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A simulation for the design of position encoding detectors for positron emission tomography Tsang, Glenn 1995

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A SIMULATION FOR T H E DESIGN OF POSITION ENCODING D E T E C T O R S FOR POSITRON EMISSION T O M O G R A P H Y G L E N N T S A N G B.Math, The University of Waterloo, 1992 A Thesis submitted in partial fulfillment of the requirements for the degree of Master of Science i n the Faculty of Graduate Studies (Department of Physics) We accept this thesis as conforming atandard THE UNIVERSITY OF BRITISH COLUMBIA April 1995 © Glenn Tsang, 1995. In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of PHYSICS The University of British Columbia Vancouver, Canada Date MARCH 31 , 1995 DE-6 (2/88) A B S T R A C T A platform developed to simulate the measured performance of position encoding multicrystal detectors for imaging applications is introduced. The platform is de-signed to treat the interactions of 7-rays in an inorganic scintillator, the geometry of the multicrystal array, as well as the propagation and detection of individual scintilla-tion photons. Energy and position spectra predicted by the simulation are compared to measured ones for the block detectors of the ECAT EXACT HR PLUS positron emission tomograph. The success of the simulation in reproducing measured results is used as a validation of the simulation. The simulation is then used in an investigation of a modification to the design of the EXACT HR PLUS that would improve significantly its Depth-Of-Interaction sensitivity. Modifications to the EXACT HR PLUS block are incorporated to the model. The simulated performances of the new design are compared to measured ones for an early prototype. An evaluation of the capacity of the new block to correct for the parallax error in Positron Emission Tomography is conducted. The outcome of this study has proved to be very encouraging and provides verification for the feasibility of the proposed correction scheme. 11 T A B L E OF C O N T E N T S ABSTRACT ii TABLE OF CONTENTS iii LIST OF TABLES v LIST OF FIGURES vii ACKNOWLEDGEMENTS x CHAPTER 1 INTRODUCTION 1 I. Physical Basis for PET 1 II. Principles of PET Imaging 9 III. Limitations to Image Resolution 14 CHAPTER 2 EXPERIMENTAL MEASUREMENTS 18 I. Block Candidate Specifications 18 II. Experimental Configuration 21 III. Raw Data Analysis 23 iii CHAPTER 3 THE MODEL 33 I. The Simulation Platform 34 II. Gamma Transport Interface 36 III. Introduction to DETECT 37 IV. Block Detector Specifications 44 V. Validation of the Simulation • 53 VI. Discussion of Systematics 61 CHAPTER 4 RADIAL ELONGATION 67 I. Radial Resolution and the Parallax Error 67 II. A Correction for DOI Blurring 71 III. Performance of New DOI Sensitive Blocks 74 CHAPTER 5 CONCLUDING REMARKS 86 BIBLIOGRAPHY 89 iv LIST OF TABLES 1.1 Physical properties of isotopes commonly used in medical applications of PET 3 1.2 Physical properties of scintillation materials used in PET 7 2.1 Dimensional specifications of the 953B and HR PLUS block detectors 20 2.2 Peak-to-valley ratios along four rows in one half of the block under study 26 2.3 Relative photopeak channels and relative FWHMs from the energy spectra for one quadrant of the HR PLUS block under study 29 2.4 The FWHMs and FWTMs for the 8 LSFs measured for the block under study 31 3.1 List of input parameters used to describe the block geometry and optical properties with DETECT 44 3.2 The measured and simulated relative photopeak channels and relative FWHMs from the energy spectra for one quadrant of the HR PLUS block under study 55 v 4.1 Spatial image resolutions for various distances from the centre of the FOV of a Siemens-CTI ECAT 953B PET scanner 70 4.2 Parameters taken from lines fitted to the Photon Count-Depth data for the 16 crystals in one quadrant of the modified block 81 vi LIST OF FIGURES 1.1 A circular ring PET tomograph system 2 1.2 Electronic band structure of inorganic crystals 6 1.3 The tube of response for two detectors in coincidence 9 1.4 The LOR coordinate system 10 1.5 Mappings of LORs to sinogram points 11 1.6 Radon transform of the tracer distribution 12 1.7 An illustration of the projection and backprojection processes 13 1.8 The three types of coincident events 16 2.1 A photograph of the Siemens-CTI ECAT 953B detector 19 2.2 Experimental hardware configuration used in the measurements of the energy and position resolutions of the block 22 2.3 The PMT configuration 23 2.4 Measured 2-D flood position calibration spectrum for the block detector under study, overlaid with its corresponding generated LUT boundaries 24 vii 2.5 Measured energy spectra for 4 crystals situated along the diagonal in one quadrant of the HR PLUS block detector under study 27 2.6 Exact locations of the four diagonal crystals of Figure 2.5 28 2.7 Profile of the fan beam produced with a gaussian fitted to the data 30 2.8 Measured LSFs for the HR PLUS block detector under study 32 3.1 Data flow diagram of the block simulation platform 35 3.2 The optical processes and their corresponding probabilities at a polished surface coated with an external diffuse reflector 41 3.3 The segmentation of the block detector 46 3.4 A study of the effects of surface finish on depth dependence 48 3.5 Dimensional outline and basing diagram for the Hamamatsu R5364 PMT 51 3.6 Typical photocathode spectral response and emission spectrum of scintillators 52 3.7 Simulated energy spectra for the 16 crystals in one quadrant of the block 54 3.8 Simulated 2-D flood position calibration spectrum for the 16 crystals in one quadrant of the block 57 3.9 Comparison between simulated and measured position map for a row of crystals at the centre of the block 58 viii 3.10 Simulated LSFs superimposed over measured LSFs for the block detector under study 60 3.11 A study of the effects of timeout 66 4.1 The effect of oblique penetration upon the spatial resolution 69 4.2 Obliquely penetrating photons in a pair of detectors in a ring tomograph gantry 71 4.3 Correction of multiple crystal penetrating 7s 73 4.4 Measured and simulated DOI responses of a crystal prior to and after surface modification 75 4.5 The simulated energy spectra for four crystals along the diagonal of a quadrant of a modified HR PLUS block detector 77 4.6 Measured depth sensitivities for the 16 crystals in one quadrant of the HR PLUS block prior to and after modification 79 4.7 Simulated depth sensitivities for the 16 crystals in one quadrant of the HR PLUS block prior to and after modification 80 4.8 Simulated transaxial coordinate distributions with several levels of DOI corrections 82 4.9 Simulated LSFs for a central column in the modified and unmodified block detectors 84 ix A C K N O W L E D G E M E N T S I would first like to thank my supervisor, Prof. Richard R. Johnson, for giving me the opportunity to work with a wonderful group and on a most worthwhile research project at TRIUMF. I would like to thank Dr. Joel G. Rogers who is a great resource of knowledge and whose feedback was always positive and never discouraging. Many thanks also to Dr. Christian Moisan, whom along with Joel Rogers, provided me with the measured data in Chapter 2. Without his help this thesis would never have come into full fruition. Under his guidance, this research was carried out very efficiently. He has taught me a great deal and was of immense help in my future academic pursuits. I am deeply indebted to him. Thanks also go out to Kenneth R. Buckley for his knowledgeable contributions to our weekly discussions. I want to thank Dr. T.A. DeVol of Clemson University for providing us with the source code and user manual for DETECT. This study would also not have been pos-sible without the contribution of Prof. S.R. Cherry of UCLA who lent us the EXACT HR PLUS prototype blocks he received from CTI. The collaboration of R. Nutt, M. An-dreaco and C. W. Williams of Siemens-CTI, where the detectors were manufactured was also much appreciated. This work would also not have been possible without the financial support provided by the Natural Sciences and Engineering Research Council via the 'Post-Graduate Scholarship A ', which was fully appreciated. Finally, I would like to give special thanks to Amy, Michelle, Dolly, the Bite Me crew (a.k.a. the Dweebs), the Nuge, Horatio, the Shikan Expressers, lady J, Greco, ollopA, the La clan and last but not least, my family. Thanks for believing in me. x i C H A P T E R 1 INTRODUCTION This chapter gives a brief overview of the basic concepts of Positron Emission Tomography (PET). For a concise and complete description of PET, the reader is suggested to research the many publications on the matter, in particular, the material presented in [1] would be an excellent start. I. Physical Basis for P E T A growing number of modalities including a technique known as Positron Emis-sion Tomography offer the promise of aiding us gain knowledge of the complex chem-istry found in the human body. The uniqueness and advantage of PET exist in the technique's ability to produce quantitative images of biochemical processes in vivo-without invasive measures. At the heart of the technique, the compounds used in the chemical processes can be synthesized from short lived radioactive isotopes of some of the basic building blocks such as carbon, nitrogen, oxygen and fluorine. Hence, drugs and organic compounds already present in the human body can be labelled by these radio-isotopes, and the resulting radiopharmaceutical can be introduced into the subject by way of a simple intravenous injection without disturbing the existing metabolism. As the body metabolizes these tagged compounds, an encircling ring of radiation detectors monitors their spatial distribution as shown in Figure 1.1. A spatial and temporal representation of the chemical distribution results. 1 Figure 1.1: A circular ring PET tomograph system [2]. As its name implies, PET images the distribution of positron emitting isotopes. The robustness of the technique lies in the fact that the aforementioned basic biolog-ical building blocks have positron emitting isotopes. These radio-isotopes are widely used in PET and can be easily produced by bombarding stable isotopes with proton beams generated in small cyclotrons. The positron range in water, the energy and half-life of the most commonly used radio-isotopes in PET are given in Table 1.1. The radionuclides are used to label organic molecules without altering their structure and composition. For example, n C labelled CO is used to measure blood volume, 1 8 F labelled 2D-deoxyglucose is used to measure glucose metabolism, 1 5 0 labelled CO2 and H 2 0 are used to study blood flow, 1 3 N labelled amino acids are used to measure regional metabolism in the brain. The radionuclides of Table 1.1 have more protons than neutrons and can emit positrons by /3+ decay: p - ^ n + e + + i/. ( l . i ) 2 Radio-Isotope Half Life Maximum Positron (min) Positron Radial Range Energy, in Water Emax (MeV) FWHM (mm) Carbon 11 20.4 0.97 1.11 Nitrogen 13 9.96 1.19 1.42 Oxygen 15 2.07 1.7 1.49 Fluorine 18 109.7 0.64 1.02 Table 1.1: Physical properties of isotopes commonly used in medical applications of PET [3]. The positron annihilates with a free electron into two photons: e + + e" -)• 7 + 7. (1.2) The positron emitted by radioactive decay has finite energy and it can lose all or part of it before annihilation with an electron. If annihilation takes place when positron momentum is zero, then conservation of energy requires that the two photons from the e+e~ annihilation will have an energy equal to the electron mass: 511 keV. Conservation of momentum requires that these photons will be emitted in exactly opposite directions in the rest frame of the positron. Photons may interact with matter in a number of ways: 1. Thomson scattering 2. Rayleigh scattering 3. Compton scattering 4. photoelectric effect 5. pair production. Since Thomson and Rayleigh Scattering dominate only at low energies relative to the electron mass, their cross-sections are small and they may be neglected for the 3 purposes of PET. Pair production requires that the 7's energy, E 7 , be larger than 1.022 MeV [4] and so is a forbidden process at the energy scale of PET. Hence, for 511 keV 7s only the Compton and photoelectric interactions are at play. The Compton interaction cross-section dominates for low Z materials, whereas, for high Z materials, the photoelectric interaction cross-section dominates. In the photoelectric effect, the 7 is absorbed by an atomic electron with the subsequent ejection of the electron from the atom. The energy of the outgoing electron is then E = h i / - B . E . , (1.3) where B.E. is the binding energy of the electron. Since a free electron cannot absorb a photon and also conserve momentum, the photoelectric effect always occurs on bound electrons with the nucleus absorbing the recoil momentum. Also, since the photon is completely absorbed by this process, materials for which the photoelectric cross-section is high will be of special interest in PET either for detection or shielding purposes. Compton scattering involves the impartation of energy to a free electron from an impinging 7 and the associated redirection of the resultant 7 with reduced energy. In typical matter, of course, the electrons are bound; however, if the 7 energy is high with respect to the binding energy, this latter energy can be ignored and the electrons can be considered as essentially free. Within the subject under study, the 7s may undergo a number of Compton scatterings before they escape its volume to reach the detector ring. The same is true when the 7s enter the detector active volume. Minimization of the acceptance to such Compton scattered 7s is of key importance in achieving good signal-to-noise ratio in PET studies. State-of-the-art PET cameras have a detection ring that incorporates scintillation counters made of inorganic crystals. Such scintillation counters exploit the abilities of certain materials to emit a small flash of light, i.e. a scintillation, when struck by 4 a charged particle or radiation. When a 7 enters a crystal, two principal processes can occur [4]. It can ionize the crystal by exciting an electron from the valence band to the conduction band, creating a free electron and a free hole. Or it can create an exciton by exciting an electron to a band, the exciton band, located just below the conduction band as shown in Figure 1.2. In this state, the electron and hole remain bound together as a pair. However, the pair can move freely through the crystal. If the crystal now contains impurity atoms, called activators, electronic levels in the forbidden energy gap can be locally created. A migrating free hole or a hole from an exciton pair which encounters an impurity centre, can then positively ionize the impurity atom. If subsequently, an electron arrives, it can fall into the opening left by the hole and make a transition from an excited state to the ground state, emitting radiation if such a de-excitation mode is allowed. This radiation lies about the visible light band and is termed a scintillation. These scintillations can be converted into photoelectrons by a photomultiplier tube (PMT), an amplifying device coupled to the back of the scintillator. The quality of PMTs is rated by a measure of their Quantum Efficiency: the proportion of incoming photons that are converted to photoelectrons to give a measurable electrical pulse. The electrical pulses can then be analyzed and counted electronically to give information concerning the incident radiation. Some of the scintillators first used in PET include CsF, BaF 2 and Nal. These were quickly abandoned in favour for BGO which is currently used extensively because of its short attenuation length for 511 keV 7s which equates to a high detection efficiency for these 7-rays. New promising materials that combine good detection efficiency and a larger scintillation yield than BGO at 511 keV are also becoming available. These include GSO, YAP, YAG, LSO and LuAP. Table 1.2 lists the most important properties of some of these scintillators. The attenuation length is defined as the distance after which the light intensity is reduced by a factor e _ 1 . The photo-fraction 5 Impurity. Traps Band Valence Band Figure 1.2: Electronic band structure of inorganic crystals [4]. is here defined as the ratio of the photoelectric cross-section to the total cross-section for 511 keV 7-rays. The photo-fraction increases rapidly with Z and the density. Photon yield is defined as the average number of scintillation photons emitted per keV of ionizing radiation. The decay time of scintillation light is defined as the time required for the emission of approximately 63% (1 — e _ 1) of the light. For better image resolution and sensitivity of the PET scanner (camera), a scin-tillator offering high atomic number (Z), high density, high light yield, short decay time and high photo-fraction is preferred. In addition, the scintillation light emitted should be of a wavelength detectable using photomultiplier tubes or photodiodes. Fi-nally, the ideal scintillation material is preferably non-hygroscopic and of course, low in cost. The position encoding multicrystal detector that is modelled in the following chap-ter utilizes BGO as its scintillation material because it fulfills all of the above criteria to a satisfactory level. BGO became commercially available in the late 1970s. The crystals are typically grown by the Czochralski method [5] in which a crystal boule is 6 generic name Nal(Tl) BGO GSO YAP LSO composition NaLTl B i 4 G e 3 0 1 2 Gd 2Si0 5:Ce YA10 3:Ce Lu2Si05:Ce atomic number 53,11 83,32,8 71,15,8 39,13,8 71,14,8 density (gm/cm3) 3.7 7.1 6.7 5.55 7.4 attenuation length (cm - 1) at 511 keV 0.34 1.1 1.4 2.2 1.2 photo-fraction (•%) 18 42 26 4.4 33 relative light yield 1.00 0.22 0.20 0.52 0.75 photon yield (per keV) 37.3 8.2 7.5 19.4 28.0 decay time (ns) 230 300 60 30 40 refractive index [6, 7] 1.85 2.14 1.9 1.95 1.82 references [6, 7, 8] [6, 7, 8] [6,7, 9] [6, 7, 10] [Uj Table 1.2: Physical properties of scintillation materials used in PET. 7 pulled from a molten mixture of bismuth oxide and germanium oxide at a rate of a few millimetres per hour. The boule can then be cut and polished using conventional methods. BGO remains very expensive at a price of $20(U.S.)/cm3 [12]. BGO is an example of a pure inorganic scintillator that does not require the presence of a trace activator element to promote the scintillation process. Instead, the luminescence is associated with an optical transition of the B i 3 + ion [5] that is a major constituent of the crystal. There is a relatively large shift between the optical absorption and emission spectra, called the Stokes shift, of the B i 3 + states. Therefore, relatively little self-absorption of the scintillation light occurs, and the crystal remains transparent to its own emission over dimensions of many centimetres, that is, BGO has a relatively long mean free path for bulk absorption. The scintillation efficiency depends strongly on the purity of the crystal, and some of the variability in the light yield reported from BGO in the past can be attributed to using crystals with different residual levels of impurities. A major advantage of BGO is its high density and the large atomic number (83) of the bismuth component. These properties result in the largest probability per unit volume of any commonly available scintillation material for the photoelectric absorption of 7-rays. Its mechanical and chemical properties make it easy to handle and use, and detectors using BGO can be made more rugged than those employing the more fragile and hygroscopic sodium iodide. Unfortunately, the light yield from BGO is relatively low, being variously reported at 10 — 25% of that of Nal(Tl). It is therefore of primary interest when the need for high 7-ray counting efficiency outweighs considerations of energy resolution as is in the case of PET. 8 II. Principles of P E T Imaging A modern 2-D PET tomograph consists of several rings of radiation detectors that encircle the subject. Within a ring, each detector can be in coincidence with a number of detectors on the opposing side of the ring. A line joining two detectors in coincidence is called a Line Of Response (LOR). In practice, however, since the detectors are finite in size, the LOR is replaced by a tube as shown in Figure 1.3. (LOR) * Figure 1.3: The tube of response for two detectors in coincidence [13]. Annihilation photons that arise within a volume element, or voxel, occur in a random order and are produced isotropically. Only those that are produced within a LOR and for which their direction of emission also lies will be detected. The number of positrons emitted from a voxel, and hence annihilations occurring in that voxel, in a given time is proportional to the tracer concentration within the voxel. Then the number of photons detected is also proportional to the tracer concentration, even though the set of possible LORs may represent a small fraction of the complete set of 9 all possible lines of emissions through the voxel. Furthermore, the number of annihi-lation photon pairs, or events, associated with each tube of response is proportional to the sum of the tracer concentration within that entire tube. This sum approximates the line integral of the tracer concentration along the LOR. During a PET scan, coincidences are detected and assigned to their appropriate LORs, the current contents of which are continually updated. This process is realized using a histogramming circuit which logs the fact that an event occurred on a LOR by adding one to a corresponding location in an array in memory. For 2-D image reconstruction, LOR indexing in its simplest form requires two indices. The rows of the array are indexed by the angle, <J3, the line makes with respect to some reference axis and the columns are indexed by the line's distance, s, from the centre of the detector ring as presented in Figure 1.4. Figure 1.4: The LOR coordinate system [13]. 10 After many hundreds of thousands of events have been collected, a pattern forms in the array that uniquely describes the distribution of radioactivity in the slice within the plane of detectors. Since a point of radioactivity will leave a pattern proportional to the sine of the angle index, this array is called a sinogram as illustrated in Fig-ure 1.5(B). Thus, a sinogram is simply a (s,</>) plot, with each LOR represented by a single point on the plot as shown in Figure 1.5(A). Figure 1.5: Mappings of LORs to sinogram points [13]. (A) A Line Of Response corresponding to one point in a sinogram. (B) The sinusoidal curve in the sinogram for a point off centre. Figure 1.6(A) presents a 1-D parallel projection which is defined to be the set of 11 all line integrals intersecting a slice of the tracer distribution at the same given angle <p = 4>0, but at different spatial positions. Therefore, the set of LORs corresponding to a parallel projection (s, <f)0) is one complete row of a sinogram as illustrated in Figure 1.6(B), and subsequent sets for different values of <p are stored as consecutive rows. Mathematically, a parallel projection (s, (p0) is the 2-D Radon transform, or identically the X-ray transform [14], of the tracer distribution f(x, y, z). Tracer D M r l b u t l o n \ 1 -OiKl ia i Figure 1.6: Radon transform of the tracer distribution [13]. (A) A 1-D parallel projection of a 2-D section. (B) A parallel projection corresponding to one row in a sinogram. In order to form an image, the inverse Radon transform must be taken. Typically, 12 the calculation used for the inverse Radon transform is the filtered backprojection (FBP) [13]. Figure 1.7 presents an illustrated comparison between the unfiltered and filtered backprojection processes. Simple backprojection involves inverse Radon transformation of the projection data. Essentially the value of each integral is spread uniformly over the coincidence chord defined by the LOR and the projections are summed over s and <p to get the activity distribution. Unfiltered backprojection pro-duces a blurred image containing artifacts. In filtered backprojection, each projection is modified using an appropriate windowed frequency ramp filter prior to backpro-jection. In this operation, each row of the sinogram is first corrected for attenuation losses and filtered with a highpass filter. FBP is very fast, but it works best when the projections are uncorrupted by noise and are comphtt. A complete projection is one which spans beyond the object on both ends, in which each measured value is exactly equal to the ray sum along the line of measurement and which satisfies all the sampling criteria set down by Shannon's sampling ihtortm [15]. Theoretically, the resulting image will accurately represent the spatial distribution of radioactivity within the subject. Figure 1.7: An illustration of the projection and backprojection processes [16]. 13 III. Limitations to Image Resolution The image resolution achievable by PET cameras is limited by several factors. These factors are broadly categorized into two divisions: the theoretical limits and the practical limits. Limitations of the first type restrict the best achievable resolu-tion under ideal circumstances, while limitations of the second type dictate the best possible resolution for individual PET cameras during standard operation. Theoretical limits are set by two properties of positron decay: the range of the positron and the angular range or acollinearity of the annihilation photons. The positrons emitted in radionuclide decay annihilate at some distance from their point of origin. This distance depends upon the energy of the emitted positron and upon the density of the medium into which the positron is released. Positrons are emitted isotropically with energies ranging from zero to the maximum Emax listed in Table 1.1 for various radionuclide decays. The average energy of the positrons is approximately 0.4 E m a x . The ranges specified in the third column of Table 1.1 corresponds to the range of positrons with energies corresponding to the average energy and not to E m a x . As already mentioned, the annihilation of a positron with an electron results in the emission of two photons at 180° to each other in the centre-of-mass. In the laboratory however, the electron-positron pair always has non-zero net momentum. As a result, the two annihilation photons are acollinear, and show finite angular distribution about 180°. The two image degrading factors described above, positron range and photon acollinearity, are intrinsic limitations upon the best resolution one may achieve under ideal circumstances. Practical limits to resolution are set by the detector size and type, the coupling ef-ficiency between the detectors and the photomultipliers that detect scintillation light 14 and inter-detector scatter as well as the linear sampling distance. Other limitations arise from restrictions on the amount of radioactivity that a subject can be given which influences the statistical quality of the image, dead-time losses, attenuation, scattering, randoms, crystal identification miscodings and body movement. Dead-time losses are mainly due to the inability of the detectors to count any interactions occurring before (until) the scintillation light of a previous event has decayed to a negligible level. Attenuation arises from the absorption of 7s by body tissue before they reach the detectors. This loss affects image uniformity and results in an under-estimation of the tracer concentration. Scatter and random coincidences cause events to be assigned to incorrect LORs. Figure 1.8 illustrates the types of coincidences in PET. Scatters occur when one 7, or both, Compton scatter and become detected. Randoms or accidentals result when 7s belonging to two separate annihilations are detected within the same coincidence time window and paired together as the same event. Crystal identification miscodings are due to the position reconstruction al-gorithms assigning scintillation events to crystals that do not match the entrance crystals of the impinging 7s. These errors are detrimental to the transverse position resolution. Finally, body movements during imaging with PET introduce image blur-ring. Even if restraining devices are employed, the body movements can still severely reduce the diagnostic value of the PET images. By far the most important factor that determines the resolution of a multiple crystal PET camera is the detector size. The width of the crystals used to detect annihilation photons determines the width of the tube of response within which the event may lie. The narrower the detectors, the thinner the tube of response, the higher the resolution. While the detector width can be made as small as possible, the detector depth must be maintained sufficiently long so that most of the 511 keV photons passing through it are stopped. In high resolution systems, the detector depth 15 true scattered random Figure 1.8: The three types of coincident events [13]. is much greater than the width of the detector. A 511 keV photon passing through a detector does not always deposit all its energy by photoelectric interaction at the first interaction point. For BGO, the probability of a photoelectric interaction at the first interaction point is about 42%. Some photons undergo Compton scatter and deflect into a neighbouring detector where they deposit all or some of the remaining energy. If such an event passed the energy threshold it would be accepted as a valid event and could be assigned to the other crystal rather than the entrance crystal. This is an unacceptable error since image reconstruction in PET relies on the detection of 7-rays impinging with their direction parallel to the long axes of the narrow crystals. The problem of crystal penetration is especially important in systems with de-tectors whose widths are significantly less than the attenuation length of a 511 keV photon. An annihilation photon created at the centre of the Field-Of-View (FOV) would be incident perpendicularly on the crystals and have a greater chance of being detected in the same crystal. Photons from an annihilation occurring at the edge of the FOV enter the crystal obliquely and may traverse many crystals before being detected. As a result of this possible mis-positioning, the resolution at the edge of the FOV is much worse compared to the resolution at the centre of the FOV. To develop 16 a remedy for this degradation, known as the radial elongation or parallax error [12], has been the first motivation for this work. This thesis describes the development of a procedure for simulating the perfor-mance of PET block detectors. The work addresses an important issue of how to effectively design block detectors for new scintillator materials, optimize block size, optimize saw-cut depths, correcting the crystal penetration problem described above, etc. This tool developed is very useful and will be of interest to the imaging and detector development community. The next chapter presents experimental measure-ments of the performances of a state-of-the-art position encoding detector for PET. A model of this detector is then developed in Chapter 3 and simulated performances are commissioned against the measured data. The investigation of a new scheme to eliminate the radial elongation problem [17], caused by crystal penetration, and its results are presented and evaluated in the fourth chapter. In Chapter 5, the validity and utility of the model are discussed and its potential explored. 17 C H A P T E R 2 E X P E R I M E N T A L M E A S U R E M E N T S Prior to the development of a model for the design of position encoding mul-ticrystal detectors for PET, measurements were undertaken to characterize the per-formances of a state-of-the-art block detector. The outline and results of these mea-surements are presented in this chapter. I. Block Candidate Specifications The block detector selected for study and simulation is the new block manu-factured by Siemens-CTI1 for the ECAT EXACT HR PLUS positron emission to-mograph. No photographs or drawings of this new block are available. However, Figure 2.1 presents a photograph of a Siemens-CTI ECAT 953B detector which is a predecessor, in its design, to the HR PLUS. The major differences between the HR PLUS block and the 953B block is that the HR PLUS detectors are considerably smaller and incorporate round PMTs whereas the PMTs used by the 953B detectors are square. Table 2.1 lists the geometrical specifications of the 953B and HR PLUS block detectors to ease the comparison. The HR PLUS block detector utilizes BGO as its scintillation material, and is of the position encoding multicrystal type with an 8x8 segmentation. This gives an ^ T I P E T Systems Inc., 810 Innovation Drive, Knoxville, T N 18 Figure 2.1: A photograph of the Siemens-CTI ECAT 953B detector [18]. 19 953B HRPLUS block dim. (mm3) 49x53x30 36x38x30 block area (mm2) 2597 1368 block segmentation 8x8 8x8 crystal dim. (mm2) 5.62x6.11 4.04x4.29 area/crystal (mm2) 34.3 21.4 saw cut widths (mm) 0.64 0.46 PMT (mm) 25.6 square 19 round reference [19] [20] Table 2.1: Dimensional specifications of the 953B and HR PLUS block detectors. array of 64 BGO crystals coupled from its back face to a 2x2 array of Hamamatsu or Burle 19 mm round PMTs via a slotted light guide. The crystals and light guide are cut from a single BGO block with the cuts running perpendicular to the PMT entrance window. The light guide is formed by extending to appropriate depths the cuts which separate the crystals. The depths of the cuts are determined by the criterion that "the probability of correctly identifying a crystal be the same for all of the crystals in the detector" [21]. The BGO block measures 36 mm transaxially by 38 mm axially across the front face and is 30 mm deep. The total scintillation light yield from interacting 7-rays is read by the assembly of four circular PMTs of 19 mm diameter. Given that each quadrant of the block is read by one of the PMTs, the granularity of the block can be characterized by its crystal to quadrant surface ratio of 1/16. Crystals have dimensions approximately of 4.04x4.29 mm2. The edge and corner crystals of the block are cut slightly smaller than the other crystals to maintain centre-to-centre crystal spacing between blocks in a ring tomograph. The crystals are coated with a highly reflective diffuse substance to maximize the light collection efficiency. 20 II. Experimental Configuration The experimental setup shown in Figure 2.2 allows for the study of the energy and position resolutions of the block [22, 23]. The benchtop setup conveniently utilizes either one block detector for single mode measurements or two block detectors for coincidence measurements. For these studies, 511 keV 7-rays from a 0.3 mCi source of 6 8 Ge are used. Figure 2.2 also shows the hardware in use for data acquisition. The sum of the four PMT signals, E 7 = A + B + C + D, (2.1) is thresholded and used to drive the trigger logic. For each trigger, the signals from the four PMTs are acquired from a 10-bit charge-sensitive analog to digital converter2 with an integration time of 1000 nsec. These digital signals are then buffered in a dual-ported memory3 before being passed on to a SUN Workstation for processing. The gains of the PMTs are balanced by requiring that the inclusive count rates be equal within statistics in each quadrant of the block when the block is exposed to a flood source. Such a uniform flood of single 7-rays at the block front face is obtained from an uncollimated point source placed approximately 40 cm from the block detector. With typically 10,000 counts per quadrant, a 1% tolerance can be achieved on the relative gain balance. Once the gains are balanced, a data run of one to five million events is acquired with the flood source to perform the block position and energy calibrations. 2Lecroy 4300B Fast Encoding/Readout A D C 3Lecroy 4302 21 3 I? p o 3 8 8 * r -ul 3 ( 5 N 8 § 0= O 3 5 s b 2 < u 2 83 °- o o , 5 1 0 3 UJ • * 0 5 s - « L J I C V E l_ E T3 0) C o , u V£ CM t-c faO u . q = O § 2 s -£ 0 to (ti o * i o E ^ . E a o T3 a) 2 1 -bo a> • , - c 22 III. Raw Data Analysis For each gamma ray interaction, the crystal of scintillation is identified by the computer using the four PMT signals. Anger camera logic is used to obtain a mea-sured position value (X 7 , Y 7 ) from these signals [22]: (B + D) - (A + C) X-y Y 7 = A + B + C + D ' (A + B ) - ( C + D) (2.2) (2-3) A + B + C + D ' where A, B, C, and D are the ADC-converted pulse heights from the four PMTs as arranged in Figure 2.3. 0 0 © 0 X Figure 2.3: The PMT configuration. The flood source measurements provide a 128x128 channel position response dis-tributions in the X 7Y 7-plane. Figure 2.4 shows the position response distribution for the HR PLUS block detector under study. A distinct peak is observed in the distri-bution for each crystal in the detector. The relationship between measured and true interaction positions is non-linear in position encoding BGO block detectors [22, 24]. A lookup table (LUT) is therefore necessary to relate a given (X 7 , Y 7 ) pair to crys-tal row and column addresses. The position calibration LUT is generated for each 23 crystal in the detector so that every possible ( X 7 , Y 7 ) pair is assigned to one crystal and no detected events are thrown away. Each region in the position calibration L U T is referred to as a Region Of Interest, or ROI. The ROIs are chosen to be four-sided polygons, since they are easy to generate, yet flexible enough to bound the 64 peaks without throwing away any events. Figure 2.4: Measured 2-D flood position calibration spectrum for the HR PLUS block detector under study, overlaid with its corresponding generated L U T boundaries. The horizontal axis corresponds to the X 7 -axis and the vertical axis corresponds to the Y-y-axis. The method used for the generation of the position calibration L U T also satisfies 24 the desired condition that the ROI boundaries follow valleys in the data set as closely as possible. First, the position of the local maximum within each of the 64 peaks is determined. The average position of each group of four nearest-neighbour local maxima is then taken to be an ROI corner. Joining these corners provides all ROIs except those along the detector edges. To obtain boundaries for edge peaks, an average position of two neighbouring edge local maxima is calculated; then a line is extended from the nearest ROI corner through this average position to the edge of the data set. The position calibration ROI boundaries are shown in Figure 2.4 where the LUT is overlaid on top of the flood spectrum. The position peaks exhibit the characteristic stretching towards the corners of the block and away from the centres of the sides of the block known as the pincushion effect. Related to this effect is the ROIs decrease in area size as the position of the crystal lies further and further away from the centre of the block. At the edge, the ROIs are thin strips and at the corners, the ROIs have the smallest areas. This geometrical trend in the variation of the ROI areas produces the contrast observed in the peaks. The closer the location of the peaks are to the edges of the plot, the darker they appear indicating a higher concentration of events. Large peak-to-valley ratios (PVRs) are desirable in the position map of the block detector. Large ratios indicate that the peaks are well separated and thus the iden-tification of the crystals of interaction is correct for a large fraction of the events. For the block under study, PVRs were found to have values ranging from 1.5 to 16.5. Table 2.2 gives the PVRs along four rows in one half of the block. The columns are numbered 0 to 7 from the left to the right of Figure 2.4. The rows are numbered 0 to 3 from the top to the centre. 25 Row Column 0 1 2 3 4 5 6 7 0 (top) 4.6 2.8 1,9 1.8 2.0 2.1 4.1 7.6 1 8.7 4.4 2.9 2.6 2.9 2.9 6.1 14.1 2 6.8 3.3 2.5 2.4 2.6 2.9 4.6 12.4 3 (centre) 8.9 4.1 2.0 1.6 1.5 2.0 6.2 16.5 Table 2.2: Peak-to-valley ratios along four rows in one half of the block under study. The high statistics data are used to characterize the uniformity of energy resolution and 7-ray detection efficiency of the detector from crystal to crystal. The position calibration LUT just discussed is used to relate (X 7 , Y 7 ) of each event to a crystal address. An energy spectrum for each crystal is then obtained by histogramming the events by total pulse height, E 7 , in accordance to their crystal address. Figure 2.5 presents the measured energy spectra for 4 crystals situated along the diagonal in one quadrant of the block. The exact locations of these crystals are presented in Figure 2.6. Each of the energy spectra consist of a gaussian-like peak, contributed by events with photoelectric interactions, with a low energy tail, known as the Compton tail, produced by Compton interactions in which the interacting 7s are redirected out of the detector block. The relative ratios of the peak to Compton tail yields range from 2 for crystals along the edge, to 9 for crystals near the centre of the block. This is expected since 7s are more likely to scatter out of the block from edge crystals than when scattering from crystals near the centre. Table 2.3 gives the relative photopeak channels for the 16 crystals in one quadrant of the block in percentages. The individual photopeak channels were normalized to 26 Scaled ^(Photoelectrons) Figure 2.5: Measured energy spectra for 4 crystals situated along the diagonal in one quadrant of the HR PLUS block detector under study. 27 Figure 2.6: Exact locations of the four diagonal crystals of Figure 2.5. the photopeak channel of the crystal located in row 3 and column 3 (B): ^ x 100%; I 3 3 (2.4) this being the crystal exhibiting the highest photopeak channel. The table also gives the FWHM as a percentage of the photopeak position for each energy distribution in parentheses: FWHM ^ x 100%. (2.5) The transverse Line-Spread-Functions (LSFs) of the block candidate are measured by stepping a fan beam across the front face of the detector and parallel to the columns of the block. A fan beam is obtained by using two 5 cm thick lead bricks separated by a 1 mm slit. Figure 2.7 shows the measured fan beam profile at the front face of the detector superimposed by a gaussian fitted to the data. The fan beam was found to have a FWHM of 2.3 mm and a F W T M of 4.7 mm. The detector under study is positioned close behind the bricks while the source is at a distance of typically 10 cm from the bricks. The response of each crystal along one column of the block is measured by counting the number of 7-rays detected in those crystals as a function 28 Column Row 1 (centre) 2 3 4 (edge) 4 (edge) A 50±2 53±2 62±3 39±2 (34±3) (32±3) (30±2) (42±5) 3 B 77±2 84±2 100±0 78±1 (26±1) (23±1) (22±1) (25±1) 2 C 73±2 81±2 93±1 71±1 (26±1) (23±1) (23±1) (26±1) 1 (centre) D 71±2 75±2 86±2 67±2 (30±1) (25±1) (25±1) (28±1) Table 2.3: Relative photopeak channels and (relative FWHMs) from the energy spec-tra for one quadrant of the HR PLUS block under study. of the fan beam slit position relative to the centre of that column. Steps of 1 mm are used to completely scan the face of the block. Finally, the response is averaged along the column of the block. Figure 2.8 presents 8 LSFs for the HR PLUS block detector, one for each column of the block, which have been superimposed relative to the centre of the block. The measured LSFs are basically narrow distributions centred at their corresponding crys-tal locations, with slight shoulders at the tenth maximum level. There are also large tails located beyond the edges of the block which are caused by inward scattering 7S. Table 2.4 gives the FWHMs and FWTMs for the 8 LSFs measured for the block under study. The measured data presented in this chapter are sufficient to fully characterize the EXACT HR PLUS block detector. In the next chapter, a model of the HR PLUS 29 Figure 2.7: Profile of the fan beam produced with a gaussian fitted to the data. block is developed and simulated results are commissioned against this data. 30 Column 0 j 1 2 3 4 5 6 7 FWHM (mm) 5.3 5.1 4.8 4.9 4.8 4.7 5.1 4.6 FWTM (mm) 13.2 12.2 10.3 11.0 9.8 9.9 11.7 13.2 Table 2.4: The FWHMs and FWTMs for the 8 LSFs measured for the block under study. 31 X (mm) Figure 2.8: Measured LSFs for the HR PLUS block detector under study. 32 C H A P T E R 3 T H E M O D E L With the price of a detector such as the EXACT HR PLUS being of the order of $600(U.S.)/in.2 just for parts [12], the search for the multicrystal geometry and signal readout of a new prototype achieving better performances is an expensive process when done on an empirical trial and error basis. A dedicated simulation platform then becomes an attractive tool to bring down the time and material invested in the development phase of the new block and to guarantee the satisfactory performance of the early prototype. Work in the past on PET simulations by Ziegler et al. [25] and Tzanakos et al. [26] has focussed on studies of the time resolution in simple monocrystal BaF2 detectors for Time-Of-Flight (TOF) PET. The energy and position responses of multicrystal array detectors found in modern PET cameras were not addressed by these authors. A platform designed to simulate the performance of position encoding multicrystal detectors has been developed and is presented in this chapter. The initial develop-ment of the platform has focussed on the simulation of the block detector that was introduced in the previous chapter: the Siemens-CTI ECAT EXACT HR PLUS block detector. This simulation platform models the interactions of 7-rays as well as optical light transport in a geometry as sophisticated as that of position encoding multicrys-tal detectors. An overview of the simulation platform is given in Section I. A brief 33 description of the treatment of 7-ray interactions is given in Section II. The pub-lic domain package used to treat the propagation of optical photons in the detector geometry is introduced in Section III. The block detector specifications to the simula-tion are then presented in Section IV. The results of the simulation are compared to the measurements of Chapter 2 as a validation of the platform in Section V. Finally, the details and results of systematic studies performed on the simulation are given in Section VI. I. The Simulation Platform Figure 3.1 shows the conceptual design of the simulation platform. The platform treats the interactions of 7-rays in an inorganic scintillator, the geometry of the multicrystal array, as well as the propagation and detection of individual scintillation photons. The geometry of the block is first used as input to the j-ray transport module. A uniform beam of 7-rays with specified energy is generated from a distant point source. Each incident photon is then tracked in the volume of the block for photoelectric and Compton interactions. For each interaction, an event scintillation vector is written to a f-ray interaction list file. The first three words are the coordinates of the interaction point within the block volume (X;, Y, , Z,). The fourth word gives the calibrated light yield, L,, at that vertex. The fifth word is a sequential index incremented for each interaction until all the energy of the incoming 7 is deposited in the block volume or until the 7 escapes from the block volume. Finally, the last two words give the block entrance coordinate, (X*, Y*), of the incoming 7. The DETECT syntax translator describes the geometry and optical properties of the block in adherence to the language syntax imposed by the light transport 34 Detector design Scintillator Active volume Gamma transport module —•^"Incident gamma ^ /scintillation vertex i event scintillation vector (Xi ,Yi ,Zi ,Li , i ) Block geometry Optical specifications D E T E C T syntax translator D E T E C T description of detector design Gamma interaction list file D E T E C T simulation driver Initialize geometry scintillation vertex i >^ Simulate Ni scintillation photons propagating from X i , Y i , Z i Event signal vector 1 r Event signal list file Figure 3.1: Data flow diagram of the block simulation platform [23]. 35 simulator DETECT [27]. Once the translation is complete, the geometry and optical properties of the detector are included in the DETECT simulation driver to initialize the block design specifications. Scintillation events from the 7-ray interaction list file are then simulated sequen-tially. The coordinates (X», Y t , Z,) of the interaction point are used to specify an infinitesimal voxel from which Lj scintillation photons are isotropically generated and tracked by DETECT. The signals collected in each of the detection elements of the detector are written in an event signal vector. Event signal vectors belonging to the same incoming 7-ray are finally merged to compute a record encoding the total energy deposited in the detector and the coordinates of the interaction. Monte Carlo information from the event scintillation vectors may also be duplicated. The resulting signal list file is then stored on disk for subsequent data analysis. II. G a m m a Transport Interface A gamma transport model, introduced in [28, 23], simulates the interactions of 7-rays in an inorganic scintillator. The active volume of the block is first used as input to the 7-ray transport mod-ule. A uniform beam of 7-rays with specified energy is generated from a distant point source. Each incident photon is then tracked in the volume of the block for photo-electric and Compton interactions. Cross-sections for these two processes are derived from GEANT [29] for the inorganic scintillator considered in the block design. Their relative ratio is used to randomly choose which of the two processes occurs, and the trajectory of the interacting 7 is randomly generated according to an exponential 36 distribution. The total cross-section at the energy of the interacting 7 determines the interaction length of the exponential. Values of the cross-sections are tabulated from 15 to 511 keV in bins of 5 keV. Tracking is stopped either by a photoelectric interaction, escape of the photon from the block volume, or by a Compton interaction leaving less than 15 keV to the recoil photon. For Compton interactions, the direction of the scattered 7 follows the Klein-Nishina angular distribution [4] and the energy of the recoil electron is assumed to be converted to light at the interaction vertex. Scattering of 7s into the block detector from the cannister or surrounding supports is not modelled. Scintillation event information including location and number of light photons emitted are produced as output. This data is required as input to the light transport module detailed in Section III. III. Introduction to D E T E C T The development of a simulation of PET multicrystal block detectors necessitated the incorporation of a program created by Dr. G.F. Knoll aptly named DETECT. Written in the PASCAL language, the program DETECT is a Monte Carlo model of the optical behaviour of scintillation detectors. It generates individual scintillation photons in specified voxels of the active volume of the detector, follows each photon in its passage through the various components and interactions with surfaces, allows for possible absorption and re-emission by a waveshifting component, and records the fate (absorption, escape, or detection) of each. The probabilities of these processes are derived from the results of multiple histories involving the simulation of many scintillation photons. Data is also recorded on the number of reflecting surfaces encountered and the photon flight time to detection. 37 DETECT allows for the specification of the geometry of a detector with a very general syntax. Any complex system may be modelled as long as it is possible to separate it into constituent parts consisting of a volume specified by multiple planar, cylindrical, conical, or spherical surfaces with arbitrary orientation. For more flexi-bility, common unphysical surfaces shared by contiguous elements may be declared as pseudo-surfaces and are ignored by the tracking. The optical behaviour of real surfaces may be specified to simulate possible reflections under polished, ground, painted, or metalized conditions. Surfaces in optical contact are treated using Fresnel's equations governing reflection and transmission as well as Snell's law of refraction. Within each optical element, bulk absorption, scattering, and wavelength shifting are simulated by specifying a mean distance of photon travel for each process. Only one wavelength shift is allowed per history, and is accompanied by appropriate changes in absorption, reflection, and scattering properties. The program is well structured using initial definition statements to specify the optical properties of all materials used in the system. A component is selected from this list of possible materials, and its geometry is delineated by specifying multiple bounding surfaces. These surfaces may be selected from previously defined planes, cylinders, cones, or spheres, and can be used in either convex or concave orientation. The optical behaviour of each surface is chosen by selecting one of a set of previously defined surface finishes, including a detector surface representing for instance the pho-tocathode of a photomultiplier tube. Surfaces may either be external, and assumed to be an interface with vacuum, or shared in common with another component. Once the geometry is defined, the scintillation voxel may be specified along with the number of photons produced within that voxel, L;. These scintillation photons are then isotropically generated and are tracked on an individual basis until they are 38 absorbed, detected, or have escaped from the system. At each photon reflection or scattering, the program logic determines the new direction of the photon, identifies the component in which it is travelling, and computes the next intersection with a surface. A random sampling is then made to determine if the photon is bulk absorbed, scattered, or wavelength shifted over this path. If none of these processes occur, the optical properties of the next surface determine whether the photon is reflected, refracted, detected, or absorbed. This process is then repeated for all subsequent paths in the history. A maximum flight time per history is specified to abort those cases in which a photon becomes internally trapped. After the specified number of histories have been completed, a report is prepared that summarizes the probability of occurrence with statistical uncertainty estimate for each of the possible fates. Data are also reported on the probability for wavelength shift, mean age, and mean number of surfaces encountered. These data are separately tabulated for all photons and for just the subset that are ultimately detected. A histogram describing the elapsed time to detection can also be generated. As mentioned, five types of optical finishes are allowed for component surfaces. A brief description of each optical finish is provided here: 1. POLISH. This option represents a polished surface that may or may not be in optical contact with another component. If no other component is specified, the surface is assumed to interface with vacuum. Photons incident on the surface are assumed to have random polarization, and are first tested for the possibility of Fresnel reflection if a change in refractive index occurs at the surface [30]. This probability is given by sin2(a — b ) tan2(a — b ) sin2(a + b ) tan2(a + b ) J 39 (3.1) where a and b are the angles of incidence and refraction, respectively. If reflec-tion is selected, the angle of reflection is set equal to the angle of incidence. If reflection does not occur, the photon is transmitted with the complementary probability of: T = 1 - R, (3.2) and assumed to follow Snell's law of refraction: « 5 & l = 5S, (3.3) sm(b) n a where n a and nj, are the refractive indices of the incident and transmitted compo-nents [30]. Depending on the refractive index change and the angle of incidence, this may result in total internal reflection of the photon back into the incident component. If the surface interfaces with vacuum and a coat of diffuse reflector has been specified by including a reflection coefficient in the specification of the surface treatment, the transmitted photon may be reflected back across the sur-face. The value of the reflection coefficient is the probability that, if a photon escapes from the surface, it is returned to the original medium by Lambertian reflection [30]. In this case, the angle of reflection is independent of the angle of incidence, and is sampled from a distribution given by 1(0) = l(a)cos(0 - a), (3.4) where a is the angle of incidence, 9 is the angle of reflection and 1(6) is the light intensity at angle 6. The photon is again refracted as it crosses the surface back into the original medium. Should the reflected photon fail to cross the surface on its first attempt, additional reflection angles are randomly selected until the reflected photon successfully re-enters the original component. Figure 3.2 illustrates the possible optical processes and their corresponding probabilities 40 for a surface that is polished and coated with a diffuse reflector with a specified reflection coefficient of r. Figure 3.2: The optical processes and their corresponding probabilities at a polished surface coated with an external diffuse reflector with a reflection coefficient of r. R and T are the probabilities for Fresnel reflection and transmission respectively. 2. GROUND. This surface specification simulates a roughened or ground optical surface. It is treated in the same way as the polished surface described above, except that the normal to the surface used to define the angles in Equations 3.1 to 3.4 is randomly distributed following a Lambertian distribution around the nominal surface normal. To prevent unrealistic cases in which a photon travel-ling at an oblique angle could arrive on the wrong side of one of these perturbed (l-r)T n N = 1 R 41 surface elements, a test is made of the dot product of the reflected photon di-rection with the nominal surface normal. For those cases in which the result is negative, a new surface normal is randomly selected until this dot product is positive. Additionally, a reflection coefficient can be specified to simulate an external diffuse reflector for those photons that pass through a rough surface. 3. PAINT. The surface is assumed to be painted with a diffuse reflecting material characterized by reflection coefficients that may be supplied for the primary and shifted wavelengths. If random sampling shows that reflection occurs, it is assumed to be Lambertian. Transmission is not allowed and so jumps in the index of refraction is of no relevance. 4. METAL. The surface is assumed to be smooth and covered with a metallized coating representing a specular reflector. Reflection coefficients may be speci-fied for the primary and shifted wavelengths. A random sampling determines whether the photon is absorbed at the surface or undergoes reflection at an angle equal to the angle of incidence. Transmission is not allowed and so jumps in the index of refraction is of no relevance. 5. DETECT. This specification represents a photocathode, photodiode, or any other photon detecting layer. As an option, the photocathode may be located on the opposite surface of a thin window of chosen refractive index. Records are kept of the number of primary and waveshifted photons that are detected, so that the results may be. easily modified to account for the spectral sensitivity of actual photocathodes at each wavelength. Some limitations are encountered when applying this program. DETECT does 42 not allow more than one finish type per surface for each component. Hence, com-ponents are required to be segmented into smaller components until this restriction is satisfied. Another important restriction to note is that a surface of one compo-nent in optical contact to a surface of a second component must either lie completely within the other surface or otherwise completely envelope it. Although defining two components with a coupling area less than either of the two surfaces in question is syntactically permissable, problems nevertheless eventually arise during photon track-ing. These run-time errors occur due to DETECT's mishandling of photons that exit one of the components in contact through the uncoupled area of the questionable surface and resulting in a position outside the second component, and in fact, outside the defined active volume. An inconvenience of DETECT is the hard-coding of the quantum efficiency in the program which does not allow for quick modifications since re-compilation is necessary. A solution to this problem is to decrease the number of scintillation photons tracked by a factor equal to the quantum efficiency desired and to hard-code the quantum efficiency in DETECT to be 100%. This solution is physically acceptable since the quantum efficiency can be thought of as a purely statistical factor. Indeed, the product of the quantum efficiency and the scintillation light yield is constrained only by the energy resolution. The solution is also compu-tationally desirable as it reduces processing time by a factor equal to the quantum efficiency. Lastly, a shortcoming of the program that has to be addressed is its lack of a parabolic surface type to model the photocathode. An approximation of the photo-cathode must be made to one of the available surface shapes and this approximation will be described in detail in Section IV. 43 IV. Block Detector Specifications Table 3.1 lists the input parameters used to describe the HR PLUS block detector geometry and optical properties with DETECT. The detector model begins with the description of the block given in Chapter 2. Additional factors in the geometry specifications as well as known optical properties of the materials involved represent the author's best understanding of the block at this point given the information made available by the fabricant. parameter Input Value BGO light yield BGO refraction index BGO scattering length BGO absorption length 8200 photons/MeV 2.14 4000 mm 4000 mm surface finish surface reflectivity block dim (mm) block segmentation adjacent crystal spacing (mm) ground 95% 36(T)x38(A)x30(H) 8x8 0.46(T)x0.46(A) PMT photocathode radius PMT window refraction index photocathode quantum eff. 7.4 mm 1.52 0.125 photon loss adjustment factor 0.47 Table 3.1: List of input parameters used to describe the block geometry and optical properties with DETECT. Figure 3.3 allows for a good visualization of the geometry used in the model of the HR PLUS block detector. As well, the material and surface treatment specifications to the simulation are given in the figure. Diagram A shows the configuration of the major components of the detector. Diagram B displays the BGO block along with the saw-cuts. The component modelling simultaneously the PMT window and the optical coupling glue is shown in Diagram C. Diagram D presents a component that models 44 the material surrounding the photocathodes within each PMT. The actual PMT is given in Diagram E where the bottom spherical surface represents the photocathode. A dummy component is specified only to satisfy one of the error checks that DETECT enforces. Finally, Diagram F illustrates the positioning of a PMT over a quadrant of the block detector. Detailed descriptions of the BGO block, the PMTs and the entrance window are given in the remainder of this section. A. The BGO Block The intrinsic properties of BGO are available in the literature. The scintilla-tion light yield of the crystal was taken from the measured value of 8200 ±750 pho-tons/MeV reported in [8]. Each time that an energy deposition occurs, the simulation generates a number of scintillation photons proportional to the energy deposited by the scintillation light yield factor. For simplicity, the number of scintillation photons generated was not allowed to fluctuate statistically. The BGO sample used in the pro-duction of the HR PLUS block is characterized by an index of refraction of 2.14 [31]. Derenzo and Riles [32] have measured a total attenuation length, Xt, of 200 mm at the peak emission wavelength in clear BGO crystals. A beam of known intensity was di-rected through BGO blocks of varying lengths and the directly transmitted intensity was measured from which the total attenuation length was determined. Nowadays, improvements in crystal growing methods provide much clearer BGO samples, and total attenuation length values are quoted to be as high as 2,000 mm [31]. To allow for bulk absorption as well as scattering in the simulation, this value was split in two components, A a and Xs. Assuming that attenuation of the scintillation light through bulk absorption and scattering are independent processes, one has: e " = x e - ^ , (3.5) 45 ground * BGO / / / / / / / / -•all values are in mm. Figure 3.3: The segmentation of the detector assembly. A: Layout of the detector assembly. B: BGO scintillator block. C: Glass plate. D: PMT holder. E: PMT. F: Overlap between the upper-right quadrant of the scintillator block and its associ-ated PMT. 4 6 from which the following relation can be derived between A t , A a and Xs: L - JL — At A a A s DETECT provides two options, polished and ground, which could potentially model the surface finish and coating of the block. To see which of the two is appro-priate, a 30 mm long crystal centred on a detection unit was simulated. The finish of the five faces of the crystal not in contact with the detection unit was considered to be either polished or ground and coated by an opaque reflector with reflection coefficient (RC) of 85% [31]. A point source was moved in steps of 1 mm across the length of the crystal, starting at 1 mm from the top down to the position of the detection unit. The simulated photon yield in the detection unit is presented as a function of the source position in Figure 3.4 for both surface finishes. The results from the simula-tion are compared to the measured dependence of the photopeak energy channel, for a crystal at the centre of a quadrant of the HR PLUS block, upon the longitudinal position of a fan beam of interacting 7-rays incident on a side face of the block [33]. All simulated data points were normalized so that the total photon yield projected at the top face of the crystals matched that projected for the measurements. The light-yield for a source at the top face was chosen for the normalization because the simu-lation is believed to be more accurate in predicting the measured photopeak channels for shallow events since the uncertainty of the saw cut depths present a less signifi-cant problem for these events. The polish finish exhibits a dependence matching well the measured data. Using the ground option with 85% RC causes a dependence of the light collection upon the longitudinal position of the source that is incompatible with measurements. However, increasing the reflection coefficient to 95% improves the performance of the ground finish to match the measured depth dependence most satisfactorily. So either polish at 85% RC or ground at 95% RC could potentially 47 (3.6) model the surface finish. c o o c o o J C CL "D O O co 1 0.8 0.6 0.4 * Measured • Polished 0.85RC 0.2 0 • i i i 1 . i i , 1 i i i , 1 , , i , l . i . , • i i i 1 0.8 0.6 0.4 0.2 0 ¥ Measured • Ground 0.85RC • Ground 0.95RC _1 1 I I 10 15 20 25 30 Depth (mm) Figure 3.4: A study of the effects of the simulated surface finish on the depth depen-dence of the collected light yield in an isolated crystal of the HR PLUS. Although polish matches the measured data, it does not exhibit a true depth dependence. The light collection is actually constant throughout most of the detec-tor's length. A modification to the standard reflector coating the HR PLUS crystals that will be presented in Chapter 4 exploits the already present depth dependence by exaggerating it. It was found that the polish option of DETECT was unable to repro-duce the measured results for this modification primarily due to its lack of true depth 48 dependence. On the basis of these comparisons, the surface finish of the HR PLUS crystals was specified to DETECT as being ground with a reflection coefficient of 95%. Intuitively, this option is the more physical surface treatment since the saws that produce the cuts leave behind a surface that is definitely non-uniform. However, a ground finish of 95% RC yields a photon count that is in disagreement with the measured FWHM of the photopeaks presented in Table 2.3. Modelling the surface will be discussed further in Section VI. To allow for saw cut recesses varying for different crystal rows or columns, the geometry specification of the block had to be partitioned into a number of components much larger than the number of crystals. The cut depths were assumed to be axially and transaxially symmetrical. In that case, a crystal is a stack of four components, one for each depth of cut of the four bounding sides. The volumes beneath the cuts had to be separated into their own components as well. Adding to this the components that define the four PMTs, a total exceeding 500 components is required for the block geometry declaration to DETECT. To acquire the capacity of determining saw cut recesses providing good crystal position identification is one of the principal aims of the simulation platform. Values of the cut depths for the simulated 2-D flood position reconstruction map were therefore inferred through optimization. The optimization process begins with setting all the cut depths to zero. Each cut is then systematically lowered starting from the outside towards the centre. The optimal depth of the cut for the simulation is reached when the peak of the position distribution for events in the crystal defined by the cut is well resolved from its inner and outer neighbours, i.e., possessing PVRs that match the measurements presented in Table 2.2. Due to the four-fold symmetry of the block, the optimization only considered crystals along the diagonal of one quadrant. The 49 search process can be further quickened by simulating a small set of point sources, equivalent to 511 keV 7-rays having a prompt photoelectric interaction, in a given crystal. Point sources were spread along the longitudinal axis of the crystal within the mean attenuation length of 511 keV photons in BGO and the average of the reconstructed positions gave a good approximation of the peak location. Even though the peak locations in the reconstruction map match well the measured locations, the PVRs may be unacceptably inconsistent with measurements. As a result, once a preliminary set of cut depths is found, a full simulation of the 16 crystals in a quadrant is still needed. Fine adjustments to the cut depths are finally made to improve the PVRs. This last step again involves using 511 keV-equivalent point sources and exploits the empirical fact that as the cut recesses are lowered, the peaks shift towards the edge of the block and become better resolved. Interactions located within the volume of the saw cuts, taken to be 0.46 mm wide [31], were ignored by the DETECT simulation driver. DETECT rejects any scintillation locations lying outside the specified geometry and automatically reports zeros for the PMT counts. Null signal events are subsequently filtered out by lower energy thresholding. Finally, the dimensions of the block detector were decreased on four sides by 0.48 mm to reproduce the volume removed to allow for the box containing the detector block. Thus the edge crystals are smaller by that amount along one side, while corner crystals are similarly reduced in volume along two sides. B. The PMTs Burle and Hamamatsu PMTs of 19 mm diameter are both used in HR PLUS prototypes, and here, the geometry of the PMTs is modelled after the Hamamatsu 50 R5364, a typical 19 mm PMT. A scaled drawing of the R5364 is given in Figure 3.5. The model of the PMT consists of a cylinder with a flat top and spherical detection surface at the bottom representing the photocathode. The inner cylindrical wall is given a 100% reflective mirror finish and the top surface is assumed to be polished with an index of refraction of 1.52, that of glass. The cylinder length of 1.5 mm and radius of 7.5 mm are taken from the Hamamatsu catalog diagram of the R5364. The radius of curvature of the photocathode is chosen such that the photocathode approaches within 0.1 mm of the top surface. <j)18.6±0.7 faceplate photocathode Figure 3.5: Dimensional outline and basing diagram for the Hamamatsu R5364 PMT [34]. 51 The average quantum efficiency of the PMTs was derived from the convolution of the normalized emission spectrum of BGO with the spectral response of bialkali pho-tocathodes. Figure 3.6 presents typical photocathode spectral responses for various PMT window materials and emission spectrums for various scintillators. An allowable range of 11% < Q.E. < 14% was determined from which the intermediate value of 12.5% was selected for the model. Figure 3.6: Typical photocathode spectral response and emission spectrum of scin-tillators [34]. A: Borosilicate Glass. B: UV Glass. C: Synthetic Silica. D: Bialkali Photocathode. E: High Temp. Bialkali Photocathode. F: Extended Green Bialkali Photocathode. 52 C. The Coupling Window To simulate the coupling window of the PMTs and the optical glue coupling the PMTs to the BGO block, four rectangular glass plates are used with dimensions of 19x18x0.5 mm3. The top and bottom faces of the plates are specified as polished. This coupling interface allows for a crosstalk of a few percent of the total signal between the PMT reading directly an edge or corner crystal and the other PMTs. The thickness of the plate is chosen to reproduce this relative signal sharing by matching the observed width of the position response function of the edge crystals. The next section presents the simulated results and a comparison with measured data is made in a validation of the platform. Section VI will discuss the systematics of the simulation including the effect of the thickness of the simulated coupling plate. V . Validation of the Simulation Results from the simulation were commissioned against block performance mea-surements acquired in Chapter 2. Figure 3.7 presents the simulated energy spectra for 16 crystals in one quadrant of the block. The lower left spectrum belongs to the crystal at the centre of the block. For each crystal, the absolute difference between the measured and simulated photopeak position and FWHM are given in parenthesis. Values are percentages derived from: /pij p>: FWHM i j FWHM i j- \ V exp 1 sim r exp r s i m / where P'-7 denotes the photopeak channel of the crystal in the ith row and ] t h column from the centre. For the simulated data, events were assigned to the crystal in which the energy weighted centroid of interaction was located. Photopeak channels were normalized to the value obtained for the crystal in the third row and third column from 53 (-3.+3) (-9.+7) F (+11-6) (-11.+5) 1. (-13. + 6) (0.+5) (-3.+4) F (-10.+5) (-2.-2) Centre C r y s t a l Jl (-11.+7) Jl (-7.+3) 1 (-4.+7) (-2.+5) L (-1.+5) E (+6.-4) J 0 200 [^(Photoelectrons) Figure 3.7: Simulated energy spectra for the 16 crystals in one quadrant of the block. The lower left spectrum is that of the crystal at the centre of the block. The num-bers in parentheses are absolute differences in percentages between the measured and simulated photopeak position and FWHM normalized to the crystal in the third row and third column from the centre. the centre. Simulated and measured data were respectively normalized to the same crystal. For both simulated and measured data, this crystal presents the maximum photopeak channel. The measured and simulated relative photopeak channels and relative FWHMs in percentages are given in Table 3.2. The simulation reproduces well the crystal-by-crystal variations of the energy spec-tra. The simulated FWHMs of the photopeaks are within the range —6% to +7% of their expected values. This indicates that the combination of the BGO light yield, 54 crystal,^ P . i / P 3 3 x 100% F W H M t j / P t j x 100% row col meas. sim. meas. sim. 1 1 71 73 30 32 1 2 75 82 25 22 1 3 86 88 25 20 1 4 67 61 28 32 2 1 73 83 26 21 2 2 81 92 23 16 2 3 93 97 23 16 2 4 71 72 26 21 3 1 77 88 26 21 3 2 84 97 23 17 3 3 100 100 22 17 3 4 78 81 25 21 4 1 50 53 34 31 4 2 53 62 32 25 4 3 62 71 30 23 4 4 39 28 42 48 Table 3.2: The measured and simulated relative photopeak channels and relative FWHMs in percentages from the energy spectra for one quadrant of the HR PLUS block under study. 55 PMT quantum efficiency and crystal surface finish reproduces effectively the mea-sured total photostatistics for incident 511 keV 7-rays in the block. The simulation also predicts the measured relative position of the peaks to within the range: —13% to +11%. Finally, the familiar Compton tail of the energy spectra is well modelled. At low energy, the contribution of events escaping the block through the back plane or from the side walls is apparent in each crystal. For crystals in the inner rows and columns, the simulated ratio of the Compton plateau to the photopeak is w ^ . As expected, the Compton tail is more prominent for crystals in the edge row and column of the quadrant as more events can scatter out of the block active volume. The Compton plateau remains however significantly underestimated by the simulation. This is not surprising, as inward scattering from the detector cannister and surrounding supports is not modelled. Figure 3.8 presents the simulated 2-D flood position calibration spectrum for 16 crystals in one quadrant of the block. The levels of the contour plot were chosen to enhance the view of all 16 crystals. The simulated flood spectrum shows 16 resolved crystal peaks in the (X 7 , Y 7 ) space. This indicates that our choice of cut depths effectively models the relative op-tical coupling of individual crystals to each of the four PMTs. Detailed features of the measured flood position calibration spectrum of the HR PLUS block [20] are also well reproduced. The simulated data show the typical pincushion deformation stretching the inner crystal peaks towards the outmost corner of the block. Furthermore, in the HR PLUS block, optical cross-talk between the PMTs through a common entrance window is more suppressed than in previous block designs [22]. As a result, peaks 56 0 0.2 0.4 0.6 0.8 1 Block Centre X R Figure 3.8: Simulated 2-D flood position calibration spectrum for the 16 crystals in one quadrant of the block. belonging either to the edge rows or columns are not bowed inwards. Figure 3.8 shows that this new feature is well reproduced by the simulation. Figure 3.9 allows a more quantitative evaluation of the simulation accuracy in reproducing the measured flood position map. The figure presents a comparison between the simulated and measured position peaks for a row of crystals at the centre of the block. Only events with more than 350 keV were selected. The integral of counts was normalized to unity in both cases. The simulated spectrum is perfectly 57 symmetrical with respect to the centre and was smoothed to avoid artifacts from low statistics. 0.25 0.5 0.75 1 Fi gure 3.9: Comparison between simulated (a) and measured (b) position map for a row of crystals at the centre of the block. The experimental spectrum displays a slight gain imbalance which pulls the peaks towards the left of the block and one can see that the simulation models well the relative features of each peak even though this gain imbalance was not simulated. For crystals in the inner columns, the simulation successfully predicts the relative yield and width of the position peaks. In both simulated and measured data, the crystals 58 in the four central columns have a relative peak intensity of 0.009, while those of the second columns from the edge have one of 0.015. At the centre of the block, the peak-to-valley ratio is slightly better in the simulated spectrum. The simulation is also slightly optimistic in its prediction of the position response of the edge crystals. It overestimates the peak yield by a factor 1.2 and places the peaks at the very edge of the position map. A possible explanation of this is the following. For events in the edge crystals, signal sharing is a few percent more important in the real block than what is predicted by the simulation. This is attributed to an overly simplistic modelling of the optical coupling between the BGO block and the PMTs. For instance, values for the coupling glue's thickness, index of refraction, as well as absorption and scattering mean free paths were not considered. At the cost of simplicity, the coupling plate used in the simulation to effectively model the PMT entrance window could have better been sectioned into separate components for the glue and the PMT window, each with their own characteristics. Figure 3.10 presents the measured and simulated LSFs for the four columns along one quadrant of the block. The peaks were re-scaled to 1 and the positions are given as distances in mm from the centre of the block along the transaxial direction X. The simulated LSFs accounted for the fan beam width by convolving the simulated results with the fan beam gaussian of Figure 2.7. The simulation closely reproduces the measured LSFs as is clearly seen in the fig-ure. The simulated peaks coincide with their measured counterparts and the FWHM of the simulated LSFs match well the FWHM of their corresponding measured LSFs. This is, however, not true for the right edge LSF. The measured right edge LSF is 59 - 5 -2.5 0 2.5 5 7.5 10 12.5 15 17.5 20 X (mm) Figure 3.10: Simulated LSFs superimposed over measured LSFs for the HR PLUS block detector under study. 60 broadened and is extended beyond the physical dimensions of the block by inward scattering 7s. Inward scattering is not modelled, thus the simulated edge LSF does not display this anomaly. The differences in the FWTMs can be attributed to crystal address miscodings associated to both the simulated and measured crystal ID LUT. VI. Discussion of Systematics Whenever one attempts to simulate a subset of reality, one must fully parameterize that subset of reality in order to satisfactorily describe it and simulate it. It is with these parameters, constrained by reality, that a simulation may have the freedom and flexibility to approach and reproduce that reality. In this section, the tweaking knobs of the model of the EXACT HR PLUS are examined, especially their influence upon the results. The BGO light yield relates the number of scintillation photons emitted at an interaction point, Lt-, to the energy deposited, E,-, by: L; = Ei x 8200 scintillation photons/MeV. (3.8) Thus, for a 511 keV 7 that is absorbed in a photoelectric interaction, approximately 4190 scintillation photons are produced. Changes in the light yield of BGO has no effect on relative crystal-by-crystal photopeak channel values, nor does it have an effect on the position reconstruction distribution. Decreasing (increasing) L,, however, will correspondingly increase (decrease) the FWHM of all the crystal energy spectra. The total mean free path, \ t , was originally set to 200 mm in accordance to [32], but was later revised to 2,000 mm with the advent of the latest reported figures [31]. This alteration greatly shifted the photopeak channels higher and forced the saw cuts deeper in order to maintain good resolution of the crystal peaks in the simulated flood 61 spectrum. The prominence of this effect is strongest as At increases from 200 mm but weakens gradually to about 1,000 mm at which the effect is barely appreciable. The principle component contributing to this phenomenon is the bulk absorption mean free path, A a . Increasing Xa allows photons to travel further, resulting in higher PMT counts as well as allowing photons to reach the distant PMTs causing the peaks in the simulated flood position spectrum to be pulled away from the edges of the block. To counteract this shift away from the block edges, saw cut recesses were required to be lowered thus trapping more photons within their quadrant of origin. The scattering mean free path, A s, does not exhibit the same strong influence on the photopeak channel nor on the flood spectrum. The reflection coefficient of the coat of diffuse reflector surrounding the BGO block was set to 95%. Decreasing this value squeezes the energy spectra to the low energy end and increases the Depth-Of-Interaction sensitivity of the block to values that disagree with experiment. At the expense of simplicity, the program DETECT could have been modified to allow for a more physical modelling of the surface finish of the HR PLUS. A more sophisticated model could have been designed with more adjustment parameters than the RC alone. However, tuning free parameters is a difficult task when the exact nature of the surface finish and coating is unknown. The most practical model then has the fewest free parameters and can easily be constrained by data. At least for the HR PLUS block, there are 3 constraints provided by experiment: 1. the measured depth dependence of the photopeak. 2. the crystal-by-crystal relative FWHM, 3. the crystal-by-crystal relative energy and position peak locations, Figure 3.4 showed that the standard ground finish with a reflection coefficient of 95% 62 convincingly reproduces the measured depth dependence of the HR PLUS blocks. In Section V, it was shown that this finish also performs well in satisfying constraint 3. However, ground alone fails to reproduce the measured relative FWHM. Adding a photon light loss adjustment parameter with a value of 0.47 improves the relative FWHMs of the simulation to acceptable values. This parameter accounts for the losses of photons due to a mismatch of the index of refraction for the crystals, the coupling grease and photomultiplier window together with the absorption of scintillation light in these materials. Although the ground option of DETECT is only an effective model of the surface finish of the blocks, it is sufficient to reproduce the data and thus sufficient for the simulation at this point. The saw cut recesses have an inherently direct impact on the flood spectrum since they are the mechanism by which the crystals may be resolved in the reconstructed X 7Y 7-space. The lowering and raising of the recesses affect the peak positions of the crystals bordered by the recesses. A raise causes the peak positions to be shifted in the direction of the recess whereas a drop shifts them away. The position peaks of the central crystals were found to have a high sensitivity to changes in the recess depths of the saw cuts that surround them. This sensitivity decreases for crystals further and further away from the centre of the block. Changes of fractions of a mm can produce noticable changes in the position peaks of the central crystals, while the outlying crystals are insensitive to changes smaller than a few mm. Only a certain range of combinations of the saw cut depths will give good resolution of the crystal peaks. Otherwise, peaks will be seen to merge and exhibit poor PVRs. If the cuts are consistently too deep then the peaks will be observed to be concentrated along the edges. If the cuts are consistently too high then the peaks will converge at the centre. Unfortunately, the optimized saw-cut depths cannot be presented here. The true recess values are CTI proprietary information and were not disclosed to the 63 author. It is believed from the accuracy of the model in predicting the position map PVRs, LSFs and photopeaks that the saw-cuts modelled in the simulation are in close agreement with reality and thus are not publishable information. Furthermore, the values of the saw-cut depths used in the simulation are of little scientific interest as they would not add to the substance of this work. The thickness of the glass plate used to model concurrently the PMT window and the coupling glue has a profound effect on the width and spread of the edge peaks of the flood spectrum. In fact, this thickness was adjusted by matching the edge peaks of the simulated flood spectrum to that measured. The thickness of the plate was optimized to 0.6 mm. Variations in thickness of the order of 0.25 mm produce observable changes in peak widths. The PMTs were initially placed at the centre of each quadrant and shifted by 0.5 mm transaxially away from the centreline. It was found that the relative photo-peak channels were better matched to measured values presented in Table 2.3 when the PMTs were further shifted axially 0.25 mm away from their respective centre-lines. The relative photopeak channels are highly dependent upon the positions of the PMTs and shifts of the order of 0.25 mm produce stark changes in these values. The Q.E. along with the BGO light yield, L, determine the FWHM of the energy distribution around the photopeak according to the rules of Poisson statistics. As far as the model is concerned, only their product can be constrained by the data. Veri-fication of this statement was achieved when it was discovered that the combination of Q.E. = 100% and L = 8200 x 0.125 produced results that are identical within statistics to results produced using a combination of Q.E. = 12.5% and L = 8200. 64 A study of the implications of setting a timeout at which the tracking of a scintil-lation photon is terminated was conducted. The purpose of a timeout is to decrease the computer processing time required for a simulation by setting a maximum flight time. This is based on the assumption that the cases that are aborted are ones in-volving photons that are internally trapped or photons that would eventually be lost and not be counted by the PMTs. Point sources of 1,000 scintillation photons were simulated at a depth of 1 mm in two crystals, one located at the centre of the block detector and one at the corner. DETECT requires the timeout to be specified as a measure of the maximum tracking distance in mm as opposed to units of time. The simulation was executed repeatedly for a wide range of values for the timeout, from distances of 100 mm to 10,000 mm. The resulting scaled total photon count for each of these simulated events is plotted against timeout in Figure 3.11. As ev-ident from the graph, for timeouts > 4,000 mm, there is insignificant change in the detected scintillation photon yield. However, at 4,000 mm, it was discovered that there is considerable difference in the position reconstruction values when compared to simulations with a timeout of 10,000 mm. This would indicate that a significant proportion of those aborted photons would have been detected by any of the other three PMTs not directly coupled to the bottom face of the crystal. A comparison between timeout values of 10,000 mm and 100,000 mm demonstrated that the differ-ence in position reconstruction values does not present a critical concern when the timeout reaches these distances. Thus a timeout value of 10,000 mm was chosen for the simulation. The full simulation of the block detector is rather CPU intensive. The worst case involves the optical tracking of 4190 scintillation photons from the photoelectric interaction of a 511 keV 7-ray. In that case, approximately 0.5 minutes of CPU time is needed on a DEC Station 3000/400. This corresponds to 2 minutes in real-time under 65 c o o c o o o CO 0.4 I 0.3 L • Centre Crystal A Corner Crystal J _ _ L 2 0 0 0 4 0 0 0 6000 8 0 0 0 10000 Timeout (mm) Figure 3.11: A study of the effects of timeout on the detected scintillation photon yield in two crystals of the block. normal system load. The four-fold symmetry of the block was exploited by folding the events into only one quadrant. A typical run consisting of a thousand interactions in each crystal of one quadrant then requires at minimum 4 days to complete when the tasks are distributed among 5 DEC Station 3000/400s. Absorbing the quantum efficiency into the generated photon counts, as described in section III, scales these figures down by a factor approximately equal to the quantum efficiency itself. 66 C H A P T E R 4 RADIAL ELONGATION Upon validation of the simulation platform through satisfactory reproduction of measured results, the platform can then be used as a research tool to investigate unan-swered questions regarding PET detectors. In the process of developing the platform, much about the physics of PET detectors was revealed, yet more are awaiting to be explored. Also the simulation platform can be exploited for numerous applications such as the proposal and investigation of alternate detector designs. One such ap-plication is studied in this chapter which concerns a modification to existing PET detectors to remove the problem of radial elongation, known as the parallax error, and hence, provide uniform radial resolution across the FOV of modern PET cam-eras. The factors determining the radial resolution in PET are first introduced in Section I along with an explanation of the nature of the radial elongation. A method that would allow correction of this systematic effect is presented in Section II. The re-sults of the simulation incorporated with the necessary modifications to the EXACT HR PLUS block design is finally presented in Section III. I. Radial Resolution and the Parallax Error The image resolution of a PET system is specified by quoting three parameters: the radial, tangential and axial resolutions. The radial resolution is measured radially across the FOV in a plane perpendicular to the longitudinal axis of the detector ring. 67 The tangential resolution is measured in the same plane tangential to a radial line at different radial distances. Finally, the axial resolution is measured along the longitu-dinal axis of the detector ring and depends upon the detector size, the inter-detector spacing in the axial direction and the collimation. This chapter will concentrate solely on radial resolution. In particular, the principal cause of its deterioration as the im-aged subject position departs from the FOV centre will be discussed and a method for its correction will be outlined. The radial resolution is determined by the width of the coincidence tubes defined by the possible detector pairs in the ring tomograph. As the imaged point moves towards the edge of the FOV, this tube becomes wider. The situation is illustrated in Figure 4.1. With reference to the figure, if the annihilation photons are emitted along the horizontal direction, they penetrate into the narrow crystals along their long axes and the width of the coincidence tube is equal to the crystal width. If the 7-rays are emitted in the vertical direction, one or both could stop in any one of five crystals that lie in their path. The coincidence tube is considerably wider in this case due to the so-called Depth-Of-Interaction (DOI) of the 7s. This results in a considerable degradation of the radial resolution at the edge of the FOV. Table 4.1 presents radial and tangential image resolutions for three distances from the centre of the FOV measured in a Siemens-CTI ECAT 953B PET scanner. It is clear that the tangential resolution suffers no degradation. However, there is a severe loss in radial resolution as the distance from the centre of the FOV increases. As the imaged source is moved towards the periphery of the FOV, the photon incident angles can have a wider and wider range. This increased obliqueness in the incident angles allows for an increase in frequency of multiple crystal penetrations producing the centre-side tail seen in the radial projection of Figure 4.1 as well as an 68 Tangential Projection Figure 4.1: The effect of oblique penetration upon the spatial resolution [12]. 69 Distance w.r.t. centre (cm) Tangential Image Res. FWHM (mm) Radial Image Res. FWHM (mm) 0.0 4.7 4.7 4.5 4.8 4.9 9.0 4.8 5.9 Table 4.1: Spatial image resolutions for various distances from the centre of the FOV measured in a Siemens-CTI ECAT 953B PET scanner [19]. offset in the peak position of the projection. This tail represents crystal miscodings and escalates in size as the radial position of the source increases. This crystal mis-coding is detrimental to image reconstruction which presumes and relies on the fact that the entrance crystal is identical to the detection crystal. Figure 4.2 shows an obliquely incident photon penetrating a distance d a into a pair of block detectors mounted on a tomograph ring gantry. From geometrical con-siderations, a relationship between the transverse displacement, Ax, the penetration distance, d a , the tomograph radius, R t o m o , and the radial distance, r, is derived: A x = c L x j ; _ 0 < r < R F O V , (4.1) tttomo where R F O V is the radius of the FOV. This equation shows explicitly that the trans-verse mis-positioning increases linearly with the radial distance, as well as the pene-tration distance, and reaches its extreme at the edge of the FOV. Hence, the radial elongation of an image depends on two factors: 7 penetration distance in BGO, d a , and the angle of incidence at the face of the crystal, 6. On average, a 511 keV 7 will penetrate BGO to a depth of about 1.1 cm. The larger the angle of incidence, the less distance a 7 needs to penetrate to enter an adjacent crystal. To get an idea of the size of incident angles encountered: a point 20 cm from 70 Figure 4.2: Obliquely penetrating photons in a pair of detectors in a ring tomograph gantry. the centre of a tomograph with a radius of 40 cm will subtend an angle of 30° when the LOR is perpendicular to the radial and the axial directions. II. A Correction for DOI Blurring There exist two basic approaches to dealing with the problem of radial elonga-tion. The first involves eliminating or minimizing radial elongation by altering the scanner gantry design or geometry. The second involves actual determination of the DOI of annihilation photons within the crystal. Examples of the first type include the insertion of lead or tungsten septa between crystals, a technique widely used in commercial PET scanners which is weakly effective. Others include implementation 71 of shorter crystals and increasing the diameter of the detector ring which consequently escalates the price for a commercial PET scanner. Unfortunately, every single one of these methods suffer the unwanted side-effect of overall reduction in sensitivity to the specific activity of the subject due to the reduction in the number of accepted events. A method [17] of the second type, which does not suffer this same set-back, is investigated in the remainder of this chapter. Figure 4.3 shows how knowledge of the DOI, Z, can allow the correction of radial elongation. With reference to the figure, an event may be initially assigned to crystal 5 in one block detector during processing. Using this crystal and the opposite crystal in coincidence, the angle of incidence, 0, may be estimated. The correction factor, Ax, can be determined from the measured Z and the estimated 6 from their trigonometric relationship: Ax = Z-tan(0). (4.2) Applying Ax reveals the true entrance crystal to be crystal 2. In this manner, radial resolution uniformity throughout the FOV could be achieved if there existed a means of determining the depth of the interaction point of the annihilation photon within a crystal. Over the past years, several attempts have been made to determine the DOI of the annihilation photon within the crystal of interaction. These include a scheme incorporating the decay time variation of scintillation light with temperature of the scintillation crystal and the application of a temperature gradient across the length of the crystal [35]. Though valid, this technique is difficult to implement in practice. Another method utilizes composite detectors made up of two or more scintillators each with different decay times [36, 37]. One method places a position sensitive photodiode along a transaxial face of a crystal [38]. A fourth method involves the 72 Back Face (PMT's) Front Face Figure 4.3: Correction of multiple crystal penetrating 7s. application of a black band around each crystal [39] thus segmenting each crystal into two depth regions. This method is rather difficult and expensive to implement in the manufacturing process. In comparison to the previous methods, the method proposed by [17] is a very cost-effective scheme. It does not require a complete re-design of the block detectors, nor does it require additional electronic hardware which would add to the commercial cost of a new tomograph. In addition, its implementation is quite simple and could easily be appended to the current existing manufacturing process. The technique involves the alteration of the standard surface treatment of all the crystals in the block to become "lossy" over a fraction, / , of its length. The exact steps and specifications of 73 this alteration are currently protected by a patent [40]. This modification induces a gradient in the light collection efficiency of the detector producing the desired DOI sensitivity. Consequently, allowing the DOI of 511 keV 7-rays to be simply estimated from the variation in the photopeak pulse height from the front to the back face of the block. The side-effect of this method is the decrease in light collection efficiency. However, this effect is not as detrimental as a reduction in overall gamma detection efficiency suffered by radial resolution improvement methods of the first type. The loss in light collection efficiency results in a deterioration of the accuracy of the usual determination of the energy as well as the x and y entrance coordinates of the 7s. The feasibility of the method, that is, the improvement in radial resolution mediated by the deterioration of the light collection efficiency, is investigated in the next section. III. Performance of New DOI Sensitive Blocks The necessary modifications to the simulation of the HR PLUS block detector were made as in [40]. Parameters of the modification were then adjusted and fine tuned until a point source test was able to reproduce measured results. Figure 4.4 shows the results of the test after the final values of the DOI modification parameters were obtained. The test involved the simulation of 29 point sources situated along the long axis of crystal B of Figure 2.6, located directly over a PMT. The point sources were spaced 1 mm apart and produced 10,000 scintillation photons each. The figure displays the peak pulse height channels for each of the 29 depths for both the modified and unmodified blocks. Experimental data [33] are concurrently displayed but at much fewer depth locations. The DOI response was experimentally measured, as in [17], from the peak pulse height as a function of the position of a fan beam of 511 keV 74 10 15 20 25 Depth ( m m ) Figure 4.4: The predictions of the simulation (histograms) for the DOI response of a crystal at the centre of a quadrant in the HR PLUS block before and after modification are compared to corresponding measured DOI responses. 7-rays incident on a side face of the block. The simulated results were normalized so that the peak pulse heights at the shallow depths matched measured values. The normalization of the modified and unmodified blocks were performed independently and the normalization factors used were found to be within 10% of each other. As evident from the figure, the simulation matches measured results exception-ally well. The simulation successfully predicts the measured DOI sensitivity [28] of S D O / = + 1 2 % and +45% observed in the crystal prior to and after its modification respectively. Most impressively, the simulation reproduces the shape of the depth performance. From the results of this test, the analysis and evaluation of the DOI 75 blurring correction method can be continued with full confidence in the implementa-tion of the modification to the HR PLUS block in the simulation. Attention was turned towards the analysis of the energy spectra for the modified block. Figure 4.5 showcases the simulated energy spectra for the four crystals A, B, C, and D of Figure 2.6 located along the diagonal of a quadrant of the modified The first row contains the results for a flood source incident on the top face. When compared to their unmodified counterparts, the peaks are noticeably smeared towards higher energies. In fact, there no longer exists a well-defined peak. These results are in agreement with what is observed in the experimentally measured data. Most remarkable, the distinct shape of the measured flood energy spectra [33] was reproduced by the simulation. For the remaining four rows, a side-incident fan beam was simulated at well-spaced depths of 1 mm, 6 mm, 15 mm and 24 mm. The energy spectra resembles in shape the energy spectra for the unmodified block with the flood source; the peaks are again sharp and well-defined. These photopeaks have been characterized by their channels relative to that of crystal B at a depth of 24 mm and by their FWHM relative to their own photopeak channels. Both these values are given in parentheses as percentages for each of the four crystals and for each of the four depths: From these values, a definite relationship between the relative FWHM and the DOI can be observed. The deeper the fan beam is placed, the smaller the relative FWHM becomes and this is the case for all crystals. More importantly, the positions of the peaks for each of the four crystals are observed to shift towards the right to higher energies as the depth increases. This is the DOI sensitivity sought by the HR PLUS block. (4.3) 76 (V o o to o o O A- Corner Crystol (10,123) 13,103) rCpS.SI) o (24,54) B-(52,28) 1 J L 1_ (65,26) (82,20) fl (100,14) JI JI (48,31) (60,29) (75,20) (86,16) JI D- Centrol Crystol (31,54) fl (40,37) (55,31) (76,21 i L ) 200 E^Photoelectrons) Figure 4.5: The simulated energy spectra for four crystals along the diagonal of a quadrant of a modified H R PLUS block. The energy spectra were taken for a flood source as well as for side-incident fan beams at four well-spread depths. 77 modification. This indicates that the modification does indeed induce a gradient in the light collection efficiency of the block detectors. Figure 4.6 and Figure 4.7 present the measured and simulated depth sensitivities, respectively, for the sixteen crystals of one quadrant of the block detector prior to modification (top line) and after modification (bottom line). Side-incident fan beams at depths of 2 mm, 12 mm, 16 mm, 20 mm, 22 mm, 24 mm, 26 mm and 28 mm were used to produce the measured results, whereas, depths of 1 mm, 6 mm, 15 mm and 24 mm were used in the simulation. The crystal-by-crystal photopeak channels are plotted against depth and these points are simply joined with straight lines. The simulation performs fairly well in reproducing the measured data. The shapes of the graphs are very close, however, the simulated data do not display the dip at the higher depths seen in the measured depth sensitivities. This is due to the crystal ID miscodings incurred by the real system, whereas the simulation presently utilizes cheat information to get perfect crystal IDs. The overall drop in the photopeak pulse heights after modification is well simulated. To quantify the induced depth sensitivities, SDOI ['28], from the modification, a value was assigned to each crystal determined by the difference in photopeak channels at the two depths of 26 mm and 0 mm relative to the photopeak channel at 26 mm: o P 2 6 — P o t A A \ SDoi = —5——• (4.4) A depth of 26 mm was chosen since at this depth the maximum pulse height is observed in the measured data. Differences between the measured and simulated depth sensitivities of the modified block range only from -0.06 to +0.171. These differences are believed to be due to crystal ID miscodings. 78 40 13 30 c c 20 o O O O < o CL o -t—> o JZ CL 10 0.456 _ J i i i i_ 0.466 -J I I L 0.430 _l I I I L 0.459 f_L 0.454 _J i i i i . .Q.483-I I I I I 0.510 J I I I L 0.556 0.437 0.497 J I I I L 0.540 J i i i i_ 0.573 0.482 _i i i i i_ 0.552 0.550 J I I I L 0.590 _l I I I u 0 15 30 Block Centre Depth-Of- lnteract ion (mm) Figure 4.6: Measured depth sensitivities for the 16 crystals in one quadrant of the HR PLUS block prior to (upper lines) and after (lower lines) modification. 79 40 "55 30 c c 20 o O 1 0 o 0 : 0.627 : 0.553 ~ I i i i I i : 0.541 : I , , , I , : 0.633 : I , , i I , ; 0.531 " I 1 1 1 I 1 " I i i i I i : _J^50JL--I i i i I i : ^C \649^ , : 1 , i i 1 , : 0.513 " I l i r I i : _ O 6 0 4 ^ : I , , , l , ; ^ O 4 8 0 _ : I , , , I , : 0.546 : 1 , , , 1 , ; 0.612 : I , , , I , ; ^ 6 0 3 ^ . : 1 , , , 1 , : 0.553 : I , , , i , : 0.627 : I , , , l , 0 15 30 Block Centre Depth-Of- lnteract ion (mm) Figure 4.7: Simulated depth sensitivities for the 16 crystals in one quadrant of the HR PLUS block prior to (upper lines) and after (lower lines) modification. 80 The simulation does well to reproduce measured results, but that is not the focus of this study. The aim is to demonstrate the feasibility of the correction method through a simulation standpoint, thus offering a second proof of feasibility as reinforcement. Up to this point, the simulation has managed to show that the DOI response for each crystal can be enhanced through a simple modification to the surface treatment of the crystals. Table 4.2 gives the photopeak channels (photoelectrons) for the modified block at depths of 0 mm and 30 mm using a line fitted to the simulated data as well as the slope, dP/dZ, of the line. Each of the 16 crystals, numbered from 0 (the block centre) to 15 (the corner of the block), is represented in the table. Crystal Po P 3 0 dP/dZ 0 45 130 2.83 1 56 136 2.67 2 65 148 2.77 3 37 102 2.17 4 59 137 2.60 5 75 147 2.40 6 78 158 2.67 7 53 113 2.00 8 65 150 2.83 9 77 158 2.70 10 78 171 3.10 ' 11 60 133 2.43 12 30 98 2.27 13 42 95 1.77 14 51 117 2.20 15 13 41 0.93 Table 4.2: Parameters taken from lines fitted to the Photon Count-Depth data for the 16 crystals in one quadrant of the modified block. To observe the scale of improvement afforded by the modification, the radial resolu-tion for an obliquely incident fan beam was investigated for uncorrected and corrected 81 DOI. The setup of Figure 4.2 was simulated along with an axially aligned fan beam produced at r = ^Ktomo and incident upon the block detector at 30° from the normal at 1 mm in from the edge of the block. Data from the simulation were analyzed to determine the transaxial resolution for this study and the results are exhibited in Figure 4.8. The radial resolution is then obtained from the transaxial resolution by simply scaling the transaxial resolution by a factor of cos 30°. i i I i i i i I i • i i i i i i i i 1 i i i i i - n T i i 1 i i i i 1 i i i i 1 i i 1 1 1 1 , 1 , , , , 1 , , , , I i , , , I i i , , I , i , ,' 1 , ±J*/t , , r , i . , , i , , k l 1 • i i t 1 i i i i 1 i t i i | 1 i I I :'l i i ij/x i i i , - 2 0 - 1 5 - 1 0 - 5 0 5 10 15 2 0 X (mm) Figure 4.8: Simulated transaxial coordinate distributions with several levels of DOI corrections: uncorrected (top), corrected (middle) and edge corrected (bottom). 82 The transaxial position centroid, X , weighted by the number of scintillation pho-tons emitted at each 7 interaction point was calculated for each incoming 7, and the events were binned according to X . The bins are 4.5 mm wide, approximately the width of a crystal in the HR PLUS block. The data was fitted with a gaussian with a FWHM of 17.1 mm as shown in the top figure. At this stage, the data is uncorrected for DOI and its analysis gives the radial resolution achievable by an unmodified block. The DOI, Z, of the gamma interactions were determined with the total photoelec-tron count, P, for each interaction and with data from Table 4.2. Z = ^ . (4.5) The table was indexed with the crystal ID obtained from the spatial centroid position. The transaxial displacement, Ax, was then calculated from: A x = Z x t a n 3 0 ° . (4.6) The transaxial position was subsequently corrected as follows: x' = x + Ax. (4.7) The corrected data was replotted in the middle figure of Figure 4.8 with the new DOI-corrected transaxial positions. Again, a gaussian was fitted to the data with a much improved FWHM of 7.5 mm. In the bottom plot of Figure 4.8, the data was further corrected for edge spill-overs. All events corrected to positions beyond the physical dimensions of the block were re-allocated to the block edge bin. A gaussian fitted to this new distribution exhibited a FWHM of 3.9 mm in accordance to that expected for normal incidence on edge crystals. Improvements are seen to range from a factor 2.28 to 4.38 in radial resolution. The results of this test demonstrate the DOI correction method's ability to improve the radial resolution of a tomograph. 83 To observe how the decrease in light collection efficiency has affected the transverse resolution at normal incidence, a comparison of the LSFs was performed between the modified and unmodified blocks for a normal incident fan beam. Figure 4.9 displays this comparison for a central column of crystals. § 1 O O c o ~0 0.8 sz Q_ ~o 8 0.6 GO 0.4 • Unmodified • Modified 0.2 h ° 2 0 - 1 5 - 1 0 - 5 0 10 15 20 X ( m m ) Figure 4.9: Simulated LSFs for a central column in the modified and unmodified block detectors. 84 The two LSFs almost coincide perfectly. The FWHMs are identical and the FWTM of the modified block is increased by 0.5 mm. These results are very en-couraging and indicate that the loss in transverse resolution due to the modification is insignificant for SDOI < 60%. 85 C H A P T E R 5 CONCLUDING R E M A R K S The initial aim of this work was to develop a model of the light transport in detectors for Positron Emission Tomography. This work has grown to also include a gamma transport model into a very concise simulation platform currently dedi-cated to the simulation of state-of-the-art multicrystal position encoding scintillation detectors. The principle feature of the platform is this interface between 7-ray in-teractions with the propagation and detection of scintillation photons. The former is determined by the bulk properties of the inorganic scintillator and allows to re-alistically model the deposition of energy in the active volume of the block through photoelectric and Compton interactions. The latter is dominated by the geometry and optical properties of the saw-cut multicrystal array as well as the PMT coupling scheme. Combining the two clearly provides an optimization tool more realistic than geometrical and optical optimization alone. The platform was put to a first test by comparing the predicted energy and position spectra to those measured for an ECAT EXACT HR PLUS block. With a simple and realistic treatment of the block geometry and optics, the relative features of the various crystal energy spectra were well reproduced. The measured crystal-by-crystal photopeak channels and energy resolutions were predicted to a good accuracy. Similarly, the simulated 2-D flood position calibration specturm reproduced detailed features of the measured one. The relative intensities of the peaks and valleys, as 86 well as the widths of the peaks in the position map were shown to agree well with measurements. Improvements upon the accuracy of the simulation platform can be achieved pri-marily through the specification of a more realistic surface treatment. This can only be obtained if exhaustive studies of the finish are performed and this requires coop-eration of the fabricants. Unfortunately, information like this are considered trade secrets and are not divulged to those outside their companies. Improvements may also be observed if the saw cuts were allowed to be asymmetric along the two axes in the simulation. Indeed, knowledge of the exact shape and depths of the saw cuts would be most helpful. Furthermore, the simulation platform would benefit greatly from a more detailed and realistic model of the BGO/PMT coupling medium. As a minor point, light leakage which has yet to be modelled, could finally be incorporated to produce a complete simulation platform. An application of the simulation platform towards the study of a block design ca-pable of simultaneously good transverse and Depth-Of-Interaction resolutions through a modification of the surface treatment of the HR PLUS was undertaken. The simula-tion platform was able to reproduce experimentally measured data to a good accuracy as well as offer proof that the DOI correction scheme is capable of improving radial resolution significantly while negligibly degrading the central resolution. Therefore the simulation was successful in verifying the feasibility of the proposed DOI correc-tion scheme. The potential of the simulation platform is opened to many possibilities. The prediction power of the platform can be exploited to investigate new designs of yet nonexistent block prototypes. Recently, fast inorganic scintillators with high light 87 yield, such as YAP [10], LSO [11] or LuAP [41] have been placed under study. A l -though the improvement in energy and time resolution is clear from measurements taken on small individual crystals, the overall performance of multicrystal arrays made of newly available scintillators is difficult to assess because of their relatively high cost per unit volume compared to the extensively used BGO. The design of multicrystal position encoding detectors based on these new high light yield inorganic scintillators would definitely be an important application. Block designs capable of simultaneously good transverse and Depth-Of-Interaction resolutions, such as the design studied in Chapter 4 and mentioned above or designs with novel readout schemes to achieve this, also deserve attention. The capacity of the platform to tackle 7-ray and light transport together makes it a good tool to study event miscoding in existing or future block designs which might lead eventually to energy and position encoding algorithms of better resolution. Finally, the platform is not restricted to multicrystal position encoding scintillation detectors nor is it restricted to PET. And so the single most important aspect of the simulation platform is its versatility and applicability which the author hopes will benefit detector physics throughout. 88 BIBLIOGRAPHY [1] Z.H. Cho, J.P. Jones and M. Singh. Foundations of Medical Imaging. John Wiley & Sons, New York, New York, U.S.A., 1993. [2] M.E. Casey. An Analysis of Counting Losses in Positron Emission Tomography. Ph.D. thesis, University of Tennessee, 1992. [3] Z.H. Cho et al. Positron Ranges Obtained from Biomedically Important Positron Emitting Radionuclides. Journal of Nuclear Medicine, Vol.16, No.12, 1975, (1174-1176). [4] W.R. Leo. Techniques for Nuclear and Particle Physics Experiments. Springer-Verlag, Berlin, Germany, 1987, (34, 50-54, 149, 157-158). [5] G.F. Knoll. Radiation Detection and Measurement. John Wiley &; Sons, New York, U.S.A., 1989, (227-230, 234-236). [6] E.J. Hoffman, M . Dahlbom, A.R. Ricci and I.N. Weinberg. Examination of the Role of Detection Systems in Quantitation and Image Quality in PET. IEEE Trans. Nucl. Sci., NS-33 420, 1986. [7] H. Ishibashi. Cerium Doped GSO Scintillator and its Applications to Position Sensitive Detectors. IEEE Trans. Nucl. Sci., NS-36 170, 1989. [8] I. Holl, E. Lorenz and G. Mageras. A Measurement of the Light Yield of Common Inorganic Scintillators. IEEE Trans. Nucl. Sci., NS-35 105, 1988. [9] K. Takagi and T. Fukazawa. Cerium activated Gd2SiOs single crystal scintilla-tors. Appl. Phys. Lett. 42, 1, 1983, (43-45). [10] S.I. Ziegler, J.G. Rogers, V. Selivanov and I. Sinitzin. Characteristics of New YAl03:Ce Compared with BGO and GSO. IEEE Trans. Nucl. Sci., NS-40 194, 1993. [11] C L . Melcher and J.S. Schweitzer. Cerium-doped Lutetium Oxyorthosilicate: A Fast, Efficient New Scintillator. IEEE Trans. Nucl. Sci., NS-39 502, 1992. [12] W.W. Moses, S.E. Derenzo and T.F. Budinger. PET Detector Modules Based on Novel Detector Technologies. Nuclear Instruments and Methods in Physics Research A 353, 1994, (189-194). 89 [13] D.W. Townsend and M. Defrise. Image Reconstruction Methods in Positron To-mography. CERN 93-02, Geneva, Switzerland, 1993, (9-11). [14] D.C. Solmon. The X-ray Transform. J. Math. Anal. Appl., 56, 1976, (61). [15] F. Natterer. The Mathematics of Computerized Tomography. John Wiley &.Sons, New York, New York, U.S.A., 1986. [16] C.J. Dykstra. The Use of Radon Transforms in Fully 3-Dimensional Positron Volume Imaging - A Feasibility Study. M.Sc. thesis, Simon Fraser University, 1986. [17] J.G. Rogers. A Method for Correcting the Depth-of-Interaction Blurring in PET Cameras. IEEE Trans. Med. Imag., MI-14 146, 1995. [18] Siemens Gammasonics, Inc. A High Performance Positron Emission Tomography System - ECAT Scanner. Order No. A91004-M2330-M005-01-4A00, 1991, (14). [19] T J . Spinks, T. Jones, D.L. Bailey, D.W. Townsend, S. Grootoonk, P.M. Bloom-field, M.C. Gilardi, M.E. Casey, B. Sipe and J. Reed. Physical Performance of a Positron Tomograph for Brain Imaging with Retractable Septa. Phys. Med. Biol., Vol. 37, No. 8, 1992, (1637-1655). [20] S.R. Cherry, M.P. Tornai, CS . Levin, S. Siegel and E.J. Hoffman. A Comparison of PET Detector Modules Employing Rectangular and Round Photomultiplier Tubes. To appear in the Record of the 1994 IEEE Medical Imaging Conference of Norfolk, Virginia. [21] M.E. Casey and R. Nutt. A Multicrystal Two-Dimensional BGO Detector System for Positron Emission Tomography. IEEE Trans. Nucl. Sci., NS-33 460, 1986. [22] J.G.Rogers, A.J.Taylor, M.F.Rahimi, R.Nutt, M.Andreaco and C.W.Williams. An Improved Multicrystal 2-D BGO Detector for PET. IEEE Trans. Nucl. Sci., NS-39 1063, 1992. [23] G. Tsang, C. Moisan and J.G. Rogers. A Simulation Platform for the Design of Position Encoding Multicrystal Detectors. Submitted to IEEE Trans. Nucl. Sci., 1995. [24] J.G. Rogers, R. Nutt, M . Andreaco and C W . Williams. Testing 144- and 256-Crystal BGO Block Detectors. IEEE Trans. Nucl. Sci., NS-41 1423, 1994. [25] S.I. Ziegler, H. Ostertag, W.K. Kuebler and W.J. Lorenz. Effects of Scintilla-tion Light Collection on the Time Resolution of a Time-Of-Flight Detector for Annihilation Quanta. IEEE Trans. Nucl. Sci., NS-37 574, 1990. [26] G. Tzanakos, M . Bhatia and S. Pavlopoulos. A Monte Carlo Study of the Timing Resolution of BaF2 for TOF PET. IEEE Nucl. Sci., NS-37 1599, 1990. [27] G.F. Knoll, T.F. Knoll and T.M. Henderson. Light Collection in Scintillation Detector Composites for Neutron Detection. IEEE Trans. Nucl. Sci., NS-35 872, 1988. 90 C. Moisan, J.G. Rogers, K.R. Buckley, T.J. Ruth, M.W. Stazyk and G. Tsang. Design Studies of a Depth Encoding Large Aperture PET Camera. Submitted to IEEE Trans. Nucl. Sci., 1994. R. Brun, F. Bruyant, M . Maire, A.C. McPherson and P. Zanarini. GEANT3. CERN Data Handling Division, DD/EE/84-1, September 1987. J.R. Meyer-Arendt. Introduction to Classical and Modern Optics. Prentice Hall, Englewood Cliffs, New Jersey, U.S.A., 1989, (289-292, 372). Private communication, CTI PET Systems, Inc., Knoxville, Tennessee, USA. S.E. Derenzo and J.K. Riles. Monte Carlo Calculation of the Optical Coupling Between Bismuth Germanate Crystals and Photomultiplier Tubes. IEEE Trans. Nucl. Sci., NS-29 191, 1982. J.G. Rogers, C. Moisan, K.R. Buckley, M.S. Andreaco, C W . Williams and R. Nutt. A Practical Block Detector for a Depth Encoding PET Camera. Sum-mary submitted to the 1995 IEEE Medical Imaging Conference of San Francisco, California. Hamamatsu Photonics K.K. Photomultiplier Tubes and Assemblies for Scintilla-tion Counting And High Energy Physics. Cat. No. TPMO0001E03, March 1994, (28, 69). J.S. Karp and M.E. Daube-Witherspoon. Depth of Interaction Determination in Na(Tl) and BGO Scintillation Crystals using a Temperature Gradient. Nuclear Instruments and Methods in Physics Research, A260, 1987, (509-517). W.H. Wong. Designing a Stratified Detection System for PET Cameras. IEEE Trans. Nucl. Sci., NS33 591, 1986. C. Carrier, C. Martel, D. Schmitt and R. Lecomte. Design of a High Resolution Positron Emission Tomograph using Solid State Scintillation Detectors. IEEE Trans. Nucl. Sci., NS35 685, 1988. S.E. Derenzo, W.W. Moses, H.G. Jackson, B.T. Turko, J.L. Cahoon, A.B. Geyer and T. Vuletich. Initial Characterization of a Position-Sensitive Photodiode/BGO Detector for PET. IEEE Trans. Nucl. Sci., NS36 1084, 1989. P. Bartzakos and C J . Thompson. A Depth Encoded PET Detector. IEEE Trans. Nucl. Sci., NS38 732, 1991. J.G. Rogers. Gamma-ray Detector for Three Dimensional Position Encoding. U.S. patent #8145143, 1994. W.W. Moses, S.E. Derenzo, A. Fyodorov, M . Korzhik, A. Gektin and B. Minkov. LuAlOz'-Ce- A High Density, High Speed Scintillator for Gamma Detection. To appear in the Record of the 1994 IEEE Medical Imaging Conference of Norfolk, Virginia. 91 

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