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Crystal design simulation for a high resolution depth encoding pet tomograph Astakhov, Vadim 2003

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C R Y S T A L DESIGN SIMULATION FOR A HIGH RESOLUTION DEPTH ENCODING PET TOMOGRAPH by V A D I M A S T A K H O V B.Sc, Novosibirsk Russian State University, 1992 M . S c , Novosibirsk Russian State University, 1995 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE F A C U L T Y OF G R A D U A T E STUDIES (Department Physics and Astronomy) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH C O L U M B I A March 2003 @ Vadim Astakhov, 2003 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 The University of British Columbia Vancouver, Canada DE-6 (2/88) Abstract Position Emission Tomography is a functional imaging modality where positron labeled radiotracers are used to investigate biological processes. The imaging process occurs via simultaneous detection of two 511keV gamma rays originating from positron annihilation. A PET camera is a y detection apparatus. Reconstruction algorithms are used to reconstruct the original radioactivity source distribution in the camera field of view (FOV) from the simultaneous detection of y rays originating from the same annihilations. PET has been extensively used to investigate function in living organism, especially in human subjects. In order to make the detection process efficient and useful, PET camera designs strive for high detection sensitivity and high resolution. One of the factors influencing the resolution is the size of the detectors. Smaller detectors lead to a better spatial resolution. On the other hand sensitivity is affected by the detector crystal composition and by the solid angle subtended by the detection apparatus. An ideal tomograph design will therefore involve small, efficient detectors placed as close as possible to the object being scanned. The work described in this thesis examines various detector crystal configurations that would lead to an optimum tomograph performance. In order to make the results of this study immediately relevant to the PET community the overall tomograph geometry was constrained to that which is currently being built by a tomograph manufacturing company CTI. This design consists of an octagonal detector configuration where each detector head is built with two layers of detector material. Such a design allows for the identification of the y depth of interaction (DOI) in the detector assembly which in turn allows to minimize the effect of the parallax error and thus contributes to an increased resolution uniformity across the camera FOV. The studies presented here examine the effect of different crystal layer configuration on resolution and sensitivity. Octagonal HRRT geometry was also compared to circular detector geometry. As part of a system design optimization, several novel methods for crystal element identification were investigated: Genetic-algorithm, neural network algorithm and "simple" geometric algorithm were tested and showed relatively equal identification performance in identifying 64x64 crystal elements of each layer. A fuzzy-logic approach for estimation of depth-of-interaction (DOI) was investigated and compared with the decay time discrimination approach. The simulation results were used to generate a Look-Up-Table (LUT) that is accessed during simulated data acquisition for an effective and quick crystal identification. ii A correct crystal identification also facilitated an improvement of the capability for accurate energy discrimination, since the detector gain and appropriate energy thresholds were considered on an element-by-element basis by accessing energy LUT. The final result of the work presented in this thesis is the determination of the effect of DOI correction on resolution uniformity for different crystal configuration. DOI correction was found to improve the resolution uniformity up to 67%. iii T A B L E OF C O N T E N T S Abstract ii Table of Contents iv List of Tables vii List of Figures ix Acknowledgements x 1. Introduction 1 1.1 Goal of this Thesis 1 1.2 Thesis outline 1 1.3 Motivation 1 1.4 Previous work 3 2. Principles of Positron Emission Tomography 4 2.1.1 Positron Emission Tomography 4 2.1.2 Positron-emitters and labeled compounds used in PET 4 2.1.3 Basics Principles of PET 6 2.1.4 Positron Decay X- 6 2.1.5 Loss of energy by the positron 7 2.1.6 Positron Annihilation 7 2.1.7 Interaction of the gamma rays 8 2.1.8 Scintillation 9 2.1.9 Coincidence detection 9 2.2 Sensitivity 10 2.3 Spatial resolution in PET 11 2.3.1 Physical limits of spatial resolution 11 2.4 Detector block and intrinsic detector resolution 12 2.5 Interaction between detector design resolution and sensitivity 18 3. Overview of the HRRT system 14 3.1 Detector Geometry 14 3.2 Simulated Scintillators 15 iv 4. Simulation study 17 4.1 GEANT overview 17 4.2 Summary of all simulated by GEANT physical processes 18 4.2.1 Simulated interactions of an 511 keV photon in scintillators material 19 4.2.2 Simulated continuing photon and recoil electron transport 19 4.2.3 Cut-off energies 19 4.2.4 Simulated creation of bremsstrahlung photons 20 4.2.5 Implemented Mechanism of scintillation 20 4.2.6 Photon Interactions NOT simulated 21 4.3 DETECT 22 4.3.1 Scintillation light propagation simulation 22 4.3.2 Scintillation of scintillation light detection by PMTs 23 4.4 Simulation of Photomultiplier tubes (PMTs) 23 4.5 Simulation of the surface of Crystals 25 4.6 Crystal position identification 25 4.6.1 Saw-cuts estimation 26 4.6.2 Transformed Anger Logic 28 4.6.3 Position Map Look-Up table 30 4.6.4 "Simplest Algorithm"(General steps) 30 4.6.5 Neural Net algorithm 31 4.6.6 "Genetic algorithm" (General steps) 32 4.6.7 Algorithm summary 34 4.7 Depth of Interaction information 35 4.7.1 Depth of Interaction Algorithm 35 4.7.2 Time discrimination Approach 35 4.7.3 Fuzzy Logic Approach 36 4.8 DOI estimation algorithms summary 43 5. Results 44 5.1 Energy Resolution 46 5.2 Detector efficiency 48 5.3 Miss position identification 48 5.4 Spatial Resolution 50 v 6. Validation of the simulations 56 7. Discussion 57 8. Conclusion 58 Bibliography 59 vi List of Tables 1. Table 1. Summary of some of the positron emitters and labeled compounds used in PET. 2. Table 2. Maximum energy imparted to the positron following the decay of its parent nucleus. 3. Table 3. Physical characteristics of simulated inorganic scintillators. 4. Table 4 Summary of algorithms. 5. TableS Data from CTI technical specification. 6. Table 6 LSO/GSO (7.5 mm-7.5 mm) Energy Resolution (%) 7. Table 7LSO/GSO (10.0 mm-10.0 mm) Energy Resolution (%) 8. Table 8 Energy resolution for different crystal configurations. 9. Table 9 Events stopped in crystals. 10. Table 10 Events correctly identified by Position Map. 11. Table 11 DOI miss position identification. 12. Table 12 HRRTgeometry LOR spatial resolution. 13. Table 13 Round geometry LOR spatial resolution. 14. Table 14 Spatial resolution for the most oblique LOR for three source positions. Data are shown in the format a/b, where a is the resolution obtained with DOI correction and b is the resolution obtained without DOI correction. All values are expressed in mm. 15. Table 15. Overall spatial resolution, for three source positions. Data are shown in the format a/b, where a is the resolution obtained with DOI correction and b is the resolution obtained without DOI correction. All values are expressed in mm. 16. Table 16. Spatial resolution for ECAT 953B. Vll List of Figures 1. Figure 1. What does a PET scan show? 2. Figure 2 Parallax errors. 3. Figure 3 Here the example of 18F-fluorodeoxyglucose molecule that mostly often used for glucose metabolism study. 4. Figure 4. Principles of the detection of coincident annihilation gamma rays. 5. Figure 5. Positron annihilation. 6. Figure 6 Compton scatter. 7. Figure 7 A schematic drawing of the detector block, top "face" view. And coordinate map setup for detector block surface. 8. Figure 8. HRRTgeometry 9. Figure 9. Detector block. 10. Figure 10. Simulated by GEANT propagation a 511kev photon through a media. 11. Figure 11. Algorithm diagram of simulation 511 keV gamma ray interaction in the detector block. 12. Figure 12 Interface between the Algorithm and the PET detector Simulator (GEANT/DETECT). 13. Figure 13 Distribution of events hitting a detector crystals. 14. Figure 14 Detected events distribution. 15. Figure 15 Position Maps. 16. Figure 16 Position Maps. 17. Figure 17 Real position map (CTI). 18. Figure 18. Signal distribution. 19. Figure 29. Chromosome in the Genetic algorithm. 20. Figure 20. Coincident-Event Correction. 21. Figure 21 Decay time distribution. 22. Figure 22 Decay time and energy distribution 23. Figure 23 Fuzzy Logic Model. 24. Figure 24 Degree of confidence. 25. Figure 25 Logical Space. 26. Figure 26 Defuzzification process. 2 7. Figure 2 7 DOI estimation dynamic generator for arbitrary number of layers. 28. Figure 28 Data flow chart for energy discrimination and spatial resolution. 29. Figure 29 Simulated LORs for the source located at center of the FOV, 5cm and 10cm off center for the octagonal geometry. 30. Figure 30LSO/LSO (15mm) 7.5mm-7.5 mm 31. Figure 31 Energy distribution for LSO/GSO(15mm) viii 32. Figure 32 Coincidence detection efficiency as a function of total crystal thickness (equal depth layers were used). Squares: LSO/GSO, triangles: LSO/LSO, x: LSO/GSO with an energy window of 350-650 keV and circles: LSO/LSO for the same energy window. 33. Figure 33. Represent source shifted Wsm left and LOR uniformly distributed over left top detector sector. 34. Figure 34. Source profiles without DOI correction. Squares: events stopped in the top layer, triangles: events stopped in the bottom layer, rombs: events, where one /-ray stopped in the top and the other one in the bottom layer and x: total profile 35. Figure 35. Source profiles after DOI correction. Squares: events stopped in the top layer, triangles: events stopped in the bottom GSO layer, rombs: events, where one y-ray stopped in the top and the other one in the bottom layer and x: total profile. ix Acknowledgements I would like acknowledging the contributions of my friends and colleagues at the Positron Emission Group at the Pacific Parkinson's Research Centre and at the TRIUMF National Laboratory for Particle and Nuclear Physics. I am very grateful to my supervisor, Dr. Vesna Sossi, for her encouragement, patience, and guidance over the course of this very exciting project. I am greatly indebted to Dr. Peter Gumplinger, his help in simulation set up was very important to start simulation study. Many thanks to Dr.Tom Ruth who involved me in this interesting project. x Chapter 1 1. Introduction 1.1 Goal of this Thesis This thesis will present a crystal design simulation for a High Resolution Research PET tomograph (HRRT) with depth of interaction encoding capability. The general goal of this research was to estimate an optimal crystal design for the HRRT in terms of detector efficiency, energy and space resolution. To reach this goal various parameters were simulated such as types of scintillation materials, geometric design and depth of interaction information. 1.2 Thesis Outline To understand the problems and the significance of the approaches applied to them, some background in PET is required. After an introduction in Chapter-1 a short summary of this will be presented in this Chapter-2, followed by a brief overview of the HRRT system in Chapter-3. Exposition of the general software parts will be described in Chapter-4. The simulation and validation of the results are present in Chapter-5 and Chapter-6. The discussion section Chapter-7 will examine the results and propose which of the detector block designs provides optimal data acquisition. The conclusion is in Chapter-8. 1.3 Motivation In order to improve the capability of PET to investigate human brain functions such as blood flow, metabolism and receptor characteristics, the spatial resolution and the y detection sensitivity has to be improved relative to what is available today. Figure 1. What does a PET scan show? Image produced by Positron Emission Group at the Pacific Parkinson's Research Centre. The brain function being studied during a PET scan determines which radiopharmaceutical is used. Oxygen-15 can be used to label oxygen gas for the study of oxygen metabolism, carbon monoxide for the study of blood volume, or water for the study of blood flow in the brain. Similarly, fluorine-18 is attached to a glucose molecule to produce FDG for use in the observation of the brain's sugar metabolism. Many more PET radiopharmaceuticals exist, and research is under way to develop still more to assist in the exploration of the working human brain. For example, dopa, a chemical active in brain cells, is labeled with positron-emitting fluorine or carbon and applied in research on the communication between certain brain cells which are diseased, as in dystonia, Parkinson's disease, or schizophrenia. 1 In order to meet this goal a new next generation high-resolution 3D-only brain PET scanner has been developed at CTI using a new scintillation material LSO (Lu(i_x) Ce 2 x (Si04)0). To improve the sensitivity in modern high-resolution brain PET system, it is necessary to reduce the detector radius to increase solid angle and place detectors as close as possible to the object being scanned. But for events coming from a source out of the detector center, a small detector radius leads to an increase of the number of events hitting the detector surface at a non-normal angle. These events have a high probability to be detected in a crystal that neighbors the one where they first hit the detector; this leads to a mis-identification of the crystal position. This is known as "parallax error" and leads to worse spatial resolution for off-center events. Figure 2 Parallax errors. Annihilations ofpositrons out off center of the detector will produce 51 lkev gamma photons that will hit the detector block at a non- normal That can leads to situations when angle a photon hitting a certain crystal will stop in a neighbor crystal. In that case, we will see the event shifted from its real position its real position. This possible shift will degrade spatial resolution for off-center regions. These mis-identifications of the position is called the parallax error. The depth of the y ray interaction (DOT) information is needed for correction of this spatial resolution degradation. This requirements is led to development of two-layer detector block designs like LSO/GSO (GSO - Gd2SiOs) where the two molecules have different scintillation decay times. The separation of the events occurring in LSO and the GSO layer by pulse shape discrimination allows discrete DOI information to be obtained. This type of tomograph is just currently being built by CPS (Knoxville, TN, USA) and it goes under the name of High resolution Research tomograph (HRRT). The HRRT is the first human size tomograph with DOI. At the time this thesis work was started the final design for HRRT was being finalized. It was therefore important to perform simulations to determine how much the DOI determination was going to improve the resolution uniformity. 2 1.4 Previous W o r k Over the past decade, new architectures and designs for high performance PET systems have been investigated. A design study of a depth encoding large aperture PET camera was performed at TRIUMF in 1995 [1]. This study showed the fundamental importance of the depth of interaction information (DOI) to correct the resolution. Simulation study for practical BGO ( Bi4(Ge04) )block detector with depth-encoding capability was performed in the works [2,3]. This study showed that up to 38% of image degradation resolution could be removed by DOI corrections. Also, DOI made the resolution nearly uniform across the useful field of view (FOV). In the last a few years new multiple layers design with new type of scintillator (LSO) was proposed. Preliminary studies were performed [4,5] where these designs have been simulated and first evaluation measurements were obtained. These results showed a significant improvement compare to old PET cameras designs. Different DOI estimation and corrections methods were investigated by various researchers around the world such as Critical On-line DOI Rebinning Method [13], Berkeley design [12] and application neuro-fuzzy approach [14] to DOI correction. None of these methods has been implemented in a human size scanner yet. 3 Chapter 2 Principles of Positron Emission Tomography 2.1.1 Positron Emission Tomography Positron Emission Tomography (PET) is an imaging modality that measures the spatial distribution of substances labeled with positron emitters. The potential for using positron emitters for imaging has been recognized since the 1950s when Brownell and Sweet localized brain tumors using the first practical positron camera. Since that time, PET has been used to visualize a host of physiological processes, including blood flow, substrate metabolism, and the binding of minute concentrations of various agents to receptor sites. PET has been used extensively in medical research, providing new insight into disorders including Parkinson's disease, epilepsy, schizophrenia, Alzheimer's disease, and depression. A better understanding of cardiac disorders and stroke has been obtained using PET. Application in neuropsychological studies are elucidating the response of the brain to various stimuli in terms of blood flow, blood volume or glucose metabolism. In addition, using animal subjects, new pharmaceuticals can be evaluated and screened in vivo. Finally, PET has been used successfully to monitor the metabolism and treatment of various cancers. PET, unlike some imaging technologies like MRI and X-rays, collects functional, rather than anatomical information which can be used to probe into the biochemical function of specific cells or to extract information about kinetic parameters of metabolic pathways. The physical basis for PET is fundamentally simple. The patient is injected with tracer amounts of a chemical compound, which have been labeled with a positron-emitting radioactive isotope. The tracer is either a close analogue of the endogenous species present in the body, and can be used to study a metabolic pathway, or has a high specificity for a particular binding site whose population is of interest. For example, glucose metabolism in the body is studied with 8F-deoxygluxose. The compound is 1 S similar to glucose except that one of the Hi has been replaced with the F isotope and 1 S one of the -OH groups is missing. The F allows the detection of the tracer in the body through the use of PET and the missing -OH traps the tracer in a specific metabolic step. Once the tracer has had time to diffuse to the organ of interest, the patient is placed in the detector and the tracer activity distribution is monitored with the scanner. 2.1.2 Positron-emitters and labeled compounds used in P E T The positron-emitters used in PET are isotopes of some of the most prevalent elements found in living systems (carbon, oxygen, and nitrogen). For this reason they can be used to synthesize various compounds that mimic the function of substances found physiologically. The most common radioisotopes used in PET, their half-lives (t 1/2), and some of the useful labeled compounds produced are listed in Table 1. These radioisotopes are typically produced in medical cyclotrons by bombarding stable isotopes with 4 positively charged particles (proton or deuterons). The nuclear reactions employed for each isotope are also listed in Table 1. Table 1. Summary of some of the positron emitters and labeled compounds used in PET. Isotope t >/2 (minutes) Nuclear Reactions used Labeled Compounds Application in PET imaging M C 20 10B(d,n)"C 14N(p,a)nC "CO u c-raclopride Cerebral blood volume Dopamine D 2 receptor 10 12C(d,n)1JN 160(p,a)!3N 1JN-amino acids 1 3 NH 3 Amino acid metabolism Cerebral blood flow 2 I4N(d,n)i:,0 15N(p,n)150 H 2 '-O, C 1 5 0 2 c 1 5 o 2 Cerebral blood flow Oxygen consumption Cerebral blood volume lop 110 180(p,n) lsF 20N(d,a)18F fluorodeoxy glucose f 8 F -fluorodopa mine Glucose metabolism Dopaminergic system The labeled compound is administered by injection or inhalation and enters the blood stream of the subject. The subsequent distribution of the substance throughout the body then closely imitates that of its naturally occurring analogue. Figure 3 Here is the example of 18F-fluorodeoxyglucose molecule that is mostly often used for glucose metabolism study. 2 fiuoro 2-deoxyD-gIucose "FOG" 5 2.1.3 Basic Principles of PET Conventional PET scanners consist of one or more rings of detectors surrounding the subject as shown in Figure 4. Each detector is composed of an arrangement of dense, high effective- atomic-number (Z) scintillation crystals optically coupled to photo-multipliers tubes (PMT). Figure 4. Principles of the detection of coincident annihilation gamma rays. Isotope distribution Channel 1 Channel 2 c o i n c i d e n c e e v e n t s After an amount of the administered positron-emitter has been distributed through the region of the body that is going to be imaged, a number of physical processes occur before the location of this activity is detected. Below, the underlying principles of PET are summarized by following these events in order, from the decay of the nucleus to the detection of coincident annihilation gamma rays. 2.1.4 Positron Decay The common feature shared between the radioisotopes listed in Table-1 is a surplus of nuclear positive charge, making the nuclei unstable. The nucleus decays to a lower energy state by emitting the net positive charge in the form of a positron. This process also results in the conversion of a proton to a neutron, and the emission of a neutrino: p+ -> n + e+ + v + energy (1) Both charge and lepton numbers are conserved in this process. The energy released is shared between the positron and the neutrino. The proportion of the total energy imparted to the positron as kinetic energy is a stochastic quantity, but it is described in terms of a maximal value, E m a x depends on the positron emitter used, as indicated in Table 2. The average value of kinetic energy received by the positron is approximately equal to 1/3 E m a x . 6 Table 2. Maximum energy imparted to the positron following the decay of its parent nucleus. Radioisotope Maximum Positron Energy (Emax) "C 0.97 Mev 1.19 Mev lio 1.70 Mev 0.64 Mev 2.1.5 Loss of energy by the positron Positrons are emitted isotropically from the parent nucleus and travel several millimeters in the medium while continually losing energy through excitation and ionization interactions. A small amount of energy (-1%) is also lost in the form of bremsstrahlung radiation. The distance over which the positron travels is dependent upon the initial kinetic energy and the density of the medium into which it is emitted. 2.1.6 Positron annihilation The probability for in-flight annihilation of positron is very low. After duration of approximately 10"9 s, the positron slows down to thermal energy and annihilates with an electron. In positron annihilation, the slow positron combines with a loosely bound electron located in one of the shells of an atom in the medium. The combined mass of the two particles (two times 511.1 Mev/c2) is entirely converted into energy, and two annihilation gamma rays are emitted. Figure 5.Positron Emission Tomography 7 The gamma rays emerge from the site of annihilation in approximately opposite directions. The line connecting two coincidently firing detectors is called "line of response"-LOR. This process must conserve both energy and momentum. It is unlikely, however, that the net momentum of the e+ e- system is zero just before the annihilation occurs, since the typical energy of the positron at this time is 10 eV, and the electron is orbiting in an atomic shell. In order to conserve momentum, the annihilation gamma rays are slightly non-collinear. The deviation of the angular separation of the gamma rays is described by a Gaussian distribution in water-equivalent materials, with a full-width-half-maximum (FWHM) of approximately 0.5° . 2.1.7 Interaction of the 511 kev gamma rays in tissue and in detector material. The 511 kev gamma rays interact with the matter through which they pass. The matter includes both the surrounding material of the subject (i.e. tissue, fat and bone) and the dense scintillation crystal in the PET detector. At the energy of 511 keV the probability for coherent (Rayleigh) scattering is negligible, and the energy is insufficient for pair or triplet production to occur. Hence, the dominant interactions of the annihilation gamma rays are Compton (incoherent) scattering and photoelectric absorption. Compton Scattering Compton scattering is an interaction that occurs between an incident photon of energy hv and an electron in the medium. Energy is transferred to the electron, and it recoils along an angle § from the initial trajectory of the photon with energy E'. The photon scatters through an angle of 0 and emerges with a reduced energy, hv' according to: hv hv' = (2) 1 + a(l-cosO) where hv a = m e c 2 Where c is the speed of light and me is electron mass. The Compton attenuation coefficient (a) is nearly independent of the atomic number of the medium in which the photon interacts (8), but increases with electron density. For photons in the energy range of interest in PET, this is the most predominant interaction in tissue. For a 511 keV photon in tissue, the relative probability for Compton scattering (given by o7p x 100%, where u. is the total attenuation coefficient) is approximately 99.7% (9). For a 511 keV photons in detection material-scintillator the relative probability for Compton is approximately 40-60 % and rapidly decreases to almost zero at an energy lower then 100-50 keV. 8 Photoelectric Absorption In photoelectric absorption, the entire energy of the incident photon is transferred to an electron ejected from the K, L, M or N shell of an atom and to either characteristic radiation photons or Auger electrons. The energy of the ejection photoelectron is hv - Ebind, where E bind is the binding energy of the shell in which it originated. The photoelectric cross section varies with photon energy according to, approximately, l/(hv) , and with atomic roughly according to Z 3 or Z 3 8 for low-Z materials (8). In the detection material-scintillatyor, the probability of photoelectric absorption is almost 100% at energy 50 keV and comes down to approximately 40-60 % at energy 511 keV. 2.1.8 Scintillation If a gamma ray escapes the volume of the patient and is incident upon a PET detector, it may deposit energy within the dense, high-Z scintillating crystal through the interactions outlined above. In LSO scintillator the relative probability of photoelectric absorption of 511 keV photon (x/u. X100%, where x is the photoelectric attenuation coefficient and where p is the total attenuation coefficient) is approximately 45%, while the relative probability for Compton scattering is approximately 55%o. The absorption of energy by the crystal is followed by scintillation, or the emission of light. This light is detected by the photomultiplier tube to which the crystal is coupled, and its detection is discussed in some details later in this chapter. 2.1.9 Coincidence detection As explained below in the discussion of PET detectors, the scintillation light produced is detected by the PMT, and converted into an electrical signal. Two gamma rays are considered to be temporally coincident if the detected signals arrive within a specified time interval, determined by the coincidence resolving time of the system (typically between 3-20 ns). Figure 6 Compton scatter. 9 The possibility of Compton scatter of the gamma rays was already discussed early. Scatter can occur both in tissue and in the detector. Because one or both gamma rays may have undergone scattering before coincidence detection, these events degrade the image quality and reduce the accuracy of quantitative measurements (10). In addition, accidental coincidences occur, corresponding to the coincident detection of two gamma rays originating from separate annihilations. Thus, the total count rate ' R ' is the sum of the true ( R t ) , scatter ( R s ) and accidental rate ( R « ) : R = R , + R s + R a (3) The true coincidence rate, R t , is equal to the product of the positron emission-rate of the nuclide (R p 0 si t rons)> the efficiencies of the detectors ( € A and G B ) , the geometric efficiency of a detector (g) and a factor accounting for attenuation of gamma rays through a thickness L of matter with an attenuation coefficient p.: Rt = Rpositrons G A G B g e (4) The detector efficiency (e) refers to the fraction of incident gamma rays that deposit a sufficient amount of energy so that they are detected. The geometric efficiency (g) is the ratio of the surface area of the detectors divided the area of a sphere of radius r equal to one-half of the detector separation. The accidental coincidence count rate for a given detector pair is determined by the coincidence resolving time of the detectors (x) and the singles count rates for the detectors ( R A s i n g and R e s m g ) : R a = 2T RAsing RBsing (5) The scatter count rate is the rate of detecting both gamma rays from the same annihilation in coincidence, where at least one gamma ray undergoes Compton scatter before detection. Unlike the accidental rate, the scatter count rate cannot be reduced by shortening the detector coincidence resolving time. The amount of scatter is highly dependent on the scanner used and on the size of the object being scanned. For a source located near the midpoint of two detectors, it is: R s = K Rpositrons G A G B g (6) The factor K is derived empirically and accounts for various geometrical and scanner design parameters. This is the factor that account for the size of the object. If the source is in the air that mean we don't have object then there is no that scatter factor. And we have just scatter in the detector material. The detection g scattered events is reduced by using energy discrimination, where the only events with energies located within specified range around the energy spectrum photopeak are accepted. 10 2.2 Sensitivity Sensitivity is defined by ratio between the number of detected events and the number of decays and it depends on several important criteria. The first is attenuation of the 511kev gamma photons in the detector and the second is the production of a detectable number of scintillation photons at a wavelength to which PMT is sensitive, following the deposition of energy by gamma rays. The time scale of this emission is also crucial, since it largely determines the count rate capabilities of the scanner, the coincidence resolving time, and in turn, the accidental coincidence rate defined in Equation 5 (Chapter 2.1). The scanner geometry also affects the sensitivity by introduction geometry factor for gamma rays paths in the detector materials. 2.3. Spatial resolution in P E T 2.3.1 Physical limits on spatial resolution Two factors, which occur in the course of positron decay and annihilation, introduce physical limits on the achievable spatial resolution of PET. The first is the finite range of the positron before annihilation. The coincidence detection of the annihilation gamma rays localizes the annihilation events, not the location of the parent nucleus, since the trajectory of the positron is undetermined. This introduces a blurring into the image, reducing the spatial resolution. On average, however, the error in estimating the location of the parent nucleus based on the annihilation location is less than the positron range, since some component of the positron trajectory likely will be along the line of response (LOR) between the two detectors (i.e. there is no error if the positron annihilates a point along the LOR). Dorenzo et al (11) have studied the point-spread function (PSF) due to positron range. This distribution is expressed by the sum of two exponentials, one accounting for small peak at a very short range where a small number of annihilation occur, and a second which introduces broad tails corresponding to the range of the majority of positrons. Because of the presence of these broad tails, the PSF is characterized by a full-width-tenth-maximum (FWTM) and by a full-width-half-maximum (FWHM). For F, which produces a comparatively low-energy positron, the resolution broadening due to positron range (12) is 0.22mm full-width-half-maximum (FWHM) and 1.09mm FWTM in water. The second limit on spatial resolution is introduced by the acollinearity of the annihilation gamma rays. Because the angular deviation of the gamma rays from 180° cannot be determined on an event-by-event basis, localizing the site of annihilation involves an assumption that the gamma rays were emitted in exactly opposite directions. The magnitude of the blurring introduced by this effect increases linearly with detector separation. In typical PET systems, the distance between opposite detectors, D, is typically -50 cm. For the arrangement of the high-resolution detectors described in this work the detector separation is D = 46.9 cm. The acollinearity of gamma rays originating 11 in the center of the ring results in a blurring of (±234.5 mm x tan (0.25°)), or ±1.0 mm, since the probability distribution of angular deviation is Gaussian with a FWHM of 0.5°. 2.4 Detector block and intrinsic detector resolution While the two effects described above impose theoretical physical limits on resolution, the intrinsic detector resolution is a factor that can be controlled to some extent in the design of the imaging system. Most modern PET system employs either individual scintillators or modular crystal detector blocks from which a number of discrete elements are cut. Scintillators produce, optical light when a high-energy particle like a 511 kev photon has interaction in the crystal. This optical light propagates through the block and is detected by photomultipliers (PMTs) The crystal matrix is coupled to PMTs through a light guide that is generally cut. These saw-cuts have different depth and they are used to selectively guide the optical photons from each crystal to the PMTs. The depth of the saw-cuts affects the light distribution for each crystal, which affects the number of scintillation photons detected by each PMT. A sub-optimal saw-cuts pattern will lead to a non-uniform distribution of signals for each crystal. That will cause poor crystal identification, since the peaks will not be easily separable. The identification of the crystal where the event occurred is based on the light distribution for each crystal. For an event that produced scintillation light detected by PMTs, we use the so called "Anger Logic" to estimate the coordinates where the event occurred (Xy Yy): Figure 7 A schematic drawing of the detector block, top "face " view and coordinate map setup for detector block surface. A xY I (A+C) - (B+D) where A, B, C and D are the amplitudes of the signal from the four PMTs corresponding to the detected event. Then all detected events are plotted on the plane (Xy Yy) that is divided on 1024x1024 bins. The number of events with discrete coordinates (Xy Yy) creates a histogram. Each peak of the histogram is associated to a specific crystal "ID". 12 The width and separation of the crystal elements determine the spatial frequency at which the radioactivity distribution is sampled. According to the Nyquist theorem, in order to detect a distribution of activity with spatial frequency of/"at a particular location in the FOV of the scanner, the lines of response must pass though that location at a frequency of at least 2/in order to avoid aliasing artifacts in the image. Unfortunately, the crystal element dimensions are not the only intrinsic factor determining the frequency response. In addition, when modular detector blocks are used there is some uncertainty in determining in which element of the block the even occurred. This ambiguity may result from several factors, including limited scinitillation photon statistics or imprecision of the readout used to provide positioning signals. An empirical study by Moses and Derenzo [12] summarized the various factors affecting the spatial resolution discussed above. According to this work, the components add in quadrature, resulting in a spatial resolution defined by the expression T: r = 1.25* V {dl2f + (0.0022 D)z +sz + (8) Here, d is the crystal element width in mm, D is the detector separation in mm, s accounts for the positron range ~0.1-0.5 mm and b accounts for the uncertainty in the determination of the location of the event within a block detector with multiple crystal elements. If crystal elements are coupled to PMTs in 1:1 ratio, b equals zero. The factor 1.25 accounts for a degradation of resolution resulting from the image reconstruction process. 2.5 Interaction between detector design resolution and sensitivity Modular detector design where a detector is cut into 8x8 smaller crystal "fingers" has higher spatial resolution compared to a big single crystal because significant part of scintillation light will propagate through the crystal where an event occurred because of internal surface reflection. That lead to more precise event-to-crystal localization. A deep crystal increases the probability of 511kev photons to have photo interaction in the detector block, which increases the detector sensitivity but at the same time decreases the spatial resolution because of "parallax error " for events coming from a source out of the detector center. To minimize this problem, novel detectors employ a two-level detector block design to extract depth of interaction (DOI) information by various methods. Using this DOI information, we improve resolution for off-center events. 13 Chapter 3 3. Overview of the High Resolution Research Tomograph (HRRT). The octagonal design corresponds to the HRRT (CPS, Knoxville, TN) (Figure 8). Figure 8. HRRT geometry Eight panel detectors consisting of 117 (9x13) HRRT fast-LSO/slow-LSO or LSO/GSO phoswhich block detectors are arranged in the gantry with a 46.9 cm distance between two opposing heads. Each of the 936(8 panels x 117 detector blocks per panel) detector blocks has 128 single detector elements (1.9x1.9x10.0 mm3) in two layers. This leads to large number of electronic channels resolving the information of the 119,808 single crystals. Each head (14,976 crystals) in the system is in coincidence with the "opposing" 5 heads. That lead to 20 unique head to head pair coincidences that is equal to 4.486* 109 Line of Responses (LOR). 3.1 Detector geometry. In all cases the block is composed of a double layer 8x8 crystal matrix (single crystal x and y dimensions 2.1x2.1mm2). The simulated block design followed that of the HRRT [4]. In the simulated configuration the double crystal matrix was coupled to four PMTs via a light guide (figure 3). Each PMT was shared by four detectors [9]. Figure 9. Detector block. put mt The light-guide coupling the detector block with the PMTs contained saw-cuts of variable depths. This design allows the light originating from each crystal to be channeled in such a way as to produce a unique energy distribution between the four PMTs. The interaction position ( X , Y ) in the x and y plane (crystal surface plane) was thus identified from the energy output of the PMTs using a modified Anger logic approach yielding the following formula (see 9). Xy'=[exp(P*Xy)-l]/[exp(P)-l] (9) Yy'=[exp(P* Yy)-1 ]/[exp(P)-1 ] where X y and Y y are the standard combinations of the PMTs energy information [10] (see eq.7, Chapter 2 ) and P is an optimization parameter. The optimum saw cut pattern was identified by varying the saw cut depths until an optimum crystal separation was achieved. 3.2 Simulated Scintillators An analysis of the physical properties of various scintillators was performed. Various compound materials were simulated by GEANT. There are several important criteria to consider in selecting a scintillator to be used in a FfRRT detector. First, the detector must be able detect higher-energy 511kev gamma rays. Scintillators with very high effective atomic numbers and densities present large cross sections for Compton and photoelectric interactions, and therefore offer higher attenuation for 51 IkeV gamma rays. A second requirement is the production of a detectable number of scintillation photons, at a wavelength to which the PMT is sensitive, following the deposition of energy by gamma rays. The time scale of this emission is also crucial, since it largely determines the count rate capabilities of the scanner, the coincidence resolving time, and in turn, the accidental coincidence rate defined in Equation 5 (Chapter 2). Finally, several issues related to bulk properties of the scintillator are important, including whether the material is hygroscopic, and whether it can be easily grown and machined for the construction of PET detector blocks. Table 3.2 summarizes the physical properties of simulated scintillators. 15 Table 3. Physical characteristics of simulated inorganic scintillators. Property B G O L S O G S O Density (g/cm3) 7.13 7.4 6.71 Effective Z 74 66 59 Linear Attenuation Coefficient u(cm"2) 0.903 0.870 0.6 Avg. number of photons per keV 6 24 9 Relative light yield (/100) 15 75 27 Scintillation decay time (ns) 300 40 60 Peak wavelength of emission (nm) 480 420 430 Refractive index 2.15 1.82 1.85 Hygroscopic No No No Chemical formula Lu(i-x) Ce2x (SiO4)0 Lu(i.x) Ce2x (SiO4)0 Gd 2Si0 5 BGO was simulated to evaluate the software and repeat previous results [3]. LSO and GSO were simulated for the detector block. Properties of this scintillator block and its dimensions were used as inputs to the detector simulation software. 16 Chapter 4 4. Simulation Study The simulation software was developed using the CERN simulation packet GEANT [5] and the program DETECT [7] developed at TRTUMF. GEANT was used to simulate the propagation of 511 kev photons through the detector block and the production of scintillation light. DETECT was used to simulate the propagation of scintillation photons through the detector block and to simulate the process of detection of these photons by photomultiplier tubes (PMTs). DETECT was modified to be applicable to this investigation. In additional to the HRRT geometry a circular scanner geometry was also simulated to investigate the effect of DOI correction on resolution as a function of scanner geometry. The same detector-to-detector distance (46.9cm) was used for both octagonal and circular design simulation. Four crystal configurations were simulated: two LSO layers 7.5 mm deep (LL_7.5), two LSO layers 10 mm deep (LL_10), a 7.5 mm deep LSO layer followed by a 7.5 mm GSO (LG7.5) layer and finally a 10 mm deep LSO layer followed by a 10 mm deep GSO layer (LG10). The LSO layer closely facing the object - top layer- was assigned a decay time centered on 33 nsec with standard deviation 5.2 nsec (fast LSO), the LSO closer to the photomultiplier tube (PMT) - bottom layer - a decay time centered on 40 nsec with standard deviation 6.4 nsec, while GSO was given a decay time centered on 60 nsec with standard deviation 12 nsec [8]. 4.1 G E A N T A GEANT based software was used to simulate a uniform beam of photons of energy 511 keV outgoing from a distant source. The source geometry could be set to be a point, a line or a flood. Each incident photon was then tracked in the volume of the detector block for photoelectric and Compton interactions. Cross sections for interactions in the scintillator material were derived using GEANT. The relative ratio of the cross-sections for the Compton and photoelectric processes was used to randomly choose which of these occurs at each interaction vertex. The distance traveled by the incident or Compton scattered photon between interactions was randomly generated according to an exponential distribution. The total cross-section at the energy of the interaction photon determined the interaction length of the photon. Values of the cross-sections were tabulated from 10-520 keV in bins of 5 keV. For Compton interactions, the angular distribution of the scattered photon follows the Klein-Nishima formula. Compton scattering of photons into the detector block from the surrounding supports like detector frame was not modeled. 17 Figure 10. Simulated by GEANT propagation a 511kev photon through a media. Scattered photon. Incident 511 kev photon \ X-ray Compton recoil electron Scintillation photons Phptoelectron Compton recoil electron At each interaction vertex, the energy lost by the absorbed or scattered photon was assumed to be converted to scintillation light. The number of scintillation photons generated at each interaction was derived from the product of the energy lost at the interaction vertex and of the light yield of LSO from [9]. The scintillation response of LSO and GSO displays a significant dependence upon the energy of the incident photons. From the recent compilation available in Ref [6], the scintillation yield of this crystal effectively doubles as incident photon energies increase from 10 to 100 keV. Above 100 keV it reaches a nominal plateau of 23000 scintillation photons/MeV which extends at least up to 1 MeV. Implementing this effect in the model only requires a correction to the number of scintillation photons associated with each interaction vertex of an energetic photon tracked in the block. This effect was already implemented for LSO and we added a correction factor for GSO that was taken directly from the experimental scintillation response curve available in Ref[6]. For each interaction vertex, /, an event scintillation vector was written to an output Interaction List File. The first three words of the vector were the coordinates of the interaction point within the detector block volume (X„ Y„ Z,). The fourth word gave the number of scintillation photons released at that vertex, N,. This number was computed as the product of the energy lost at the vertex and the known light yield per unit of deposited energy for the scintillator material. The last word of the vector was a sequential index incremented for each interaction until the tracking was stopped. Tracking was stopped either by a photoelectric interaction, escape of the photon from the block volume, or by a Compton interaction leaving less than 10 keV to the outgoing photon. 4.2. Summary of all physical processes simulated bv GEANT A Monte-Carlo approach was used to simulate different physical processes involved in the propagation of 511kev photons through scintillation media. As mentioned before the distance traveled by the photon between interactions was randomly 18 generated according to an exponential distribution. The total cross-section of all processes at the energy of the interaction photon determines the interaction length in the of the exponent of the distribution. The probability of each type of interaction was calculated on the base of the cross-section of a particular reaction at a given energy. The type of a interaction was chosen for each point interaction by a Monte-Carlo generator implementing these probabilistic functions. 4.2.1 Simulated interactions of a incident 511 kev photon in scintillator material • Rayleigh scattering (negligible for hv> 100 keV) • Photoelectric effect dominant for hv < 50 keV and important for hv < 90 keV • Compton (incoherent) scattering important for hv > 50 keV and only interaction for 200 keV<hv<511 keV 4.2.2 Simulated continuing photon and recoil electron transport Ionisational losses • In media, photons and recoil electrons interact "continuously", losing energy via inelastic collisions with atomic electrons. It is impractical to simulate each collision individually. 2500 elastic scattering events occur as an electron slows from 511 keV to 250 keV in LSO/GSO. Therefore so called "condensed history" technique was used: continuous photon or electron path (the distance traveled by the incident or Compton scattered photon or electron between interactions that randomly generated according to an exponential distribution) is divided into steps, whereby each step results in a small energy loss due to the soft non-elastic interactions with orbital electrons in crystal media. This lost energy (calculated by GEANT functions) depends on the energy of the particle and electron density of the media that this particle is traversing. • Electrons and photons were transported until their energies reach a threshold D (see 4.2.3). • For interactions above D, i.e. creation of knock-on electrons or "delta rays", secondary particles were followed individually. 4.2.3 Cut-off energies The user specifies individual photon and electron cut-off energies below which the histories are terminated; the particles are assumed to be stopped and the rest of the energy is assumed to be converted to scintillation light. The selection of these parameters is crucial in determining the time required for the simulation. We chose the following parameters that were set as thresholds "D" for photons and recoil electrons . • Photon Cut-off D ~ 10 keV (mean free path = 0.02 mm in LSO/GSO) • Electron Cut-off D ~ 100 keV (free path < 0.02 mm in LSO/GSO) 19 4.2.4. Simulated creation of bremsstrahlung photons GEANT library for simulation of bremsstrahlung photons was used to take into account bremsstrahlung photons during the simulation of the correct energy released by the propagated high energy photons and recoil electrons. 4.2.5 Implemented mechanism of scintillation The mechanism of scintillation is understood in terms of the energy levels of the crystal lattice. Individual molecules do not scintillate - this behavior is a property seen only in the crystal as a whole. LSO scintillates weakly in its pure state, and require the introduction of impurities (e.g. Tl is introduced into scintillator to facilitate scintillation at more practical temperatures). This produces activator centers between the valence and conduction bands of the lattice, which trap excited electrons. These activator sites are responsible for the light-emitting effect when the energy is absorbed. The recoil electron provides this energy as it slows down in the crystal. When a valence electron absorbs energy, it may be excited to the conduction band, leaving a hole in the valence band. Subsequently, the electron may be trapped in an activator center before returning to the lower energy state. Three different types of activator centers exist within scintillators: 1) quenching centers, from which electrons return to the valence band by emitting heat, but not light. This transition is of no use for the detection of gamma rays in PET; 2) phosporescence centers in which electrons may be trapped until they absorb an additional amount of energy, then return to the valence band with the emission of a photon. Although light is produced during this transition, it occurs over a long time scale, and results in an "after-glow" which is a undesirable characteristic in PET scintillators; 3) fluorescence centers, from which the electron returns to the ground state very rapidly with the emission of a photon. Because of the useful short time scale of this transition (scintillators commonly used have decay times of 0.8 ns to 300 ns, depending on the crystal), growers of scintillation crystal aim to increase the number of this particular type of activator center. These centers were accurately simulated rather then assuming that each point of interaction produces scintillation light. The type of center for each interaction was randomly chosen based on the relative ratios of cross-sections of these processes at a certain energy. Cross-sections were estimated by GEANT physical package. 20 One extra part in the mechanism of scintillation is the presence of excitons, which consist of hole-electron pairs that remain bound (although the electron remains in an excited state). As a neutral association, an exciton moves though the crystal lattice until it is trapped by an activator center. This process was not taken into account during simulations because of its low probability (-137 times less) compared to the over considered processes. 4.2.6 Photon Interactions NOT Simulated • Double Compton effect (rare) • Thompson scattering "low energy Compton" no energy transferred to the electron • Photonuclear reactions (y, n) and (y, p) cross sections are very low in typical geometries ( « 10%) Figure 11. Algorithm diagram of simulation 511 keV gamma ray interaction in the detector block. Yes 4. Transport taking into account detector geometry Terminate history 1. Source: 2. Pick up Place initial energy, position, photons — • direction and - • parameters on geometry of top of stack current particle Yes 3. Determine DISTANCE to next interaction 5. Determine INTERACTION: -PE - Compton - Rayleigh 6. Determine energies and directions of resultant particles, store on stack. 21 4.3 D E T E C T DETECT [4] is a Monte Carlo simulator capable of realistically modeling the optics of scintillation detectors. The program isotropically generates a number of scintillation photons from a volume element of the detector where the y ray has an interaction and tracks them individually throughout their passage within the components of the detector as well as their interactions with the surface. 4.3.1 Scintillation light propagation simulation. Optical tracking of a scintillation photon was pursued until it is either absorbed or reaches a detection element or escaped from the detector volume. The geometry of the detector can be described with a very general syntax. The components of the detector are specified by connecting surfaces that are planar, cylindrical, conical, or spherical. The optical properties of an individual surface may be specified as polished, ground, painted, or metalized. Each of these options uses a different model to treat the transmission and reflection of light at surface boundaries, with a reflection coefficient as the only free parameter. Once the geometry and the optics of the detector are described by DETECT's syntax, they are passed to the DETECT Simulation Driver to declare the detector design specifications. Scintillation events are then simulated sequentially by reading the Interaction List File produced by GEANT. The coordinates (X„ Y„ Zt) of the interaction point specify a volume element from which N, scintillation photons are isotropically generated and tracked by DETECT. For each interaction vertex, the numbers of photoelectrons generated in each of the detection elements of the detector are written as an event signal vector along with the coordinates and index of the scintillation. Index "0" for the first interaction of an incoming y, index "1" for the second interaction of this y, index "3" for the third and so on. Event signal vectors belonging to the same incoming photon are combined to compute a final record encoding its total number of detected scintillation photons and centroid-"energetic center of mass" of an interaction, both derived, as in measurements, from photoelectron numbers in the detection units. These records are written into a file as a Signal List File for subsequent analysis. All optical photons tracked by the program are assumed to have one of the two possible wavelengths: one value characterizing the primary scintillation light, and the second representing the light re-emitted by a wavelength shifting component. The second component appears when the scintillation light hits the surface. Part of this light will absorb, another part will reflected and the other part will re-emitted with different wavelength. This optical processes simulated by Frenal optical scatter is implemented in DETECT. Scintillation photons are generated at the y interaction point. Each photon is assumed to be emitted in a random direction selected from an isotropic distribution. 22 Photons are assumed to travel at the speed of light in vacuum, c, divided by the refractive index of the optical medium in which they are moving. The path lengths to possible bulk absorption, scattering, or wavelength shifting are randomly sampled from exponential distribution characterized by the corresponding mean free paths. If absorption is indicated as the first interaction and it occurs before the next surface intersection, the history is terminated. If scattering is indicated, the photon is redirected at a random scattering angle selected from an isotropic distribution. If wavelength shifting is indicated, the direction of the shifted photon is also randomly chosen from an isotropic distribution, and further interaction probabilities and reflection coefficients of a crystal are set to new values appropriate to the shifted wavelength. Only one such shift is allowed per history and is accompanied by appropriate changes in absorption, reflections and scattering properties. The number of scintillation photons reaching the photo detectors depends upon the positions when they were generated, their interactions with the crystal's material and the surface coat. To simulate that, we used the approach described in previous work [3]. The attenuation of scintillation photons was split into two components to account for the scattering processes, which change their direction, and for absorption that terminates them. The strength of each process was specified by independent values for the scattering and absorption lengths, Xs and Xa respectively. From measurements taken on a 10mm thick, well polished LSO crystal [10], the total attenuation length, Xt, of LSO takes a value of 138 mm at its peak emission wavelength. This result gives a useful constraint on Xs and Xa through the relationship: \IXt = + \lla • Relative values of Xs and Xa were constrained to a total attenuation length of 138 mm. The best match between this experimental measurement and DETECT simulation of 10mm LSO block was obtained using the GROUND finish parameter of DETECT with a reflection coefficient (RC ) value of 97 to 98% along with Xs =138 mm and Xa = oo. These parameters were used for the all-subsequent simulations. This simulation does agree with work [3]. 4.3.2 Simulation of scintillation light detection by PMTs. To simulate efficiency for detection of scintillation photons by PMTs, the PMT quantum efficiency (Q) was obtained from the spectral response of Hamamatsu bialkali photo cathodes by the emission spectrums of LSO Q=22.5%, GSO Q=22% [8]. 4.4 Simulation of Photomultiplier tubes (PMTs) The PMT geometry and quantum properties were taken into account during the simulation of propagation optical photons through the detector. The geometry of the PMTs was modeled after a typical 19mm phototube, the Hamamatsu R5364. The PMT sensitive window was modeled as a cylinder, 7.4 mm in radius, with a flat entrance face and bounded by spherical surface representing the 23 photocathode. The entrance surface was assumed to be polished with an index of refraction of 1.52, and its inner cylindrical wall was given a 100% reflective finish. The window maximum thickness and radius were taken from the Hamamatsu catalogue. The radius of curvature of the photocathode was chosen such that the window thickness was 0.1 mm at the center. The optical glue connecting each PMT to the LSO/GSO or LSO/LSO was simulated by a 0.6 mm thick polished layer with an index of refraction of 1.52. This glue layer produces a cross talk of a few percent of the total signal between the PMT directly coupled to an edge or corner crystal and the other PMT's. The secondary emitter material used for the dynode is typically a p-type semiconductor TiO(Cs). Optical properties for both materials were simulated by DETECT. The statistical noise associated with the amplification of the photoelectrons from the photo cathode through the dynode chain is significant and was modeled by adding to the number of photoelectrons collected in each phototube a random Gaussian component of standard deviation given by ax = <X(1 +v(M)), (10) where X can be the energy signals A, B, C or D from PMTs (A,B,C,D), and where v(M) is the variance in the amplification of photoelectrons through the dynode chain of the phototube with a multiplication factor M. If 5, is the multiplication factor of the fh dynode, the mean value and relative variance of the photoelectron amplification factor of a tube with n stages are then: n M=nbj (ii) J=I v(M)= I Tll/oj (12) /=/ j=i The value of v(M ) is therefore determined to leading order by the amplification factor at the first dynode stage of the tube. In our simulation we kept a value of v(M) = 0.07 that was obtained from assuming a multiplication factor of 8/ =15. These values were taken from the work described in [3]. 24 4.5 Simulation of the surface of Crystals PET detector block is finished by applying some kind of reflector to the surfaces of the individual scintillator elements. For the PET block, the purpose of adding a reflecting material is two-fold. First, if scintillation photons are incident at the side-walls of the crystal element at an angle less than the critical angle for Total Internal Reflection (TIR), some fraction will be transmitted outside of the crystal volume (according to the Fresnel laws of reflection and transmission). In order to recover this light, it is desirable to reflect these photons back inside the scintillators. Second, if the scintillation photons escape from the volume of a crystal element, it is possible that they may enter an adjacent element. This effect would undermine the original motivation behind cutting the crystal; optimally the light should be channeled directly to the photocathode of the PMT. By covering individual elements with a reflector that is opaque to the scintillation light, adjacent elements are optically isolated. An 0.25mm thick SiO reflecting material have been used in simulations with a refraction coefficient n=1.5. It was necessary to simulate a very thin, uniform glue layer between the crystals and the PS-PMT window. A thicker layer would result in the increased spreading of light due to refraction as the light was transmitted from the crystal through the coupling material. This is especially important if the indices of refraction of the coupling material and the scintillator differ significantly. I repeated simulation experiment [3] to see whether DETECT's polished or ground surface finish models fit to the HRRT block. A 20-mm-long crystal centered on a detection unit was simulated. The finish of the five faces of the crystal not in contact with the detection unit were considered to be either polished or ground and coated by an opaque reflector with a reflection coefficient (RC) that was allowed to vary. The reflection coefficient of the reflector was varied from 0.94 to 0.99 until the simulation reproduced the actual depth dependence of the signal intensity at RC=0.98 of the photopeak measured [11] in a crystal at the center of a quadrant of the HRRT block. The simulation showed the same results as simulation described in work[3]. 4.6 Crystal position identification As was mentioned before, common PET systems use Anger logic to estimate the position in the detector block where 511 kev photon is stopped. This logic tries to estimate the position on the basis of signals coming from PMTs by the well-known formula: Xy=(A+B-C-D)/(A+B+C+D); Yy=(B+D-A-C)/(A+B+C+D) This approach produces very high non-linearity between actual and calculated crystal position and poor ability to separate signals coming from crystals close to the corners of a block. 25 This non-linearity can be minimized: • by optimization of saw-cuts pattern and / or • by using another logic for the coordinate estimation. I investigated both options. 4.6.1 Saw-cuts estimation The cut depths that provide good crystal position identification for our model of the HRRT block were searched as described below. An automated interface was developed to perform this simulation study. Figure 12 Interface between the Algorithm and the PET detector Simulator (GEANT/DETECT). Algorithm Initialize Initial Saw-cuts ff Evaluate Operate New Saw-cuts Detected events peak Parameter Detector Simulation Oiitnnt file i Detector Simulation i Inniit file The search process began by setting all the cut depths to zero in the Initialize block of the saw-cuts estimation Algorithm and in the DETECT input file. Each cut was then systematically deepened {Operate block), starting with the edge crystal and working toward the center. Due to the four-fold symmetry of the block model, this search only considered the crystals along the diagonal of one quadrant. For each cut depth and each crystal, a large number of events was simulated at the transverse center of the crystal and 1.1 cm from its front face. Each event simulates the detection of a 511 keV photon by a single photoelectric interaction at a depth equal to the 26 mean attenuation length of LSO. The mean coordinates of the resulting event energy centroid in the (XY, Y r) plane were used to estimate its location. The search (Evaluate block) for the cut depth was stopped when the cluster's mean position was resolved from its on-diagonal neighbor, in other words when they were further from each other than the signals full-width-on-half-maximum along X T and Y Y . Several saw-cuts patterns were estimated for diagonal crystals. On Figure 13 we can see the distribution of the number of detected events in Anger X y Y y coordinates, entering the surface of detector block at various diagonal crystals of top-right quadrant of the detector block. A collimated source was simulated to let 511 kev photons hit each crystal at the center. Detected position for a particle was estimated on base of Anger X Y Y Y coordinates. All detected events were presented as a histogram over the X coordinate (the total range from 0 tol) with bin size 0.02, events with a different Y coordinate but same X were grouped into the same bin. Crystal 1.1 represent crystal at the center of detector block, Crystal 2.2 and Crystal 3.3 are the diagonal crystals and the Crystal 4.4 is a crystal at the corner of the block. Figure 13 Distribution of events hitting a detector crystals. Crystal 1.1 Crystal 2.2 Crystal 3.3 Crystal 4.4 II. 1' ll \ 1 Saw-cuts patter: 1.5; 2.0; 2.5; 5.5. Crystal 1.1 Crystal 2.2 Crystal 3.3 Crystal 4.4 : . I I i i j _ J Saw-cuts patter: 2.0; 2.7; 3.0; 5.5. Crystal 1.1 Crystal 2.2 Crystal 3.3 Crystal 4.4 i I I i i • H M I B H H I I U Saw-cuts patter: 2.2; 2.5; 3.8; 5.5. Saw-cuts patters represented for left-top quadrant of the detector block. I used one quadrant in my simulation because of detector block symmetry. Numbers measure distance from the light guide bottom surface (attached to PMT) to saw-cuts ends. Numbers start from detector block center (1.1) toward to the edge crystal (4.4). 27 Here on Figure 14 the distribution of detected events from flood source in Anger X Y Y y coordinates. Each distribution represents the spatial distributions of events detected in the top-right quadrant of the detector block for different saw-cuts patterns. Figure 14 Detected events distribution. Crystals (11,12,12,14 ) (21,22,23,24) (31,32,33,34) (41,42,43,44) . . i i i i i i i i i i i i i i i i i uiiiii . i l l ' . - - dlllUijli.. J : s llllUl.Ulllu.ljlllljl.u..^ l | ll u . u l l l J I U .ilJltjIl.... Saw-cuts patter: 1.5; 2.0; 2.5; 5.5. 1 '• ^ U l b , . alb, 1 • | Jill., lllllmlll J i . u . u . u i J ] u . _ » . » u . J l l i u l L u Saw-cuts patter: 2.0; 2.7; 3.5; 5.5. . M. uuMjlbjullliiib. u.iillllllii.u.al . ' ^.uiuibiiu.mlUliii.u..dillli.,.. . . ._ u..uiiiii^^....iiiiiiiiiuiiLli J . i l l l L U - b - < illlliulllli Saw-cuts patter: 2.2; 2.5; 3.8; 5.5. Saw-cuts patter for all eight crystal: 5.5; 3.8; 2.5; 2.2; 2.5; 3.8; 5.5 (mm) showed the best crystal separation. The one remaining problem is the singular point for the very edge (4.4) crystal at the corner of the detector block. All events which were stopped in that crystal had Anger coordinates X Y ~ 1 Y y ~ l , because almost all the light from these events was guided to one PMT by the saw-cut 5.5mm deep going through all glass light guide. In our block design, where PMTs were mounted at the detector block corner and used for detecting light from four connected detector blocks, this factor lead to poor resolution of events which have interaction in the corner crystals of neighbor blocks. To eliminate this problem we needed to shift distribution for crystal (4,4) to the middle of X Y plane. Physically that means that light coming from that crystal should be shared by neighboring crystal. We can just make saw-cut for edged crystal less deep but that lead to bad separation of crystal (4.4) with neighbors. To eliminate this problem, I performed a simulation study to find new "Transformed Anger Logic". 4.6.2 Transformed Anger Logic. The basic idea was to find an analytic transformation of Anger X Y plane to reach a relatively uniform distribution of signals (peaks) coming from all crystals. An exponential transformation was chosen. The new coordinates were calculated as follows: Xy'=[exp(P*Xy)-l]/[exp(P)-l]; Yy'=[exp(P*Yy)-l]/[exp(P)-l]; 28 Where Xy,Yy are the common Anger coordinates and P is an optimization parameter. The simulation was repeated for various values of "P" and saw-cuts patter following the previously described algorithm. Position maps for different "P" are shown in Figure 15. Figure 15 Position Maps. Once a preliminary set of cut depths was found, using the diagonal crystal, fine adjustments to the cut depths were made to improve the separation of the event clusters for the off-diagonal crystals. This last step exploited the empirical fact that as the cut depth of a crystal increased, the mean position of the peak shifted toward the edge of the (Xv Y Y )plane and its width decreased. The optimal saw-cuts pattern that overcome the problem with the corner crystals was estimated to be {3.5; 1.5; 1.0; 0.5; 1.0; 1.5; 3.5} millimeters depth and optimal value of the parameter "P" was found to be 2. That is different from saw-cuts pattern estimated for original Anger approach. The Figure 16 shows the improvement of the position map after applying the optimal saw-cuts pattern and the transformation parameter "P". The figure represents the top-right quadrant (4x4) of a detector block (8x8) and the detected events distribution which is plotted in X Y Y Y coordinates. 29 A visual comparison with the example of a real position map form CPS Figure 17 shows a good match in terms of uniformity of peak distribution. Figure 17 Real position map (CTI) 4.6.3 Position Map Look-Up table In order to decode the crystal element in which a gamma ray interacted, different decoding schemes were simulated, based on the premise that each crystal element gives rise to a different distribution of scintillation light over the PMTs. A number of events occurring in a single crystal element gives rise to a distribution of (Xy Yy) signals that are characteristic of that element. Therefore, localizing an event to a single element involves specifying boundaries around each distribution in (XY Y Y ) signal-space, and then determining to which distribution an acquired (Xy Y y ) pair corresponds. The set of specifying boundaries will define the position map look-up table. Several algorithms of crystal identification have been examined. They include "simple algorithm", neural-network approach and genetic algorithm. 4.6.4 "Simplest algorithm"(General steps) This algorithm was developed in the Matlab programming environment[16]. Matlab is a programming language and set of libraries of functions optimized for analytical problems. Our simple algorithm had several steps of data preparation which were easily performed by Matlab built in functions. First step was the determination of the maximum of all peaks on (Xy YY ) plane. They were estimated and recorded by applying Matlab function-"max()" to 2-dim matrix describing plane (Xy Y y ) . Then the shape of each peak was smoothed by applying "Match filter" and "spline" interpolation function from Matlab software package. After that 64 peaks with the highest maximums were chosen. Minimum points were estimated between each four neighbor peaks by "Matlab" function - "min()" for 2-dim matrix describing surface between this four neighbors. Also the minimum points were estimated for each pair of peaks. 30 At the end, these points were connected in X and Y directions. That separates (Xy Y Y ) plane into 64 four-sided polygons. Each polygon was associated with relative crystal ID. The set of specifying boundaries defines the position map look-up table. Figure 18 illustrates the signals distributed in the (XY Y y ) plain. Each peak is bounded by a four-sided polygon estimated by the "Simplest Algorithm". Figure 18. Signal distribution. 4.6.5 "Neural Net" algorithm" A three-layer divergent feed forward artificial neural-net (ANN) was proposed for crystal identification [16]. The mathematical model of ANN is : Output(m)=F*Input(n), 'lnput(n)' is a n-dimensional vector of input values ("input neurons"), 'Output(m)' is a m-dimensional vector of output values ("output neurons") and F' is non-linear transformation ("hidden layer neurons") 'F' ~ l/(l+exp[W*Input(n)]), where W - is a matrix of real numbers. The first time all parameters of W-matrix are set randomly. Then we start the "training" of the ANN: Every time when we send data to ANN, we calculate 'Output' and compare to the known 'Output' for the 'Input' (50w#?wf-difference between them). The parameters of the W are adjusted by the so called "learning Back-Propagation" algorithm: AW(new)=n*60w^+^*AW(last), The parameters n- learning rate and u-momentum are manually set. 31 The training is stopped after ANN starts to produce results with error less then some threshold convenient for the specified task. After that ANN can be used to predict unknown 'Outputs' for new 'Input' vectors. In our case, the first level was accepting four PMT signals as input, intermediate two-units neuron the so called hidden level and 16-units output level was identified crystal ID. The learning algorithm was error back-propagation. Training was performed for 300 cycles. The respective values for momentum and learning rate were 0.5 and 0.2. CrystalJD_16 4.6.6 "Genetic algorithm"(General steps) In the first step, the maximums of all peaks on (XY Y Y ) plane were estimated and recorded by applying "Matlab" function-"max()" to 2-dim matrix describing plane (XY YY). Then the shape of each peak was smoothed by applying "Match filter" and "spline" interpolation function from "Matlab" software package. Then (XY YY) plane was divided on 256x256 bins called "genes". A set of genes around peak was defined as a "chromosome". All "Chromosomes" were chosen to cover (XY YY) plane without overlap and a "chromosome" could not have "holes" inside. The algorithm lets two neighbor "Chromosomes" randomly exchange their neighbor "genes" residing at the shared bounder on the each step. That process is called "mutation". If mutation decreases the "Fitness function" (see below) then "Chromosomes" keep the new genes otherwise the mutation is rejected. 32 Figure 19. Chromosome in the Genetic algorithm. The "Fitness function" can be defined in many ways. I defined it as the ratio: FF = 2; ( N i a - N i r )2 /64 , where N j a - number of simulated events associated to "chromosome" relative to crystal "i" (associated to be stopped in the crystal "i"). N j r - number of simulated events stopped in the crystal "i" Sum done over all 64 "chromosomes" those are relative to 64 crystal elements. This "Fitness function" is a measure of how correctly events are associated with the crystal where they deposited most of their energy. CONVERGENCE CRITERION: The idea is to get "chromosomes" that represent crystal elements. The algorithm was stopped when "Fitness function" did not decrease its value by more then 1 % from its previous absolute value over the last 1000 iterations. Then the bins relative to each chromosome were collected in the look up table. 33 4.6.7 "Algorithm summary" To compare different algorithm, I used the "Average mis-identification factor" (AMF) defined as: AMF = l/64*(2j ABS(N i a - N i r) / N i r )*100%, where the sum was done over all 64 crystals. Nja - number of simulated events correctly associated to be stopped in the crystal "i" by the chosen algorithm). Nj r - number of simulated events stopped in the crystal "i" Although the crystal elements were well separated and quite easily identified, it was necessary to address concerns related to the reproducibility and Stability of the Position Look-Up-Tables The requirement was tested by repeating simulation of the same detector blocks six times, and regenerating the crystal identification for each simulation. As we can see from Table 4, the "Simplest" scheme results in AMF lower -5% compare to Neural Network approach. Table 4 Summary of algorithms. Simple Algorithm Neural Network Algorithm Genetic Algorithm Crystal Correctly detected events Correctly detected events Correctly detected events (1,1) 88% 85% 72-92% (2,2) 89% 87% 77-95% (3,3) 90% 89% 83-90% (4,4) 92% 90% 88-98% The "Genetic Algorithm" algorithm was rejected because of non-robustness. The produced position maps were different for different simulation . There are several probable reasons for that failure. One is that "fitness function" and "convergence criterion" is not optimal and do not lead the system to the global minimum (in terms of "fitness function") but fall to a local minimums. Or system of mutated "chromosomes" does not have global maximum at all. The investigation of this approach was left for the future. 34 4.7 Depth of Interaction information As was mentioned in chapter 2, the "parallax error" for events coming from off-center of field of view degrades the spatial resolution. Events hitting the crystal surface at a non-normal angle have a high probability to escape this crystal and to be detected in a neighbor crystal. This leads to a wrong identification of the crystal where the y ray interacted. To decrease this position mis-identification depth of interaction information was used. Basically idea behind of DOI determination is to estimate the layer where each particle from a coincident pair deposits most of its energy. Then the most likely interaction point within that crystal is set as a point where a particle was stopped. The line connecting two such points from opposite detectors is called "line of response" (LOR). This is a line that approximates the path of a coincident 51 lkev photon. In case of a system without DOI correction, events which deposited most of their energy at the top or bottom part of a crystal will not be separated and will be electronically assigned to have the same LOR. This will obviously degrade the resolution for events coming from the edges of the field of view, as it is shown on Figure 20. Figure 20. Coincident-Event Correction. This effect is worse for longer crystal. The shorter crystal the less significant this effect is, but the detection efficiency is less as well. To keep good detector efficiency a two levels crystal design is proposed for detector block. DOI gives as ability to discriminate events occurring in different crystal layers and to assign them to different LORs, thus leading a resolution improvement. 4.7.1 Depth of Interaction Algorithms Different DOI algorithms were investigated during simulation study. The goal was to estimate an effective and robust algorithm for discrimination of events stopped in different detector block layers. 4.7.2 Time discrimination approach Generation of scintillation light occurs not immediately in the crystal but over some period of time. This time, called "decay time", depends on the type of scintillation 35 material. The simplest way to discriminate events in the two layers is to set a time threshold. Scintillation light generated in period of time less then the threshold is assigned to one layer and the light generated after the threshold time period to another layer. Figure 21 Decay time distribution. The Figure 21 shows simulated particles as a function of their decay time. The decay time was simulated by a Gaussian distribution for each interaction with a mean and a standard deviation specific to each scintillation material. The values were taken from experimental data [8]. Then the decay times for each interaction were weighted by the deposited energy end averaged for each events. Table5 Data from CTI technical specification Scintillator Mean (ns) Sigma (ns) Fast LSO 33 5.2 Slow LSO 40 6.4 LYSO 42.7 4.6 GSO 60 12 4.7.3 Fuzzy Logic approach. This simplest approach still has some uncertainties because generation of light is a statistical Poisson process and the threshold approach does not allow to discriminate events in the tails of the distributions. A more sophisticated approach was investigated. The basic idea was to use information about deposited energy of detected scintillation photons together with scintillation decay time information. This joint information may provide additional information on the location in the crystal where most of the gamma photon energy was deposited. The distribution of events for LSO/GSO block as a function of measured decay time and detected energy is shown on Figure 22. 36 Figure 22 Decay time and energy distribution 4 0 60 " n » scintillation photons The decay time distribution and the energy distribution were simulated for both layers of a detector block. Probabilistic predicates were associated to each time and energy values: 1. Event with decay time f, has probability P(tj)Lso to occur in LSO. 2. Event with decay time U has probability P(tj)cso to occur in LSO. 3. Event with decay time E, has probability P(EJ)LSO to occur in LSO. 4. Event with decay time Et has probability P(EJ)GSO to occur in LSO. A fuzzy logic model was developed to formalize the probabilistic logic. Model: First Step Each particle stopped in the detector block must fit to one of the two linguistic sentences: "A particle deposited/stopped most of its energy in the first LSO layer" or "A particle deposited/stopped most of its energy in the second LSO layer. In other words, the sentence "A particle deposited most of its energy in the first LSO layer" can be "true" or "false". But in reality, a particle can deposit its energy over both layers and the sentence "a particle deposited most of its energy in ... layer" does not have to be exactly "true" or "false". We can say that this sentence can be "true" partly, say 30% or 70%>. This approach is known as a "fuzzy logic" approach [16]. To specify the degree of "truth" for the sentence, the so called "member function"-M was defined for each layer. This function formalized the degree of being true for the sentence "a particle stopped in first LSO layer". M equal "1" means "true" and M equal "0" means "false" but M can accept any value between 1 and 0. I defined "M" to be dependent from experimental parameters such as: decay time ( M(t) ) and detected scintillation photons yield normalized to a maximum number 511*23 (100%) scintillation photons that can be produced by 51 lkev photon in the LSO layer of the detector block (M(N%)). Different shapes of M(t) and M(N%) were simulated. Figure 23 shows a graphical representation of the "M" function. 37 Figure 23 Fuzzy Logic Model. GSO M(t) GSO M(N%) 30 33 40 50 60 ns Decay time 100% Scintillation LSO M(t) LSO M(N%) 30 33 40 50 60 Decay time ns 100% Scintillation photons The "M" function discriminates the events with "low", medium" and "high" degree of confidence to be stopped in a certain layer. For example, an event with decay time between 30-33 ns has a "low" or "medium" confidence to be assigned to LSO and a "high" or "medium" confidence to be assigned to LSO. At the same time, if the number of detected scintillation photons is close to a maximum value we have "high" confidence to assign this event to LSO. Probability to predict event in specified layer (LSO) was analyzed by Fuzzy logic approach. Fuzzy logic rules for GSO layer are complementary to the LSO layer and can be repeated analogically. Events with some measured "decay time" are represented by a 3-dim vector in the 3-dim logical space (low, med, high) of "member function". For example, measured decay time 33 ns let us conclude that we have a "medium" degree of confidence to have a particle stopped in LSO with a confidence factor -0.3 or in other words we have "medium" degree of confidence to have the particle stopped in GSO with 38 a confidence factor -0.7. "High" and "Low" degree of confidence have zero confidence factors in both cases by default. For both LSO and GSO see Figure 23. The same can be repeated for measured "detected scintillation photons" and we have a 6-dim logical space for two measurable parameters. The second step is to project this 6-dim space into a 3-dim logical space (Figure 25) that formalizes the degree of our confidence to have a particle in LSO layer. Figure 24 Degree of confidence. Degree of confidence to have particle stopped in first LSO Second step (projection 6-dim space to 3-dim space) is the extraction of fuzzy rules that emulate human decision process. Decay Time Low Med High Energy Low Low Low Med Low Med High Med Med High High High The table represents logical sentences like: IF (Time=High) AND (Energy=Low) THEN (Confidence Degree = Med); That is equivalent of the sentence: "If measured time give us "high" confidence degree with some confidence factor and the measured "scintillation photons yield" gives us "low" confidence degree with some confidence factor, then we have "medium" Confidence Degree with some Confidence Factor to have a particle deposit most its energy in the first LSO layer". The confidence Factor can be calculated by "fuzzy calculus" from confidence factors for 3-dim logical vectors representing measurable values "decay time" and "detected scintillation photons yield". Let's take an event with decay time "X" and scintillation photons yield "Y". The degree of truth to have this event stopped in LSO will be described in a 3-dim logical space by three 3-dim vectors (Figure 26): 39 Figure 25 Logical Space. A M Low Med High \ A. / 0.7 X Time A M Med 0.8 ft* Detected Scintillation Photons Y M(X)=0.0*Low+0.3*Med+ 0.7*High M(Y)=0.0*Low+0.2*Med+ 0.8*High Each confident degree has confident factor - W. W (Low [M (X)])=0.0; W(Med[M(X)])=0.3; W(High[M(X)])=0.7. The projection to the 3-dim logical "Confidence" space is performed by calculating all the possibilities (IF (Time=High) AND (Energy=Low) THEN (Confidence factor = High).). The "Confidence factor-C" is min{ W(...[M(X)]), W(...[M(X)]), .... } of confident factors at a considered confidence degrees ("high", "medium" or "low"). At the end, we have a confidence degree vector with a set of possible confidence factors for each basic degree. Fuzzy logic averaging is performed by operator maximization of coefficients relative to basic confidence logical vectors. Low * C(l) * Low * Max{ C(l), C(2),..} Low * C(2) High* C(i) High*Max{ C(i), C(i+1),...} We have obtained a 3-dim logical vector -(A,B,C), where the letters represent weights at the basic logical vectors "Low", "Med", "High". The last step is "Defuzzification process, where we are estimating numerical value for "Confidence factor"(Fz'gz/re 26). 40 Figure 26 Defuzzification process. • Degree of confidence Degree of confidence to have event in LSO Logical weights bounded logical volume (presented on the graph as a 2-dim shape), numerical value of "Confidence to have event in LSO" is calculated as the projection of the graph center of the mass on the Degree of confidence coordinate. Each event was assigned to occur in a certain layer on the bases of the values of all predicates. For example, if event has. value of predicates "low" to occur in LSO and "high" or "medium" to occur in GSO then this event was assigned to GSO. Each simulated Fuzzy rule strategy was compared with truly simulated events for each layer. The block scheme of developed software for DOI estimation using fuzzy logic is represented on Figure 27. 41 V, Figure 27 DOI estimation dynamic generator for arbitrary number of layers. Crisp Input Variables Input Membership Functions - — 7 Vi v2 TNB NS PB Fuzzy Rules Sets of output membership functions p(X)f FS„ FS l n FS 12 I U ( X ) I FS 2 n I _ES. F S a i I s2 V 2 I v2 NB NS PB NB S, s2 Sj-i NS s2 s2 Si xeX I I > FUZZIFICATION Fuzzified Input Variables V > RULE EVALUATION > DEFUZZIFICATION • j I p(X) FSi g I FSn I" SOMF FS 2 Fuzzified Output Variables 42 4.8 DOI estimation algorithms summary. To compare DOI estimation algorithms, mis-position identification was calculated for each algorithm. Miss position identification factor (MIF) was calculated the same way as for position map: MIF = ABS(N i a - N i r ) / N i r * 100%, where Nj a - is a number of simulated events correctly associated by chosen algorithm to be stopped in the layer of detector block. Njr - is a number of simulated events stopped in the layer of detector block. The fuzzy logic approach did not show any improvement compared to simple decay time discrimination. I choose time discrimination approach for simulation because of less analytical overhead. 43 Chapter 5 Results This chapter presents the results of the simulation study. Figure 28 shows the full data pathway, which represents the simulation steps and the phases of data processing. Digitized signal from electronics let us calculate energy deposited in a detector block by a couple of 51 lkev photons originated from an annihilation event in the source. The energy passed through energy look up table that discriminate event within and out of specified energy window. Also electronics signals let us calculate X Y , Y Y coordinates of each event for a pair of 51 lkev photons by Anger approach. This coordinates passed through position map to assign the event to a crystal. Crystal id information, depth of interaction information with detector efficiency information let us estimate spatial resolution of the system. Figure 28 Data flow chart for energy discrimination and spatial resolution. (LUT - look up table) Digitized signal from pair of coincidence events 'a' and 'b' X a , Y a Xb, Yb Position Map LUT, b Position Map LUT, a Energy resolution Efficiency LUT Spatial Resolution A. Energy resolution and detection efficiency. The energy resolution was evaluated on a crystal-by-crystal basis by histogramming the event energy information obtained from the PMTs for each crystal separately. The energy resolution was evaluated as the ratio between the full width at half maximum (FWHM) of the peak corresponding to those events where the entire 511keV were released in the crystal and the position of the peak on the energy spectrum. In a realistic 44 scanning situation, events are only accepted in a pre-set energy window so as to minimize the detection of those events where Compton scattering occurred. Therefore the detection efficiency was evaluated in two ways: all events were accepted- in this case the detection efficiency must approximate that obtained from analytical calculations - only events in which the energy fell in the 350-650kev range were accepted. In all cases detection efficiency was estimated as the ratio between the number of simulated incident y rays and the number of detected y rays. B. Implementation of DOI correction and LOR binning. Coincident events were simulated by emitting two y rays in opposite directions hitting two detectors at predefined angles. For each y ray the corresponding crystal X and Y position was identified using the modified Anger logic. DOI information was then used to identify if the interaction occurred in the top or bottom layer of each crystal. The event was then assigned to that LOR connecting the most likely interaction points in the identified crystal layer. C. Determination of position resolution. For the octagonal geometry the following y incidence angles were simulated: 90° (orthogonal direction), 75°, 60°, 45° and 30°. These incidence angles simulated y rays being emitted in the directions shown in figure 4 for three source positions (center, 5cm and 10 cm off center) for the octagonal geometry. Figure 29 Simulated LORs for the source located at center of the FOV, 5 cm and 10cm off center for the octagonal geometry. The corresponding incidence angles for the source located at the same positions in the FOV in the circular geometry were 90°, 84°, 80° , 78°, 77°, 74°, 70° For each detected y pair the LOR was defined following the procedure described above. Position resolution for a particular y pair direction was calculated as the FWHM of the source profile in the direction orthogonal to that of the y pair emission. The source profile was obtained by assigning the number of events in each LOR to the profile bin corresponding to the intersection between the LOR and the profile direction. The bin size was chosen to be half of the crystal-to-crystal distance (1.05 mm). The overall position resolution for a particular source location was calculated from a profile obtained as detection efficiency, weighted average of the individual profiles. 45 5.1 Energy Resolution. The energy resolution of a detector block was estimated on the base of the crystal-by-crystal energy resolution and then averaging. The number of events detected in a particular crystal was plotted as a function of detected scintillation photons, which are related to the energy deposited for that event. Energy distribution for each crystal was normalized to 511 keV. The maximum of the peak was set to 51 lkev for each crystal. Then energy resolution was defined as a full width at half maximum of an energy distribution peak. Here Figure 30 and Figure 31 show the two examples of energy resolution of one quadrant (x4 crystals) for two block designs. Figure 30 Energy distribution for LSO/LSO (15mm) 7.5mm-7.5 mm W ] ffl E E E E J. fQ tVitl 20M 0 ItiMI 3QM 0- TGB0 SOW ft ~ 100 EOOO. An LSO/LSO two layer detector block has one observed photopeak, compared to the layers LSO/GSO detector block, which has two energy peaks. One comes from LSO and another from GSO crystal because LSO and GSO have different scintillation yield. Figure 31 Energy distribution for LSO/GSO(l 5mm) LSO/GSO (7.5 mm-7.5 mm) LSO layer z J C F A o pa ic i)< ic * J ; ^ Iii : ^ : u « c c o fir a « u o GSO /aver LL 'LA 'LA.'IU. *ia I •^i s» 3« ( ' ]ic ac ": u aic *: » ro \LL. . :Li ! :Luk. I '^ l ' The results of simulation study for detector block with two layers of different scintillation materials are grouped in the Table 6 and Table 7 that show the energy resolution for each of the 16 crystals in the top-right quadrant of detector block for the configurations LSO/GSO (7.5 mm-7.5 mm) and LSO/GSO (10 mm-10 mm) respectively. 46 Table 6 LSO/GSO (7.5 mm-7.5 mm) Energy Resolution (%) LSO GSO 23.4 19.1 26.6 19.4 23.3 23.9 33.4 24.3 14.8 17.8 21.0 27.3 18.5 22.3 26.3 34.2 19.5 16.1 23.5 19.0 24.4 20.3 29.5 23.8 17.6 17.1 19.9 20.3 22.2 21.4 24.9 25.4 Table 7LSO/GSO (10.0 mm-10.0 mm) Energy Resolution (%) LSO GSO 16.1 16.6 23.1 16.8 20.8 21.4 29.8 21.7 12.8 15.4 18.2 23.7 16.5 19.9 23.5 30.6 16.9 14.0 20.4 16.5 21.8 18.1 26.3 21.3 15.3 14.8 17.2 17.6 19.7 19.1 22.2 22.7 The energy resolution averaged over all 64 crystals is summarized in Table 8. Four crystal configurations were simulated: two LSO layers 7.5 mm deep (LL7.5), two LSO layers 10 mm deep (LL10), a 7.5 mm deep LSO layer followed by a 7.5 mm GSO (LG_7.5) layer and finally a 10 mm deep LSO layer followed by a 10 mm deep GSO layer (LG_10). Table 8 Energy resolution for different crystal configurations. Crystal configuration Top layer Bottom layer LL 10 18.4% 20.4% LL 7.5 19.5% 21.1% LG 10 18.6% 24.3% LG 7.5 19.7% 25.2% As we can see the energy resolution is slightly worse for GSO. The reason is low statistics of scintillation photons for events stopped in GSO. The energy resolution was better for longer crystals. That is due to higher photo-fraction compare to a scatter fraction in the longer crystals. Also energy resolution seems better for Top-level. Particles stopped in bottom level can be scattered in top-level before. That leads to a broader energy distribution for particles stopped in bottom level. The simulated data were in good agreement with CPS data for energy resolution (-20-25%) [8]. 47 5.2 Detector efficiency. One of the questions was how the sensitivity changes as a function of crystal material depth. Simulation was done for LSO/LSO and LSO/GSO crystal blocks from 1-cm to 3-cm long then simulated results were compared with following theoretical calculations. LSO/LSO £LSO/LSO = 1 - exp(-dLsoA-Lso) LSO/GSO L^SO/GSO = 1 - expC-dLsoA-Lso) + exp(-dLsoA-Lso)[l- exp(-dGsoA<;so)] = 1- exp(-dLso/^LSo) exp(-dosoAoso) where X,LSO =1.155 & XGSO =1.43 Simulated data and theoretical calculation showed identical results. The detection efficiency for coincidence imaging is shown in Figure 33, calculated without and with energy thresholds. Figure 32 Coincidence detection efficiency as a function of total crystal thickness (equal depth layers were used). Squares: LSO/GSO, triangles: LSO/LSO, x: LSO/GSO with an energy window of350-650 keVand circles: LSO/LSO for the same energy window. o.o 0.5 1.0 1.5 2.0 2.5 3.0 3.5 LSO/LSO and LSO/GSO equal Thickness 5.3 Position mis-identification. The total position mis-identification is a combination of two factors: mis-position identification of position map and DOI algorithm. Tables 9 and 10 show a simulated example for the configuration LG-10, top-right quadrant of the detector block. 48 Table 9 Real number of events stopped in the left top 4x4 crystals of the detector block Events stopped in LSO crystals Events stopped in GSO crystals 545 571 524 589 195 181 190 176 526 539 554 538 201 221 225 219 571 580 545 557 235 213 193 180 590 570 543 553 231 212 254 203 Table 10 Events correctly identified by Position Map. 466 489 446 509 165 150 157 146 446 458 477 458 166 182 185 181 487 497 468 486 204 176 159 149 503 488 467 474 194 178 208 168 The "Simplest algorithm" for position map was chosen, so simple rectangular drawn around the local maxima of the flood position response were used. The rectangular boundaries along Xy and Yy were found by requiring each row and column to include four local peaks. This was done using the LSO block response. The same boundaries were then adopted to mask the GSO spectrum. As we can see, 511 keV photons were detected uniformly over the crystals with mean 556 detected events per LSO crystal (standard deviation 24 events) and mean 208 detected events per GSO crystal (standard deviation 22 events). The sub-set of events non-correctly identified was approximately 13% less. As we can see, we've got -10-13% for LSO/LSO and 13-15% for LSO/GSO mis-identified events to neighbor's crystals for crystals, which identified on base of position map and decay time discrimination. There are a few reasons for that. First of all, particles that had scatter interaction in the first LSO layer and then stopped in the second layer have a broader energy distribution that leads to a decrease of Anger-logic coordinates precision and worse mis-position identification event-to-crystal by Position Map. These tables present a quantitative analysis of the position encoding accuracy predicted by our simulation for the LSO and GSO HRRT blocks. The analysis is presented considering all detected events inclusively. The position mis-identification in the x and y direction averaged over 64 crystals has been found to be comparable for the 7.5/7.5 and 10/10 mm configurations. It was approximately 12% for the LSO/LSO case and approximately 14%) for the LSO/GSO case and fairly uniform across crystals. The higher miss-identification obtained in the latter case is due to the GSO lower light output and consequent less precise position identification by the PMT energy information. Depth of interaction miss position identification depends on material to use for detector block layers. Table 11 summarize the results of simulation for a few configurations. 49 Table 11 DOI miss position identification. Detector block design DOI miss position % Fast LSO/Slow LSO 12 LSO/GSO 1.67 The depth of interaction position misidentification was found to be 12% for LSO/LSO crystal configurations and 1.67% for LSO/GSO configurations. The lower DOI mis-identification for LSO/GSO is due to a better decay time separation between LSO and GSO compared to the fast-LSO and slow-LSO. 5.4 Spatial Resolution. To simulate a space resolution and resolution uniformity, three position of the point source were examined: at the center of the detector, 5cm and 10cm shift from the center. A collimated point source was set on the middle of the line connecting the centers of two blocks. A collimator was used to allow 511 kev photons to hit just one column of the block detector (8crystals). Each of 8 columns of the block was examined. Spatial distributions were estimated for all configurations for real coordinates of a particle energy centroid in the detector block and for position map coordinates obtained using Anger logic. Space coordinates (Xy, Yy, Zy) of a particle were estimated from Anger logic coordinates and depth of interaction information. The "position map" was used to estimate crystal "ID". The Zy coordinate was set to the crystal depth corresponding to the most likely point of interaction. Algorithm for correction on base of "Depth of Interaction Information" was applied. If coordinates (XyYy) were somewhere in the crystal than coordinates were automatically set to the center of the crystal. Decay time discriminator discriminated particles that were stopped in the top or bottom layer. Then the Z-coordinate position of the particle in the crystal was assigned to the point in the crystal where probability for a particle to be stopped was maximum. Then centers of coincidence pair were connected by a line and source position was estimated as the intersection of this the line with line that was perpendicular to the line connecting the centers of blocks surfaces. After that, space distribution of the source was build and fitted by Gauss distribution and space resolution was estimated as full width of distribution at half maximum. Simulations were performed for four lines of response (LOR) uniformly distributed over one sector of detector. One sector simulation gave us all needed information because of detector symmetry. 50 Figure 33. Source shifted 10cm left and LOR distributed over the left top detector sector. Distribution of detected coincidence events represents the line-spread function of the source for particular line of response (LOR). Full width of this distribution at half maximum (FWHM) gives us spatial resolution of the detector for this particular LOR. Spatial resolution and detected efficiency for each LOR of lOcm-shifted source are presented in the Table 12 for HRRT geometry and in the Table 13 for round geometry. Table 12 HRRT geometry LOR spatial resolution in centimeters (cm). LL 10 Effic. Sp. Res. Sp.Res. without DOI LORI 0.308 2.81 6.49 LOR2 0.64 2.23 3.46 LOR3 0.63 2.25 3.4 LOR4 0.69 1.4 1.4 LL 7.5 Effic. Sp. Res. Sp.Res. without DOI LORI 0.307 2.5 4.9 LOR2 0.508 2.0 2.6 51 L0R3 0.5 2.1 2.6 L0R4 0.55 1.4 1.4 LG 10 Effic. Sp. Res. Sp.Res. without DOI LORI 0.283 3.375 7.5 LOR2 0.6025 2.79 4.01 LOR3 0.602 2.8 4.0 LOR4 0.6528 1.5 1.5 LG 7.5 Effic. Sp. Res. Sp.Res. without DOI LORI 0.283 3.0 5.68 LOR2 0.4722 2.48 3.016 LOR3 0.46 2.5 3.02 LOR4 0.5039 1.5 1.5 Table 13 Round geometry LOR spatial resolution in centimeters (c LL 10 Effic. Sp. Res. Sp.Res. without DOI LORI 0.308 2.81 6.49 LOR2 0.64 2.23 3.46 LOR3 0.63 2.25 3.4 LOR4 0.69 1.4 1.4 LL 7.5 Effic. Sp. Res. Sp.Res. without DOI LORI 0.307 2.5 4.9 LOR2 0.508 2.08 2.6 LOR3 0.548 1.7 2.1 LOR4 0.623 1.4 1.4 LG 10 Effic. Sp. Res. Sp.Res. without DOI 52 LORI 0.283 3.4 7.5 LOR2 0.602 2.79 4.0 LOR3 0.654 2.0 2.5 LOR4 0.672 1.5 1.5 LG 7.5 Effic. Sp. Res. Sp.Res. without DOI LORI 0.283 3.0 5.7 LOR2 0.472 2.48 3.02 LOR3 0.504 1.9 2.2 LOR4 0.522 1.5 1.5 The spatial resolutions with and without the DOI correction are summarized in Table 14 for the most oblique LORs for each crystal and tomograph geometry configuration simulated. Table 14. Spatial resolutions for the most oblique LOR for three source positions. Data are shown in the format a/b, where 'a' is the resolution obtained with DOI correction and 'b' is the resolution obtained without DOI correction. All values are expressed in mm. Geometry Crystal config. 0cm 5 cm 10cm Octagonal LL_7.5 LL_10 LG_7.5 LG_10 2.1/2.6 2.2/3.5 2.5/3.0 2.8/4.0 2.3/3.0 2.5/3.5 2.8/3.2 3.1/5.6 2.5/4.9 2.8/6.5 3.0/5.7 3.4/7.5 Circular LL_7.5 LL_10 LG_7.5 LG_10 1.4/1.4 1.4/1.4 1.6/1.6 1.6/1.6 2.1/2.6 2.2/3.5 2.5/3.0 2.8/4.0 2.5/4.9 2.8/6.5 3.0/5.7 3.4/7.5 The effect of DOI correction on spatial resolution is also shown in Figure 34 and Figure 35. The figures illustrate the separate components of the source profiles due to those events where both y rays stop in the top layer, those events where both y rays stop in the bottom layer and events where one y ray stopped in the top and the other in the bottom layer. The total profile is obtained by summing the three contributions. The 53 resolution improvement obtained after DOI correction can be visualized by comparing the total profiles in the two figures. Figure 34. Source profiles without DOI correction. Squares: events stopped in the top layer, triangles: events stopped in the bottom layer, rhombs: events, where one y-ray stopped in the top and the other one in the bottom layer and x: total profile 500 in 0.25 2.25 4.25 6.25 8.25 Source position profile in mm Figure 35. Source profiles after DOI correction. Squares: events stopped in the top layer, triangles: events stopped in the bottom GSO layer, rhombs: events, where one y-ray stopped in the top and the other one in the bottom layer and x: total profile 600 i 0.25 2.25 4.25 6.25 8.25 Source position profile in mm The detector resolution was calculated as a average of weighted sum of resolutions at each LOR. Weights were defined as detector efficiency for a chosen LOR. The overall position resolution results are summarized in Table 15, which shows the spatial resolutions (full width half maximum (FWHM)) obtained for a point source located at three radial positions: center, 5 cm off-center and 10 cm off-center for all simulated configurations. The resolution data are presented in the format 2.06/3.25 (mm) where the first number represent spatial resolution with DOI correction in millimeters and the second without DOI correction. 54 Table 15. Overall spatial resolution for three source positions. Data are shown in the format a/b, where 'a' is the resolution obtained with DOI correction and 'b' is the resolution obtained without DOI correction. All values are expressed in mm. Geometry Crystal config. Ocm 5 cm 10cm Octagonal LL_7.5 LL_10 LG_7.5 LG_10 1.9/2.0 2.0/2.6 2.1/2.4 2.3/3.0 1.9/2.2 2.0/2.7 2.1/2.5 2.4/3.1 2.1/2.9 2.3/3.6 2.5/3.4 2.8/4.1 Circular LL_7.5 LL10 LG_7.5 LG_10 1.4/1.4 1.4/1.4 1.6/1.6 1.6/1.6 1.9/2.2 2.0/2.6 2.2/2.5 2.3/3.0 2.0/2.9 2.1/3.3 2.3/3.3 2.3/3.7 55 Chapter 6 Validation of the simulations. To validate the simulation studies we simulated the BGO tomograph of the Siemens ECAT 953B [3] at UBC campus. The evaluation was performed to address the question of how well simulation compare to real data. The ECAT 953B was installed in 1991. This tomograph consists of a cylinder of bismuth germanate (Bi4Ge30i2) detectors arranged in ring. ECAT 953B has 16 rings, each of 384 detectors of size 56.45x56.45x30mm. Crystal size is 6.25x6.25x30 mm. The detectors are grouped into blocks of 8x8, mounted on a set of four photomultiplier tubes. The crystal hit in an 8x8 block is obtained by combining the readings from the four photomultipliers, using Anger logic. The ring diameter is 76cm, with an axial length of 10.8cm. Spatial resolution was measured for 1.372 MBq point source at the center of the field of view and at 5, 10 and 15 cm off center. The bin size of the sinogram was 3.1 cm. The experimental and simulated data represented in the Table 16. Table 16. Spatial resolution for ECAT 953B. Off center Experimental data Simulated data 0cm 4.4 4.3 5cm 5.1 4.8 10cm 5.8 5.6 15cm 5.8 5.6 Experimental data were compared with simulation and showed close values. 56 Chapter 7 Discussion. Detection efficiency is one of the most important parameters in PET imaging, since it has direct bearing on the image quality and on the feasibility of imaging protocols. An obvious method to maximize crystal efficiency is to increase detector thickness and/or to decrease the detector-to-detector distance. However both these approaches degrade the resolution and resolution uniformity since they increase the parallax error. The trade-off between detection efficiency and resolution uniformity can be evaluated by comparing the results shown in figure 5 (efficiency) and table 6 (resolution). For example, by increasing the overall crystal depth from 1.5 to 2.0 cm, an approximately 15-20% gain in coincidence detection efficiency can be achieved. However resolution simulation also show that such detection efficiency increase is obtained at a significant cost of resolution uniformity degradation. Simulations results show that the resolution uniformity degradation without DOI is more severe for circular tomograph geometry compared to the octagonal one. This is due to the fact that in the octagonal geometry, in contrast to the circular geometry, the parallax error affects also the resolution at the center of the FOV: the direction of the emitted y rays is not always perpendicular to the detector surface. In terms of absolute resolution values, DOI correction is shown to have a bigger impact in the case of the circular geometry. In general the LSO/LSO crystal configuration versus LSO/GSO showed better spatial resolution. The reason for this is likely the fact that the GSO second layer has a higher ratio between the Compton cross section for non-monoenergic gamma photons compared to LSO. Therefore the probability that most of the y ray energy will be released in a crystal neighboring the one that the y ray first hit, is higher for GSO compared to LSO thus degrading the spatial resolution. The resolution data presented in this simulation are slightly better than the measured values reported for the present HRRT [4]. (Spatial resolution <2.4mm in center and <2.8 mm at 10cm off center) . This could possibly be attributed to the fact that the decay time difference between the front and back layer in the tomograph might not be as uniform as the one used in the simulation. Another possible source of discrepancy could be due to the parameters used in modeling the light transport in the crystals: although the parameters used were previously validated [15], small changes in the manufacturing process might introduce some variations in the light transport process. 57 Chapter 8 Conclusion We developed a simulation tool that let us simulate detector design on a system level such as detector geometry and detector block design as well as on physical level like simulation of physical properties of material which are used for the detector build. Depth of interaction mechanism was implemented for the DETECT program. Simulation studies have shown that even in a detector configuration where the total crystal thickness is 1.5 cm there is significant resolution and resolution uniformity degradation due to the parallax error for a tomograph with a relatively small detector-to-detector distance. The degradation pattern differs as a function of tomograph geometry: in the case of the octagonal geometry the resolution degradation affects the entire FOV, while in the case of the circular geometry the resolution is preserved in the center of the FOV and is progressively degraded off center, leading to a greater resolution non-uniformity. In both cases DOI correction is necessary to correct for the parallax error. The DOI correction scheme based a two-layer crystal configuration, where the two crystal layers are characterized by two different scintillation decay times has been shown to provide an effective correction method. It must however be noted that these results have been achieved with simulations where the same decay time distribution was used for all crystals made from the same scintillation material. In practice, this might not be always feasible, and/or the decay time differences between the two crystal layers might be different than those used here. The present simulations could easily be extended to include those instances, which were beyond the scope of this work. Our simulation shown that the implementation of the DOI correction significantly improves resolution uniformity across the field of view (FOV), and in the case of an octagonal tomograph design, also improves the resolution in the FOV center. DOI corrections were found to improve the resolution by up to 23% in the center and by up to 37% at 10cm off center for the octagonal tomograph configuration and up to 38% at 10cm off center for the circular tomograph design. 58 Bibliography I] W.F. Jones, M.E. Casey, A. van Lingen, and B. Bendriem. "LSO PET/SPECT Spatial Resolution: Critical On-line DOI Rebinning Methods and Results". CTI PET Systems Inc., Free University Hospital (Amsterdam, Netherlands) 2] W. W. Moses, R. H. Huesman and S. E. Derenzo."A new algorithm for using Depth-of-Interaction measurement information in PET data acquisition." J. Nucl. Med., 32 995 (1991). 3] C. Moisan, J. G. Rogers, K. R. Buckley, T. J. Ruth, M. W.Stazyk, G. Tsang. "Design Studies of a Depth Encoding Large Aperture PET Camera."IEEE Trans. Nucl. Sci., vol. 42, no. 4, 1995 4] K. Wienhard, M. Schmand, M.E. Casey, K.Baker. "The ECAT HRRT: Performance and First Clinical Application of the New High Resolution Research Tomograph." IEEE Trans. Nucl. Sci., vol. 49, no. 1, 2002 5] "GEANT" CERNProgram Library Long Writeup W5013. 6] M.E. Casey, L. Eriksson, M. Schmand, M.S. Andreaco, M. Paulus, M. Dahlbom, and R. Nut."Investigation of LSO crystals for high resolution positron emission tomography," IEEE Trans. Nucl. Sci., vol. 44, no. 3, pp. 1109-1113, 1997. 7] A. Levin* and C. Moisan, "A More Physical Approach to Model the Surface Treatment of Scintillation Counters and its Implementation into DETECT." 1996 IEEE Nuclear Science Symposium of Anaheim, pp. 702-706, November 1996.[TRIUMF preprint PP-96-64] 8] M. Schmand, L. Eriksson, M.E. Casey, M.S. Andreaco, C. Melcher, K. Wienhard, G. Flugge, R. Nutt. "Performance results of a new DOI detector block for a High Resolution PET - LSO Research Tomograph HRRT." IEEE Trans. Nucl. Sci., vol. 42, no. 4, 2002 9] W.H. Wong, "Designing a stratified detection system for PET cameras," IEEE Trans. Nucl. Sci., vol. 33, no. I, pp. 591-196, February 1986. 10] M.E. Casey and R. Nutt, "Amulti-crystal two dimensional BGO detector system for positron emission tomography," IEEE Trans. Nucl. Sci., vol 33, pp. 460-463, 1986. II] C L . Melcher and J.S. Schweitzer, "Cerium-doped lutetium oxy-orthosilicate: A fast, efficient new scintillator," IEEE trans, Nucl. Sci., vol. 39, no. 4, pp 502-505, 1992. 12] Patrick Rommel G. Virador, William W. Moses, and Ronald Huesman1 (Professors Thomas F. Budinger and Stephen E.Derenzo) "Design of a High-Resolution, High-Sensitivity PET Camera for Human Brains and Small Animals." US Department of Energy and National Institute of Health. [\3] Bronstein, M. Zibulevsky, Yahoshua Y. Zeevi . "Optimal non-linear estimation of photon coordinates in PET". Israel Institute of Technology, Haifa. 14] Joel G. Rogers, Christian Moisan, Emile M. Hoskinson, Mark S. Andreaco, C. W. Williams, Ronald Nutt. "A Practical Block Detector for a Depth-En 59 

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