the PET scanner. The total, true, random, and scatter count rates are plotted in Figure 29 as functions of time into the experiment, and in Figure 30 as functions of activity. The true count rate is plotted as a function of time into experiment in Figure 31, and as a function of activity in Figure 32. As can be observed in Figures 31 and 32, the true count rate reaches a maximum of 83.86 kcps at an activity of 9.738 mCi. The noise equivalent count rate is plotted as a function of time into experiment in Figure 33 and as a function of activity in Figure 34. Notice that the noise equivalent count rate also reaches a maximum and drops again as activity increases. The maximum NEC rate is 31.04 kcps at 5.876 mCi. As shown in the figures, the total, true, and scatter count rates will all rise with activity 300 250 STotal \-- True S200 -- Random o \ Scatter 1504 0 100 0 01 0 100 200 300 400 500 600 Time [min] FIG. 29. Count rates versus time into experiment 300 250- Total -U- True S200 -- Random Scatter S150 o 100 50 50- 6 8 Activity [mCi] 10 12 14 FIG. 30. Count rates versus activity 83.86 kcps 0 100 200 300 400 500 600 Time [min] FIG. 31. True count rate versus time into experiment 83.86 kcps at 9.738 mCi 700 800 12 14 0 2 4 6 8 10 Activity [mCi] FIG. 32. True count rate versus activity I kcps 0 100 200 300 400 500 600 700 Time [min] FIG. 33. Noise Equivalent Count rate versus time into experiment 31.04 kcps at 5.876 mCi 12 14 2 4 6 8 10 Activity [mCi] FIG. 34. Noise Equivalent Count rate versus activity and tend to reach a maximum value before they start to decrease. The total peak count rate is not shown here because higher activities are needed to reach the maximum. The shape of these count rate curves is consistent with the behavior of the radiation detector that experiences paralyzable dead time and pulse pile up. The count rate increases, starting at low activities, until a maximum count rate is reached and then starts to decrease as the activity continues to grow. The random coincidence count rate, on the other hand, starts very small at low activities and continues to increase with increasing activity. As expected, the continued increase in randoms reflects the fact that random coincidence rate is proportional to the square of the activity. A fact that is also illustrated by the fitted function [see Equation 13]. Figure 35 below shows the fitted randoms as a function of activity. The constant [a] of Equation 13 was found to have a value of 1028 cps and the fit had an R-squared value of 0.976. 180 140 ---- 120 fitted equation S100 -randoms rate E/ 0 60 - 40- - 20--- 0 0 2 4 6 8 10 12 14 Activity [mCi] FIG. 35. Randoms count rate and its fitted equation versus activity Spatial Resolution The parameters of the fitted Gaussian equation, Equation 14, of the final sum projection are given in Table 3 for each acquisition. TABLE 3. Parameters of fitted curves Acquisition a b [cps] [cps] 770.44 615.53 481.14 371.08 284.04 217.22 165.78 126.57 97.07 74.19 57.03 43.89 33.87 26.17 20.22 15.75 12.24 9.51 7.44 5.76 4.54 22650.41 23107.80 22195.66 20343.87 17966.86 15439.59 12959.74 10691.55 8712.33 7001.64 5593.31 4433.85 3506.68 2763.25 2163.63 1700.44 1331.75 1039.44 813.13 628.40 494.66 Xo [cm] -0.0013 -0.0025 -0.0022 -0.0031 -0.0029 -0.0023 -0.0029 -0.0031 -0.0021 -0.0027 -0.0028 -0.0023 -0.0028 -0.0021 -0.0026 -0.0034 -0.0018 -0.0018 -0.0034 -0.0031 -0.0022 C R-squared [cm] 0.35401 0.34175 0.33342 0.32724 0.32270 0.31935 0.31679 0.31486 0.31341 0.31233 0.31169 0.31111 0.31039 0.31002 0.30961 0.30964 0.30946 0.30927 0.30927 0.30926 0.30935 0.888 0.923 0.944 0.957 0.965 0.971 0.975 0.977 0.979 0.980 0.981 0.982 0.983 0.983 0.983 0.984 0.984 0.984 0.984 0.984 0.984 Figures 36 through 56 [Appendix K] show the fitted final sum projections. Figures 36 and 56 are shown on the next page to illustrate the differences in the fit between the first and last acquisitions. This approach of fitting the final sum projections to a gaussian function plus a constant is not very accurate. The fitting parameters would differ when a better fit is introduced. Therefore, the FWHM would change as well. Due to this inaccurate curve fitting, the technique employed here remains subject to error. However, it still provides a rough estimation of the resolution to at least get an overall picture of the imaging system behavior. The calculated FWHMs are given in Table 4. The adjusted x 104 * Final Sum Projection Fitted Gaussian -15 -10 -5 0 5 10 15 Distance along transverse axis [cm] FIG. 36. Final sum projection of acquisition 1 and its fitted gaussian curve *Final Sum Projection Fitted Gaussian FIG. 56. Final sum projection of acquisition 21 and its fitted gaussian curve 500 7 400 - 300 rE 0 8 200 100 0" -15 -10 -5 0 5 10 15 Distance along transverse axis [cm] TABLE 4. FWHM calculated from the fitted final sum projections Acquisition measured FWHM adjusted FWHM [mm] [mm] 1 8.32 7.68 2 8.03 7.37 3 7.84 7.15 4 7.69 6.99 5 7.58 6.88 6 7.5 6.79 7 7.44 6.72 8 7.4 6.67 9 7.37 6.63 10 7.34 6.61 11 7.32 6.59 12 7.31 6.57 13 7.29 6.55 14 7.29 6.55 15 7.28 6.54 16 7.28 6.54 17 7.27 6.53 18 7.27 6.53 19 7.27 6.53 20 7.27 6.53 21 7.27 6.53 FWHMs are plotted as functions of time into experiment and activity as shown in Figures 57 and 58, respectively. The degradation in the spatial resolution can be seen as the FWHM increases from 6.53 mm to 7.68 mm as the activity increases. After the 16th acquisition, no further improvement is seen in the spatial resolution. The FWHM has a constant value of 6.53 mm as activity continues to decay going from the 17th acquisition to the 21st acquisition. The spatial resolution of the PET scanner atlO cm from the center in the transverse direction is 5.9 mm, as measured by a point source. This value was provided by Philips-ADAC Medical Systems. The different value obtained for the resolution, therefore, could have been a result of the different approach used to calculate the resolution of the camera and the finite line source diameter involved in the 8 7.8 7.6 7.4 7.2 7 6.8 6.6 6.4 0 8r 7.8 7.6 7.4 7.2 7- 6.8 6.6 6.4 100 200 300 400 500 600 700 Time [min] FIG. 57. Adjusted FWHM as a function of time into experiment 0 2 4 6 8 10 12 14 Activity [mCi] FIG. 58. Adjusted FWHM as a function of activity measurement. The point to keep in mind here is that the spatial resolution improves as the activity decreases. Therefore, the administration of entailed high activity levels to the patients will not only increase their radiation absorbed dose, but also degrades the spatial resolution. Assessment of the spatial resolution is also obtained from the modulation transfer functions. Figure 59 shows the modulation transfer functions of acquisitions 1, 5, 10, and 21. For the first acquisition, at an activity equal to 12.536 mCi, the gain at a 1 acquisition 1 0.9 acquisition 5 0.8 acquisition 10 acquisition 21 0.7 21 0.6 - S 0.5 - 0.4 A -1 0.3 - 0.1 0 0. -- -- 4"^ ^ - - --- ---- - 0 0.5 1 1.5 2 2.5 v [cm-1] FIG. 59. Modulation transfer function of acquisitions 1, 5, 10, and 21 spatial frequency of 0.1 cm-1 is only 24.3%. This gain increases in the fifth acquisition, which has an activity of 4.564 mCi, to 42.6% at the same spatial frequency. Figures 60 through 80, in Appendix L, are the modulation transfer functions for each acquisition. Acquisitions 10, 15, and 21 also show the increase in gain to 52.8%, 56%, and 56.5%, respectively, at 0.1 cm- spatial fequency. The MTF Figures also illustrate the same trend of increasing gain across the range of frequencies as activity decays over time. For example, at a frequency of 0.5 cm-1, the gain is only 15 % at the first acquisition, but 25.8% at the fifth, and 32.3%, 34.3%, and 34.6% for the tenth, fifteenth, and twenty-first acquisitions, respectively. No significant increase in the gain was noticed going from the seventeenth acquisition to the twenty-first. The gain for the latter acquisitions was almost constant and had a value of about 34.6 %. This also explains the constant values obtained for the spatial resolution at the same last five acquisitions. There was a ripple, or dip, present in each modulation transfer function. This rippling effect is believed to originate from the background counts that were included when computing the Fourier transforms. Therefore, exclusion of the background by subtracting the random and scatter events from the total events would eliminate or reduce the rippling significantly because the modulation transfer functions would now be created based on the true events only. Improvements in the gain, in this case, would also reflect the improvement in image contrast. The rippling in the MTF curves was first thought to be an artifact due to the truncation of counts located 12 cm away from the center of the image. However, the same effect was still seen in the MTFs, even without the truncation. CHAPTER 5 DISCUSSION Some recent studies have investigated the counting rate performance of state-of- the-art dedicated PET systems using different size phantoms.6' 7, 9 The most common of these phantoms were the 19-cm and the 70-cm phantoms. A longer phantom of 70-cm length was considered in order to study the effects of out-of-field activity on the count rates detected. The scatter fraction calculated from the last acquisition was 40.3%. The scatter fraction is not representative of a realistic body size. However, it is useful for standard evaluations among scanners of different configurations. On the other hand, a concern with using the line source to measure the counting rate performance was whether or not it yields counting rates that are clinically relevant. A recent study7 shows that there is good correlation between the measured counting rates for the 70-cm line source and whole-body studies. It is important in comparing performance between scanners to look both at the peak true and NEC counting rates and at the activity levels where these peak rates occur. A highly sensitive system may saturate at a relatively low activity level, but the counting rate at this level may be higher than that of a system with lower sensitivity at a higher activity level. The NEC rate peaked at 31.04 kcps, at an activity of 5.876 mCi, it then decreased with increasing activity. If the 70-cm phantom were considered to mimic the body of an adult, then an activity more than the 5.876 mCi should not be used because any increase in activity above that point would decrease the count rate due to pileup in the detector. According to Philips-ADAC Medical Systems, the peak NEC rate was 33 kcps at an activity of 0.25 LICi/mL or 5.5 mCi for the 22 liters phantom. The 22 liters is the volume of the 70-cm polyethylene phantom. In comparison, the manufacturer's NEC rate was higher than the one calculated in this experiment because the value provided by the manufacturer was based solely on rough hand calculations. Moreover, the peak true count rate was 84 kcps at an activity of 0.4 LICi /mL or 8.8 mCi for the same phantom. This value was also based on rough hand calculations. These values were considered to be suggestive and not authoritative, as pointed out by the manufacturer. The spatial resolution was degraded when activity increased, as can be seen from the FWHM plot as a function of activity. This should be taken into consideration when administering a radiopharmaceutical to the patient. The improvement in the FWHM is only 1.15 mm from the first to the last acquisition. The FWHM, where the peak NEC occurred, was 7mm. The true count rate at that resolution is 72.46 kcps. Better resolution can be obtained at lower activities, but the count rate would be noisier. A compromise, therefore, should be made between the resolution and the activity. The spatial resolution of the last five acquisitions remained the same as indicated by the FWHM value of 6.53 mm. This means that no further improvement in spatial resolution is obtained beyond the seventeenth acquisition. One could argue that the limit on the transverse spatial resolution of the scanner at 4.5 cm radial distance from the center of the transverse FOV is 6.53 mm. Further discussion on the modulation transfer functions that describes the spatial resolution of an imaging system was previously included in the results chapter of this paper. MTFs illustrated that the gain improves as activity decreases. Significant improvements in gain were shown up to the sixteenth acquisition, after which the gain remains unchanged. In the MTF plots, the curve extending from the zero frequency to the region, where the little dip exists, is due to the scatter and random components. This portion of the plot corresponds to the background and appears to occur within the frequency range of zero to 0.125 cm-1. The other portion of the plot, extending from 0.18 cm-1 to the Nyquist frequency, 2.22 cm-1, is due to the true events. Activity outside the field of view was shown to greatly affect the count rate performance, especially the scatter component, of the scanner as well as the spatial resolution. The random coincidence counts were mostly affected by the dead time of the radiation detector. The expected trend of the random events, as illustrated by the fitted equation in Figure 35, is a quadratic increase as the activity increases. However, the calculated random events continued to increase then their increase slows down as indicated by the intersection of the fitted and the measured random curves. This is due to the increased dead time of the radiation detector because of the more activity, and hence more counts, introduced. Each undetected event will only increase the dead time and therefore, some random counts will be lost. Finally, the spatial resolution of the PET scanner degrades as more activity is administered to patients. In addition, more counts are lost at the same high activities. The optimal activity that should be administered to patients is the one at which the peak noise equivalent count rate occurs. Because very high activities will result in more counts being lost because of system dead time and counts pile up. CHAPTER 6 CONCLUSIONS The 3-dimensional scanning system had a high scatter fraction of 40%, compared to the scatter fraction obtained using a smaller phantom. This high value was expected because of the long phantom employed in the measurements and the 3-dimensional type system involved. The scattered events from outside the field of view contributed to this increase in the scatter fraction as opposed to smaller phantoms, typically 19-cm phantoms. The longer phantom provides a more clinically reasonable distribution of activity outside the scanner. This out-of-field activity can impact the counting rate performance as well as the spatial resolution. Future work involving the same measurements discussed in this investigation could be in the form of utilizing an even longer phantom, possibly 150 cm, which would mimic the length of an average person. This would more closely approximate a true whole-body clinical scan. The scatter fraction would also be expected to increase even more. However, the increased value would only mean that more scattered coincidences would be eliminated from the data reconstructed, and therefore, improve the overall quality of the clinical images. In addition, the NEMA protocol may be revised to include the new longer phantom, if proved to be better than the existing ones. The peak noise equivalent count rate was determined to be 31.04kcps. This peak occurred at an activity of 5.876 mCi. The full width at half maximum decreased from 7.68 mm to 6.53 mm as activity decreased, indicating the improvement in spatial resolution as activity decays over time. The improvement in resolution was supported by the modulated transfer functions. The study shows that both the count rate capabilities and the spatial resolution of the 3-dimensional dedicated positron emission tomograph will degrade at high levels of activity. This is important when determining the amount of radionuclide tracer that should be administered for a PET scan. The amount of 18F radioactivity administered to patients undergoing a PET scan is usually about 6 mCi at the start of the acquisition and after enough time is allowed for uptake of the radionuclide in tissue. Notice that this value is too close to the peak NEC rate that occurs at 5.9 mCi. This amount will still introduce some amount of noise in the image, but it results in an ideal imaging condition. The NEC rate is the ratio between the net trues and prompts. Therefore, the best imaging condition is reached when acquisitions are performed using an activity corresponding to the peak NEC rate. In conclusion, to allow for more accurate imaging of patients, a compromise between the acceptable noise equivalent count rate and adequate spatial resolution is required. APPENDIX A NEMA 2001 (SECTION 4) NEMA Standards Publication NU 2-2001 Performance Measurements of Positron Emission Tomographs Published by National Electrical Manufacturers Association 1300 N. 17m Street, Suite 1847 Rosslyn, VA 22209 Copyright 2001 by the National Electrical Manufacturers Association. All rights including translation into other languages, reserved under the Universal Copyright Convention, the Berne Convention for the Protection of Literary and Artistic Works, and the International and Pan American Copyright Conventions. NU 2-2001 Page 11 Section 4 SCATTER FRACTION, COUNT LOSSES, AND RANDOMS MEASUREMENT 4.1 GENERAL The scattering of gamma rays emitted by the annihilation of positrons results in falsely located coincidence events. Variations in design and implementation cause positron emission tomographs to have different sensitivities to scattered radiation. The measurements of count losses and random rates express the ability of a positron emission tomograph to measure highly radioactive sources with accuracy. The measurement of noise equivalent count rates is based on work described in Strother, S.C., Casey, M.E. and Hoffman, E.J., Measuring PET Scanner Sensitivity: Relating Count-Rates to Image Signal-to-Noise Ratios Using Noise Equivalent Counts, IEEE Trans Nucl Sci, NS- 37(2):783-788, 1990. 4.2 PURPOSE The first purpose of this procedure is to measure the relative system sensitivity to scattered radiation. Scatter is expressed by the scatter fraction, SF, for the entire tomograph. The second purpose of this procedure is to measure the effects of system dead-time and the generation of random events at several levels of source activity. The true event rate is the total coincident event rate minus the scattered event rate and minus the randoms event rate. The method of randoms estimation includes misplaced events as part of the randoms measurement. Furthermore, shifts in the energy peak may influence the results for the randoms, since it is assumed that the scatter fraction is constant as a function of countrate. 4.3 METHOD The test phantom is a solid right circular cylinder composed of polyethylene with a specific gravity of 0.96 0.01, with an outside diameter of 203 3 mm (8"), and with an overall length of 700 5 mm. A 6.4 0.2 mm (1/4") hole is drilled parallel to the central axis of the cylinder, at a radial distance of 45 1 mm. For ease of fabrication and handling, the cylinder may consist of several segments that are assembled together during testing. However, in both design and assembly of the completed phantom one must insure a tight fit between adjacent segments, as even very small gaps will allow narrow axial regions of scatter-free radiation. The test phantom line source insert is a clear polyethylene or polyethylene coated plastic tube at least 800 mm in length, with an inside diameter of 3.2 0.2 mm (1/8") and an outside diameter of 4.8 0.2 mm (3/16"). The central 700 5 mm of this tube will be filled with a known quantity of activity and threaded through the 6.4 mm hole in the test phantom. To begin the test, a source of relatively high activity is placed in the field of view of the positron emission tomograph. Regular measurements are then taken while the activity in the phantom decays over several half-lives. A decrease in the ratio of the random event rate to the true event rate accompanies the activity decay, eventually falling to less than 1%. In addition, the efficiency of the system in processing coincident events improves as the activity decays, until count losses may be effectively neglected. Thus by waiting long enough one obtains a measurement of the coincidence count rate that is effectively free from both randoms and processing losses. By NU 2-2001 Page 12 extrapolating this true rate back to higher activity levels and comparing it to the measured rate one may estimate count losses suffered by the system at higher activity levels. The accuracy of this technique depends critically on adequate statistics being gathered at sufficiently low levels of activity. This may require repeated measurements at the lower count rates. Center of cylinder at center of FOV 145mm Figure 4-1 POSITIONING OF PHANTOM 4.3.1 Symbols Scatter fraction (SF) a dimensionless ratio of scattered coincidence events to the sum of scattered and true coincidence events in a defined ROI of the scanner field-of-view. 4.3.2 Radionuclide The radionuclide used for this measurement shall be 18F. The amount of radioactivity shall be great enough to allow the following two rates to be measured: a. Rt.peak peak true count rate b. RNEC.peak peak noise equivalent count rate Recommendations for the initial activity required to meet these objectives will be supplied by the manufacturer. The initial activity in the phantom shall be determined from the activity injected into the phantom as measured in a calibrated dose calibrator. 4.3.3 Source distribution The central 700 5 mm of the test phantom line source insert shall be filled with water well mixed with the measured amount of radioactivity and sealed at both ends. This line source shall be inserted into the hole of the test phantom such that the region of activity coincides with the 70 cm length of the phantom. The test phantom with line source is mounted on the standard patient table supplied by the manufacturer and rotated such that the line source insert is positioned nearest to the patient bed (see Figure 4-1). The phantom is centered in the transverse and axial fields-of-view to within 5 mm. 4.3.4 Data collection Data shall be acquired at intervals more frequent than half the radionuclide half-life, T1/2, until true events losses are less than 1.0%, and the random rates are less than 1.0% of true rates. The durations of the individual acquisitions, Tacqj, shall be less than one-fourth of T112. Acquisitions shall be fully tomographic; therefore, rotating scanners must rotate to provide complete and uniform angular sampling for each acquisition. In the case of rotating scanners, the acquisition time Tacq shall include the time required to rotate the detectors. NU 2-2001 Page 13 It is essential to the accurate estimation of system dead-time losses that sufficient statistics be acquired with count loss rates and random rates both below 1.0% of true rates. Each acquisition should contain a minimum of 500,000 prompt counts. It is also important that the measurements around the peak count rate be done with sufficient frequency so that the peak rate can be accurately determined. Therefore, it is expected that manufacturers will recommend a protocol for their scanners that includes starting activity, acquisition times, and acquisition durations. 4.3.5 Data processing For tomographs with an axial field of view of 65 cm or less, sinograms shall be generated for each acquisition j of slice i. For tomographs with an axial field of view greater than 65 cm, sinograms shall be generated for each acquisition for slices within the central 65 cm. No corrections for variations in detector sensitivity or detector motions such as wobble, randoms, scatter, dead-time, or attenuation shall be applied to the measurements. Real-time subtraction of random events shall not be done. The sinograms must contain the total acquired counts of the scanner without corrections; i.e., the sinograms must contain true, random, and scatter counts. Furthermore, scanners with the capacity for direct measurement of random rates may not use these measurements in the estimation of random count rates detailed below. Oblique sinograms are collapsed into a single sinogram for each respective slice (by single-slice rebinning) while conserving the number of counts in the sinogram. 4.4 ANALYSIS All pixels in each sinogram i of acquisition j located farther than 12 cm from the center of the phantom shall be set to zero. For each projection angle a within the sinogram, the location of the center of the line source response shall be determined by finding the pixel having the greatest value. Each projection shall be shifted so that the pixel containing the maximum value is aligned with the central pixel of the sinogram. After alignment, a sum projection shall be produced such that a pixel in the sum projection is the sum of the pixels in each angular projection having the same radial offset as the pixel in the sum projection: C(rj = IC(r -rmax().),Oj a Where: a. r is the pixel number in a projection, b. is the projection number in the sinogram (i.e., the sinogram row), and c. rm,(a) refers to the location of the maximum value in projection . The counts CLqjand CRj the left and right pixel intensities at the edges of the 40 mm wide strip at the center of the sinogram, shall be obtained from the sum projection (see Figure 4-2). Linear interpolation shall be employed to find the pixel intensities at 20 mm from the central pixel of the projection. The average of the two pixel intensities CL.,i and CR.j shall be multiplied by the number of pixels, including fractional values, between the edges of the 40 mm wide strip, and the product added to the counts in the pixels outside the strip, to yield the number of random plus scatter counts C+,,ij for the slice i of acquisition j. NU 2-2001 Page 14 Maximum -pixel scattered Scattered counts radial distance from maximum pixel Figure 4-2 INTEGRATION OF BACKGROUND COUNTS INSIDE AND OUTSIDE 40mm STRIP The total event count CTOTr.i is the sum of all pixels in the sum projection for slice i of acquisition j. The average activity A,,j for each acquisition j shall be calculated (see Section 1.2). 4.4.1 Scatter fraction The final acquisitions j' of the sequence with count loss rates and random rates below 1.0% of the trues rate shall be used to determine the scatter fraction. For these acquisitions, it is assumed that C,si.r has a negligible number of random counts and consists only of scatter counts, and likewise, CTOT.i. consists only of true and scatter counts. The scatter fraction SF, for each slice is calculated by summing over the low activity acquisitions as follows: SCr+s,i,rj SFi =- I CTOT,i,j The system scatter fraction SF is computed as the weighted average of the SF, values as follows: Cr+s,i.f SF= i' ,CTOT,i,J 4.4.2 Total event rate measurement For each acquisition j, the total event rate Rror.ij for each slice i is computed as: CTOT,ij RTOTjij = Tacqj Tacqj Where Tacj is the acquisition time. The system total event rate RTorj is computed as the sum of RTroT, over all slices i. 4.4.3 True event rate measurement For each acquisition j, the true event rate R,,i for each slice i is computed as: NU 2-2001 Page 15 S(CToT,J Cr+s,ij) Rt= Tac Where Taj is the acquisition time. The system true event rate Rt is computed as the sum of Rij over all slices i. 4.4.4 Random event rate measurement For each acquisition j, the random event rate R,ij for each slice i is computed as: Rr,,j = RTOTi,j -1 The system random event rate Rrj is computed as the sum of Rrij over all slices i. 4.4.5 Scatter event rate measurement For each acquisition j, the scatter event rate Rs,i for each slice i is computed as: R [j ^(SFi The system scatter event rate Rsj is computed as the sum of Rij over all slices i. 4.4.6 Noise equivalent count rate measurement For each acquisition j, on all systems except those which perform direct randoms subtraction, the noise equivalent count rate RNECj for each slice i is computed as: R2 RNEC)J j R--- RNEC RTOT,j Systems that use direct randoms subtraction should instead compute RNEC.Ijfor each slice i as: Rt2 RNECij = - RNEC = RTOTi,j +Rr,i The system noise equivalent count rate RNECJ is computed as the sum of RNEC.J over all slices i. 4.5 REPORT 4.5.1 Count rate plot For the system, plot the following five quantities as a function of the average effective radioactivity concentration a,,j, as defined in Section 1.2, where the volume V is the total volume of the cylindrical phantom (22,000 cm3): a. Rt, system true event rate b. Rr.j system random event rate c. Rsj system scatter event rate d. RNECJ system noise equivalent count rate e. RTOTJ system total event rate 48 NU 2-2001 Page 16 4.5.2 Peak count values Report the following values, derived from the above plot: a. Rpeak peak true count rate b. RNECpeak peak noise equivalent count rate c. at,peak the activity concentration at which Rtpeak is reached d. aNECpeak the activity concentration at which RNECpeak is reached 4.5.3 System scatter fraction Report the value of SF. APPENDIX B HOW TO COPY SPECIFIC FILES TO A CD * Go to File Management * Double click a patient's name * Press and hold control key, then click with the mouse on the desired files to highlight the files you want to copy (usually ----.scn). Add up the sizes of the files picked to make sure they add up to less than 700 MB (CD capacity) * Go to File, then Copy File (s). A window will pop up * Type in the directory to which the files will be copied: /sunO/patient/dicom. This will take a minute * Open an x-term window * Type: cd /sunO/patient/dicom * Then type: Is. This will list the files you copied to the dicom directory. * Right click with the mouse on the background, just as you would do to open an x-term window. * Go to Utilities, Makeimage, then Burn CD. You're done until this point * However, you will need to empty the dicom directory if you want to burn more files again. So, go back to Utilities and click Clean DICOM Directory. APPENDIX C A SHELL SCRIPT FOR SCALING FACTORS This program extracts the scaling factors information found in the subheaders of each sinogram. To run the program, the following commands should be used. Notice that the name of this file is scnscl.sch and therefore the commands used refer to this file. The first command to be used is chmod 555 scnscl.sch, followed by ./scnscl.sch filename.scn 2. The filename here refers to the Fourier rebinned file. The output from this program will consist of three columns indicating frame number, slice index, and scaling factor, respectively. #! /bin/csh -f # First parameter is name of .scn file. # The second parameter is number of first slice in each frame. set scnfilename = $1 set firstslice = $2 set nslices = 45 set nframes = 21 set slicethickness = 4 set i = 1 set f = 1 while ( $f <= $nframes while ( $i <= $nslices @ s = ( $i 1 ) $slicethickness + $firstslice echo -n $f $s " sbhdrdmp $1 $s 0 $f | fgrep scnscl I awk '{ print $4 }' @ i = $i +1 end @ f = $f + 1 set i = 1 end APPENDIX D MATLAB CODE FOR ORGANIZING ACQUISITIONS %*************************************************************************************************** % ACQUISITIONS CODE %* The acquisitions' raw data acquired on the PET scanner are all saved in one huge %* scan file (xxx.scn). The interpolated scan file (xxx_int.scn) is generated by the %* reconstruction program on the scanner. This interpolated file has 45 slices per tilt %* angle. There are 7 tilt angles per acquisition. Moreover, 192 projection angles and %* 256 linear samples are present for each acquisition. Notice that the Fourier Rebinned %* file doesn't have any tilt angles involved. %* This program extracts the sinograms from the scan file. The extracted sinograms of %* each acquisition are output to different acquisition files numbered 1 through 21. The %* main header in the file, the subheader of each sinogram, and the directory record %* header are all extracted and are not included in the output acquisition files. %*************************************************************************************************** fid=fopen('p311 sOcou_FORE.scn','rb','ieee-be') %------------------------------------------------------------------------------------------------------------------- % Input parameters %------------------------------------------------------------------------------------------------------------------- nray=256; nang=192; nslc=45; nacq=21; mainH=512; %main header subH=512; %sub header DR=1:1:31; %directory records Tnslc=945; %Total number of slices %----------------------------------------......................................................................... sinog_num=0:1:(Tnslc-1); nbytes=nray*nang*2; %sinogram size scn=intl6(zeros(nray,nang,nslc,nacq)); %----------------------------------------......................................................................... % Reading sinograms from file into scn matrices %.....................................---------------------------------------..........................................................................--------------------------------------------------------------------------- j=1; k=1; for i=1:1:Tnslc nbytes_skip=mainH+DR(j)*512+subH+(nbytes+subH)*sinog_num(i); % # of bytes to skip status=fseek(fid,nbytes_skip,-1); % returns 0 if success position=ftell(fid); % tells indicator position in the file scn(:,:,i)=fread(fid,[nray nang],'intl6'); if( i== j*31) %to add bytes of new directory record j=j+1; end end %----------------------------------------------------------------------------- % Output acquisitions only. These acquisitions are free of any header information %----------------------------------------------------------------------------- acquisition=zeros(nray,nang); START=1; END=45; for acq=1:1:nacq fname=sprintf('acquisitions\\acq%d',acq); fid=fopen(fname,'wb','ieee-be'); for i=START:1:END acquisition=scn(:,:,i); fwrite(fid,acquisition(:),'intl6'); acquisition=zeros(nray,nang); end fclose(fid) START=START+45; END=END+45; end % END OF CODE %************************************************************************************************** APPENDIX E MATLAB CODE FOR SCALING FACTORS %*************************************************************************************************** %* SCALING FACTORS CODE %* This program organizes the scaling factors in a format suitable for %* processing in the Data Processing Code. load ('sclfact.txt') global ScalingFactors k=1; forj=1:1:26 for i=1:1:45 ScalingFactors(i,j)=sclfact(k,3); k=k+l; end end clear % END OF CODE %************************************************* APPENDIX F MATLAB CODE FOR DATA PROCESSING AND ANALYSIS %************************************************************************************************** %* The source files used in this program contain data acquired from the dedicated %* 3-D PET scanner in the Radiology Department at Shands Medical Plaza at the %* University of Florida. This program processes the acquired data according to %* section 4 of NEMA 2001 Standards. %************************************************************************************************** %* NEMA NU 2-2001 DATA PROCESSING CODE global ScalingFactors global FinalSumProj global nray nang nslcs nacq global mm_pxl cm_pxl CenterPixel pxl_L pxl_R global sino ssino global xx_R xx_L npxl_fourcm x_cm global ScatF TRandScat TCounts %----------------------------------------......................................................................... % Input Parameters describing the sinogram size and the number of acquisitions involved %----------------------------------------......................................................................... nray=256; %number of linear samples or rays nang=192; %number of projection angles nslcs=45; %number of slices in an acquisition nacq=21; %number of acquisitions mm_pxl=576/256; % sampling distance (i.e., # of mm per pixel) %----------------------------------------......................................................................... CenterPixel=nray/2; cm_pxl=mm_pxl/10; % cm per pixel pxl_twelvecm=12/cm_pxl; % # of pixels in 12 cm pxl_twelvecm=ceil(pxltwelvecm); % rounds the number to the next integer pxlL=CenterPixel-pxltwelvecm-1; % pixel # located 12 cm to the left side of the center % of the phantom pxl_R=CenterPixel+pxl_twelvecm+l; % pixel # located 12 cm to the right side of the %center of the phantom %----------------------------------------......................................................................... xx_R=round(CenterPixel + 20/mm_pxl); %pixel # located 2 cm to the right from the Central pixel xx_L=round(CenterPixel 20/mm_pxl); %pixel # located 2 cm to the left from the central pixel npxl_fourcm=ceil(40/mm_pxl); % # of pixels in 4 cm (40 mm) strip %----------------------------------------------------------------------------- % Converting x-axis to cm scale %----------------------------------------------------------------------------- ncm=CenterPixel*cm_pxl; % number of cm from pixel 1 to central pixel (128) x_cm=-(ncm-cmpxl):cm_pxl:ncm; % new scale in cm %----------------------------------------------------------------------------- sino=zeros(256,192,45); initializingg zero matrices for the sinograms ssino=zeros(256,192,45); initializingg zero matrices for the shifted sinograms FinalSumProj=zeros([1 nray nacq]); initializingg a zero matrix for the Final Sum Projection for acq=1:1:nacq global ScalingFactors global FinalSumProj global nray nang nslcs global mm_pxl cm_pxl CenterPixel pxl_L pxl_R global sino ssino global xx_R xx_L npxlfourcm x_cm global ScatF TRandScat TCounts %----------------------------------------------------------------------------- % open and read from each acquisition %----------------------------------------------------------------------------- disp('opening acquisition'),disp(acq) % to keep track of the code execution process fname=sprintf('acquisitions\\acq%d',acq); fid=fopen(fname,'rb','ieee-be'); %----------------------------------------------------------------------------- % Notice: the 'ieee-be' defines the machine format used to read the acquisitions data. % Here the data was acquired at a Unix Sun Computer, so the data is saved as big % endian which is the format associated with Unix Operating Systems. PC or Windows % Operating Systems, on the other hand, are of little endian type. Therefore, when % reading Unix data on a PC it is essential to define the machine format before reading % the data. This was found to be better than swapping the data before using it in PC % environment, because the swapping process might introduce some unwanted % data embedded within the original data. %----------------------------------------------------------------------------- disp('reading sinograms') % to keep track of the code execution process slc_num=0:1:(nslcs-1); % # slices to skip I=zeros(nray,nang,nslcs); % Initializing zero matrices for i=1:1:nslcs NumBytesSkip=(nray*nang*2)*slcnum(i); %number of bytes to skip status=fseek(fid,NumBytesSkip,-1); %returns 0 if success position=ftell(fid); %tells indicator position in the file 1(:,:,i)=fread(fid,[nray,nang],'intl6'); %sinogram images end fclose(fid); %------------------------------------------------------------------------------------------------------------------- % Multiplying by the scaling factors to correct for the number of counts in each pixel of each % sinogram %----------------------------------------------------------------------------- disp('multiplying by the scaling factors') % to keep track of the code execution process for i=1:1:nslcs I(:,:,i)=ScalingFactors(i,acq).*l(:,:,i); end disp('open and read done') % to keep track of the code execution process % 4.4 ANALYSIS (NEMA 2001) %0 %------------------------------------------------------------------------------------------------------------------- % Calculating the sum projection of each slice i in each acquisition j and the final sum % projection of each acquisition. %----------------------------------------------------------------------------- disp('Analysis begins') %----------------------------------------------------------------------------- % One sinogram is processed at a time and the results are loaded into the appropriate matrix %----------------------------------------------------------------------------- % Initialization u=1; % index for ray sinogram=zeros([nray nang]); shiftedsinogram=zeros([nray nang]); shiftedsinogramtranspose=zeros([nang nray]); MaxCount=zeros([1 nray]); ray=zeros; angle=zeros; SumProj=zeros([1 nray nslcs]); finalsumprojection=zeros([1 nray]); for i=1:1:nslcs % Notice that there are 2 mid-points, 128 & 129 pxls sino(:,:,i)=l(:,:,i); % saves all sinograms of one acquisition only sinogram=l(:,:,i); sinogram(l:pxl_L,:)=0; % pxls 12cm to the left from center of phantom are zeroed. sinogram(pxl_R:nray,:)=0; % pxls 12cm to the right from center of phantom are zeroed. %.----------------------------------------------------------------------- % Finding the pixel of maximum count and the location of that pixel within the %sinogram slice %.----------------------------------------------------------------------- MaxCount=max(sinogram); % returns max number of counts in each proj angle for phi=1:1:nang if (MaxCount(phi) == 0) continue; elseif (MaxCount(phi) > 0) r=find (sinogram(:,phi)==MaxCount(phi)); % Tells location of PixelMaxCount. % Returns the row pixel location. if(length(r)==1) % if only one value returned (ideal case) ray(u)=r; elseif(length(r)>1) % if more than one value returned ray(u)=min(r); % Notice that in the same projection angle there % may be more than one pixel that has max counts. % Only the nearest pixel to the center of the line % source is taken as the pixel of maximum count. end angle(u)=phi; % tells location of PixelMaxCount % returns the column pixel location=projection angle u=u+1; % increment index u end end % check if sizes of row and clmn arrays are the same if(size(ray) ~= size(angle)) 'sizes not equal' end %.-------------------------------------------------------------------- % Shifting the pixels around until the pixel of maximum count aligns with the %central pixel (i.e., pixel #128) %-------------------------------------------------------------------------------------------------- for j=1:1 :length(angle) ProjAngle=angle(j); PixelMaxCount=ray(j); offset=CenterPixel-PixelMaxCount; for r=(pxl_L+1):1:(pxl_R-1) r_new=r+offset; if(r_new < 1) r_new=1; elseif(r_new > nray) r_new=nray; end shiftedsinogram(r_new, ProjAngle)=sinogram(r, ProjAngle); end end %This is based on zeroing of the array before and assumes that %the difference is less than 79 pixels or 17.78 cm shiftedsinogram(1:48,:)=0; shiftedsinogram(208:nray,:)=0; ssino(:,:,i)=shiftedsinogram; %.-------------------------------------------------------------------- % Saving all important data each in the appropriate matrix %---------------------------------------------------------------------------------------------------------- ShiftedSinogram(:,:,i)=shiftedsinogram; % shifted proj, result is straight line shiftedsinogramtranspose=shiftedsinogram'; SumProj(:,:,i)=sum(shiftedsinogramtranspose);% 1-D profile 'sum in every column' finalsumprojection=finalsumprojection+SumProj(:,:,i); %.----------------------------------------------------------------------- % Calculating the scatter + Random counts and the total counts per slice %.----------------------------------------------------------------------- y=zeros([1 nray]); y=SumProj(:,:,i); C_R=y(xxR); C_L=y(xx_L); avg=(C_R+C_L)/2; sum_R=sum(y((xx_R+1):nray)); % counts in pixels outside the strip sum_L=sum(y(1:(xx_L-1))); %counts in pixels outside the strip sum_RL=sum_R+sum_L; %total counts in pixels outside the strip C_rand_scat=(avg*npxl_fourcm)+sum_RL; C_r_sj(i)=C_rand_scat; % random plus scatter counts for acquisition j C_TOTj(i)=sum(y); % Total counts per acquisition j %.----------------------------------------------------------------------- % re-zeroing the matrices to avoid data overlap sinogram=zeros([nray nang]); % zero out the sinogram matrix shiftedsinogram=zeros([nray nang]); % zero out the shifted sinogram matrix shiftedsinogramtranspose=zeros([nang nray]); MaxCount=zeros([1 nang]); ray=zeros; angle=zeros; r=zeros; u=1; % index for row end % closes the nslices loop disp('Calculating Random and Scatter, and Total counts') TRandScat(acq)=sum(C_r_s); %Total Randoms and scatters in an acq TCounts(acq)=sum(C_TOTj); % Total counts in an acq ScatF(acq)=TRandScat(acq)/TCounts(acq); %------------------------------------------------------------------------------------------------------------- % output counts per acquisition to a text file %------------------------------------------------------------------------------------------------------------- SliceNum=1:1:45; Datal =[SliceNum;C_r_sj; CTOTj]; Data2=[TRandScat(acq);TCounts(acq)]; fname=sprintf('Output\\DataCounts%d.mw',acq); fid=fopen(fname,'w'); fprintf(fid,'acq '); fprintf(fid,'%2.0f\n', acq); fprintf(fid,'Slice\t RandScat\t Total\n'); fprintf(fid,'%2.0f\t %10.2f\t %10.2f\n',Datal); fprintf(fid,'\n\n'); fprintf(fid,'Total '); fprintf(fid,'%10.2f %10.2f,Data2); fclose(fid); %.------------------------------------------------------------------------- % In slight deviation from NEMA 2001, the sum projections of all sinograms in a % particular acquisition are summed to create a final sum projection that describes % the behavior of the PET scanner at that particular activity. It is this final sum % projection that is used to calculate the different count rates following NEMA 2001 % standards. %-------------------------------------------------------------------------- FinalSumProj(:,:,acq)=finalsumprojection; % final sum projection of each acquisition %.------------------------------------------------------------------------- % output Final Sum Projections data to a text file %.------------------------------------------------------------------------- T_acqj=20*60; Data3=[x_cm;FinalSumProj(:,:,acq)/T_acqj]; fname=sprintf('Output\\FinalSumProjection%d.mw',acq); fid=fopen(fname,'w'); fprintf(fid,'acq '); fprintf(fid,'%2.0f\n',acq); fprintf(fid,'x (cm)\t Count Rate [cps] \n'); fprintf(fid,'%2.3f\t %10.2f \n',Data3); fclose(fid); clear %clears all but the global variables end %ends the nacq loop global cm_pxl xcm CenterPixel global nacq FinalSumProj %---------------------------------------------------------------------------- % Plot Final Sum Projections (cps) vs. radial distance (cm) %---------------------------------------------------------------------------- T_acqj=20*60; % N=1; fig=1; % forj=1:1:nacq % fname=sprintf('Final Sum Projection: acq%d',j); % figure(fig),subplot(2,1,N),plot(xcm,FinalSumProj(:,:,j)/T_acqj),title(fname) % xlabel('Distance along transverse axis [cm]'),ylabel('Count Rate [cps]') % grid % N=N+1; % if(j==fig*2) % fig=fig+1; % N=1; % end % end %----------------------------------------------------------------------------------------------------------------- % To place each profile in a separate figure %----------------------------------------------------------------------------------------------------------------- fig=l; forj=1:1:nacq figure(fig),plot(x_cm,FinalSumProj(:,:,j)/T_acqj) xlabel('Distance along transverse axis [cm]'),ylabel('Count Rate [cps]') grid fig=fig+1; end %*END OF CODE************************************************************************************************* %* END OF CODE %************************************************* APPENDIX G MATLAB CODE FOR CONTINUING PROCESSING AND ANALYSIS %/************************************************************************************************** %* NEMA NU 2-2001 DATA PROCESSING CODE (Continue ...) * %************************************************************************************************** global FinalSumProj mm_pxl global ScatF TRandScat TCounts T_acqj=20*60; %acquisition time in seconds C_r_sj=TRandScat; C_TOTj=TCounts; %---------------------------------------------------------------------------- % 4.4.1 SCATTER FRACTION (NEMA 2001) %----------------------------------------------------------------------------- % The final acquisition j' of the sequence with count loss rates and random rates below % 1% of the trues rate is used to determine the scatter fraction. For this acquisition, it is % assumed that C_r+s,j' has a negligible number of random counts and consists only of % scatter counts, and likewise, C_TOT,j' consists only of true and scatter counts. The % scatter fraction of the final acquisition is used as the system's SF for it is assumed % that the randoms rate is again negligible in that acquisition and therefore the C_r+s,j' % consists only of scatter counts. %----------------------------------------........................................................................ %----------------------------------------........................................................................ % 4.5.3 SYSTEM SCATTER FRACTION (NEMA 2001) %----------------------------------------........................................................................ SF =0.403; %----------------------------------------........................................................................ % 4.4.2 TOTAL EVENT RATE MEASUREMENT (NEMA 2001) %.....................................---------------------------------------.........................................................................--------------------------------------------------------------------------- R_TOTj = C_TOTj ./ T_acqj; %------------------------------------------------------------------------------------------------------------------ % 4.4.3 TRUE EVENT RATE MEASUREMENT (NEMA 2001) %.....................................---------------------------------------.........................................................................--------------------------------------------------------------------------- R_tj = (C_TOTj C_r_sj) T_acqj; %------------------------------------------------------------------------------------------------------------------ % 4.4.4 RANDOM EVENT RATE MEASUREMENT (NEMA 2001) %........-----------------------------------------............................................................... R_rj = R_TOTj (R_tj ./(1-SF)); %---------------------------------------------------------------------------- % 4.4.5 SCATTER EVENT RATE MEASUREMENT (NEMA 2001) %------------------------------------------------------------------------------------------------------------------ R_sj = R_tj .* (SF/(1-SF)); %------------------------------------------------------------------------------------------------------------------ % 4.4.6 NOISE EQUIVALENT COUNT RATE MEASUREMENT (NEMA 2001) %------------------------------------------------------------------------------------------------------------------ R_NECj = (R_tj.2) ./ R_TOTj; % 4.5 REPORT %/ %------------------------------------------------------------------------------------------------------------------ % 4.5.1 COUNT RATE PLOT (NEMA 2001) %---------------------------------------------------------------------------- %calculated initial activity (mCi) at the beginning of each acquisition is Act=[12.536 9.738 7.564 5.876 4.564 3.545 2.754 2.139 1.662 1.291 1.003 0.779 0.605 ... 0.470 0.365 0.284 0.220 0.171 0.133 0.103 0.080]; % mCi; %time into experiment (minutes): time=[0 40 80 120 160 200 240 280 320 360 400 440 480 520 560 600 640 680 720 760 800]; figure(1), plot(time, R TOTj1 000,'m*-',time, Rtj/1000,'rs-',time, Rrj/1000,'bd-',time,R_sj/1 000,'gx-') xlabel('Time [min]'),ylabel('Count Rate [kcps]') legend('Total','True','Random','Scatter') figure(2) plot(Act,R TOTj/1 000,'m*-',Act,R_tj1/000,'rs-',Act,R_r-j 1 000,'bd-',Act,R_s-j 1000,'gx-') xlabel('Activity [mCi]'),ylabel('Count Rate [kcps]') legend('Total','True','Random','Scatter') figure(3) plot(time,R_NECj/1000,'k.-') xlabel('Time [min]'),ylabel('NEC Count Rate [kcps]') figure(4) plot(Act,RNECj/1000,'k.-') xlabel('Activity [mCi]'),ylabel('NEC Count Rate [kcps]') %----------------------------------------....................................................................... % output count rates to a text file %.......----------------------------------------------.......................................................... Data=1:1:nacq; 63 countdata = [Data; C_TOTj; R_TOTj; R_tj; R_rj; R_sj; R_NECj; Act]; fid=fopen('Output\\CoincidenceCounts.txt','w') fprintf(fid,'Acquisition Total Paired Count Rate Count Rate Count Rate Count Rate Count Rate Average\n') fprintf(fid,' Events Total True Random Scatter NEC Activity\n') fprintf(fid,' (counts/20min) (cps) (cps) (cps) (cps) (cps) (mCi) \n') fprintf(fid,'%2.0f fclose(fid); %12.0f %12.2f %12.2f %12.2f %12.2f %10.2f %1.4f\n',countdata) %************************************************************************************************** %* END OF CODE * %************************************************* APPENDIX H MATLAB CODE FOR CURVE FITTING %*************************************************************************************************** %* CURVE FIT CODE %------------------------------------------------------------------------------------------------------------------- % Fitting the Final Sum Projections to Gaussian Functions using the Non-Linear Least % Squares Method %----------------------------------------------------------------------------- global FinalSumProj global nacq cm_pxl CenterPixel T_acqj=20*60; ncm=CenterPixel*cm_pxl; % number of cm from pixel 1 to central pixel (128) x_cm=-(ncm-cm_pxl):cm_pxl:(ncm); % new scale in cm forj=1:1 :nacq Total(:,:,j)=FinalSumProj(:,:,j)/T_acqj; end model = fittype('a*exp(-0.5*(x-b)^2/c^2)+d'); % the gaussian equation used to fit the data opts = fitoptions('Method','NonlinearLeastSquares'); opts.Lower = [-Inf -Inf 0 -Inf]; forj=1:1 :nacq a_strt=max(Total(:,:,j)); % Normalizes the gaussian function b_strt=1; %the mean c_strt=1; % the standard deviation d_strt=1; % the offset opts.StartPoint=[a_strt b_strt c_strt d_strt]; % starting points of statistical constants [f,gof] = fit(x_cm',Total(:,:,j)',model,opts); a(j)=f.a; b(j)=f.b; c(j)=f.c; d(j)=f.d; R_squared(j)=gof.rsquare; G{j}=f; % Gaussian fitted equation J 70------------------------------ ------- ---------------------------------------------------------------------- % Calculating the FWHM of the fitted curves. FWHM = 2.35 sigma %------------------------------------------------------------------------------------------------------------------- forj=1:1:nacq FWHM(j)=c(j)*2.35*10; % FWHM (mm) FWHMa()=sqrt(FWHM(j)^2-3.2^2); %adjusted FWHM (mm) end %----------------------------------------------------------------------------- % Output the FWHM values to a text file %------------------------------------------------------------------------------------------------------------------- acqs=1:1:nacq; values=[acqs; FWHM; FWHMa]; fid=fopen('Output\\FW HM.txt','w'); fprintf(fid,'acq\t FWHM(mm)\t FWHMa(mm)\t \n'); fprintf(fid,'%2.0f\t %3.2f\t %3.2f\n',values); fclose(fid); %----------------------------------------------------------------------------- % Output the coefficients values, for each fitted gaussian function, to a text file %----------------------------------------------------------------------------- coeff=[a; b; c; d; R_squared]; fid=fopen('Output\\FittedFunctionsCoeff.txt','w'); fprintf(fid,'\t a\t\t b\t\t c\t\t\t d\t R_squared \n'); fprintf(fid,'%7.2f %1.4f %1.5f %7.2f %1.3f\n',coeff); fclose(fid); %----------------------------------------------------------------------------- % Output fitted curves to a text file %------------------------------------------------------------------------------------------------------------------- forj=1:1:nacq f=G{j}; %fitted gaussian equation for i=1:1:length(x_cm) fg(i)=f(x_cm(i)); end g=[x_cm;fg]; fname=sprintf('Output\\FittedGaussian%d.mw',j); fid=fopen(fname,'w'); fprintf(fid,'acq '); fprintf(fid,'%2.0f\n',j); fprintf(fid,'x (cm)\t Count Rate [cps] \n'); fprintf(fid,'%2.3f\t %10.2f\n',g); fclose(fid); end %----------------------------------------------------------------------------- % Plot the FWHM as a function of activity and time into experiment %----------------------------------------------------------------------------- figure(30),plot(Act,FWHMa) xlabel('Activity [mCi]'), ylabel('FWHM [mm]') figure(31 ),plot(time, FWHMa) xlabel('Time [min]'), ylabel('FWHM [mm]') %------------------------------------------------------------------------------------------------------------------- % Plotting the fitted functions along with the Final Sum Projections %------------------------------------------------------------------------------------------------------------------- % N=1; % fig=1; % forj=1:1:nacq % fname=sprintf('Final Sum Projection: Acq%d',j); % figure(fig),subplot(2,1,N),plot(G{j},x_cm,Total(:,:,j)),title(fname) % xlabel('Distance along transverse axis [cm]'),ylabel('Counts') % legend('Final Sum Projection','Gaussian'),grid % N=N+1; % if(j==fig*2) % fig=fig+1; % N=1; % end % end fig=1; forj=1:1:nacq figure(fig),plot(G{j},x_cm,Total(:,:,j)) xlabel('Distance along transverse axis [cm]'),ylabel('Count Rate [cps]') legend('Final Sum Projection','Fitted Gaussian'),grid fig=fig+1; end %----------------------------------------------------------------------------- % Fitting the Randoms rate curve to power equation using the Non-Linear Least Squares % Method %----------------------------------------------------------------------------- model = fittype('powerl'); opts = fitoptions('Method','NonlinearLeastSquares'); opts.Lower = [-Inf-lnf]; opts.Upper = [Inf 2]; opts.StartPoint=[1 2]; %R_rj=R_rj/1000; % kcps [Rand,gof,out] = fit(Act',R_rj',model,opts); %----------------------------------------------------------------------------- % Output fitted randoms curve to a text file %----------------------------------------------------------------------------- for i=1:1:length(Act) R(i)=Rand(Act(i)); end r=[Act; R]; fid=fopen('Output\\FittedRandoms.mw','w'); fprintf(fid,'Act (mCi)\t Count Rate [kcps] \n'); fprintf(fid,'%2.3f\t %10.2f\n',r); fclose(fid); %.---------------------------------------------------------------------------- % Plotting the fitted function along with the Randoms count rate %.---------------------------------------------------------------------------- figure(fig),plot(Rand,Act,R_rj) %title('Random count rate and fitted equation vs. activity') xlabel('Activity [mCi]'), ylabel('Count Rate [cps]') legend('Randoms rate','Fitted equation'),grid %*************************************************************************************************** %* END OF CODE * %*************************************************************************************************** APPENDIX I MATLAB CODE FOR PRODUCING MTF CURVES %*************************************************************************************************** %* Modulation Transfer Function (MTF) CODE %* The Fourier Transform of the Final Sum Projection is computed to produce the %* Modulation Transfer Function (MTF) for each acquisition. %------------------------------------------------------------------------------------------------------------------- % Fourier Transform of Final Sum Projections %------------------------------------------------------------------------------------------------------------------- global nray MTF =zeros([1 nray nacq]); %Modulation Transfer Function MTFn=zeros([1 nray nacq]); %Normalized Modulation Transfer Function MTFa=zeros([1 nray/2 nacq]); %Adjusted Modulation Transfer Function forj=1:1 :nacq fsp=FinalSumProj(:,:,j); %fsp=final sum projection FSP=fftshift(fft(fsp)); MTF(:,:,j)=abs(FSP); %magnitude of FT{fsp} end %----------------------------------------------------------------------------- % Normalized Modulation Transfer Functions (MTFn) %------------------------------------------------------------------------------------------------------------------- forj=1:1 :nacq MAX=max(MTF(:,:,j)); MTFn(:,:,j)=MTF(:,:,j)/MAX; end %----------------------------------------------------------------------------- % Adjusted Modulation Transfer Functions (MTFa) %------------------------------------------------------------------------------------------------------------------- forj=1:1 :nacq MTFa(:,:,j)=MTFn(:,(nray/2 + 1):nray,j); end %----------------------------------------------------------------------------- % Plotting the adjusted Modulation Transfer Functions (MTFa) %----------------------------------------------------------------------------- N=nray %# of pixels, so 1mm/pxl dx=cm_pxl % sampling distance: i.e., # of cm per pixel dnu=1/(N*dx) %sampling distance in frequency domain nu_min=-(N*dnu/2) nu_max=(N*dnu/2)-dnu nu=numin:dnu:nu_max; % frequency domain (1/cm) nul=nu((nray/2 + 1):nray); % frequency domain used for the plots %fig=1; % subfig=1; % forj=1:1:nacq % figure(fig),subplot(2,1,subfig),plot(nul,MTFa(:,:,j)) % title(['Modulation Transfer Function: Acq 'int2str(j)]) % xlabel('\nu [cm^-^l]'),ylabel('Gain'),grid % subfig=subfig+1; % if(j==fig*2) % fig=fig+1; % subfig=1; % end % end fig=1; forj=1:1:nacq figure(fig),plot(nul,MTFa(:,:,j)) xlabel('\nu [cm^-^l]'),ylabel('Gain') h = gca; set(h,'YGrid','on'); fig=fig+1; end %----------------------------------------------------------------------------- % Plotting the adjusted Modulation Transfer Functions (MTFa) of selected acquisitions % on ONE plot %----------------------------------------------------------------------------- figure(20) plot(nul,MTFa(:,:,1),nu1,MTFa(:,:,5),nul,MTFa(:,:,10),nu1,MTFa(:,:,21)) xlabel('\nu [cm^-^l]'),ylabel('Gain') h = gca; set(h,'YGrid','on'); %title('Modulation Transfer Functions of selected acquisitions') legend('acquisition 1','acquisition 5','acquisition 10','acquisition 21') %----------------------------------------------------------------------------- % Output MTF curves to text files %----------------------------------------------------------------------------- for acq=1:1:nacq mtf=[nul;MTFa(:,:,acq)]; fname=sprintf('Output\\MTF%d.mw',acq); fid=fopen(fname,'w'); fprintf(fid,'acq '); fprintf(fid,'%2.0f\n',acq); fprintf(fid,'frequency (1/cm)\t Gain \n'); fprintf(fid,'%2.3f\t\t %10.5f \n',mtf); fclose(fid); end %----------------------------------------------------------------------------- % Plotting ALL of the adjusted Modulation Transfer Functions (MTFa) on ONE plot %------------------------------------------------------------------------------------------------------------------- % forj=1:1:nacq % ModTranFuna(j,:)=MTFa(:,:,j); % fname=sprintf('s%d',j); % ACQMTFa{j}=fname; % end % figure(21) % plot(nul,ModTranFuna),legend(ACQMTFa) %.---------------------------------------------------------------------------- % Plotting ALL of the Modulation Transfer Functions (MTF) on ONE plot %------------------------------------------------------------------------------------------------------------------- % forj=1:1:nacq % ModTranFunNorm(j,:)=MTFn(:,:,j); % flname=sprintf('s%d',j); % ACQMTFn{j}=flname; % end % figure(22) % plot(nu,ModTranFunNorm),legend(ACQMTFn) % xlabel('\nu [cm^-^l]'),ylabel('Relative intensity') % title('Norm MTFs') %* END OF CODE %************************************************************************************************** APPENDIX J FINAL SUM PROJECTIONS x 104 2 o. 0 . 1.5 0 1 0.5 -15 -10 -5 0 5 10 Distance along transerse axis [cm] FIG. 8. Final sum projection of acquisition 1 15 20 -15 -10 -5 0 5 10 Distance along transverse axis [cm] FIG. 9. Final sum projection of acquisition 2 -15 -10 -5 0 5 10 Distance along transverse axis [cm] FIG. 10. Final sum projection of acquisition 3 x104 15 20 x 104 2.5 2- 0 o 1.5 o 0 15 20 -15 -10 -5 0 5 10 Distance along transverse axis [cm] FIG. 11. Final sum projection of acquisition 4 -15 -10 -5 0 5 10 Distance along transverse axis [cm] FIG. 12. Final sum projection of acquisition 5 x 104 2.5 2 0 1.5 O 0 0 15 20 x 104 2 1.5 O 1 o 0 15 20 -15 -10 -5 0 5 10 Distance along transverse axis [cm] FIG. 13. Final sum projection of acquisition 6 ------ ------- -------h- -- ------^- -----~ -15 -10 -5 0 5 10 Distance along transverse axis [cm] FIG. 14. Final sum projection of acquisition 7 18000 16000 14000 7 12000 0, a 10000 CO 8000 000 o 6000 4000 2000 14000 12000 10000 CO 8000 - 6000 4000 2000 15 20 15 20 -15 -10 -5 0 5 10 Distance along transverse axis [cm] FIG. 15. Final sum projection of acquisition 8 I-- -15 -10 -5 0 5 10 Distance along transverse axis [cm] FIG.16. Final sum projection of acquisition 9 12000 10000 8000 6000 4000 2000 10000 8000 0 6000 | 4000 0 0) 2000 15 20 15 20 -15 -10 -5 0 5 10 Distance along transverse axis [cm] FIG. 17. Final sum projection of acquisition 10 -+-------\-------- ------- -------+-------{- --1- --- --1- --- --------H------+-------1- - -15 -10 -5 0 5 10 Distance along transverse axis [cm] FIG. 18. Final sum projection of acquisition 11 8000 7000 6000 & 5000 o O 4000 0 3000 2000 1000 6000 5000 W 4000 O 3000 0 0 2000 1000 15 20 15 20 -15 -10 -5 0 5 10 Distance along transverse axis [cm] FIG. 19. Final sum projection of acquisition 12 ----------------^- --- --- --------------- -------------- - ---- ---- --- ---- --- -------"------- - -15 -10 -5 0 5 10 Distance along transverse axis [cm] FIG. 20. Final sum projection of acquisition 13 5000 4000 0 3000 I 2000 o 0 1000 4000 3500 3000 & 2500 0 O 2000 = 1500 1000 500 15 20 15 20 -15 -10 -5 0 5 10 Distance along transverse axis [cm] FIG. 21. Final sum projection of acquisition 14 -1-- -- ------- I------ F ------ T------^-I-- -15 -10 -5 0 5 10 Distance along transverse axis [cm] FIG. 22. Final sum projection of acquisition 15 3000 2500 2000 1500 1000 2500 2000 0 1500 S1000 0 0) 15 20 15 20 -15 -10 -5 0 5 10 Distance along transverse axis [cm] FIG. 23. Final sum projection of acquisition 16 -15 -10 -5 0 5 10 Distance along transverse axis [cm] FIG. 24. Final sum projection of acquisition 17 1800 1600 1400 7 1200 a 1000 800 S600 0 600 15 20 1400 1200 1000 O 800 S600 8 15 20 -15 -10 -5 0 5 10 Distance along transerse axis [cm] FIG. 25. Final sum projection of acquisition 18 -15 -10 -5 0 5 10- Distance along transverse axis [cm] FIG. 26. Final sum projection of acquisition 19 1200 1000 800 700 , 600 0. a 500 400 0 0 300 15 20 15 20 -15 -10 -5 0 5 10 Distance along transverse axis [cm] FIG. 27. Final sum projection of acquisition 20 I - -15 -10 -5 0 5 10 Distance along transverse axis [cm] FIG. 28. Final sum projection of acquisition 21 700 600 500 400 CO 300 o 0 200 100 W 400 o c 300 0 0 200 15 20 15 20 APPENDIX K FITTED FINAL SUM PROJECTIONS x104 * Final Sum Projection Fitted Gaussian 2 1.5 ------P-------1-- ------- ----- F- ------P------- ------- ------ 1.5 - 1 0.5----- 0 -20 -15 -10 -5 0 5 10 15 Distance along transverse axis [cm] FIG. 36. Final sum projection of acquisition 1 and its fitted gaussian curve x 104 * Final Sum Projection Fitted Gaussian C0 1.5 0 1 0.5 -20 -15 -10 -5 0 5 10 15 Distance along transverse axis [cm] FIG. 37. Final sum projection of acquisition 2 and its fitted gaussian curve 83 X 104 2.5 --- Final Sum Projection Fitted Gaussian 2 15-------f------- ------- -------[ ------------- ------- ------ 0 1.5 --- 0 o * 0 I 0.5 - -20 -15 -10 -5 0 5 10 15 Distance along transverse axis [cm] FIG. 38. Final sum projection of acquisition 3 and its fitted gaussian curve x104 2.5 - Final Sum Projection Fitted Gaussian 2 0 1.5 1.5 -------f------- ------- -------[------^------- ------- ------ 0 .5 -- + -^------- -------^ ---------- ------ ------- i 1 0 . 0.5 . -20 -15 -10 -5 0 5 10 15 Distance along transverse axis [cm] FIG. 39. Final sum projection of acquisition 4 and its fitted gaussian curve x 104 2, * Final Sum Projection Fitted Gaussian -20 -15 -10 -5 0 5 10 15 Distance along transverse axis [cm] FIG. 40. Final sum projection of acquisition 5 and its fitted gaussian curve 18000 16000 14000 7 12000 S10000 8000 o 6000 * Final Sum Projection Fitted Gaussian 4000 ------- 2000 -------------------------- ------------------------ 0 -20 -15 -10 -5 0 5 10 15 Distance along transverse axis [cm] FIG. 41. Final sum projection of acquisition 6 and its fitted gaussian curve * Final Sum Projection Fitted Gaussian 14000 12000 10000 8000 6000 4000 2000 0 -2 12000 10000 8000 6000 4000 2000 0 * Final Sum Projection Fitted Gaussian .0 -15 -10 -5 0 5 10 15 Distance along transverse axis [cm] FIG. 42. Final sum projection of acquisition 7 and its fitted gaussian curve -20 -15 -10 -5 0 5 10 15 Distance along transverse axis [cm] FIG. 43. Final sum projection of acquisition 8 and its fitted gaussian curve * Final Sum Projection Fitted Gaussian -20 -15 -10 -5 0 5 10 15 Distance along transverse axis [cm] FIG. 44. Final sum projection of acquisition 9 and its fitted gaussian curve * Final Sum Projection Fitted Gaussian 10000 8000 6000 4000 2000 8000 7000 6000 & 5000 | 4000 0 3000 2000 1000 -20 -15 -10 -5 0 5 10 15 Distance along transverse axis [cm] FIG. 45. Final sum projection of acquisition 10 and its fitted gaussian curve |

Full Text |

PAGE 1 COUNT RATE AND SPATIAL RESOLU TION PERFORMANCE OF A 3-DIMENSIONAL DEDICATED POSITR ON EMISSION TOMOGRAPHY (PET) SCANNER By RAMI RIMON ABU-AITA A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2004 PAGE 2 Copyright 2003 by Rami Rimon Abu-Aita PAGE 3 To my parents PAGE 4 ACKNOWLEDGMENTS I would like to thank several people who contributed to the completion of this work. First, I would like to express my sincere thanks to Dr. David R. Gilland for his support, encouragement, guidance, and most of all his trust, patience and, sense of humor throughout the last two years of graduate work at the University of Florida. I would also like to thank Professor Wesley E. Bolch, Dr. David E. Hintenlang, and Dr. Manuel M. Arreola for their quality teaching and for taking the time to be on my supervisory committee. The committee provided invaluable comments that truly made this work complete. I also thank the faculty, staff, and students of the Nuclear and Radiological Engineering Department. I also thank Michael Nicole, Mike Frame, and the technologists of the Nuclear Medicine Department of Shands Radiology for their willingness to share time and facility. Without their assistance, this investigation would not have been possible. Moreover, I thank Ben Chase, Ed Sokolowski, and Renate Gill of Philips/ADAC Medical Systems for providing assistance and guidance that lead to the success of this investigation. I am very grateful to my parents, Rimon and Khitam Abu-Aita, and my brothers, Wadie, Giovanni, and Marco, for their love and support and for accepting a long-term separation that lasted about five and a half years. I truly thank my parents for giving me the opportunity to pursue higher education in the United States. I thank both of my iv PAGE 5 uncles, Awni and Issa Abu-Aita, for their support and encouragement. Finally, I thank God for giving me the strength to finish my work. v PAGE 6 TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iv LIST OF TABLES...........................................................................................................viii LIST OF FIGURES...........................................................................................................ix ABSTRACT.....................................................................................................................xiii CHAPTER 1 INTRODUCTION........................................................................................................1 2 BACKGROUND..........................................................................................................5 Count Loss and Dead Time..........................................................................................5 The Effects of High Count Rate on Spatial Resolution................................................8 Random and Scatter Coincidence Events.....................................................................8 Noise Equivalent Count Rate......................................................................................10 3 MATERIALS AND METHODS...............................................................................12 Camera Description....................................................................................................12 Phantom Description..................................................................................................13 Allegro Data Acquisition and Processing...................................................................14 Allegro Data Acquisition Parameters..................................................................15 Allegro Data Processing Parameters...................................................................16 File Structure.......................................................................................................16 NEMA Data Processing and Analysis Using MATLAB...........................................18 Count Rates.........................................................................................................18 Spatial Resolution................................................................................................22 4 RESULTS...................................................................................................................24 Count Rate..................................................................................................................25 Spatial Resolution.......................................................................................................31 vi PAGE 7 5 DISCUSSION.............................................................................................................37 6 CONCLUSIONS........................................................................................................40 APPENDIX A NEMA 2001 (SECTION 4)........................................................................................42 B HOW TO COPY SPECIFIC FILES TO A CD..........................................................49 C A SHELL SCRIPT FOR SCALING FACTORS.......................................................50 D MATLAB CODE FOR ORGANIZING ACQUISITIONS........................................51 E MATLAB CODE FOR SCALING FACTORS.........................................................53 F MATLAB CODE FOR DATA PROCESSING AND ANALYSIS...........................54 G MATLAB CODE FOR CONTINUING PROCESSING AND ANALYSIS.............61 H MATLAB CODE FOR CURVE FITTING................................................................64 I MATLAB CODE FOR PRODUCING MTF CURVES............................................68 J FINAL SUM PROJECTIONS....................................................................................71 K FITTED FINAL SUM PROJECTIONS.....................................................................82 L MODULATION TRANSFER FUNCTIONS............................................................93 REFERENCES................................................................................................................104 BIOGRAPHICAL SKETCH...........................................................................................106 vii PAGE 8 LIST OF TABLES Table page 1 Initial conditions of image acquisitions...................................................................24 2 Processed coincidence count rates of all acquisitions..............................................26 3 Parameters of fitted curves.......................................................................................31 4 FWHM calculated from the fitted final sum projections.........................................33 viii PAGE 9 LIST OF FIGURES Figure page 1 Pulse before and after shaping...................................................................................6 2 Possible coincidence events.......................................................................................9 3 Module Assembly....................................................................................................12 4 Phantom utilized in the experiment..........................................................................13 5 Illustration of sinograms shifting process................................................................20 6 Example of determining random and scatter coincidences......................................20 7 Final sum projections of selected acquisitions.........................................................25 8 Final sum projection of acquisition 1.......................................................................71 9 Final sum projection of acquisition 2.......................................................................72 10 Final sum projection of acquisition 3.......................................................................72 11 Final sum projection of acquisition 4.......................................................................73 12 Final sum projection of acquisition 5.......................................................................73 13 Final sum projection of acquisition 6.......................................................................74 14 Final sum projection of acquisition 7.......................................................................74 15 Final sum projection of acquisition 8.......................................................................75 16 Final sum projection of acquisition 9.......................................................................75 17 Final sum projection of acquisition 10.....................................................................76 18 Final sum projection of acquisition 11.....................................................................76 19 Final sum projection of acquisition 12.....................................................................77 20 Final sum projection of acquisition 13.....................................................................77 ix PAGE 10 21 Final sum projection of acquisition 14.....................................................................78 22 Final sum projection of acquisition 15.....................................................................78 23 Final sum projection of acquisition 16.....................................................................79 24 Final sum projection of acquisition 17.....................................................................79 25 Final sum projection of acquisition 18.....................................................................80 26 Final sum projection of acquisition 19.....................................................................80 27 Final sum projection of acquisition 20.....................................................................81 28 Final sum projection of acquisition 21.....................................................................81 29 Count rates versus time into experiment..................................................................27 30 Count rates versus activity.......................................................................................27 31 True count rate versus time into experiment............................................................28 32 True count rate versus activity.................................................................................28 33 Noise Equivalent Count rate versus time into experiment.......................................29 34 Noise Equivalent Count rate versus activity............................................................29 35 Randoms count rate and its fitted equation versus activity......................................30 36 Final sum projection of acquisition 1 and its fitted gaussian curve.........................82 37 Final sum projection of acquisition 2 and its fitted gaussian curve.........................82 38 Final sum projection of acquisition 3 and its fitted gaussian curve.........................83 39 Final sum projection of acquisition 4 and its fitted gaussian curve.........................83 40 Final sum projection of acquisition 5 and its fitted gaussian curve.........................84 41 Final sum projection of acquisition 6 and its fitted gaussian curve.........................84 42 Final sum projection of acquisition 7 and its fitted gaussian curve.........................85 43 Final sum projection of acquisition 8 and its fitted gaussian curve.........................85 44 Final sum projection of acquisition 9 and its fitted gaussian curve.........................86 45 Final sum projection of acquisition 10 and its fitted gaussian curve.......................86 x PAGE 11 46 Final sum projection of acquisition 11 and its fitted gaussian curve.......................87 47 Final sum projection of acquisition 12 and its fitted gaussian curve.......................87 48 Final sum projection of acquisition 13 and its fitted gaussian curve.......................88 49 Final sum projection of acquisition 14 and its fitted gaussian curve.......................88 50 Final sum projection of acquisition 15 and its fitted gaussian curve.......................89 51 Final sum projection of acquisition 16 and its fitted gaussian curve.......................89 52 Final sum projection of acquisition 17 and its fitted gaussian curve.......................90 53 Final sum projection of acquisition 18 and its fitted gaussian curve.......................90 54 Final sum projection of acquisition 19 and its fitted gaussian curve.......................91 55 Final sum projection of acquisition 20 and its fitted gaussian curve.......................91 56 Final sum projection of acquisition 21 and its fitted gaussian curve.......................92 57 Adjusted FWHM as a function of time into experiment..........................................34 58 Adjusted FWHM as a function of activity...............................................................34 59 Modulation transfer function of acquisitions 1, 5, 10, and 21.................................35 60 Modulation transfer function of acquisition 1..........................................................93 61 Modulation transfer function of acquisition 2..........................................................94 62 Modulation transfer function of acquisition 3..........................................................94 63 Modulation transfer function of acquisition 4..........................................................95 64 Modulation transfer function of acquisition 5..........................................................95 65 Modulation transfer function of acquisition 6..........................................................96 66 Modulation transfer function of acquisition 7..........................................................96 67 Modulation transfer function of acquisition 8..........................................................97 68 Modulation transfer function of acquisition 9..........................................................97 69 Modulation transfer function of acquisition 10........................................................98 70 Modulation transfer function of acquisition 11........................................................98 xi PAGE 12 71 Modulation transfer function of acquisition 12........................................................99 72 Modulation transfer function of acquisition 13........................................................99 73 Modulation transfer function of acquisition 14......................................................100 74 Modulation transfer function of acquisition 15......................................................100 75 Modulation transfer function of acquisition 16......................................................101 76 Modulation transfer function of acquisition 17......................................................101 77 Modulation transfer function of acquisition 18......................................................102 78 Modulation transfer function of acquisition 19......................................................102 79 Modulation transfer function of acquisition 20......................................................103 80 Modulation transfer function of acquisition 21......................................................103 xii PAGE 13 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science COUNT RATE AND SPATIAL RESOLUTION PERFORMANCE OF A 3-DIMENSIONAL DEDICATED POSITRON EMISSION TOMOGRAPHY (PET) SCANNER By Rami Rimon Abu-Aita May 2004 Chair: David Gilland Major Department: Nuclear and Radiological Engineering The count rate and spatial resolution performance of a state-of-the-art 3-dimensional dedicated positron emission tomograph at Shands Medical Plaza were assessed. A 70-cm-long polyethylene phantom and a line source were both utilized to better study the effect of activity from outside the field of view on the counting rate and spatial resolution of the scanner. A series of 21 tomographic images were acquired with a fluorine-18 radionuclide. The computed true, scatter, random, and noise equivalent count rate coincidences were plotted as functions of activity, as described by NEMA 2001 protocol. The spatial resolution was studied by calculating the full width at half maximum (FWHM) of the fitted final sum projections. The FWHM decreases over time as activity decays, and is constant for the last 5 acquisitions with activities ranging from 0.22 mCi for the 17 th acquisition to 0.08 mCi for the 21 st acquisition. Moreover, the modulation transfer function was computed to completely describe the spatial resolution at varying xiii PAGE 14 activities. It was concluded that both the count rate and the spatial resolution were affected by the increase in activity, especially at high counts. It was found that the peak NEC rate occurred at 31.04 kcps at an activity of 5.876 mCi. The scatter fraction of the system was found to be about 40% indicating the disadvantageous high scatter coincidence events due to the contribution of scatter from out of the field of view. Finally, the peak NEC rate is a useful measure of the amount of activity that should be employed in a clinical patient scan. Also the trends of the true, scatter, and random count rate curves show the behavior of the scanner studied. xiv PAGE 15 CHAPTER 1 INTRODUCTION There is increasingly wide PET application for clinical diagnosis, due to the improvement in the diagnostic accuracy of this imaging modality through the development of new data acquisition and processing systems and the introduction of new positron-emitting radiopharmaceuticals. Positron emission tomography (PET) with fluorine-18 fluoro-2-deoxy-glucose ( 18 F-FDG), a glucose analog, is a unique biologic imaging tool that can be used to obtain functional information from the living human body. After administration of the 18 F-FDG radiopharmaceutical to a patient, it takes part in physiologic processes. The unstable, neutron-deficient nuclide decays by emitting a positron, which will annihilate with a nearby electron to create two 511 keV photons that are emitted essentially back-to-back at an angle of 180 degrees. Detection of the two photons within a narrow time window is called a coincidence event, where the assumption is that the photons originate from a single annihilation occurring along the line-of-response defined by the direction of the photons. The radioactive decay is continuously taking place and can be detected from the emitted gamma rays, allowing the monitoring of the distribution of the tracer concentration. The widely used 18 F-FDG has an advantage of becoming trapped in the tissue and reaching a near equilibrium state approximately 45 to 60 minutes after injection, therefore keeping the tracer concentration almost constant with time. 1 Oncology imaging using PET with 18 F-FDG has proven to be effective in evaluating the extent of disease for several types of cancer. 2 FDG-PET is useful for defining the degree 1 PAGE 16 2 of malignancy and for differentiating recurrent tumors from necrosis after therapy in patients with brain tumors. 3 Data acquisition is substantially different in PET than it is in planar nuclear medicine and single photon emission computed tomography (SPECT). In PET, annihilation coincidence detection (ACD) is used in lieu of absorptive collimation to determine the directionality of the detected photons. Because of this technique, the sensitivity, the rate at which the system detects counts per unit of activity, of PET is substantially higher than in single-photon nuclear imaging. ACD also avoids the degradation of spatial resolution with distance from the detector. 4 There are two basic ways of collecting data in PET. These are referred to as 2-dimensional (2D) and 3-dimensional (3D) PET. Some PET scanners can acquire data in both 2D and 3D modes whereas others can only acquire data in 3D mode, like the dedicated PET scanner considered in this investigation. With 2D PET, thin rings (~ 1 mm thick) of lead or tungsten, known as septa, are often placed between the detector rings to reduce the interplane scatter. These septa are more like the antiscatter grids used in radiography. They do not provide spatial definition but simply reduce the amount of interplane scatter incorporated into the data acquisition. On the other hand, the absence of septa in 3D PET increases the sensitivity by a factor of 4-6 but also increases the scatter fraction as well as the random coincidences from activity that is out of the field of view. 5 The increased sensitivity to true events is partially offset by the higher sensitivity to scattered and random coincidences. Consequently, the overall advantage of 2D versus 3D PET whole-body imaging is unclear. 6 PAGE 17 3 In the last 11 years, whole-body 18 F-FDG studies have become the predominant type of PET study performed by most centers. 7 Therefore, it is more relevant than before to measure the performance of the PET scanners under conditions that better represent whole-body studies. The phantom that was used for evaluating the performance of the scanner was a 19-cm-long phantom. The phantoms volume was more comparable to the head volume, whereas the body is much larger. Nowadays, the most significant change in testing the PET counting rate performance is the change from the 19-cm-long phantom to a 70-cm-long phantom, while keeping the diameter (20 cm) of both phantoms unchanged. The 70-cm phantom is a better approximation to the activity distribution in whole-body studies, because the effects of out-of-field activity (OFA) are included in the measurements. The 19-cm, on the other hand, continues to be used to test the performance of scanners used primarily for brain imaging, particularly for scanners that are dedicated brain imaging instruments. Most patient PET studies are not performed under conditions of low counting rate losses or negligible random rates. At higher activity levels, coincidence events are lost because of system dead time, whereas the rate of random coincidences rises. 6-11 It is necessary to measure the counting rate performance, both dead-time losses and random events, as a function of activity to understand the scanners behavior for a wide range of scanning conditions. Different studies have examined the axial and transverse spatial resolution within the field-of-view (FOV) measured with a point source. 8, 9, 12 These studies, however, do not take into account the effect of varying activities on the resolution. The increase in count rate increases the probability that pulses from multiple coincidences will be PAGE 18 4 integrated together to cause spatial distortion and degrade the spatial resolution. The consequences of high count rate on spatial resolution are therefore investigated. The aim of this study was to assess the count rate capabilities of the 3D dedicated PET scanner and determine the effects of the high count rate on the systems spatial resolution. To quantify the changes in spatial resolution, the full width at half maximum (FWHM) of line source images and modulation transfer functions (MTF) were computed from images of a decaying line source in the 70 cm cylindrical phantom. The practicality of obtaining both count rate and spatial resolution data in a single acquisition was also determined. PAGE 19 CHAPTER 2 BACKGROUND Count Loss and Dead Time An incident 511-keV annihilation photon that is absorbed in the scintillator generates a pulse of light that is converted into an electronic signal and amplified by the photomultiplier tubes (PMTs). The outputs from the photomultipliers are used to localize the incident photon and as a measure of the energy to reject photons that have scattered before reaching the detector. The overall count rate performance of the detectors is dependent on a number of factors such as pulse pile-up and system dead time or pulse resolving time. Pile-up within the crystal occurs when two photons from different annihilations arrive so closely spaced in time that they cannot be distinguished as two separate photons. The light output of such an event is the sum of the two photon energies that will, in general, exceed the upper energy level discriminator (ULD), and therefore be discarded. Both photons are subsequently lost. This type of signal pile up is called post-pulse pile-up because a pulse is received while another is being integrated. A second source of count rate loss, known as pre-pulse pile-up, occurs when the positional and energy determination of a photon is still in process when a second photon arrives. Since the detection system is dead when the second photon arrives, the count will be lost. Such a process has the characteristics of a saturating or paralyzing system. A paralyzable system is one for which each event introduces a dead time whether or not that event actually was counted. Thus an event occurring during the dead time of a preceding event would not be counted but still would 5 PAGE 20 6 introduce its own dead time during which subsequent events could not be recorded. To reduce pulse pile-up at high count rates, scintillators with short decay times are essential. The decay constant for the Gadolinium Oxyorthosilicate (GSO) scintillation crystal considered in this study is 56 nanoseconds. Scintillation light of GSO is emitted faster than Sodium Iodide (NaI(Tl)). NaI(Tl) has long been used as a scintillation crystal in most nuclear medicine imaging systems. NaI has a large decay constant of 230 nanoseconds. To further reduce pile-up and improve the high-count rate performance of the system, the amplifier shapes the pulse therby reducing it to 100 nanoseconds [Fig. 1]. FIG. 1. Pulse before and after shaping Because of dead time losses, the observed counting rate is less than the true counting rate where the latter is the counting rate that would be recorded if there was no dead time and is proportional to the source activity. At low activity, an increase in count rate is proportional to the increase in activity. However, at high activity levels, the proportionality is lost, as the system is unable to handle the increasing count rate. At sufficiently high activity levels, the count rate actually decreases with increasing activity 0R tR PAGE 21 7 as the system becomes paralyzed. The relationship between the observed and true counting rates and the paralyzable dead time is given as follows: (1) tRt0eRR Notice that the observed counting rate rises to a maximum value given by 718.21Rmax0 (2) Then the observed counting rate actually decreases with a further increase in true counting rate. This is because additional events serve only to extend the already long dead time intervals without contributing to additional events in the observed counting rate. The equation for the degradation of count rate due to pre-pulse dead time can be extended to coincidence counts by noting that both annihilation photons must be detected. If only one of the paired photons is detected, a coincidence event will not be registered but instead will be lost. Therefore, the coincidence count rate, can be expressed as cR (3) where f is the fraction of the true single event rate, which will create coincidence events. Note that for the coincidence case, refers to the rate at which photons interact with the scintillation crystal, and is assumed equal at both detectors. tttR2tRRtcefReefRRtRtR Similarly, the equation for the degradation of coincidence count rate due to post-pulse dead time is given as (4) )(R2tR2R2tcntnttefReefRR where is the integration time and the sum ( n n ) is the effective dead time constant. PAGE 22 8 The Effects of High Count Rate on Spatial Resolution As pulses arrive after the integration process has begun, the extraneous light may be integrated with the original pulse. The extra signal contains additional spatial information that, when integrated with the spatial information of the original signal, may misplace the gamma rays origin in the patient. This misplacement will create spatial distortion and worsen spatial resolution. The inaccurate positional information is reduced by the post-pulse energy discrimination levels. An alternative to energy thresholds employed by the detection system, local centroid algorithm, 13 and hence local centroid position, is used. The local centroid algorithm ensures that the PMTs that may receive light from a pile up event are excluded from the position calculation. Random and Scatter Coincidence Events In addition to detection of the true coincidence events, PET imaging with coincidence detection can result in two other undesirable types of events; scatter and random events. Scattered coincidences occur when one or both of the gamma rays undergo a Compton scatter interaction inside the body. This process changes their direction and reduces the energy of the photon. The change in direction results in misidentification of the gamma-ray origin as shown in Figure 2B. The fraction of gamma rays which get scattered depends on the scattering media and path length through the body. Therefore, the contribution from scattered events is more evident in abdominal imaging than in brain imaging. Although the gamma rays that are scattered have their energy reduced below 511 keV, the energy resolution of most PET systems is insufficient to use this as an effective means of scatter rejection. Thus, many of the scattered events are accepted and subsequently lead to falsely positioned data despite the scatter correction techniques. 14, 15 PAGE 23 9 FIG. 2. Possible coincidence events; a graphical representation of true (A), scatter (B), and random (C) events. 9 Random, or accidental, coincidences arise when two photons from different annihilations are detected within the coincidence window and recorded as a coincidence. This situation may arise either when the partner photons are scattered out of the FOV [Fig. 2C], or when the two uncorrelated photons simply arrive more closely in time than the true coincidence. The random coincidence rate increases with the singles rate on the detectors because the probability that two uncorrelated photons will arrive within coincidence window increases. The random coincidence rate is proportional to the square of the activity and is, therefore, a particular problem for high count rate studies. The randoms rate (R) is strictly related to the singles counting rate (S1 and S2) of each detector and to the coincidence time window width ( c ) by the following relation 1, 9 21cS.S..2R (5) Therefore, a quadratic increase in random events will be observed by increasing the radioactivity, whereas a decrease will result from reducing c Random coincidences contribute to the background in the image, which can lead to a loss of image contrast. PAGE 24 10 Efforts have been made to increment the net trues counting rate without incrementing random coincidences, by implementing faster systems with narrower coincidence time windows. The coincidence time window for the GSO crystal is 8 nanoseconds. Noise Equivalent Count Rate The relative noise in an acquisition is the standard deviation of the total counts divided by the mean value N of the total counts. Because counts in a nuclear medicine acquisition follow a Poisson distribution, the relative noise is simply 1/ N When random and scatter coincidence events are removed to determine the true count rate, the mean counts are reduced but the standard deviation remains the same assuming no change in randoms or scatters. The distribution is no longer Poisson and the relative noise is now N /(N-A), where A is a constant representing the random and scatter coincidence events that were removed from the total counts. The noise equivalent count (NEC) rate is a useful parameter indicating the noise component of PET data. It is a useful predictor of the PET scanner performance because it combines the effects of signal and noise in the system in a single metric. NEC rate provides a quantitative framework in which to analyze design options that may increase true, scattered, and random coincidences, by varying degrees, to assess their impact on imaging performance. The NEC rate represents the ratio between net trues and prompts. The NEC rate follows Poisson statistics and has a relative noise equal to the relative noise of the true events after the random and scatter coincidence events have been removed. Setting the relative noise equal to 1/ NEC then the NEC rate is given by the following equation: N)AN(NEC2 (6) PAGE 25 11 It is worth mentioning that the best imaging condition is reached when acquisitions are performed by placing an activity in the FOV corresponding to the peak NEC rate. PAGE 26 CHAPTER 3 MATERIALS AND METHODS Camera Description The camera used was a Philips Allegro Positron Emission Tomography (PET) system. The system is equipped with 420 Photo-Multiplier Tubes (PMTs) and 28 modules. Full-ring detection geometry is defined by placing the modules side by side, coupled to the PMTs closely packed in a hexagonal array. Each module is populated with an array of 638 Gadolinium Oxyorthosilicate (GSO) crystals, totaling 17,864 crystals. Each crystal has dimensions of 4x6x20 mm. The modules are arranged around the patient port with the GSO crystals facing the patient. The 420 PMTs are mounted on the opposite side of the 28 modules which collect light from the scintillation crystals [Fig. 3]. By surrounding the patient with 28 position-sensitive segments, data can be acquired simultaneously along any parallel line and at any right angle. FIG. 3. Module Assembly 12 PAGE 27 13 The axial field-of-view (FOV) of the camera is 18 cm and the transverse FOV is 57.6 cm. The system operates in 3D mode, and therefore no physical collimators are involved. The scanner houses a 137 Cs source, 662 keV, for generating transmission images during patient scanning. The transmission data is used to correct the emission images for photon attenuation in the patients body. Transmission scans are not used in this investigation because, according to the National Electrical Manufacturers Association 16 (NEMA), attenuation corrections should not be employed when measuring the count rate capability of the scanner. NEMA is further discussed in the following sections. Phantom Description The phantom considered here is described in section 4 of the National Electrical Manufacturers Association (NEMA) NU 2-2001 document of performance standards for PET scanners. 16 A list of the NEMA protocol is provided in Appendix A. The test phantom [Fig. 4(A), and 4(B)] is a solid circular cylinder, as shown below, A B FIG. 4. Phantom utilized in the experiment; (A) Phantom and line source, (B) Cross section of the cylindrical phantom showing line source inserted 4.5 cm off center. PAGE 28 14 composed of polyethylene with an outside diameter of 20 cm and with an overall length of 70 cm. A 6.4 mm hole is drilled parallel to the central axis of the cylinder, at a radial distance of 4.5 cm. The cylinder consists of 4 segments that are assembled together during testing. The assembly of the completed phantom was checked to insure a tight fit between adjacent segments, as even very small gaps will allow narrow axial regions of scatter-free and attenuation-free radiation. The test phantom line source insert is a clear polyethylene coated plastic tube that is 80 cm in length, with an inside diameter of 3.2 mm and an outside diameter of 4.8 mm. The central 70 cm of this tube was filled with an 18 F-FDG initial activity of 19.26 millicuries (mCi) at 3:30 PM and threaded through the 6.4 mm hole in the test phantom. Notice that the phantom in this experiment is longer than the axial FOV of the scanner, which allows studying of the effects of out-of-field activity on the count rate. For consistency, the phantom was rotated such that the line source is at the lowest position (i.e., nearest to the patients table), because the measured result will depend on the relative orientation of the line source and the table. Allegro Data Acquisition and Processing The phantom was centered in the 18 cm field of view of the scanner and the dynamic NEMA-Countloss protocol was used to acquire data at different activities. Each acquisition lasted for 20 minutes. A delay time of 20 minutes was used between acquisitions. The first acquisition began at 4:38 PM with a calculated initial activity of 12.536 mCi. A total of 21 tomographic acquisitions were taken every 40 minutes. Table I, in the results chapter, lists the initial conditions of the image acquisitions, including the times of the acquisitions, the initial activities at the beginning of each acquisition, and the time into the experiment. A complete description on how to acquire data using the Allegro PET scanner is given in the following sections. PAGE 29 15 Allegro Data Acquisition Parameters The Allegro workstation consists of two monitors; one for acquisition, and another for processing. File Management was chosen from the acquisition monitor. Invoked within the File Management is the Acquisition menu that is used to setup the parameters for the acquisition. By choosing Set Up Acquisition, one can enter all information pertinent to that acquisition, such as study date, phantom weight, etc. The information required here are easy to understand and require no previous experience. Notice that the directory where the output raw data will reside does not need to be specified. After the acquisition is finished, the raw data will be transferred to the patient directory automatically and will be easily identified. However, the filename should be entered to be able to identify the correct file. For further clarification on the acquisition parameters and any other software setup information, one can refer to the Allegro users manual. 17 The dynamic DefaultNEMA_Countloss is selected as the acquisition protocol. Typically, acquisitions are performed using a preset protocol. In this experiment, however, it was necessary to change some of the acquisition parameters, which is done by selecting to edit the protocol. The editing process consists of specifying the acquisition time and adding more frames (or acquisitions) at the end until a small activity is left in the line source. The number of frames added depends on the starting activity and the acquisition time chosen. The durations of all the frames could be added to get a good prediction of what the last frame should be. After acquiring, a file containing the raw data is generated and saved in the patients directory. This file is further processed as shown in the following section. PAGE 30 16 Allegro Data Processing Parameters The raw data file, the output at the end of the last acquisition, is then processed on the processing monitor. To do so, the NEMA_Countloss reconstruction protocol is selected by highlighting the file, and choosing Petview, Reconstruct Sinogram, and Research Protocol, respectively. The reconstruction protocol is then edited to generate the Fourier rebinned 18, 19 data to be use in the data analysis programs. The parameters of NEMA_Countloss reconstruction protocol are already set correctly as specified by the NEMA standards for testing the count rate performance of positron emission tomographs. Examples of these parameters are the background subtraction, attenuation correction, and decay correction, which are all turned off during the reconstruction. However, the user needs to save the Fourier rebinned (FORE) sinograms using the FT Output Sinogram option, found in the advanced parameters menu. A filename can also be specified for the rebinned data. Notice that in the advanced parameters, there is an option provided by the manufacturer to turn the Single Slice Rebinning 18 (SSRB) on. The SSRB, however, was not used because it is believed that this rebinning algorithm is not functioning at all. Despite consultation with the software experts at Philips, this matter has not been resolved. The manufacturer, however, suggested using the FORE for this experiment. File Structure The reconstruction program operates on the raw data to produce an interpolated scan file. The structure of this interpolated file is as follows. It has a main header that is 512 bytes long. As a confirmation, the headers display acquisition parameters and patient information. The main header contains information such as the date of the experiment or patient scan, the patient ID, isotope used, etc. The main header is followed by a directory record, which is 512 bytes long. Every directory record is followed by 31 slices. If the PAGE 31 17 file contains more than 31 slices, a new directory record is created after slice number 31. Sinograms and directory records are added to the file as needed until all data are stored. Each slice or sinogram within the file has a subheader. The size of the subheader is also 512 bytes followed by data. The data in the sinogram are always written as signed short (2 bytes) variables. The sinograms are treated as 256 by 192 arrays. The overall structure of the Fourier rebinned file is the same as the interpolated file. However, the interpolated file contains 45 tilted slices at each tilt angle. There are 7 tilt angles for each sinogram. Thus, there are 7 tilted slices at each position along the axis parallel to the axial field-of-view, including the zero tilted sinogram or direct sinogram. These oblique sinograms are collapsed into a single sinogram for each respective slice using the FORE algorithm to produce only direct sinograms suitable for NEMA processing. Therefore, reducing the total number of slices from 315 to 45 slices per acquisition. The Fourier rebinned file and the interpolated file are both output by the reconstruction protocol. To recap, in the interpolated file each event is binned into 4-dimensional projection coordinates; Transverse distance r, projection angle tilt or off-plane angle, and axial distance y. FORE is then employed to convert these 4-dimensional sinograms into 2-dimensional sinograms by collapsing the oblique sinograms into a single sinogram for each respective slice while conserving the number of counts in the sinogram. The resulting sinogram is a 256 by 192 pixel matrix with a signed short format. The pixels linear size is 2.25 mm. No corrections for dead time, attenuation, or random events are employed during the acquisitions. A total of 45 sinograms per acquisition are obtained in order to span the length of the line source in the18 cm field of view. PAGE 32 18 In addition, the Fourier rebinned file is then copied to a specified directory to be burned on a CD. Appendix B provides more information on how to copy and burn files to a CD. All pixels in a sinogram are multiplied by a scaling factor to correct the number of counts in each pixel. This is done when running the MATLAB codes, discussed in the following section. The scaling factors are obtained using a shell script provided by the software engineer at Philips Medical Systems [appendix C], which is a small program that can be run in a Unix environment. The output from this program is three columns. The first is frame number, running from 1 to 21. The second is a slice index, typically running from 1 to 45. The third value is the sinogram slice scale factor. To run this program in Unix, one should refer to the commands shown in appendix C. NEMA Data Processing and Analysis Using MATLAB Count Rates MATLAB codes, written by the author of this paper, were employed to perform the NEMA data analysis for processing the sinograms data obtained after the experiment. These codes are given in appendices D through I and will be briefly discussed here. Notice that comments are embedded within each code to make them easy to understand and user friendly. These MATLAB codes need to be run successively. Because the output from the first program [appendix F] will become the input to the second program [appendix G] and the output from the second program will become the input to the third program [appendix H], and so on. Appendix D shows the code for extracting the sinograms from the raw data file. The extracted sinograms of each acquisition are output MATLAB: The language of technical computing by MathWorks, Inc. 2002 PAGE 33 19 to different acquisition files numbered 1 through 21. The main header of the original file, the subheader of each sinogram, and the directory records are all not included in the output acquisition files. Therefore, the output data are only the sinograms data needed for processing. This step of reorganizing the data such that no header information is embedded within the actual sinogram data is significant for it reduces the amount of code necessary in processing the sinograms. Appendix E has the scaling factors code. This is a simple piece of code written to organize the scaling factors in a format suitable for processing when used by the programs to follow. Again, the order in which these programs are run is important and follows the order shown in the appendices. Section 4 of the NEMA standards publication NU 2-2001 for performance measurements of positron emission tomographs summarizes the steps used to determine scatter fraction, count losses, and randoms measurements. The MATLAB codes in appendices F and G were both used for further processing of the data. In these codes, pixels that were 12 cm or further away from the center of the phantom were set to zero counts for each sinogram. For each projection angle within a sinogram, the location of the line sources center was determined as the pixel where the highest count value occurred. The count data within that projection angle were then shifted in the transverse direction so that the center of the line source aligned with the center of the sinogram in the transverse direction [Fig. 5]. After the shift, each sinogram was compressed into a onedimensional profile, or sum projection, by summing the pixel values along the angular direction. PAGE 34 20 A B FIG. 5. Illustration of sinograms shifting process; (A) Sinogram, (B) Shifted sinogram. In slight deviation from NEMA, these profiles were then summed over all sinograms for the particular acquisition to create a final sum projection to describe each acquisition. It was from these final sum projections, rather than the sum projections of the individual slices, that the count rates were determined. From these count rates, the activity where the peak rate occurs was found. Figure 6 illustrates how the random and scatter coincidence events were determined for each acquisition. The counts at the pixels 2 cm to either side of the center of the final FIG. 6. Example of determining random and scatter coincidences; counts versus radial distance from the center pixel. 16 sum projection were averaged and then multiplied by the number of pixels equal to a distance of 4 cm. This product was added to counts outside the 4 cm strip to create the PAGE 35 21 random plus scatter counts for acquisition j, C. To separate the scatter and random components, the final acquisition j, where the random coincidence rate and the loss in count rate due to dead time are both presumed less than 1%, was considered. With the dead time equation and equations for coincidence count rates j,sr 7, 11 it was verified that the dead time losses and the random count rates in the final acquisition would both be less than one percent. Letting the random coincidence counts be negligible, the scatter and random counts, C were divided by the total counts of acquisition j, to calculate the scatter fraction SF: 'j,sr 'j,TOTC ''j,TOTj,srCCSF (7) This scatter fraction was constant for all acquisitions because the geometry and attenuation media remained constant. Variations in design cause PET scanners to have different sensitivities to scattered radiation. The scatter fraction is measured at a sufficiently low counting rate that random coincidences, dead-time effects, and pileup are negligible. The total event rate, R TOT,j for each acquisition is the total counts for that acquisition divided by the acquisition time T acq, j : j,acqj,TOTj,TOTTCR (8) The true event rate, free of scatter or random coincidences, is j,acqj,srj,TOTj,tTCC R (9) The scatter and random count rates can be found by using the following equations: j,tj,sRSF1SF R (10) PAGE 36 22 SF1RRRj,tj,TOTj,r (11) The noise equivalent count rate was determined by the following equation: j,TOT2j,tj,NECRRR (12) To determine if the random coincidence count rate increases with the square of the activity, a function was fitted to the plot of computed random count rates as a function of activity [appendix H]. The fitted function was of the following form: F(x; a) = ax 2 (13) where x is the activity in millicuries (mCi), and [a] is a constant. Spatial Resolution The final sum projection of each acquisition was fitted to a Gaussian function plus a constant, equation 14, using the nonlinear least squares method in MATLAB [appendix H]. To further verify the fitted curves, the final sum projection of each acquisition was plotted on the software package Kaleida Graph 3.5 and a function was fit to the plot. An online curve-fitting tool was also used for the same purpose. The fitted function that was always considered was a gaussian plus a constant. The parameters of the fitted functions were all the same within three significant figures. The Kaleida Graph parameters, however, were slightly different. The FWHMs, calculated based on the fitted parameter of the standard deviation, are, therefore, only rough calculations due to errors that are likely to be associated with the fitting process. A trial version of this software package was used. http://zunzun.com: An interactive 2-D and 3-D data-modeling tool. PAGE 37 23 The spatial resolution was then determined by calculating the full width at half maximum (FWHM) for each acquisition using equation 15. The standard deviation in equation 15 is obtained from the fitted functions. The FWHM was adjusted for the finite diameter of the line source (3.2 mm) using equation 16. This equation was based on the practice of squaring components of resolution, adding the squared components together, and then taking the square root of this sum to determine the final resolution. F(x; a, b, x 0 ) = a + be (14) 220/)xx(5.0 FWHM = 2.35 (15) FWHM a (mm) = [(FWHM(mm)) 2 (3.2) 2 ] 1/2 (16) To further explore the spatial resolution of an imaging system, the modulation transfer function (MTF) is preferred. By using MATLAB [appendix I], an MTF was calculated for each acquisition by computing the discrete Fourier transform of the final summed projection of each acquisition. The MTFs were each normalized to a value of unity at the zero frequency. This analysis on the spatial resolution is confined to only the resolution in the transverse direction because of the type of geometry and analysis followed. PAGE 38 CHAPTER 4 RESULTS Table 1 lists the initial conditions of the image acquisitions, including the times of the acquisitions, the initial activities at the beginning of each acquisition, and the time into the experiment. TABLE 1. Initial conditions of image acquisitions Acquisition Start time End time Time into Initial experiment activity (min) (mCi) 1 4:38 PM 4:58 PM 0 12.536 2 5:18 PM 5:38 PM 40 9.738 3 5:58 PM 6:18 PM 80 7.564 4 6:38 PM 6:58 PM 120 5.876 5 7:18 PM 7:38 PM 160 4.564 6 7:58 PM 8:18 PM 200 3.545 7 8:38 PM 8:58 PM 240 2.754 8 9:18 PM 9:38 PM 280 2.139 9 9:58 PM 10:18 PM 320 1.662 10 10:38 PM 10:58 PM 360 1.291 11 11:18 PM 11:38 PM 400 1.003 12 11:58 PM 12:18 AM 440 0.779 13 12:38 AM 12:58 AM 480 0.605 14 1:18 AM 1:38 AM 520 0.470 15 1:58 AM 2:18 AM 560 0.365 16 2:38 AM 2:58 AM 600 0.284 17 3:18 AM 3:38 AM 640 0.220 18 3:58 AM 4:18 AM 680 0.171 19 4:38 AM 4:58 AM 720 0.133 20 5:18 AM 5:38 AM 760 0.103 21 5:58 AM 6:18 AM 800 0.080 24 PAGE 39 25 Count Rate The final sum projections of acquisitions 1, 5, 10, 15, and 21 are shown in Figure 7. -20 -10 0 10 20 0 0.5 1 1.5 2 2.5 x 10 4 Distance alon g transverse axis Count Rate [cps] ac q uisition 1 ac q uisition 5 acquisition 10 acquisition 15 ac q uisition 21 FIG. 7. Final sum projections of selected acquisitions Figure 7 illustrates the changes in count rate, width of the peak, and background level due to random and scattered events. The background decreases over time indicating that the random and scatter coincidences together are decreasing. The final sum projections of all acquisitions are shown in Figures 8 through 28 [Appendix J]. Notice that the y-axis of Figures 8 through 28 is changed to reflect the changes in the count rates for each separate acquisition. The total paired events, and total, true, scatter, random, and NEC count rates are all shown in Table 2. The scatter fraction was calculated to be 0.403. This scatter fraction was computed from the last acquisition. PAGE 40 26 TABLE 2. Processed coincidence count rates of all acquisitions Acquisition Total Paired Count Rate Count Rate Count Rate Count Rate Count Rate Events Total Trues Randoms Scatter ter NEC NEC [ ] [ ] [cps] [cps] [cps] [cps] [cps] [cps] [cps] [cps] [cps] [cps] 1 1 343.88343.88 286564.15286564.15 83176.5483176.54 147239.97147239.97 56147.65 56147.65 24142.3624142.36 2 294.66 245551.51 83854.61 105091.52 56605.37 28635.93 3 246.74 205616.61 79714.53 72091.44 53810.64 30904.15 4 203.00 169163.43 72462.72 47785.34 48915.37 31040.07 5 164.77 137305.24 63534.72 30881.93 42888.59 29399.17 6 132.65 110538.73 54306.58 19572.93 36659.22 26680.29 7 105.81 88176.77 45378.88 12165.25 30632.65 23353.58 8 83.89 69904.97 37285.31 7450.51 25169.15 19886.92 9 66.32 55268.48 30308.12 4501.10 20459.25 16620.37 10 52.03 43355.99 24302.21 2648.76 16405.01 13622.05 11 40.83 34022.45 19392.04 1539.97 13090.44 11053.03 12 31.92 26602.83 15351.38 888.63 10362.82 8858.64 13 24.95 20795.78 12124.09 487.43 8184.27 7068.43 14 19.49 16244.02 9545.14 255.52 6443.37 5608.81 15 15.17 12638.47 7469.23 127.20 5042.04 4414.25 16 11.88 9898.20 5868.92 67.52 3961.77 3479.85 17 9.27 7724.92 4594.00 29.78 3101.14 2732.04 18 7.22 6016.78 3584.41 12.74 2419.63 2135.36 19 5.65 4705.65 2802.28 11.71 1891.66 1668.80 20 4.37 3640.10 2166.92 10.41 1462.76 1289.95 21 3.44 2866.02 1706.42 7.70 1151.91 1016.00 10 6 Figures 29 through 34 illustrate the counting rate performance of the PET scanner. The total, true, random, and scatter count rates are plotted in Figure 29 as functions of time into the experiment, and in Figure 30 as functions of activity. The true count rate is plotted as a function of time into experiment in Figure 31, and as a function of activity in Figure 32. As can be observed in Figures 31 and 32, the true count rate reaches a maximum of 83.86 kcps at an activity of 9.738 mCi. The noise equivalent count rate is plotted as a function of time into experiment in Figure 33 and as a function of activity in Figure 34. Notice that the noise equivalent count rate also reaches a maximum and drops again as activity increases. The maximum NEC rate is 31.04 kcps at 5.876 mCi. As shown in the figures, the total, true, and scatter count rates will all rise with activity PAGE 41 27 0 100 200 300 400 500 600 700 800 0 50 100 150 200 250 300 Time [min]Count Rates [kcps] TotalTrueRandomScatter FIG. 29. Count rates versus time into experiment 0 2 4 6 8 10 12 14 0 50 100 150 200 250 300 Activity [mCi]Count Rates [kcps] TotalTrueRandomScatter FIG. 30. Count rates versus activity PAGE 42 28 0 100 200 300 400 500 600 700 800 0 20 40 60 80 100 120 Time [min]True Count Rate [kcps] 83.86 kcps FIG. 31. True count rate versus time into experiment 0 2 4 6 8 10 12 14 0 20 40 60 80 100 120 Activity [mCi]True Count Rate [kcps]83.86 kcps at 9.738 mCi FIG. 32. True count rate versus activity PAGE 43 29 0 100 200 300 400 500 600 700 800 0 5 10 15 20 25 30 35 Time [min]NEC Count Rate [kcps] 31.04 kcps FIG. 33. Noise Equivalent Count rate versus time into experiment 0 2 4 6 8 10 12 14 0 5 10 15 20 25 30 35 Activity [mCi]NEC Count Rate [kcps] 31.04 kcps at 5.876 mCi FIG. 34. Noise Equivalent Count rate versus activity PAGE 44 30 and tend to reach a maximum value before they start to decrease. The total peak count rate is not shown here because higher activities are needed to reach the maximum. The shape of these count rate curves is consistent with the behavior of the radiation detector that experiences paralyzable dead time and pulse pile up. The count rate increases, starting at low activities, until a maximum count rate is reached and then starts to decrease as the activity continues to grow. The random coincidence count rate, on the other hand, starts very small at low activities and continues to increase with increasing activity. As expected, the continued increase in randoms reflects the fact that random coincidence rate is proportional to the square of the activity. A fact that is also illustrated by the fitted function [see Equation 13]. Figure 35 below shows the fitted randoms as a function of activity. The constant [a] of Equation 13 was found to have a value of 1028 cps and the fit had an R-squared value of 0.976. 0 2 4 6 8 10 12 14 0 20 40 60 80 100 120 140 160 180 Activity [mCi]Randoms Count Rate [kcps] fitted equationrandoms rate FIG. 35. Randoms count rate and its fitted equation versus activity PAGE 45 31 Spatial Resolution The parameters of the fitted Gaussian equation, Equation 14, of the final sum projection are given in Table 3 for each acquisition. TABLE 3. Parameters of fitted curves Acquisition a b X 0 R-squared [cps] [cps] [cm] [cm] 1 770.44 22650.41 -0.0013 0.35401 0.888 2 615.53 23107.80 -0.0025 0.34175 0.923 3 481.14 22195.66 -0.0022 0.33342 0.944 4 371.08 20343.87 -0.0031 0.32724 0.957 5 284.04 17966.86 -0.0029 0.32270 0.965 6 217.22 15439.59 -0.0023 0.31935 0.971 7 165.78 12959.74 -0.0029 0.31679 0.975 8 126.57 10691.55 -0.0031 0.31486 0.977 9 97.07 8712.33 -0.0021 0.31341 0.979 10 74.19 7001.64 -0.0027 0.31233 0.980 11 57.03 5593.31 -0.0028 0.31169 0.981 12 43.89 4433.85 -0.0023 0.31111 0.982 13 33.87 3506.68 -0.0028 0.31039 0.983 14 26.17 2763.25 -0.0021 0.31002 0.983 15 20.22 2163.63 -0.0026 0.30961 0.983 16 15.75 1700.44 -0.0034 0.30964 0.984 17 12.24 1331.75 -0.0018 0.30946 0.984 18 9.51 1039.44 -0.0018 0.30927 0.984 19 7.44 813.13 -0.0034 0.30927 0.984 20 5.76 628.40 -0.0031 0.30926 0.984 21 4.54 494.66 -0.0022 0.30935 0.984 Figures 36 through 56 [Appendix K] show the fitted final sum projections. Figures 36 and 56 are shown on the next page to illustrate the differences in the fit between the first and last acquisitions. This approach of fitting the final sum projections to a gaussian function plus a constant is not very accurate. The fitting parameters would differ when a better fit is introduced. Therefore, the FWHM would change as well. Due to this inaccurate curve fitting, the technique employed here remains subject to error. However, it still provides a rough estimation of the resolution to at least get an overall picture of the imaging system behavior. The calculated FWHMs are given in Table 4. The adjusted PAGE 46 32 -20 -15 -10 -5 0 5 10 15 20 0 0.5 1 1.5 2 2.5x 104 Distance along transverse axis [cm]Count Rate [cps] Final Sum ProjectionFitted Gaussian FIG. 36. Final sum projection of acquisition 1 and its fitted gaussian curve -15 -10 -5 0 5 10 15 0 100 200 300 400 500 600 Distance along transverse axis [cm]Count Rate [cps] Final Sum ProjectionFitted Gaussian FIG. 56. Final sum projection of acquisition 21 and its fitted gaussian curve PAGE 47 33 TABLE 4. FWHM calculated from the fitted final sum projections Acquisition measured FWHM adjusted FWHM [mm] [mm] 1 8.32 7.68 2 8.03 7.37 3 7.84 7.15 4 7.69 6.99 5 7.58 6.88 6 7.5 6.79 7 7.44 6.72 8 7.4 6.67 9 7.37 6.63 10 7.34 6.61 11 7.32 6.59 12 7.31 6.57 13 7.29 6.55 14 7.29 6.55 15 7.28 6.54 16 7.28 6.54 17 7.27 6.53 18 7.27 6.53 19 7.27 6.53 20 7.27 6.53 21 7.27 6.53 FWHMs are plotted as functions of time into experiment and activity as shown in Figures 57 and 58, respectively. The degradation in the spatial resolution can be seen as the FWHM increases from 6.53 mm to 7.68 mm as the activity increases. After the 16th acquisition, no further improvement is seen in the spatial resolution. The FWHM has a constant value of 6.53 mm as activity continues to decay going from the 17th acquisition to the 21st acquisition. The spatial resolution of the PET scanner at10 cm from the center in the transverse direction is 5.9 mm, as measured by a point source. This value was provided by Philips-ADAC Medical Systems. The different value obtained for the resolution, therefore, could have been a result of the different approach used to calculate the resolution of the camera and the finite line source diameter involved in the PAGE 48 34 0 100 200 300 400 500 600 700 800 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8 Time [min]FWHM [mm] FIG. 57. Adjusted FWHM as a function of time into experiment 0 2 4 6 8 10 12 14 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8 Activity [mCi]FWHM [mm] FIG. 58. Adjusted FWHM as a function of activity measurement. The point to keep in mind here is that the spatial resolution improves as the activity decreases. Therefore, the administration of entailed high activity levels to the PAGE 49 35 patients will not only increase their radiation absorbed dose, but also degrades the spatial resolution. Assessment of the spatial resolution is also obtained from the modulation transfer functions. Figure 59 shows the modulation transfer functions of acquisitions 1, 5, 10, and 21. For the first acquisition, at an activity equal to 12.536 mCi, the gain at a 0 0.5 1 1.5 2 2.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 [cm-1]Gain acquisition 1acquisition 5acquisition 10acquisition 21 1 5 10 21 FIG. 59. Modulation transfer function of acquisitions 1, 5, 10, and 21 spatial frequency of 0.1 cm -1 is only 24.3%. This gain increases in the fifth acquisition, which has an activity of 4.564 mCi, to 42.6% at the same spatial frequency. Figures 60 through 80, in Appendix L, are the modulation transfer functions for each acquisition. Acquisitions 10, 15, and 21 also show the increase in gain to 52.8%, 56%, and 56.5%, respectively, at 0.1 cm -1 spatial fequency. The MTF Figures also illustrate the same trend of increasing gain across the range of frequencies as activity decays over time. For example, at a frequency of 0.5 cm -1 the gain is only 15 % at the first acquisition, but PAGE 50 36 25.8% at the fifth, and 32.3%, 34.3%, and 34.6% for the tenth, fifteenth, and twenty-first acquisitions, respectively. No significant increase in the gain was noticed going from the seventeenth acquisition to the twenty-first. The gain for the latter acquisitions was almost constant and had a value of about 34.6 %. This also explains the constant values obtained for the spatial resolution at the same last five acquisitions. There was a ripple, or dip, present in each modulation transfer function. This rippling effect is believed to originate from the background counts that were included when computing the Fourier transforms. Therefore, exclusion of the background by subtracting the random and scatter events from the total events would eliminate or reduce the rippling significantly because the modulation transfer functions would now be created based on the true events only. Improvements in the gain, in this case, would also reflect the improvement in image contrast. The rippling in the MTF curves was first thought to be an artifact due to the truncation of counts located 12 cm away from the center of the image. However, the same effect was still seen in the MTFs, even without the truncation. PAGE 51 CHAPTER 5 DISCUSSION Some recent studies have investigated the counting rate performance of state-of-the-art dedicated PET systems using different size phantoms. 6, 7, 9 The most common of these phantoms were the 19-cm and the 70-cm phantoms. A longer phantom of 70-cm length was considered in order to study the effects of out-of-field activity on the count rates detected. The scatter fraction calculated from the last acquisition was 40.3%. The scatter fraction is not representative of a realistic body size. However, it is useful for standard evaluations among scanners of different configurations. On the other hand, a concern with using the line source to measure the counting rate performance was whether or not it yields counting rates that are clinically relevant. A recent study 7 shows that there is good correlation between the measured counting rates for the 70-cm line source and whole-body studies. It is important in comparing performance between scanners to look both at the peak true and NEC counting rates and at the activity levels where these peak rates occur. A highly sensitive system may saturate at a relatively low activity level, but the counting rate at this level may be higher than that of a system with lower sensitivity at a higher activity level. The NEC rate peaked at 31.04 kcps, at an activity of 5.876 mCi, it then decreased with increasing activity. If the 70-cm phantom were considered to mimic the body of an adult, then an activity more than the 5.876 mCi should not be used because any increase in activity above that point would decrease the count rate due to pileup in the detector. According to Philips-ADAC Medical Systems, the peak NEC rate was 33 37 PAGE 52 38 kcps at an activity of 0.25/mL or 5.5 mCi for the 22 liters phantom. The 22 liters is the volume of the 70-cm polyethylene phantom. In comparison, the manufacturers NEC rate was higher than the one calculated in this experiment because the value provided by the manufacturer was based solely on rough hand calculations. Moreover, the peak true count rate was 84 kcps at an activity of 0.4 Ci Ci /mL or 8.8 mCi for the same phantom. This value was also based on rough hand calculations. These values were considered to be suggestive and not authoritative, as pointed out by the manufacturer. The spatial resolution was degraded when activity increased, as can be seen from the FWHM plot as a function of activity. This should be taken into consideration when administering a radiopharmaceutical to the patient. The improvement in the FWHM is only 1.15 mm from the first to the last acquisition. The FWHM, where the peak NEC occurred, was 7mm. The true count rate at that resolution is 72.46 kcps. Better resolution can be obtained at lower activities, but the count rate would be noisier. A compromise, therefore, should be made between the resolution and the activity. The spatial resolution of the last five acquisitions remained the same as indicated by the FWHM value of 6.53 mm. This means that no further improvement in spatial resolution is obtained beyond the seventeenth acquisition. One could argue that the limit on the transverse spatial resolution of the scanner at 4.5 cm radial distance from the center of the transverse FOV is 6.53 mm. Further discussion on the modulation transfer functions that describes the spatial resolution of an imaging system was previously included in the results chapter of this paper. MTFs illustrated that the gain improves as activity decreases. Significant improvements in gain were shown up to the sixteenth acquisition, after which the gain remains unchanged. In the MTF plots, the curve extending from the zero frequency to the PAGE 53 39 region, where the little dip exists, is due to the scatter and random components. This portion of the plot corresponds to the background and appears to occur within the frequency range of zero to 0.125 cm -1 The other portion of the plot, extending from 0.18 cm -1 to the Nyquist frequency, 2.22 cm -1 is due to the true events. Activity outside the field of view was shown to greatly affect the count rate performance, especially the scatter component, of the scanner as well as the spatial resolution. The random coincidence counts were mostly affected by the dead time of the radiation detector. The expected trend of the random events, as illustrated by the fitted equation in Figure 35, is a quadratic increase as the activity increases. However, the calculated random events continued to increase then their increase slows down as indicated by the intersection of the fitted and the measured random curves. This is due to the increased dead time of the radiation detector because of the more activity, and hence more counts, introduced. Each undetected event will only increase the dead time and therefore, some random counts will be lost. Finally, the spatial resolution of the PET scanner degrades as more activity is administered to patients. In addition, more counts are lost at the same high activities. The optimal activity that should be administered to patients is the one at which the peak noise equivalent count rate occurs. Because very high activities will result in more counts being lost because of system dead time and counts pile up. PAGE 54 CHAPTER 6 CONCLUSIONS The 3-dimensional scanning system had a high scatter fraction of 40%, compared to the scatter fraction obtained using a smaller phantom. This high value was expected because of the long phantom employed in the measurements and the 3-dimensional type system involved. The scattered events from outside the field of view contributed to this increase in the scatter fraction as opposed to smaller phantoms, typically 19-cm phantoms. The longer phantom provides a more clinically reasonable distribution of activity outside the scanner. This out-of-field activity can impact the counting rate performance as well as the spatial resolution. Future work involving the same measurements discussed in this investigation could be in the form of utilizing an even longer phantom, possibly 150 cm, which would mimic the length of an average person. This would more closely approximate a true whole-body clinical scan. The scatter fraction would also be expected to increase even more. However, the increased value would only mean that more scattered coincidences would be eliminated from the data reconstructed, and therefore, improve the overall quality of the clinical images. In addition, the NEMA protocol may be revised to include the new longer phantom, if proved to be better than the existing ones. The peak noise equivalent count rate was determined to be 31.04kcps. This peak occurred at an activity of 5.876 mCi. The full width at half maximum decreased from 7.68 mm to 6.53 mm as activity decreased, indicating the improvement in spatial resolution as activity decays over time. The improvement in resolution was supported by 40 PAGE 55 41 the modulated transfer functions. The study shows that both the count rate capabilities and the spatial resolution of the 3-dimensional dedicated positron emission tomograph will degrade at high levels of activity. This is important when determining the amount of radionuclide tracer that should be administered for a PET scan. The amount of 18 F radioactivity administered to patients undergoing a PET scan is usually about 6 mCi at the start of the acquisition and after enough time is allowed for uptake of the radionuclide in tissue. Notice that this value is too close to the peak NEC rate that occurs at 5.9 mCi. This amount will still introduce some amount of noise in the image, but it results in an ideal imaging condition. The NEC rate is the ratio between the net trues and prompts. Therefore, the best imaging condition is reached when acquisitions are performed using an activity corresponding to the peak NEC rate. In conclusion, to allow for more accurate imaging of patients, a compromise between the acceptable noise equivalent count rate and adequate spatial resolution is required. PAGE 56 APPENDIX A NEMA 2001 (SECTION 4) 42 PAGE 57 43 PAGE 58 44 PAGE 59 45 PAGE 60 46 PAGE 61 47 PAGE 62 48 PAGE 63 APPENDIX B HOW TO COPY SPECIFIC FILES TO A CD Go to File Management Double click a patients name Press and hold control key, then click with the mouse on the desired files to highlight the files you want to copy (usually ----.scn). Add up the sizes of the files picked to make sure they add up to less than 700 MB (CD capacity) Go to File, then Copy File (s). A window will pop up Type in the directory to which the files will be copied: /sun0/patient/dicom. This will take a minute Open an x-term window Type: cd /sun0/patient/dicom Then type: ls. This will list the files you copied to the dicom directory. Right click with the mouse on the background, just as you would do to open an x-term window. Go to Utilities, Makeimage, then Burn CD. Youre done until this point However, you will need to empty the dicom directory if you want to burn more files again. So, go back to Utilities and click Clean DICOM Directory. 49 PAGE 64 APPENDIX C A SHELL SCRIPT FOR SCALING FACTORS This program extracts the scaling factors information found in the subheaders of each sinogram. To run the program, the following commands should be used. Notice that the name of this file is scnscl.sch and therefore the commands used refer to this file. The first command to be used is chmod 555 scnscl.sch, followed by ./scnscl.sch filename.scn 2. The filename here refers to the Fourier rebinned file. The output from this program will consist of three columns indicating frame number, slice index, and scaling factor, respectively. #! /bin/csh -f # First parameter is name of .scn file. # # The second parameter is number of first slice in each frame. # set scnfilename = $1 set firstslice = $2 set nslices = 45 set nframes = 21 set slicethickness = 4 set i = 1 set f = 1 while ( $f <= $nframes ) while ( $i <= $nslices ) @ s = ( $i 1 ) $slicethickness + $firstslice echo -n $f $s " sbhdrdmp $1 $s 0 $f | fgrep scnscl | awk '{ print $4 }' @ i = $i + 1 end @ f = $f + 1 set i = 1 end 50 PAGE 65 APPENDIX D MATLAB CODE FOR ORGANIZING ACQUISITIONS %*************************************************************************************************** % AQUISITIONS CODE %*************************************************************************************************** %* The acquisitions' raw data acquired on the PET scanner are all saved in one huge %* scan file (xxx.scn). The interpolated scan file (xxx_int.scn) is generated by the %* reconstruction program on the scanner. This interpolated file has 45 slices per tilt %* angle. There are 7 tilt angles per acquisition. Moreover, 192 projection angles and %* 256 linear samples are present for each acquisition. Notice that the Fourier Rebinned %* file doesn't have any tilt angles involved. %*************************************************************************************************** %* This program extracts the sinograms from the scan file. The extracted sinograms of %* each acquisition are output to different acquisition files numbered 1 through 21. The %* main header in the file, the subheader of each sinogram, and the directory record %* header are all extracted and are not included in the output acquisition files. %*************************************************************************************************** fid=fopen('p311s0_cou_FORE.scn','rb','ieee-be') %------------------------------------------------------------------------------------------------------------------% Input parameters %------------------------------------------------------------------------------------------------------------------nray=256; nang=192; nslc=45; nacq=21; mainH=512; %main header subH=512; %sub header DR=1:1:31; %directory records Tnslc=945; %Total number of slices %------------------------------------------------------------------------------------------------------------------sinog_num=0:1:(Tnslc-1); nbytes=nray*nang*2; %sinogram size scn=int16(zeros(nray,nang,nslc,nacq)); %------------------------------------------------------------------------------------------------------------------% Reading sinograms from file into scn matrices %------------------------------------------------------------------------------------------------------------------j=1; k=1; for i=1:1:Tnslc nbytes_skip=mainH+DR(j)*512+subH+(nbytes+subH)*sinog_num(i); % # of bytes to skip status=fseek(fid,nbytes_skip,-1); % returns 0 if success position=ftell(fid); % tells indicator position in the file 51 PAGE 66 52 scn(:,:,i)=fread(fid,[nray nang],'int16'); if( i== j*31) %to add bytes of new directory record j=j+1; end end %------------------------------------------------------------------------------------------------------------------% Output acquisitions only. These acquisitions are free of any header information %------------------------------------------------------------------------------------------------------------------acquisition=zeros(nray,nang); START=1; END=45; for acq=1:1:nacq fname=sprintf('acquisitions\\acq%d',acq); fid=fopen(fname,'wb','ieee-be'); for i=START:1:END acquisition=scn(:,:,i); fwrite(fid,acquisition(:),'int16'); acquisition=zeros(nray,nang); end fclose(fid) START=START+45; END=END+45; end %************************************************************************************************** % END OF CODE %************************************************************************************************** PAGE 67 APPENDIX E MATLAB CODE FOR SCALING FACTORS %*************************************************************************************************** %* SCALING FACTORS CODE %*************************************************************************************************** %* This program organizes the scaling factors in a format suitable for %* processing in the Data Processing Code. %*************************************************************************************************** load ('sclfact.txt') global ScalingFactors k=1; for j=1:1:26 for i=1:1:45 ScalingFactors(i,j)=sclfact(k,3); k=k+1; end end clear %************************************************************************************************** % END OF CODE %************************************************************************************************** 53 PAGE 68 APPENDIX F MATLAB CODE FOR DATA PROCESSING AND ANALYSIS %************************************************************************************************** %* The source files used in this program contain data acquired from the dedicated %* 3-D PET scanner in the Radiology Department at Shands Medical Plaza at the %* University of Florida. This program processes the acquired data according to %* section 4 of NEMA 2001 Standards. %************************************************************************************************** %************************************************************************************************** %* NEMA NU 2-2001 DATA PROCESSING CODE %************************************************************************************************** global ScalingFactors global FinalSumProj global nray nang nslcs nacq global mm_pxl cm_pxl CenterPixel pxl_L pxl_R global sino ssino global xx_R xx_L npxl_fourcm x_cm global ScatF TRandScat TCounts %------------------------------------------------------------------------------------------------------------------% Input Parameters describing the sinogram size and the number of acquisitions involved %------------------------------------------------------------------------------------------------------------------nray=256; %number of linear samples or rays nang=192; %number of projection angles nslcs=45; %number of slices in an acquisition nacq=21; %number of acquisitions mm_pxl=576/256; % sampling distance (i.e., # of mm per pixel) %------------------------------------------------------------------------------------------------------------------CenterPixel=nray/2; cm_pxl=mm_pxl/10; % cm per pixel pxl_twelvecm=12/cm_pxl; % # of pixels in 12 cm pxl_twelvecm=ceil(pxl_twelvecm); % rounds the number to the next integer pxl_L=CenterPixel-pxl_twelvecm-1; % pixel # located 12 cm to the left side of the center % of the phantom pxl_R=CenterPixel+pxl_twelvecm+1; % pixel # located 12 cm to the right side of the %center of the phantom %------------------------------------------------------------------------------------------------------------------xx_R=round(CenterPixel + 20/mm_pxl); %pixel # located 2 cm to the right from the Central pixel xx_L=round(CenterPixel 20/mm_pxl); %pixel # located 2 cm to the left from the central pixel npxl_fourcm=ceil(40/mm_pxl); % # of pixels in 4 cm (40 mm) strip 54 PAGE 69 55 %------------------------------------------------------------------------------------------------------------------% Converting x-axis to cm scale %------------------------------------------------------------------------------------------------------------------ncm=CenterPixel*cm_pxl; % number of cm from pixel 1 to central pixel (128) x_cm=-(ncm-cm_pxl):cm_pxl:ncm; % new scale in cm %------------------------------------------------------------------------------------------------------------------sino=zeros(256,192,45); %initializing zero matrices for the sinograms ssino=zeros(256,192,45); %initializing zero matrices for the shifted sinograms FinalSumProj=zeros([1 nray nacq]); %initializing a zero matrix for the Final Sum Projection for acq=1:1:nacq global ScalingFactors global FinalSumProj global nray nang nslcs global mm_pxl cm_pxl CenterPixel pxl_L pxl_R global sino ssino global xx_R xx_L npxl_fourcm x_cm global ScatF TRandScat TCounts %------------------------------------------------------------------------------------------------------------------% open and read from each acquisition %------------------------------------------------------------------------------------------------------------------disp('opening acquisition'),disp(acq) % to keep track of the code execution process fname=sprintf('acquisitions\\acq%d',acq); fid=fopen(fname,'rb','ieee-be'); %------------------------------------------------------------------------------------------------------------------% Notice: the 'ieee-be' defines the machine format used to read the acquisitions data. % Here the data was acquired at a Unix Sun Computer, so the data is saved as big % endian which is the format associated with Unix Operating Systems. PC or Windows % Operating Systems, on the other hand, are of little endian type. Therefore, when % reading Unix data on a PC it is essential to define the machine format before reading % the data. This was found to be better than swapping the data before using it in PC % environment, because the swapping process might introduce some unwanted % data embedded within the original data. %------------------------------------------------------------------------------------------------------------------disp('reading sinograms') % to keep track of the code execution process slc_num=0:1:(nslcs-1); % # slices to skip I=zeros(nray,nang,nslcs); % Initializing zero matrices for i=1:1:nslcs NumBytesSkip=(nray*nang*2)*slc_num(i); %number of bytes to skip status=fseek(fid,NumBytesSkip,-1); %returns 0 if success position=ftell(fid); %tells indicator position in the file I(:,:,i)=fread(fid,[nray,nang],'int16'); %sinogram images end fclose(fid); PAGE 70 56 %------------------------------------------------------------------------------------------------------------------% Multiplying by the scaling factors to correct for the number of counts in each pixel of each % sinogram %------------------------------------------------------------------------------------------------------------------disp('multiplying by the scaling factors') % to keep track of the code execution process for i=1:1:nslcs I(:,:,i)=ScalingFactors(i,acq).*I(:,:,i); end disp('open and read done') % to keep track of the code execution process %================================================================== % 4.4 ANALYSIS (NEMA 2001) %================================================================== %------------------------------------------------------------------------------------------------------------------% Calculating the sum projection of each slice i in each acquisition j and the final sum % projection of each acquisition. %------------------------------------------------------------------------------------------------------------------disp('Analysis begins') %------------------------------------------------------------------------------------------------------------------% One sinogram is processed at a time and the results are loaded into the appropriate matrix %------------------------------------------------------------------------------------------------------------------% Initialization u=1; % index for ray sinogram=zeros([nray nang]); shiftedsinogram=zeros([nray nang]); shiftedsinogramtranspose=zeros([nang nray]); MaxCount=zeros([1 nray]); ray=zeros; angle=zeros; SumProj=zeros([1 nray nslcs]); finalsumprojection=zeros([1 nray]); for i=1:1:nslcs % Notice that there are 2 mid-points, 128 & 129 pxls sino(:,:,i)=I(:,:,i); % saves all sinograms of one acquisition only sinogram=I(:,:,i); sinogram(1:pxl_L,:)=0; % pxls 12cm to the left from center of phantom are zeroed. sinogram(pxl_R:nray,:)=0; % pxls 12cm to the right from center of phantom are zeroed. %---------------------------------------------------------------------------------------------------------% Finding the pixel of maximum count and the location of that pixel within the %sinogram slice %---------------------------------------------------------------------------------------------------------MaxCount=max(sinogram); % returns max number of counts in each proj angle for phi=1:1:nang if (MaxCount(phi) == 0) continue; PAGE 71 57 elseif (MaxCount(phi) > 0) r=find (sinogram(:,phi)==MaxCount(phi)); % Tells location of PixelMaxCount. % Returns the row pixel location. if(length(r)==1) % if only one value returned (ideal case) ray(u)=r; elseif(length(r)>1) % if more than one value returned ray(u)=min(r); % Notice that in the same projection angle there % may be more than one pixel that has max counts. % Only the nearest pixel to the center of the line % source is taken as the pixel of maximum count. end angle(u)=phi; % tells location of PixelMaxCount % returns the column pixel location=projection angle u=u+1; % increment index u end end % check if sizes of row and clmn arrays are the same if(size(ray) ~= size(angle)) 'sizes not equal' end %--------------------------------------------------------------------------------------------------% Shifting the pixels around until the pixel of maximum count aligns with the %central pixel (i.e., pixel # 128) %-------------------------------------------------------------------------------------------------for j=1:1:length(angle) ProjAngle=angle(j); PixelMaxCount=ray(j); offset=CenterPixel-PixelMaxCount; for r=(pxl_L+1):1:(pxl_R-1) r_new=r+offset; if(r_new < 1) r_new=1; elseif(r_new > nray) r_new=nray; end shiftedsinogram(r_new,ProjAngle)=sinogram(r,ProjAngle); end end %This is based on zeroing of the array before and assumes that %the difference is less than 79 pixels or 17.78 cm shiftedsinogram(1:48,:)=0; shiftedsinogram(208:nray,:)=0; ssino(:,:,i)=shiftedsinogram; %--------------------------------------------------------------------------------------------------% Saving all important data each in the appropriate matrix PAGE 72 58 %---------------------------------------------------------------------------------------------------------ShiftedSinogram(:,:,i)=shiftedsinogram; % shifted proj, result is straight line shiftedsinogramtranspose=shiftedsinogram'; SumProj(:,:,i)=sum(shiftedsinogramtranspose);% 1-D profile 'sum in every column' finalsumprojection=finalsumprojection+SumProj(:,:,i); %---------------------------------------------------------------------------------------------------------% Calculating the scatter + Random counts and the total counts per slice %---------------------------------------------------------------------------------------------------------y=zeros([1 nray]); y=SumProj(:,:,i); C_R=y(xx_R); C_L=y(xx_L); avg=(C_R+C_L)/2; sum_R=sum(y((xx_R+1):nray)); % counts in pixels outside the strip sum_L=sum(y(1:(xx_L-1))); %counts in pixels outside the strip sum_RL=sum_R+sum_L; %total counts in pixels outside the strip C_rand_scat=(avg*npxl_fourcm)+sum_RL; C_r_s_j(i)=C_rand_scat; % random plus scatter counts for acquisition j C_TOT_j(i)=sum(y); % Total counts per acquisition j %--------------------------------------------------------------------------------------------------------% re-zeroing the matrices to avoid data overlap sinogram=zeros([nray nang]); % zero out the sinogram matrix shiftedsinogram=zeros([nray nang]); % zero out the shifted sinogram matrix shiftedsinogramtranspose=zeros([nang nray]); MaxCount=zeros([1 nang]); ray=zeros; angle=zeros; r=zeros; u=1; % index for row end % closes the nslices loop disp('Calculating Random and Scatter, and Total counts') TRandScat(acq)=sum(C_r_s_j); %Total Randoms and scatters in an acq TCounts(acq)=sum(C_TOT_j); % Total counts in an acq ScatF(acq)=TRandScat(acq)/TCounts(acq); %------------------------------------------------------------------------------------------------------------% output counts per acquisition to a text file %------------------------------------------------------------------------------------------------------------SliceNum=1:1:45; Data1=[SliceNum;C_r_s_j;C_TOT_j]; Data2=[TRandScat(acq);TCounts(acq)]; PAGE 73 59 fname=sprintf('Output\\DataCounts%d.mw',acq); fid=fopen(fname,'w'); fprintf(fid,'acq '); fprintf(fid,'%2.0f\n',acq); fprintf(fid,'Slice\t RandScat\t Total\n'); fprintf(fid,'%2.0f\t %10.2f\t %10.2f\n',Data1); fprintf(fid,'\n\n'); fprintf(fid,'Total '); fprintf(fid,'%10.2f %10.2f',Data2); fclose(fid); %-------------------------------------------------------------------------------------------------------------% In slight deviation from NEMA 2001, the sum projections of all sinograms in a % particular acquisition are summed to create a final sum projection that describes % the behavior of the PET scanner at that particular activity. It is this final sum % projection that is used to calculate the different count rates following NEMA 2001 % standards. %-------------------------------------------------------------------------------------------------------------FinalSumProj(:,:,acq)=finalsumprojection; % final sum projection of each acquisition %------------------------------------------------------------------------------------------------------------% output Final Sum Projections data to a text file %------------------------------------------------------------------------------------------------------------T_acq_j=20*60; Data3=[x_cm;FinalSumProj(:,:,acq)/T_acq_j]; fname=sprintf('Output\\FinalSumProjection%d.mw',acq); fid=fopen(fname,'w'); fprintf(fid,'acq '); fprintf(fid,'%2.0f\n',acq); fprintf(fid,'x (cm)\t Count Rate [cps] \n'); fprintf(fid,'%2.3f\t %10.2f \n',Data3); fclose(fid); clear %clears all but the global variables end %ends the nacq loop global cm_pxl x_cm CenterPixel global nacq FinalSumProj %----------------------------------------------------------------------------------------------------------------% Plot Final Sum Projections (cps) vs. radial distance (cm) %----------------------------------------------------------------------------------------------------------------T_acq_j=20*60; % N=1; fig=1; PAGE 74 60 % for j=1:1:nacq % fname=sprintf('Final Sum Projection: acq%d',j); % figure(fig),subplot(2,1,N),plot(x_cm,FinalSumProj(:,:,j)/T_acq_j),title(fname) % xlabel('Distance along transverse axis [cm]'),ylabel('Count Rate [cps]') % grid % N=N+1; % if(j==fig*2) % fig=fig+1; % N=1; % end % end %----------------------------------------------------------------------------------------------------------------% To place each profile in a separate figure %----------------------------------------------------------------------------------------------------------------fig=1; for j=1:1:nacq figure(fig),plot(x_cm,FinalSumProj(:,:,j)/T_acq_j) xlabel('Distance along transverse axis [cm]'),ylabel('Count Rate [cps]') grid fig=fig+1; end %************************************************************************************************** %* END OF CODE %************************************************************************************************** PAGE 75 APPENDIX G MATLAB CODE FOR CONTINUING PROCESSING AND ANALYSIS %************************************************************************************************** %* NEMA NU 2-2001 DATA PROCESSING CODE (Continue ...) %************************************************************************************************** global FinalSumProj mm_pxl global ScatF TRandScat TCounts T_acq_j=20*60; %acquisition time in seconds C_r_s_j=TRandScat; C_TOT_j=TCounts; %-----------------------------------------------------------------------------------------------------------------% 4.4.1 SCATTER FRACTION (NEMA 2001) %------------------------------------------------------------------------------------------------------------------% The final acquisition j' of the sequence with count loss rates and random rates below % 1% of the trues rate is used to determine the scatter fraction. For this acquisition, it is % assumed that C_r+s,j' has a negligible number of random counts and consists only of % scatter counts, and likewise, C_TOT,j' consists only of true and scatter counts. The % scatter fraction of the final acquisition is used as the system's SF for it is assumed % that the randoms rate is again negligible in that acquisition and therefore the C_r+s,j' % consists only of scatter counts. %-----------------------------------------------------------------------------------------------------------------%-----------------------------------------------------------------------------------------------------------------% 4.5.3 SYSTEM SCATTER FRACTION (NEMA 2001) %-----------------------------------------------------------------------------------------------------------------SF =0.403; %-----------------------------------------------------------------------------------------------------------------% 4.4.2 TOTAL EVENT RATE MEASUREMENT (NEMA 2001) %-----------------------------------------------------------------------------------------------------------------R_TOT_j = C_TOT_j ./ T_acq_j; %-----------------------------------------------------------------------------------------------------------------% 4.4.3 TRUE EVENT RATE MEASUREMENT (NEMA 2001) %-----------------------------------------------------------------------------------------------------------------R_t_j = (C_TOT_j C_r_s_j) ./ T_acq_j; %-----------------------------------------------------------------------------------------------------------------% 4.4.4 RANDOM EVENT RATE MEASUREMENT (NEMA 2001) %-----------------------------------------------------------------------------------------------------------------R_r_j = R_TOT_j (R_t_j ./ (1-SF)); 61 PAGE 76 62 %-----------------------------------------------------------------------------------------------------------------% 4.4.5 SCATTER EVENT RATE MEASUREMENT (NEMA 2001) %-----------------------------------------------------------------------------------------------------------------R_s_j = R_t_j .* (SF/(1-SF)); %-----------------------------------------------------------------------------------------------------------------% 4.4.6 NOISE EQUIVALENT COUNT RATE MEASUREMENT (NEMA 2001) %-----------------------------------------------------------------------------------------------------------------R_NEC_j = (R_t_j.^2) ./ R_TOT_j; %================================================================= % 4.5 REPORT %================================================================= %-----------------------------------------------------------------------------------------------------------------% 4.5.1 COUNT RATE PLOT (NEMA 2001) %-----------------------------------------------------------------------------------------------------------------%calculated initial activity (mCi) at the beginning of each acquisition is Act=[12.536 9.738 7.564 5.876 4.564 3.545 2.754 2.139 1.662 1.291 1.003 0.779 0.605 0.470 0.365 0.284 0.220 0.171 0.133 0.103 0.080]; % mCi; %time into experiment (minutes): time=[0 40 80 120 160 200 240 280 320 360 400 440 480 520 560 600 640 680 720 760 800]; figure(1), plot(time,R_TOT_j/1000,'m*-',time,R_t_j/1000,'rs-',time,R_r_j/1000,'bd-',time,R_s_j/1000,'gx-') xlabel('Time [min]'),ylabel('Count Rate [kcps]') legend('Total','True','Random','Scatter') figure(2) plot(Act,R_TOT_j/1000,'m*-',Act,R_t_j/1000,'rs-',Act,R_r_j/1000,'bd-',Act,R_s_j/1000,'gx-') xlabel('Activity [mCi]'),ylabel('Count Rate [kcps]') legend('Total','True','Random','Scatter') figure(3) plot(time,R_NEC_j/1000,'k.-') xlabel('Time [min]'),ylabel('NEC Count Rate [kcps]') figure(4) plot(Act,R_NEC_j/1000,'k.-') xlabel('Activity [mCi]'),ylabel('NEC Count Rate [kcps]') %----------------------------------------------------------------------------------------------------------------% output count rates to a text file %----------------------------------------------------------------------------------------------------------------Data=1:1:nacq; PAGE 77 63 countdata = [Data; C_TOT_j; R_TOT_j; R_t_j; R_r_j; R_s_j; R_NEC_j; Act]; fid=fopen('Output\\CoincidenceCounts.txt','w') fprintf(fid,'Acquisition Total Paired Count Rate Count Rate Count Rate Count Rate Count Rate Average\n') fprintf(fid,' Events Total True Random Scatter NEC Activity\n') fprintf(fid,' (counts/20min) (cps) (cps) (cps) (cps) (cps) (mCi) \n') fprintf(fid,'%2.0f %12.0f %12.2f %12.2f %12.2f %12.2f %10.2f %1.4f\n',countdata) fclose(fid); %************************************************************************************************** %* END OF CODE %************************************************************************************************** PAGE 78 APPENDIX H MATLAB CODE FOR CURVE FITTING %*************************************************************************************************** %* CURVE FIT CODE %*************************************************************************************************** %------------------------------------------------------------------------------------------------------------------% Fitting the Final Sum Projections to Gaussian Functions using the Non-Linear Least % Squares Method %------------------------------------------------------------------------------------------------------------------global FinalSumProj global nacq cm_pxl CenterPixel T_acq_j=20*60; ncm=CenterPixel*cm_pxl; % number of cm from pixel 1 to central pixel (128) x_cm=-(ncm-cm_pxl):cm_pxl:(ncm); % new scale in cm for j=1:1:nacq Total(:,:,j)=FinalSumProj(:,:,j)/T_acq_j; end model = fittype('a*exp(-0.5*(x-b)^2/c^2)+d'); % the gaussian equation used to fit the data opts = fitoptions('Method','NonlinearLeastSquares'); opts.Lower = [-Inf -Inf 0 -Inf]; for j=1:1:nacq a_strt=max(Total(:,:,j)); % Normalizes the gaussian function b_strt=1; % the mean c_strt=1; % the standard deviation d_strt=1; % the offset opts.StartPoint=[a_strt b_strt c_strt d_strt]; % starting points of statistical constants [f,gof] = fit(x_cm',Total(:,:,j)',model,opts); a(j)=f.a; b(j)=f.b; c(j)=f.c; d(j)=f.d; R_squared(j)=gof.rsquare; G{j}=f; % Gaussian fitted equation j 64 PAGE 79 65 end %------------------------------------------------------------------------------------------------------------------% Calculating the FWHM of the fitted curves. FWHM = 2.35 sigma %------------------------------------------------------------------------------------------------------------------for j=1:1:nacq FWHM(j)=c(j)*2.35*10; % FWHM (mm) FWHMa(j)=sqrt(FWHM(j)^2-3.2^2); %adjusted FWHM (mm) end %------------------------------------------------------------------------------------------------------------------% Output the FWHM values to a text file %------------------------------------------------------------------------------------------------------------------acqs=1:1:nacq; values=[acqs; FWHM; FWHMa]; fid=fopen('Output\\FWHM.txt','w'); fprintf(fid,'acq\t FWHM(mm)\t FWHMa(mm)\t \n'); fprintf(fid,'%2.0f\t %3.2f\t %3.2f\n',values); fclose(fid); %------------------------------------------------------------------------------------------------------------------% Output the coefficients values, for each fitted gaussian function, to a text file %------------------------------------------------------------------------------------------------------------------coeff=[a; b; c; d; R_squared]; fid=fopen('Output\\FittedFunctionsCoeff.txt','w'); fprintf(fid,'\t a\t\t b\t\t c\t\t\t d\t R_squared \n'); fprintf(fid,'%7.2f %1.4f %1.5f %7.2f %1.3f \n',coeff); fclose(fid); %------------------------------------------------------------------------------------------------------------------% Output fitted curves to a text file %------------------------------------------------------------------------------------------------------------------for j=1:1:nacq f=G{j}; %fitted gaussian equation for i=1:1:length(x_cm) fg(i)=f(x_cm(i)); end g=[x_cm;fg]; fname=sprintf('Output\\FittedGaussian%d.mw',j); fid=fopen(fname,'w'); fprintf(fid,'acq '); fprintf(fid,'%2.0f\n',j); fprintf(fid,'x (cm)\t Count Rate [cps] \n'); fprintf(fid,'%2.3f\t %10.2f\n',g); fclose(fid); end %------------------------------------------------------------------------------------------------------------------% Plot the FWHM as a function of activity and time into experiment %------------------------------------------------------------------------------------------------------------------figure(30),plot(Act,FWHMa) xlabel('Activity [mCi]'), ylabel('FWHM [mm]') PAGE 80 66 figure(31),plot(time,FWHMa) xlabel('Time [min]'), ylabel('FWHM [mm]') %------------------------------------------------------------------------------------------------------------------% Plotting the fitted functions along with the Final Sum Projections %------------------------------------------------------------------------------------------------------------------% N=1; % fig=1; % for j=1:1:nacq % fname=sprintf('Final Sum Projection: Acq%d',j); % figure(fig),subplot(2,1,N),plot(G{j},x_cm,Total(:,:,j)),title(fname) % xlabel('Distance along transverse axis [cm]'),ylabel('Counts') % legend('Final Sum Projection','Gaussian'),grid % N=N+1; % if(j==fig*2) % fig=fig+1; % N=1; % end % end fig=1; for j=1:1:nacq figure(fig),plot(G{j},x_cm,Total(:,:,j)) xlabel('Distance along transverse axis [cm]'),ylabel('Count Rate [cps]') legend('Final Sum Projection','Fitted Gaussian'),grid fig=fig+1; end %------------------------------------------------------------------------------------------------------------------% Fitting the Randoms rate curve to power equation using the Non-Linear Least Squares % Method %------------------------------------------------------------------------------------------------------------------model = fittype('power1'); opts = fitoptions('Method','NonlinearLeastSquares'); opts.Lower = [-Inf -Inf]; opts.Upper = [ Inf 2]; opts.StartPoint=[1 2]; %R_r_j=R_r_j/1000; % kcps [Rand,gof,out] = fit(Act',R_r_j',model,opts); %------------------------------------------------------------------------------------------------------------------% Output fitted randoms curve to a text file %------------------------------------------------------------------------------------------------------------------for i=1:1:length(Act) R(i)=Rand(Act(i)); end r=[Act;R]; fid=fopen('Output\\FittedRandoms.mw','w'); PAGE 81 67 fprintf(fid,'Act (mCi)\t Count Rate [kcps] \n'); fprintf(fid,'%2.3f\t %10.2f\n',r); fclose(fid); %------------------------------------------------------------------------------------------------------------------% Plotting the fitted function along with the Randoms count rate %------------------------------------------------------------------------------------------------------------------figure(fig),plot(Rand,Act,R_r_j) %title('Random count rate and fitted equation vs. activity') xlabel('Activity [mCi]'), ylabel('Count Rate [cps]') legend('Randoms rate','Fitted equation'),grid %*************************************************************************************************** %* END OF CODE %*************************************************************************************************** PAGE 82 APPENDIX I MATLAB CODE FOR PRODUCING MTF CURVES %*************************************************************************************************** %* Modulation Transfer Function (MTF) CODE %*************************************************************************************************** %*************************************************************************************************** %* The Fourier Transform of the Final Sum Projection is computed to produce the %* Modulation Transfer Function (MTF) for each acquisition. %*************************************************************************************************** %------------------------------------------------------------------------------------------------------------------% Fourier Transform of Final Sum Projections %------------------------------------------------------------------------------------------------------------------global nray MTF =zeros([1 nray nacq]); %Modulation Transfer Function MTFn=zeros([1 nray nacq]); %Normalized Modulation Transfer Function MTFa=zeros([1 nray/2 nacq]); %Adjusted Modulation Transfer Function for j=1:1:nacq fsp=FinalSumProj(:,:,j); %fsp=final sum projection FSP=fftshift(fft(fsp)); MTF(:,:,j)=abs(FSP); %magnitude of FT{fsp} end %------------------------------------------------------------------------------------------------------------------% Normalized Modulation Transfer Functions (MTFn) %------------------------------------------------------------------------------------------------------------------for j=1:1:nacq MAX=max(MTF(:,:,j)); MTFn(:,:,j)=MTF(:,:,j)/MAX; end %------------------------------------------------------------------------------------------------------------------% Adjusted Modulation Transfer Functions (MTFa) %------------------------------------------------------------------------------------------------------------------for j=1:1:nacq MTFa(:,:,j)=MTFn(:,(nray/2 + 1):nray,j); end %------------------------------------------------------------------------------------------------------------------% Plotting the adjusted Modulation Transfer Functions (MTFa) %------------------------------------------------------------------------------------------------------------------N=nray %# of pixels, so 1mm/pxl dx=cm_pxl % sampling distance: i.e., # of cm per pixel dnu=1/(N*dx) %sampling distance in frequency domain 68 PAGE 83 69 nu_min=-(N*dnu/2) nu_max=(N*dnu/2)-dnu nu=nu_min:dnu:nu_max; % frequency domain (1/cm) nu1=nu((nray/2 + 1):nray); % frequency domain used for the plots % fig=1; % subfig=1; % for j=1:1:nacq % figure(fig),subplot(2,1,subfig),plot(nu1,MTFa(:,:,j)) % title(['Modulation Transfer Function: Acq int2str(j)]) % xlabel('\nu [cm^-^1]'),ylabel('Gain'),grid % subfig=subfig+1; % if(j==fig*2) % fig=fig+1; % subfig=1; % end % end fig=1; for j=1:1:nacq figure(fig),plot(nu1,MTFa(:,:,j)) xlabel('\nu [cm^-^1]'),ylabel('Gain') h = gca; set(h,'YGrid','on'); fig=fig+1; end %------------------------------------------------------------------------------------------------------------------% Plotting the adjusted Modulation Transfer Functions (MTFa) of selected acquisitions % on ONE plot %------------------------------------------------------------------------------------------------------------------figure(20) plot(nu1,MTFa(:,:,1),nu1,MTFa(:,:,5),nu1,MTFa(:,:,10),nu1,MTFa(:,:,21)) xlabel('\nu [cm^-^1]'),ylabel('Gain') h = gca; set(h,'YGrid','on'); %title('Modulation Transfer Functions of selected acquisitions') legend('acquisition 1','acquisition 5','acquisition 10','acquisition 21') %------------------------------------------------------------------------------------------------------------------% Output MTF curves to text files %------------------------------------------------------------------------------------------------------------------for acq=1:1:nacq mtf=[nu1;MTFa(:,:,acq)]; fname=sprintf('Output\\MTF%d.mw',acq); fid=fopen(fname,'w'); fprintf(fid,'acq '); fprintf(fid,'%2.0f\n',acq); fprintf(fid,'frequency (1/cm)\t Gain \n'); PAGE 84 70 fprintf(fid,'%2.3f\t\t %10.5f \n',mtf); fclose(fid); end %------------------------------------------------------------------------------------------------------------------% Plotting ALL of the adjusted Modulation Transfer Functions (MTFa) on ONE plot %------------------------------------------------------------------------------------------------------------------% for j=1:1:nacq % ModTranFuna(j,:)=MTFa(:,:,j); % fname=sprintf('s%d',j); % ACQMTFa{j}=fname; % end % figure(21) % plot(nu1,ModTranFuna),legend(ACQMTFa) %------------------------------------------------------------------------------------------------------------------% Plotting ALL of the Modulation Transfer Functions (MTF) on ONE plot %------------------------------------------------------------------------------------------------------------------% for j=1:1:nacq % ModTranFunNorm(j,:)=MTFn(:,:,j); % flname=sprintf('s%d',j); % ACQMTFn{j}=flname; % end % figure(22) % plot(nu,ModTranFunNorm),legend(ACQMTFn) % xlabel('\nu [cm^-^1]'),ylabel('Relative intensity') % title('Norm MTFs') %************************************************************************************************** %* END OF CODE %************************************************************************************************** PAGE 85 APPENDIX J FINAL SUM PROJECTIONS -20 -15 -10 -5 0 5 10 15 20 0 0.5 1 1.5 2 2.5x 104 Distance along transverse axis [cm]Count Rate [cps] FIG. 8. Final sum projection of acquisition 1 71 PAGE 86 72 -20 -15 -10 -5 0 5 10 15 20 0 0.5 1 1.5 2 2.5 3x 104 Distance along transverse axis [cm]Count Rate [cps] FIG. 9. Final sum projection of acquisition 2 -20 -15 -10 -5 0 5 10 15 20 0 0.5 1 1.5 2 2.5x 104 Distance along transverse axis [cm]Count Rate [cps] FIG. 10. Final sum projection of acquisition 3 PAGE 87 73 -20 -15 -10 -5 0 5 10 15 20 0 0.5 1 1.5 2 2.5x 104 Distance along transverse axis [cm]Count Rate [cps] FIG. 11. Final sum projection of acquisition 4 -20 -15 -10 -5 0 5 10 15 20 0 0.5 1 1.5 2x 104 Distance along transverse axis [cm]Count Rate [cps] FIG. 12. Final sum projection of acquisition 5 PAGE 88 74 -20 -15 -10 -5 0 5 10 15 20 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 Distance along transverse axis [cm]Count Rate [cps] FIG. 13. Final sum projection of acquisition 6 -20 -15 -10 -5 0 5 10 15 20 0 2000 4000 6000 8000 10000 12000 14000 Distance along transverse axis [cm]Count Rate [cps] FIG. 14. Final sum projection of acquisition 7 PAGE 89 75 -20 -15 -10 -5 0 5 10 15 20 0 2000 4000 6000 8000 10000 12000 Distance along transverse axis [cm]Count Rate [cps] FIG. 15. Final sum projection of acquisition 8 -20 -15 -10 -5 0 5 10 15 20 0 2000 4000 6000 8000 10000 Distance along transverse axis [cm]Count Rate [cps] FIG.16. Final sum projection of acquisition 9 PAGE 90 76 -20 -15 -10 -5 0 5 10 15 20 0 1000 2000 3000 4000 5000 6000 7000 8000 Distance along transverse axis [cm]Count Rate [cps] FIG. 17. Final sum projection of acquisition 10 -20 -15 -10 -5 0 5 10 15 20 0 1000 2000 3000 4000 5000 6000 Distance along transverse axis [cm]Count Rate [cps] FIG. 18. Final sum projection of acquisition 11 PAGE 91 77 -20 -15 -10 -5 0 5 10 15 20 0 1000 2000 3000 4000 5000 Distance along transverse axis [cm]Count Rate [cps] FIG. 19. Final sum projection of acquisition 12 -20 -15 -10 -5 0 5 10 15 20 0 500 1000 1500 2000 2500 3000 3500 4000 Distance along transverse axis [cm]Count Rate [cps] FIG. 20. Final sum projection of acquisition 13 PAGE 92 78 -20 -15 -10 -5 0 5 10 15 20 0 500 1000 1500 2000 2500 3000 Distance along transverse axis [cm]Count Rate [cps] FIG. 21. Final sum projection of acquisition 14 -20 -15 -10 -5 0 5 10 15 20 0 500 1000 1500 2000 2500 Distance along transverse axis [cm]Count Rate [cps] FIG. 22. Final sum projection of acquisition 15 PAGE 93 79 -20 -15 -10 -5 0 5 10 15 20 0 200 400 600 800 1000 1200 1400 1600 1800 Distance along transverse axis [cm]Count Rate [cps] FIG. 23. Final sum projection of acquisition 16 -20 -15 -10 -5 0 5 10 15 20 0 200 400 600 800 1000 1200 1400 Distance along transverse axis [cm]Count Rate [cps] FIG. 24. Final sum projection of acquisition 17 PAGE 94 80 -20 -15 -10 -5 0 5 10 15 20 0 200 400 600 800 1000 1200 Distance along transverse axis [cm]Count Rate [cps] FIG. 25. Final sum projection of acquisition 18 -20 -15 -10 -5 0 5 10 15 20 0 100 200 300 400 500 600 700 800 900 Distance along transverse axis [cm]Count Rate [cps] FIG. 26. Final sum projection of acquisition 19 PAGE 95 81 -20 -15 -10 -5 0 5 10 15 20 0 100 200 300 400 500 600 700 Distance along transverse axis [cm]Count Rate [cps] FIG. 27. Final sum projection of acquisition 20 -20 -15 -10 -5 0 5 10 15 20 0 100 200 300 400 500 600 Distance along transverse axis [cm]Count Rate [cps] FIG. 28. Final sum projection of acquisition 21 PAGE 96 APPENDIX K FITTED FINAL SUM PROJECTIONS -20 -15 -10 -5 0 5 10 15 20 0 0.5 1 1.5 2 2.5x 104 Distance along transverse axis [cm]Count Rate [cps] Final Sum ProjectionFitted Gaussian FIG. 36. Final sum projection of acquisition 1 and its fitted gaussian curve -20 -15 -10 -5 0 5 10 15 20 0 0.5 1 1.5 2 2.5 3x 104 Distance along transverse axis [cm]Count Rate [cps] Final Sum ProjectionFitted Gaussian FIG. 37. Final sum projection of acquisition 2 and its fitted gaussian curve 82 PAGE 97 83 -20 -15 -10 -5 0 5 10 15 20 0 0.5 1 1.5 2 2.5x 104 Distance along transverse axis [cm]Count Rate [cps] Final Sum ProjectionFitted Gaussian FIG. 38. Final sum projection of acquisition 3 and its fitted gaussian curve -20 -15 -10 -5 0 5 10 15 20 0 0.5 1 1.5 2 2.5x 104 Distance along transverse axis [cm]Count Rate [cps] Final Sum ProjectionFitted Gaussian FIG. 39. Final sum projection of acquisition 4 and its fitted gaussian curve PAGE 98 84 -20 -15 -10 -5 0 5 10 15 20 0 0.5 1 1.5 2x 104 Distance along transverse axis [cm]Count Rate [cps] Final Sum ProjectionFitted Gaussian FIG. 40. Final sum projection of acquisition 5 and its fitted gaussian curve -20 -15 -10 -5 0 5 10 15 20 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 Distance along transverse axis [cm]Count Rate [cps] Final Sum ProjectionFitted Gaussian FIG. 41. Final sum projection of acquisition 6 and its fitted gaussian curve PAGE 99 85 -20 -15 -10 -5 0 5 10 15 20 0 2000 4000 6000 8000 10000 12000 14000 Distance along transverse axis [cm]Count Rate [cps] Final Sum ProjectionFitted Gaussian FIG. 42. Final sum projection of acquisition 7 and its fitted gaussian curve -20 -15 -10 -5 0 5 10 15 20 0 2000 4000 6000 8000 10000 12000 Distance along transverse axis [cm]Count Rate [cps] Final Sum ProjectionFitted Gaussian FIG. 43. Final sum projection of acquisition 8 and its fitted gaussian curve PAGE 100 86 -20 -15 -10 -5 0 5 10 15 20 0 2000 4000 6000 8000 10000 Distance along transverse axis [cm]Count Rate [cps] Final Sum ProjectionFitted Gaussian FIG. 44. Final sum projection of acquisition 9 and its fitted gaussian curve -20 -15 -10 -5 0 5 10 15 20 0 1000 2000 3000 4000 5000 6000 7000 8000 Distance along transverse axis [cm]Count Rate [cps] Final Sum ProjectionFitted Gaussian FIG. 45. Final sum projection of acquisition 10 and its fitted gaussian curve PAGE 101 87 -20 -15 -10 -5 0 5 10 15 20 0 1000 2000 3000 4000 5000 6000 Distance along transverse axis [cm]Count Rate [cps] Final Sum ProjectionFitted Gaussian FIG. 46. Final sum projection of acquisition 11 and its fitted gaussian curve -20 -15 -10 -5 0 5 10 15 20 0 1000 2000 3000 4000 5000 Distance along transverse axis [cm]Count Rate [cps] Final Sum ProjectionFitted Gaussian FIG. 47. Final sum projection of acquisition 12 and its fitted gaussian curve PAGE 102 88 -20 -15 -10 -5 0 5 10 15 20 0 500 1000 1500 2000 2500 3000 3500 4000 Distance along transverse axis [cm]Count Rate [cps] Final Sum ProjectionFitted Gaussian FIG. 48. Final sum projection of acquisition 13 and its fitted gaussian curve -20 -15 -10 -5 0 5 10 15 20 0 500 1000 1500 2000 2500 3000 Distance along transverse axis [cm]Count Rate [cps] Final Sum ProjectionFitted Gaussian FIG. 49. Final sum projection of acquisition 14 and its fitted gaussian curve PAGE 103 89 -15 -10 -5 0 5 10 15 0 500 1000 1500 2000 2500 Distance along transverse axis [cm]Count Rate [cps] Final Sum ProjectionFitted Gaussian FIG. 50. Final sum projection of acquisition 15 and its fitted gaussian curve -15 -10 -5 0 5 10 15 0 200 400 600 800 1000 1200 1400 1600 1800 Distance along transverse axis [cm]Count Rate [cps] Final Sum ProjectionFitted Gaussian FIG. 51. Final sum projection of acquisition 16 and its fitted gaussian curve PAGE 104 90 -15 -10 -5 0 5 10 15 0 200 400 600 800 1000 1200 1400 Distance along transverse axis [cm]Count Rate [cps] Final Sum ProjectionFitted Gaussian FIG. 52. Final sum projection of acquisition 17 and its fitted gaussian curve -15 -10 -5 0 5 10 15 0 200 400 600 800 1000 1200 Distance along transverse axis [cm]Count Rate [cps] Final Sum ProjectionFitted Gaussian FIG. 53. Final sum projection of acquisition 18 and its fitted gaussian curve PAGE 105 91 -15 -10 -5 0 5 10 15 0 100 200 300 400 500 600 700 800 900 Distance along transverse axis [cm]Count Rate [cps] Final Sum ProjectionFitted Gaussian FIG. 54. Final sum projection of acquisition 19 and its fitted gaussian curve -15 -10 -5 0 5 10 15 0 100 200 300 400 500 600 700 Distance along transverse axis [cm]Count Rate [cps] Final Sum ProjectionFitted Gaussian FIG. 55. Final sum projection of acquisition 20 and its fitted gaussian curve PAGE 106 92 -15 -10 -5 0 5 10 15 0 100 200 300 400 500 600 Distance along transverse axis [cm]Count Rate [cps] Final Sum ProjectionFitted Gaussian FIG. 56. Final sum projection of acquisition 21 and its fitted gaussian curve PAGE 107 APPENDIX L MODULATION TRANSFER FUNCTIONS 0 0.5 1 1.5 2 2.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 [cm-1]Gain FIG. 60. Modulation transfer function of acquisition 1 93 PAGE 108 94 0 0.5 1 1.5 2 2.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 [cm-1]Gain FIG. 61. Modulation transfer function of acquisition 2 0 0.5 1 1.5 2 2.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 [cm-1]Gain FIG. 62. Modulation transfer function of acquisition 3 PAGE 109 95 0 0.5 1 1.5 2 2.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 [cm-1]Gain FIG. 63. Modulation transfer function of acquisition 4 0 0.5 1 1.5 2 2.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 [cm-1]Gain FIG. 64. Modulation transfer function of acquisition 5 PAGE 110 96 0 0.5 1 1.5 2 2.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 [cm-1]Gain FIG. 65. Modulation transfer function of acquisition 6 0 0.5 1 1.5 2 2.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 [cm-1]Gain FIG. 66. Modulation transfer function of acquisition 7 PAGE 111 97 0 0.5 1 1.5 2 2.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 [cm-1]Gain FIG. 67. Modulation transfer function of acquisition 8 0 0.5 1 1.5 2 2.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 [cm-1]Gain FIG. 68. Modulation transfer function of acquisition 9 PAGE 112 98 0 0.5 1 1.5 2 2.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 [cm-1]Gain FIG. 69. Modulation transfer function of acquisition 10 0 0.5 1 1.5 2 2.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 [cm-1]Gain FIG. 70. Modulation transfer function of acquisition 11 PAGE 113 99 0 0.5 1 1.5 2 2.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 [cm-1]Gain FIG. 71. Modulation transfer function of acquisition 12 0 0.5 1 1.5 2 2.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 [cm-1]Gain FIG. 72. Modulation transfer function of acquisition 13 PAGE 114 100 0 0.5 1 1.5 2 2.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 [cm-1]Gain FIG. 73. Modulation transfer function of acquisition 14 0 0.5 1 1.5 2 2.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 [cm-1]Gain FIG. 74. Modulation transfer function of acquisition 15 PAGE 115 101 0 0.5 1 1.5 2 2.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 [cm-1]Gain FIG. 75. Modulation transfer function of acquisition 16 0 0.5 1 1.5 2 2.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 [cm-1]Gain FIG. 76. Modulation transfer function of acquisition 17 PAGE 116 102 0 0.5 1 1.5 2 2.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 [cm-1]Gain FIG. 77. Modulation transfer function of acquisition 18 0 0.5 1 1.5 2 2.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 [cm-1]Gain FIG. 78. Modulation transfer function of acquisition 19 PAGE 117 103 0 0.5 1 1.5 2 2.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 [cm-1]Gain FIG. 79. Modulation transfer function of acquisition 20 0 0.5 1 1.5 2 2.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 [cm-1]Gain FIG. 80. Modulation transfer function of acquisition 21 PAGE 118 REFERENCES 1 Schulthess Gustav K. Clinical molecular anatomic imaging; PET, PET/CT, and SPECT/CT. Lippincott Williams & Wilkins, PA (2003). 2 Rigo P., Paulus P., Kaschten BJ, Hustinx R., Bury T., Jerusalem G., Benoit T., Foidart-Willems J., Oncological applications of positron emission tomography with fluorine-18 fluorodeoxyglucose, Eur J Nucl Med 23, 1641-1674 (1996). 3 Fukuchi K., Hayashida K., Moriwaki H., Fukushima K., Kume N., Katafuchi T., Sago, M., Takamiya M., Ishida Y., Brain fluorine-18 fluorodeoxyglucose imaging with dual-head coincidence gamma camera: comparison with dedicated ring-detector positron emission tomography, Am J Neuroradiol 21, 99-104 (2000). 4 Bushberg J. T., Seibert J. A., Leidholdt E. M., Boone J. M., The essential physics of medical imaging, Lippincott Williams & Wilkins, PA (2002) 5 Fahey F. H., Data acquisition in PET imaging, J Nucl Med Technol 30, 39-49 (2002). 6 Lartizien C., Comtat C., Kinahan P. E., Ferreira N., Bendriem B., Trebossen R., Optimization of injected dose based on noise equivalent count rates for 2and 3-dimensional whole-body PET,J Nucl Med 43, 1268-1278 (2002). 7 Daube-Witherspoon M. E., Karp J. S., Casey M. E., DiFilippo F. P., Hines H., Muehllehner G., Simcic V., Stearns C. W., Adam L., Kohlmyer S., Sossi V., PET performance measurements using the NEMA NU 2-2001 standard, J Nucl Med 43, 1398-1409 (2002). 8 Adam L., Karp J. S., Daube-Witherspoon M. E., Smith R. J., Performance of a whole-body PET scanner using curve-plate NaI(Tl) detectors, J Nucl Med 42, 1821-1830 (2001). 9 Tarantola G., Zito F., Gerundini P., PET instrumentation and reconstruction algorithms in whole-body applications, J Nucl Med 44, 756-769 (2003). 10 Strother S.C., Casey M.E., Hoffman E.J., Measuring PET scanner sensitivity: relating countrates to image signal-to-noise ratios using noise equivalent counts, IEEE Trans Nucl Sci 37, 783-788 (1990). 104 PAGE 119 105 11 Vandenberghe S., DAsseler Y. D., Koole M., Walle R. Van de, Lemahieu I., Dierckx R. A., Physical evaluation of 511 keV coincidence imaging with a gamma camera, IEEE Trans Nucl Sci 48, 98-105 (2001). 12 Kunze W., Baehre M., Richter E., PET with a dual-head coincidence camera: spatial resolution, scatter fraction, and sensitivity, J Nucl Med 41, 1067-1074 (2000). 13 Mankoff G., Muehllehner G., The high count rate performance of a two-dimensionally position sensitive detector for positron emission tomography, Phys Med Biol 34, 437-456 (1989). 14 Cherry S. R., Meikle S. R., Hoffman E. J., Correction and characterization of scattered events in three-dimensional PET using scanners with retractable septa, J Nucl Med 34, 671-678 (1993). 15 Bailey D. L., Meikle S. R., A convolution-subtraction scatter correction method for 3D PET, Phys Med Biol 39, 411-424 (1994). 16 National Electrical Manufacturers Association. NEMA standards publication NU 2-2001: performance measurements of positron emission tomographs. National Electrical Manufacturers Association, VA (2001). 17 Philips Medical Systems. Allegro users manual: a manual describing how to use the Allegro imaging system. Philips, PA (2003). 18 Daube-Witherspoon M. E., Muehllehner G., Treatment of axial data in three-dimensional PET, J Nucl Med 28, 1717-1724 (1987). 19 Defrise M., Kinahan P. E., Townsend D. W., Michel C., Sibomana M., Newport D. F., Exact and approximate rebinning algorithms for 3-D PET data, IEEE Trans Med Imag 16, 145-158 (1997). PAGE 120 BIOGRAPHICAL SKETCH Rami R. Abu-Aita was born in Beit Jalla, Palestine, on November 22, 1979. Rami is the son of Rimon and Khitam Abu-Aita. Rami attended Bethlehem University, located in the city of Bethlehem, in 1997. After one year at Bethlehem University, he transferred to Francis Marion University in Florence, South Carolina. Rami graduated from Francis Marion University in December of 2001 with Bachelor of Science degrees in both computational physics and health physics. In January 2002, he enrolled in the medical physics program at the University of Florida and since then has been pursuing his Master of Science for which this thesis is a partial requirement. 106 |