A Simulator For Small

Positron Emission Tomography Cameras

Aaron H. Steinrnan

A thesis submitted in conformity with the requirements for the degree of Master of Applied Science

Graduate Department of Electrical and Computer Engineering University of Toronto

O Copyright by Aaron H. Steinrnan 1997

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A Simulator for Small Positron Emission Tomography Cameras

Master of Applied Science, 1997

Aaron W. Steinman

Graduate Department of Electrical and Computer Engineering

University of Toronto

Abstract

Previous research indicates a need for a dedicated small positron emission tomognphy (PET)

carnera for use in animal experiments in radiopharamaceutical developments. This thesis is the

creation of a Monte Carlo simulator to be used as a tool in the design of such a carnera. The

simulator models the physics of PET. such that a vancty of source distributions and camera

codigurations cm be simulated and compared. This simulator provides a fast, first-order camera

performance cornparison, to identiQ weaknesses and strengths in a camera configuration. The

performance of the sirnulator was validated by comparing it with the actuai PET camera at the Clarke

Institute. The simulator successfully passed the resolution, scatter, and count rate tests. Future work

includes upgnding the simulator to provide in-depth second-order camcra performance comparisons.

Acknowledgments

AAer a few years working on this thesis, there are a lot of people who have given me a lot of support and encouragement. From al1 of these people, there are a few who stand out, and 1 would like to acknowledge their contribution.

First and foremost, 1 would like to th& Dr. Houle for his constant support and understanding throughout the project. He was my source of insights and direction. His teaching style let me explore, and 1 have gained insights throughout my investigation.

Dr. Soy gave me direction. Along with Dr. Houle, Dr. Joy made sure that 1 knew what I wanted out of this project, and made sure that 1 continued to go in that direction.

My good fiiends, Sunil and Drew, were always ready to ofFer support, either with a fnendly laugh, or with in-depth computer debugging. Either way, 1 appreciate their Mendship.

Doug Hussey was very helpfbl with running the PET camera so I could ver@ my thesis. His patience with my countless requests to re:reconstruct the image with different parameters made rny verification possible.

My parents were very supportive of my work, both when things were going well and when things did not look too good. They gave me strength to continue with my work, and this completed thesis is a tribute to their love.

Finally, my wife Lisa was everything -- understanding, patient, supportive, and kind. Without whose support this thesis would not have been.

Thank you to al1 who helped me.

Table Of Contents

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Abstract ii

Acknowledgments ............................................................ iii

Chapter A . Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-1 1 . 0 Introduction and Scope of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A- l 2.0 Need For Dedicated Small Animal PET Cameras . . . . . . . . . . . . . . . . . . . . . . . . A-3

. Chapter B Theory of Positron Emission Tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B- 1 1.0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-l 2.0Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B I

2.1 Definition of "Source" . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-2 2.2 Positron Range, Positron Annihilation, and Photon Generation . . . . . . . . . B-2 2.3 Photon Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-3

3.0 Principles of Positron Emission Tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-5 3.1 Coincidence Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-5

3.1.1 Lines Of Response (LOR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-5 3.1.2 Undesirable Coincidences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-7

3.2Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-7 3.3 Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-8 3.4Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-9

4.0 PET Camera . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-9 4.1 introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-9 4.2 Scintillation Crystds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B- 10

4.2.1 Photon Interaction .................................... Bo10 4.2.2 Light Guides ........................................ B-10 4.2.3 Ideal Properties ...................................... B- 1 1 4.2.4 Available Crystd Materials ............................ B-11

4.3 Photodetectors .............................................. B-12 4.3.1 Principles of Photodetection ............................ B- 12 4.3.2 Common Methods of Photodetection ..................... B- 12 4.3.3 Depth of Interaction .................................. B-12

4.4 Detector Unit ............................................... B-13 4.4.1 One-to-One vs . Block Coupling ......................... B-13 4.4.2 Block Detectors ...................................... B-13

............................. 4.4.3 Puise Pileup and Deadtime B-13 5.0 Reconstruction Algorithm ........................................... B- 14

5.1 Iterative algorithrns .......................................... B-14 5.7 Analytic algorithms ......................................... .B-14

Chapter C . Simulation Implementation Construction Methods . . . . . . . . . . . . . . . . . . . . . . . C-1 1.0 introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-1 2.0 General Simulation Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-4

2.1 Monte Carlo Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-4 2.2 Random Number Generators (uniform) . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-5 2.3 Special Functions ............................................. C-6

2.3.1 Random SinKos Pair Generator . . . . . . . . . . . . . . . . . . . . . . . . . . C-6 2.3.2 Quadratic Equation Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-8 2.3.3 Fast Square Root ...................................... C-9 2.3.4 Generate Random 3-D Unity Vectors . . . . . . . . . . . . . . . . . . . . . C- 11 2.3.5 Normal Distribution Sarnpling . . . . . . . . . . . . . . . . . . . . . . . . . . C- 12

2.4 Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C- 13 3.0 Source Sirnulator Module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-14

3.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-1 4 3.2 Create Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-14 3.3 Source Simulator Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C- 15

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4XGAM C-16 3.5 Source Simulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C- 16

3 S.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C- 16 3.5.2 Positron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-17

3 S.2.1 Positron Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . C- 17 3 . 5.2.2 Positron Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C- 18 3.5.2.3 Positron Annihilation . . . . . . . . . . . . . . . . . . . . . . . . . . C-21

3.5.3Photon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-21 3.5.3.1 Photon Generation and Non-collinearity . . . . . . . . . . . C-21 3 S.3.2 Photon Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-24 3.5.3.3 Photon Exiting Source ......................... C-31

3.5.4 Photon File Format ................................... C-32 4.0 Canera Simulator Module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-34

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-34 4.2 Carnera Simulation Cod~guration .............................. C-34

4.2.1 Source Parameters .................................... C-34 4.2.2 Camera Parameters ................................... C-35 4.2.3 Simulation Parameters ................................ C-36

4.3 Canera Simulator ........................................... C-38 4.3.1 Introduction ......................................... (2-38 4.3.2 Non-temporal Simulation .............................. C-39 4.3.3 Temporal Simulation ................................. C-39 4.3.4 Load Event Data ..................................... C-41

....................... 4.3.5 Identiming Crystal of Intersection C-42 . .......................... 4.3 5. 1 Get Intersection Point C-44

4.3.5.2 Miss Septa .................................. C d 4

4.3 5 3 Detennine Surface Of Intersection . . . . . . . . . . . . . . . . C-45 4.3.5.4 Crystal Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . (2-45 4.3.5.5 Photodetector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-46 4.3 S.6 Energy Window .............................. C-47 4.3 S.7 Deadtirne . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-47

4.3.6 Coincidence Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-47 4.3 .6.1 Determining Coincidence . . . . . . . . . . . . . . . . . . . . . . C-48 4.3.6.2 Deadtirne ................................... C-SI

4.4 Camera Module's Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-51 5.0 Reconstruction Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-52

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.0Display C-53

Chapter D . Simulator Testing and Verifkation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-1 1.OIntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-i 2.0 Testing Linked Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-1

2.1 Testing Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D- 1 2.2 Display-Reconstruction Linked . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-2 2.3 Display-Reconstruction-Camera Modules Linked . . . . . . . . . . . . . . . . . . . D-2 2.4 Display-Reconstruction-Camera-Source Modules Linked . . . . . . . . . . . . D-2

3.0 Clarke Institute's PET Camera's Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-3 4.0 Camera Standards and Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-6

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-6 4.2 Spatial Resolution ........................................... D-6

4.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-6 4.2.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-7 4.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-8 4.2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-9

4.3Scatter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-9 4.3.1 Introduction ......................................... D-9 4.3.2 Methods ........................................... D-10 4.3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D-11 4.3.4 Discussion ......................................... D-11

4.4 Count Rate Losses and Randorns ............................... D- 12 4.4.1 Introduction ........................................ Do12 4.4.2 Method ........................................... D-12 4.4.3 Results ............................................ D-14 4.4.4Discussion ......................................... D-16

. Chapter E Conclusion and Future Work ......................................... E-1 1 . 0 Conclusion ........................................................ E- 1 2.0FutureWork ....................................................... E-2

. Chapter F Literanire Cited .................................................... F-1

Chapter A - Motivation

1.0 Introduction and Scope of Thesis

This thesis is a component of a larger project whose objective is the realization of a positron

emission tomography (PET) carnera dedicated to the irnaging of small animais, such as rats, for use

in radiopharmaceutical development. Specifically, this thesis consists of the design, implementation,

and verification of a computer simulator which models the physics of PET in a Monte Car10 format.

The simulator's primary purpose is to aid in the determination of the configuration of the "ideal"

small animal PET camera through the simulation of a variety of camera configurations.

Animal research is an integrai part of radiopharmaceutical development (Marriott et al., 1994;

Miyaoka et al., 199 1 ; Rajeswamn et al., 1992). Animai biodistribution and kinetic studies are used

to determine a compound's properties as well as its usefulness for clinical and research applications

(Cutler et al., 1992; Miyaoka et al., 199 1 ; Watanabe et al., 1992). Conventional methods of animal

studies obtain their data by dissection and radiation counting in the regions of interest (Watanabe et

ai., 1992). However, these sacrifice and dissection methods are iderior to imaging using a dedicated

animal PET camera for a vaiety of reasons: animal PET imaging can reveal a non-uniform

distribution within an organ without previous knowledge of that organts biodistribution (Marriott et

al., 1994); requires fewer animals; is more cost effective (ignoring the start-up costs of the animal

PET carnera); and will generate reproducible, and hence superior, results (Steinman, 19%).

The design implementation of an animal camera requires multiple stages. The fust stage is

needr assessment, which determines the need for dedicated small animai PET cameras as weIl as the

parameters of an "ideal" animal camera. This was performed by Steinman (1995) for his B.A.Sc.

thesis. The second stage is simzriation, which provides theoretical optimization of the design and

evaluates the performance of the system (Rowe & Dai, 1992; Thompson et ai., 1992). The purpose

of this thesis is the creation of a computer suilulator, which, for future work, will be the tool used

to provide the design optimization of the animal PET camera. The fmal stage is conîn?rction, which

is comprised of two future sub-stages: the construction of a single detector module and, once

successfully tested, the construction of a fidi prototype ring of detector modules.

The animal camera simulator is designed to be a tool for analyzing various detector

configurations using application specific simulated animal sources. incorponting the physics of

positron emission tomography with Monte Cm10 techniques. The simulator is divided into two

independent software modules: the Source Emission and the Camera Detection. Data generated fiom

the Source Emission module c m be used repeatedly as the starting point for difierent camera

configurations in the Camera Detection module. As well, each module is comprised of nurnerous

sub-modules in order to facilitate the implementation of future developments

Emphasis is placed on modeling complex source and camera geometries. However, this

thesis does not simulate in depth crystal interactions. One important and unique feature of the

software simulator is its ability to generate " temporalt' simulations. In PET, the positron, and hence

the corresponding annihilation photons, are generated based on the source's radioactivity distribution,

and are emitted according to Poisson statistics. The carnera detection module generates a random

time interval between annihilation events, which forms the basis for random coincidences and

detector unit deadtime.

The simulator's verifkation process is twofold. First, each stage in the construction of the

simulator is tested and compared with theoretical results. If the stage contains an aigorithm that is

available from the Literature, the results of the aigorithm are additionaily compared with their

original articles. Once every simulation stage is deemed correct, the entire simulator as a whole is

verified. To this end, the Scantronix-PC2048 PET Carnera at the Clarke Institute of Psychiatry was

simulated, and the simulated results are compared with the corresponding experirnental resuits.

This thesis will begin by examinhg the motivation for the construction of a simulation tool

for use in the design of a dedicated small animal PET camera, briefly touching on the need for such

a camera. Chapter B will summarize the principles of positron emission tomography, concentrating

on the theory which will be used in the simulation. Chapter C will describe the methods used in the

implementation of the simulation; including the theoretical basis for the construction of the simulator

and the necessary validation tests for each stage of construction. Chapter D will discuss the

validation of the thesis, including comparing the actuai performance of a PET camera with its

simulated performance. Finaily, Chapter E will nunmarize and present the conclusion to diis thesis

as well as directing the readerrs attention to M e r research.

2.0 Need For Dedicated Small Animal PET Cameras

Positron emission tomography ailows the memurement of complete regional tissue tracer

time courses in individual animals without sacrifice and dissections (Rajeswaran et al., 1992). This

approach requires less animais and generates reproducible results from the same animal (Steiman,

1995), eliminating inter-animai variation present with conventional methods. Moreover, animai PET

imaging is capable of revealing a non-unifonn distribution within an organ without previous

knowledge of that organ's biodistribution (Marriott et al., 1994).

Currently, no carnera exists that meets the specifications of an ideal small animal PET canera

(Steinman, 1995). This " ideal" animal camera requires 1 -2mm full-width-half-maximum (F WHM) uniform spatial resolution to get a crisp image of small animal organs and minimize partial volume

effects (Hichwa, 1994; Miyaoka et al., 1991). A small ring radius is desired to increase sensitivity

and spatial resolution (Steinman, 1995). As well, the increase in random and scatter coincidences

caused by the srnall radius will be offset by the increase in true coincidences from small animal

imaging (Ingvar et al., 199 1; Miyaoka et al., 1991). However, the small ring radius causes an

increase in the radial elongation artifact (Rajeswaran et ai., 1992), because more annihilation photons

penetrate adjacent crystals before interacting and being detected (Moses & Derenzo, 1994). Depth

of interaction can be used to eliminate this artifact (Steinman, 1995). As well, minimization of scan

time, which is necessary for kinetic studies of tracer uptake, occurs with a full ring detector

(Steinman, 1995). Ignonng the cost, the investigation (Steinman, 1995) proposed an ideal animal

camera with an LSO scintillation crystal. This would either be one-to-one coupled with avalanche

photodiodes or be cut in a comb-slit style and coupled to a position sensitive photomultiplier tube.

The investigation determined that the resolution of contemporary "human" cameras is too limited

for the size of the animals' anatomy (the rat cerebellum for example), and a dedicated animal camera

is needed.

Chapter B - Theory of Positron Emission Tomography

1.0 Introduction

Positron emission tomography (PET) is an important imaging modality which provides

insight into the complex function of the human body. The extemal PET carnera "images" the

biodistribution of injected pharmaceuticals. These compounds are labeled with short lived

radioisotopes of the basic building blocks of life, such as carbon, nitrogen, oxygen, and fluorine.

Thus, both drugs and organic compounds already present in the subject can be labeled. When the

resulting ndiophamiaceutical is injected into the subject in trace amounts, the existing metabolism

is not disturbed (Tsang, 1995). Common PET applications are neural and cardiac fùnctional

imaging, as well as pharmaceutical development.

PET imaging begins with the injection of the radiopharmaceutical into a subject. n i e body

metabolizes the injected radiopharrnaceuticd and the radioisotopes emit positrons as the compound

moves dong the path of biodistribution. The positrons move a short distance (a few millirnetres)

in the tissue (Derenzo, 1986) and are annihilated by nearby free electrons. This annihilation process

creates two high energy photons (both 5 1 1 keV) which are emitted in opposite directions. These

photons exit the body and are detected by the extemal carnera. The cornputer system attached to the

carnera calculates the location of the positron-electron annihilation, and outputs this location in a

displayed image.

This chapter outlines the theory of PET, following the path from compound injection to

image reconstruction. Section 3.0 examines how the radioactivity interacts with the object being

imaged, including positron generation and tracking the annihilation photon as it attempts to leave

the source. The pinciples of PET are discussed in section 3.0, with emphasis on the detection of

the two coincident photons by an ided carnera. Section 4.0 outlines the components of the PET

camera. The effects of the detection crystals, which are used to convert the high energy photons into

light that can be detected by photodetectors, as well as examining the detector unit, formed through

the interfacing of crystals and photodetectors will be examined. Finally, section 5.0 describes the

two families of reconstruction algorithms, which take the coincident information and create the £inal

image.

2.0 Source

2.1 Defiition of "Source"

Throughout this thesis, the terni "source" will refer to the object being "imaged".

Specifically, there are three properties to the source. First, the source contains the positron

emitting radionuclides. Second, the attenuation of the positron's motion (range) as well as the

positron-electron annihilation occur within the source. Third, the source contains the materials

which potentially interact with the photons generated from the positron-electron annihilation.

Once a photon leaves the imaged object, it has left the source, and will either hit the camera and

be detected, or will miss the camera and be lost. The term "sub-source" refers to a homogeneous

material within the object itself. Examples of sub-sources are bones, organs and blood vessels.

The sub-source contains at least one of the aforementioned properties of the source.

2.2 Positron Range, Positron Annihilation, and Photon Generation

Radionuclides used in PET decay by ernitting positrons. These positrons travel a short

distance in the tissue, losing their kinetic energy in a number of ionizing events with surrounding

atorns . Eventually , the positrons interact with electrons and become annihilated. This short distance before annihilation is called positron range, and is dependent on the type of positron

emitting isotope (Cho et al., 1993) and the tissue material (Derenzo, 1986). The physical

properties of the most commonly used PET radioisotopes are listed in table B-1. The positron

range has a bi-exponentially probabiiity disrribution (Derenzo, 1986). The value of the "positron

Table B-1: Physical properties of common PET radio-isotopes (Cho et al., 1993) Il I 1 L I I

Radio-Isotope

Carbon 1 1 ('i C) Nitrogen 13 (': N)

Oxygen 15 ('i 0)

Hal f-life (min)

20.3

10.0

2.0

Maximum Positron Energy (MeV)

Positron Radial Range in Water ( mm)

0.959

1.197

1.738

0.111

O. 142

O. 149

range" in table B-1 is defmed as the distance, in rnillimetres, that corresponds to the probability

of fifty percent survival.

The product of the positron-electron annihilation is the sirnultaneous production of two

photons with SllkeV energy (Leo, 1994). Through the conservation of momentum, these

photons travel in opposite directions (180 O f 0.5 O F W H M ) , approximating collinearity (Cho et al.,

1993; Leo, 1994; Thompson et al., 1992). Perfect collinearity will occur only if the positron and

electron either have opposite momentum or annihilate at rest. Figure B-l shows the effect of non-

collinearity. The photons are emitted randornly in an isotropic marner (4x steradian) (Leo,

1994). Both the positron range and non-collinearity are intrinsic limitations to image resolution.

D ETECTO R BANK -

D ETECTO R BANK

Annihilation Evcnt Dcscribcd by: C-U. C - W . C-X. C-Y. C-2. but only C-X i i Coflnrar

1

0.8 Q, n 5 0.6 z pl

0.4 m - 8 0.2

O -1 .O -0.5 0.0 0.5 1 .O

Offset Angle (degrees)

Figure B-1: Effect of non-collinearity in coincidence detection.

2.3 Photon Interactions

Each annihilation photon may interact with matter before it escapes the source. The three

types of interactions are incoherent (Compton) scattering, photoelectric effect, and coherent

(Rayleigh) scattering (Chan t Doi, 1988; Lux & Koblinger, 1991; Williamson, 1988). However,

B -4

the photon energy in PET (51 1keV) is too low for pair production, which is an interaction where

a photon whose energy exceeds 1.022MeV disappears in the field of the nucleus, and the result

of the interaction, creates a positron-electron pair (Lux & Koblinger, 1991).

In Compton scattering, a photon collides with an atom and a secondary photon of lesser

energy as well as an electron are ejected (Chan & Doi, 1988; Lux & Koblinger, 1991;

Williamson, 1988). This new photon moves through the source in a different direction than the

original photon. Compton's relationship (cited in Chan & Doi, 1988) between the deflection angle

and the new energy of the scattered photon is:

This equation assumes that the electron was initially free and stationary. The differential cross-

section of a Compton collision for a free electron is given by the Klein-Nishina formula (cited in

Chan & Doi, 1988):

Where r, is the classical electron radius. The probability of Compton scattering can be expressed

as the product of the aforementioned Klein-Nishina differential cross-section formula and the

incoherent-scattering function (Chan & Doi, 1988). This factor includes the effect of electron

binding on the incoherent scattering's differential cross-section (Chan & Doi, 1988).

In the photoelectric effect, a photon is absorbed by an atom, effectively ending its history

before it can escape from the source. The energy fiom the absorption of the photon is used by the

atom to eject an electron fiom its orbit ( Lux & Koblinger, 1991; Williamson, 1988).

In coherent or Rayleigh scattering, the photon undergoes an elastic collision with the atom,

with no resultant energy loss (Chan & Doi, 1988; Lux & Koblinger, 1991; Williamson, 1988).

However, the high photon energy level in PET does not allow for Rayleigh scattering to occur

(Chan & Doi, 1988; Lux & Koblinger, 1991; Williamson, 1988).

In positron emission tomography, the most cornmon photon interaction is Compton

scattering, followed by the photoelectric effect. The determination of whether or not an

interaction occurs is based on the source matenai's fiee path length; this is the inverse of the linear

attenuation coefficient and is dependent on the energy of the photon. In fact, the photons may

undergo multiple Compton scatters before they either escape the source or are absorbed through

photoelectric effect.

3.0 Principles of Positron Emission Tomography

3.1 Coincidence Detection

3.1.1 Lines Of Response (LOR) Collinearity of the ...............................................................................................................................

Volume of annihilation photons foms the i

basis of PET irnaging (Cho et +~f:-rv : .nu A

al., 1993). The positron source f &&& un + - is placed between two D e t e c m 1 D e t e c t o r 2

coincidence detectors. If b0th of Tracer Dis t r ibu t ion f(x,y J I

these detectors simultaneously ............................................................................................................................. *.: detect an annihilation photon, Figure B-2: Volume of Response

then an annihilation event

occurred somewhere along the iine joining the two detectors. This line is called the line of

response (LOR). However, the fuiite size of the detectors cause the LOR to have depth, as well

as length and width. Thus, the LOR is acniaily a volume of response, as shown in figure B-2.

Annihilation photons are detected only if the LOR volume encloses the positron-electron

annihilation as well as the photons' direction of emission.

With a single ring camera, one image (a "slice") is produced. With multi-ring PET, slices

are produced for each ring, as weiI as "inter-ring" slices, which have the LOR between two

neighboring rings. In true 3D PET, the LORs connect detectors from one ring to any other ring.

This increases the sensitivity of the camera because there are more LORs, and thus a greater

fraction of annihilation photons are detected (Guerrero et al., 1994).

For single plane PET, the LORs are referenced by two parameters: the angle between the

LOR and a fixed reference, as well as the distance between the centre of the detector ring and the

LOR. This is s h o w in figure B-4a. The "sinogram" is a rnap of every LOR, with the horizontal

axis representing distance, and the vertical axis representing angle (Cho et al., 1993). Figure B-4

shows a sinograrn of a single LOR as well as a sinogram of al1 the LORS which travel through a

point off centre. The sinogram shows the importance of angular sampling (see section 3.2): blank

horizontal lines appear for every angle that is not sampled. The set of parallel LORs are

represented as a horizontal line in the sinogram and is a 1D projection of the 2D slice of the

object. These projections are converted into the desired image of the object by the reconstruction

algorithms described in section 5 .O.

1 300

' d, . - d: Linc O f R c s p o n s c i l

for d A - d, - 740 ' 4 -- t " - $ , , C l i , L / d . 120 1

1 I / 6 0 d A

'--

O ' 1

-r O X '

r

r *

Y c) d ------a 360 '

I i

Lincs O f Rcrponsc 240 CI - 2

' r n I rn

=. ---+--. -r O r r'

................................................................................ ...................................................... :""""""""'.'.. -..--*-.* .-..------+--.**...*..-.-. : Figure B-3: a) LOR referenced; b) sinogram of (a); c) LORs through a point off centre; d) sinogram of (c)

B-7

3.1.2 Undesirable Coincidences

There are three different types of coincidences that can be detected: m e , scattered and

random (see figure B-5). True coincidences (figure B-5a) are the desired events as they represent

the only information useful for image reconstruction (Cho et al., 1993). Both the randorn and

.............................................................................................................................................................................................. i a ) Truc b) Scattcrcd C ) Random

C o i n c i d c n c c A - B C o i n c i d c n c c A-C Coinc idcncc A - D

is maximized when linear sampling fulfils the Nyquist sampling criterion. This criterion States

that the linear sarnpling distance must be less than half of the desired spatial resolution to fulfd the

linear sampling requirement (Cho et al., 1993). In ring PET design, as opposed to parallel plate

(planar) PET, the sampling distance corresponds to one-fourth the detector width (Cho et al.,

1993). There are camera geometry dependent factors which affect linear sampling. In polygonal

camera geometry (see section 4.11, translational motions need to be added to obtain the desued

linear sampling (Cho et al., 1993). The most common geometry is circular, and iü linear

sampling can be increased by introducing a wobbling motion (Palmer et al., 1985). However, this

movement creates a non-uniformity in linear sampling, reducing its efficiency (Palmer et al.,

1985). More uniform methods include dichotomie motion (two half rings rotating back and forth)

(Cho et al., 1993) and clam motion (two half rings, attached at a pivot) (Lecomte et al., 1994).

Linear smpling becornes compromised for obliquely incident photons intersecting with the

scintillation crystals (section 4.2). These photons enter one crystal, but spi11 into adjacent crystals

where they interact with the crystal and become detected (Cho et al., 1993). This problem can

be aileviated by either increasing the ring diameter or by increasing the detector width, which will

both limit spatial resolution (Cho et al., 1993).

3.3 Resolution

There are three types of resolution considered in PET: spatial, temporal and energy.

Spatial resolutiondeterrnines the rninimurnobject size that can be identified in PET. For a given

detector size or width, maximum resolution in spatial sampling can be achieved by fulfilling

Nyquist sampling cntenon (Cho et ai., 1993). Thus, high spatial resolution is achieved by using

small crystals, although eficiency and light yield are sacrificed (Cho et al., 1993; Utchida et al.,

1986). The distance the positron travels before annihilation (positron range) is approximately one

to two millimetres, depending on the positron emitting isotope (see table B-1). Increased positron

range reduces spatial resolution, and sets f ~ t e limits on the spatial resolution (Budinger et al.,

1991). Temporal resolution determines the maximum time pemiitted to have a coincident event

between detectors. Energy resolution represents the ability of the scintillator crystal and

photodetector to correlate the height of the photodetector's output pulse with the initial energy of

the incident photon. This is related to the crystal's intrinsic energy resolution and energy

conversion efficiency, as well as the photodetector's quantum and collection efficiency

(Harnmarnatsu, 1994). The narrower the crystal width, the greater the loss in energy resolution

caused by numerous reflections off of the crystal sides (Yamashita et al., 1990).

Resolution is very important because low resolution reconstnictions with a high number

of detected events fails to show "hot spots" as accurately as high resolution with a low nurnber of

detected events (Budinger et al., 1991). As well, Compton scatter and pulse pileup in the detector

reduce resolution (Miyaoka et al., 199 1). The resolution degradation from the non-collinearity

of the annihilation photons decreases with ring diarneter (Moses & Derenzo, 1994).

3.4 Sensitivity

Sensitivity is the camera's capability to detect m e coincidences for a given amount of

radioactivity (Cho et al., 1993). This is determined by the detector's efficiency, which is

dependent on the crystal scintillator's capacity to stop annihilation photons. The greater the

sensitivity of the camera, the lower the dose requirements (Budinger et al., 1991). Sensitivity is

also related to camera radius - m e coincidences are inversely proportional to the camera radius, whereas random and scatter coincidences are inversely proportional to the radius squared

(Miyaoka et ai., 199 1).

4.0 PET Camera 4.1 Introduction

The PET camera consists of detector units arranged in a geomeuical pattern. The choice

of system geometry determines the fundamentai system performance (Cho et al., 1993). There

are three types of geometry: planar, polygonal, and circular ring. Planar geomeny consists of two

detector planes facing each other (Cho et al., 1993). Angular rotation is required to fuifiill angular

sampling . Polygonal systems are typicaliy comprised of hexagonal or octagonal detector planes, and require only simple translations and rotations to hilfill both uniform Iinear and angular

sampling (Cho et al., 1993). However, polygonal systems have limited eficiency , especidy

towards the periphery of the image (Cho et al., 1993). Circular ring systems provide both

uniformity and symmetry (Cho et al., 1993). Linear sarnpling problems have been resolved by

incorporating wobbling and dichotomic sarnpling schernes (Cho et al., 1993). Imaging in the axial

direction is accomplished through the incorporation of extra "rings" (see section 3.1.1).

The detector units are compnsed of scintillation crystals coupled to photodetectors. Section

4.2 investigates the effects of the detection crystals, which are used to convert the high energy

photons into light that can be detected by photodetectors, as described in section 4.3. The

interfacing of crystals and photodetecton form the detector unit, which is examined in section 4.4.

4.2 Scintillation Crystals

4.2.1 Photon Interaction

Once an annihilation photon enters the crystal, it will either pass through the crystal and

be lost, or it will ionize an atom in the crystal by interacting with and exciting an elecuon, leaving

a hole. Subsequendy, another electron will drop from its excited state to the ground state CO fil1

that hole (Leo, 1994), emitting light photons in 47r steradian. These light photons either interact

with the photodetector, leave the crystal, or reflect back into the crystal (Leo, 1994). Each photon

which leaves the crystal reduces the canera's efficiency and sensitivity (Leo, 1994). Blurring,

and subsequent reduced spatial resolution, occurs when a light photon leaves one crystal and enten

into another (Thompson, 1990). There are two types of reflection which partialiy counter photon

loss: extemal and internai. External reflectors redirect the escaping light back into the crystal.

Total interna1 reflection of the crystal can be increased by minimizing the index of refraction of

the medium surroundhg the crystai 00, 1994).

4.2.2 Light Guides

Non-uniform sensitivity of photodetectors (see section 4.3), especially photomultiplier

tubes (PMTs), c m be overcome by light guides (Hilal et al., 1989). Light guides also are used

in block detectors to control the distribution of light to the PMTs. The crystal is cut at various

depths, leaving a light guide at the bottom, such that each individual cut crystal can be identified

by a unique set of PMT signal combinations, all having the same probability of being detected

B-1 1

(Rogers et al., 1992). Greater depth in the crystal cuts corresponds to the increased efîiciency

(Thompson et al., 1992), but decreased performance because of crystal attenuation (Rogers et al.,

1992), radial blurring (Thompson et al., 1992), and the loss of spatial resolution (Thompson,

1990). Although the light guide reduces the dependence of the light distribution on depth of

interaction, it does this by undesirably maxunizing the spread of light (Siegel et al., 1995).

4.2.3 Ideal Properties

The ideal properties of a crystal scintillator for PET are: high photon stopping power

(density), high light yield, fast decay time (Zhang et al., 1994), low cost, environmentally stable,

non-temperature dependence, and easy coupling to the photomultiplier tube (S teinman, 1995).

4.2.4 Available Crystal Materials

Bismuth germinate @GO) is currentiy the preferred crystal used in PET because of its high

blocking power and moderate cost (Miyaoka & Lewellen, 1994). However, BGO has a long

decay constant and a relatively low light output (see Table B-2). Lutetium oxyorthosilicate (LSO)

Table B-2: Physical properties of the two main scintillation rnatenals used in PET (Daghighian et al., 1993) Il I 1 1 1 11

1

is a superior crystal for PET (Daghighian et ai., 1993). However, LSO is currently too costly for

use is PET applications (Miyaoka & Lewelien, 1994). Both BGO and LSO are environrnentally

stable. Daghighian et al. (1995) performed a computer simulation comparing these two crystal

types and they discovered that the sensitivity of both crystals is similar. However, the low light

output of BGO was the cause for failure to detecr some photons because the crystai output fell

below the electronic noise level. The difference in decay constants will cause a significant changz

in detector deadtirne.

BGO

LSO

crystal

7.13 g/cm3

7.4g/cm3

density

0.2

1 .O

relative light intensity

300ns

1211s

decay constant 1 peak emission I

48Onm

42Onm

I

4.3 Photodetectors

4.3.1 Principles of Photodetection

Photodetectors convert the scintillator crystai7s light output into measurable electric current

(Leo, 1994). High resolution PET systems require that the photodetector has good sensitivity,

stability , timing propenies (Hayashi, 1989). Physically , the photodetector must have an

appropriate size and shape to fit together in closed packed detector rings (Hayashi, 1989),

otherwise the carnera rotation would be necessary in order to fulfil the anguiar sampling

requirement. With 1: 1 crystal/photodetector coupling, the photodetector is the limiting factor for

spatial resolution.

4.3.2 Common Methods of Photodetection

The photomultiplier tube (PMT) is the most common rnethod of photodetection. However,

there is a physical size constraint which limits the building of very small photomultiplier tubes

(Hayashi, 1989), so block coupling is cornmon with PMTs. Although most PMTs are rectangular,

Wong et al. (1995) proposed using circular PMTs to reduce the camera cost. A position sensitive

PMT (PS-PMT) determines the location (x, y coordinates) of the light photon's interaction on the

surface of the PMT, and hence the crystal of interaction. Avalanche photodiodes (APD) are solid

state amplifiers, and are small enough to have the desired 1:l crystal1APD coupiing without

sacrificing spatial resolution (Hichwa, 1994). However, APDs are very temperature dependent,

and have limited availability because of their high cost (Lecomte et al.. 1994).

4.3.3 Depth of Interaction

Knowledge of the depth of interaction of the photon in the crystal is used to enlarge the

field of view without enlarging the ring radius, and helps fiIfil the image reconstruction sampling

requirements, such that detector motion becomes unnecessary (Miyaoka et al., 1991). As well,

it reduces radial blurring from off-axis crystal penetration (Derenzo et al., 1989; Hayashi, 1989)

by calculating which crystal the annihilation photon would have reacted with if it had interacted

at the front face of the block (Rogers, 1995). Detector units incorporating depth of interaction are

cornplex, costiy, and not commonly used (Rogers, 1995).

4.4 Detector Unit

4.4.1 One-to-One vs. Block Coupling

One-to-one coupling of the crystal and photodetector is the simplest, most reliable coupling

scheme (Hayashi, 1989), with potentially the best temporal and energy resolutions, and the

shortest deadtirne (Thompson, 1990), and thus the highest count rate çapability (Hayashi, 1989).

However, the physical size limitation on the PMT makes 1: 1 coupling to srna11 crystals difficult.

Thus, group coupling of multiple crystals to PMTs (blocks) are used (Hayashi, 1989). Block

detectors are more cost effective than 1: 1 coupling (Thompson, 1990). As well, they have better

spacial resolution, because they incorporate crystals that are much smaller than the smallest crystal

used in 1 : 1 coupling (Hayashi, 1989).

4.4.2 Block Detectors

A typical block contains numerous crystals and four PMTs. A modified version of 2-D

Anger-type logic is used to identiQ the crystal of interaction (Cutler et al., 1992). A weighted

sum of the four PMTs yield a X and a Y coordinates, which is the address on a look-up table that

matches the X, Y address to its correspondhg crystal (Cutler et al., 1992; Rogers et al., 1992).

Ligb guides can be used in block detectors to increase the uniformity of sensitivity (Hilal et al.,

1989). A common problem with block detectors is photon scattering behveen crystals (Thompson,

1990). This can be countered by ensuring that the crystal scintillator is thick and dense enough

to absorb al1 Compton scattered rays before they leave the block (Thompson, 1990), or by depth

of interaction (Derenzo et al., 1989; Hayashi, 1989). An alternative to individual crystals in a

block is a scintillator crystal cut in an offset comb-slit pattern. This method provides superior

resolution uniformity over a wider range of incident angles than discrete crystals (Yarnashita et

al., 1990).

4.4.3 Pulse Pileup and Deadtime

h l s e pileup occurs when two or more Light photons strike a photodetector within its

electronic integrarion t h e (Germano & H o m , 1990). This may cause a mispositionhg of an

event if the photodetector simultaneously processes both light photons as one event and hence

B-14

d e t e d e the incorrect crystal of interaction (Germano & Hoffman, 1990). Or, if these signals

sum to an energy greater than the detector's energy window, the desired event will be discarded

(Gerrnano & Hofhan, 1990). In block detectors, every crystal becornes "dead" when a photon

interacts with one of the crystals (Gerrnano & Hoffinan, 1990; Thornpson & Meyer, 1987). A

coincident pair of detectors are "live", and hence cm make a coincident detection, only if neither

of them are "dead" (Thompson & Meyer, 1987). Less activity will reduce the nurnber of signals,

and hence the number of pulse pileups, increasing "live" tirne.

5.0 Reconstruction Algorithm

5.1 Iterative algorithms

The farnily of iterative algorithms is based on the principle of minimizing the difference

between measured and calculated projections of the image. Each aigorithm has a different method

to iteratively employ this difference of image projections (Nuyts et al., 1993). In general, iterative

algorithms require vast quantities of memory for projection data and the system point spread

function, as well as long cornputation times (Chen et al., 1991). Since its introduction in 1982,

modified versions of Shepp and Vardi's EM algorithm have been the predorninant iterative

algorithms in the Literature.

5.2 Analytic algorithms

The farnily of analytic algorithms is based on filtered Fourier backprojections, and is the

standard reconstruction algorithm currendy used in most PET centres (OISullivan et al., 1993).

These algorithms are derived from the projection slice theorem, which stares that the Fourier

transform of a projection of data (from an object) in the tirne domain is equivalent to a slice

through the origin in Fourier "k-space" (Cho et al., 1993; Joy, 1994). The slice angle is

perpendicular to the projection angle. Thus, the more angles sampled in the scan, the better the

k-space picture. This method requires a complete set of projection data with sufficiently fme

angular and linear sarnpling (Brooks et al., 1979, cited in Tanaka, 1987).

An inverse Fourier transform will recreate the scanned object in the standard space domain.

Since a11 of the slices intersect the ongin in k-space, al1 of the data near the origin will have an

increased weight in the inverse Fourier transform. A filter is typically employed to "even" the

weight in an inverse Fourier transform; the process is called filtered back projection (Joy, 1994).

Although these algorithm are computationally efficient (Tanaka, l987), noise cannot be effciently

suppressed without sacrificing other aspects of image quality (Ouyang et al., 1994).

Chapter C - Simulation Implementation: Construction Methods

1.0 Introduction

The construction of a Positron Emission Tomography cornputer simulator requires a multi-

stage approach. Before any of the physical processes are modeled, efficient fundamental routines

must be constructed and tested. These routines, described in section 2.0, include a random

nurnber generator, random sampling from a normal probability distribution, a fast square root

algorithm, a quadratic equation solver, a random sine-cosine pair generator, and a random 3D

unity vector generator. They are used throughout the sirnulator and forrn the backbone of the

simulation.

The Monte Carlo simulation (see section 2.1) models the physical processes associated with

PET and assigns a probability distribution for each of these processes. A flow chart of the

sirnulator is shown in figure C-1. As can be seen, the sirnulator is divided into four independent

modules: source ernission, camera detection, reconstruction and display. The source emission

module, section 3.0, simulates the generation of the positrons in the source to the escaping of the

photons from the source. The sub-modules include Create-Source, XGAM, XGAM-to-Material,

Source-Shulator-Configuration, and the Source-Simulator. The carnera detection module, section

4.0, simulates the escaped photons interacting with the crystals, and the subsequent activity of the

photodetectors and the coincident circuits. The sub-modules include Camera-Simulator-

Configuration, Carnera-Simularor. The reconstruction module, section 5.0, reconstructs the

results from the camera detection module into a ready to display image. The display module,

section 6.0, displays the recomtructed image on the screen.

The Create-Source sub-module defines the geometry of the source to be simulated. This

source may be as simple as a uniform cylindrical water phantom to as cornplex as a rat's brain,

which is a combination of sub-sources comprised of spheres, cylinders and rectangular prisrns of

various sizes and materiais. The Source-Simulator-Configuration sets the Source-Simulator's

operating parameters. The Source-Simulator creates an event history data Ne for each

Uscr Dched Source

hrnmctcn

Uscr Dcfincd Simulation hnrncrcrs

- ' -!!-- SOURCE

CmAI'E -> suurce SOURCE Dar* fi& SIMULATOR CONFIG.

XGAM

v Photon

l n /o. Fife

v Photun Dda files

(Jilr euch mb-suit ne)

~V~ User Dcfincd

CAMERA Source Rndio;ictivity D h i b ~ t i o n . Y SIM-OR -, camrrn ~ h ~ ~ f a h CAMERA C?nictn Prinrnctcrs,

CONFIG. Cunf ip r r~ iun Fife SIMULATOR Simulation hnmctcrs

Figure C-1: Flow chart of the simulation

sub-source. A single event begins with positron generation in the sub-source. The event will

track the positron to annihilation. After the positron annihilation, the event tracks the two

annihilation photons as they travel through the source, potentially undergoing photon interactions

such as photoelectric effect and Compton scattering. The event concludes when both photons

either are photoelectrically absorbed or escape the source. The event history file contains millions

of these events. Every part of the source ernission module was created for this thesis with the

exception of the photon interaction material-dependent parameters were obtained through the

National Institue of Science and Technology's XGAM (Berger, 1988) material pararneter

generation software.

The Carnera-Simulator-Conf~guration sub-module defmes the geometry of the detector unit,

incorporating the important parameters of camera radius, crystal properties , number of rings, and

type of detector unit. As well, this sub-module defmes the radioactivity injected into the source

as well as the radioactive distribution arnong the sub-sources. The simulation parameters, such

as scan length, are also defined in this sub-module. The Camera-Sirnulator creates a file of

projection data for each irnaged angle in the camera's geometry. A single event begins with a

single photon that has escaped the source. The event tracks the photon to the detector ring (if it

intersects). At the detector ring, the photon is tracked entering the crystal, where a probability

to scintillation and probability to inter-crystal scattering are calcuiated. An ideal photodetector

detects the crystal scintillation and sends the results through an ideal coincident circuit, which

determines the line of coincidence between a set of detector pairs. The set of al1 parallel lines of

coincidence forrn the projection data for the specific angle of the parallel lines. The

Reconstruction-Configuration defmes the reconstruction parameters, and the Reconstruction-

Algorithm takes the projection data and reconstructs the original source through the use of a

filtered backprojection. The Display sub-module takes the reconsuucted image and displays it

through a MATLAFI program. Unique to this simulation is the ability to become temporal. Each

event in the source emission event history file is based on the source's radioactivity, and is emitted

according to Poisson statistics. This temporal nature is incorporated into the detector unit

deadtirne, ie the time when the detector unit is busy processing a photon and camot acknowledge

the presence of another photon. Every part of the detector detection module was created for this

thesis. However, this module can be improved in the following manners: in depth crystal

simulations can easily be included through the incorporation of Dr. G. F. Knoll's 'DETECT" crystal

simulator (cited in Tsang, 1995). The effects of non-idealized photodetectors can be added based

on the appropriate specifications. The effects of the electronics for the coincidence circuit would

be similar for each detector confiiguration, and thus would not affect the cornparisons of

configurations.

2.0 General Simulation Considerations

2.1 Monte Carlo Theory

The application of the Monte Carlo method to a physical system is the construction of a

stochastic model in which the expected value of a (combination of) random variable(s) is

equivalent to the value of the physical quantity to be determineci (Chan & Doi, 1988; Lux &

Koblinger, 1991 ; Raeside, 1976; Thompson et al., 1992; Wiiliarnson, 1988). This model can be

constructed in two ways: analog simulations, which have a 1: 1 correspondence of the actual

physical process; and non-analog simulations, which deviate from this 1: 1 correspondence (Lux

& Koblinger, 1991). Non-analog methods are refmed because they Save computer tirne (Lux &

Koblinger, 1991) by minimizing the sarnpling variance through clever statistical sarnpling

(Raeside, 1976; Wiiliarnson, 1988). The Source Simulator (3.0) is an analog simulator, whereas

the Camera Simulator (4.0) incorporates aspects of both analog and non-analog sirnulators.

The key to creating the stochastic model is by generating the appropriate probability

distribution which describes the physical process. Random sampling of each probability

distribution accurately describes the physical process, provided that the distribution of the random

"sarnpler" is unity (Lux & Koblinger, 1991; Raeside, 1976; Williamson, 1988). Aside from the

importance of selecting an appropnate random number generator (section C-2.2), other tools must

be created to improve the efficiency of the simulation as well as minimizing calculation tirne.

These tools required for this sirnulator include a random sine-cosine pair generator (as opposed

taking the sine and cosine of a random number), a quadratic equation solver, a fast square root

algorithm, and a 3-D unity vector generator (see section C-2.3).

2.2 Random Nwnber Generators (uniform)

A rnethod of generating or obtaining random numbers is needed whenever simulating a

system or process which has inherent randorn components (Law & Kelton, 1991). The three

methods for generating random numbers are: (1) sampling frorn tables generated from sampled

numbered balls in a well-stirred um; (2) monitoring the output of some physical process or device;

and (3) calculation using a specific mathematical algorithm (Law & Kelton, 199 1 ; Raeside, 1976).

The first two methods are slow and rcquire excess memory; they are unacceptable for typical

Monte Carlo simulations. In order to make the third method acceptable, Law and Kelton (199 1)

outline the four properties necessary for an ideal pseudo-randorn number generator:

1. The numbers produced must be distributed uniformiy on [O, 11, and they must not exhibit any correlation with each other.

2. The generator must be fast and avoid the need for a lot of storage. 3. The grnerator must be able to exactly reproduce a given Stream of numbers. 4. The generator must be able to produce several independent streams.

Most of the cornmon pseudo-random number generators satisQ the final three

aforementioned properties (Law & Kelton, 1991), but rnany of these generators, including several

in actual use and provided with some cornputer systems, fail to satisfy the uniformity and

independence criteria, and simulations based on these generators would yield erroneous results

(Law & Kelton, 1991; Raeside, 1976). The majority of pseudo-random generators currently in

use are linear congruentiai generators (LCGs) (Law & Kelton, 1991). A pseudo-random sequence

of integers, Z,, &, , &, . . . , are LCG defmed by the recursive formula: 7 = (a Zi.i + C) (mod m) (Knuth (1981) cited in Law & Kelton, 1991). There are many different tests that are used to prove

the acceptabili~ of the generator, however it is best that the generator be tested in a manner that

is consistent with its intended use (Law & Kelton, 1991).

For this thesis, the random number generator that cornes with VaxC was examined. The

generator is a multiplicative LCG with a period of 2' (VaxCRTL). As more than two million random numbers wiii be needed for this thesis md because it contains the inherent flaws of LCGs

(especially multiplicative), this gewrator is not sufîicient. A random number generator written

by George Marsaglia and Arif Zaman (1987) has a period of 2'' and is completely portable

between systems. The algorithm is a combination of a Fibonacci sequence (with lags of 97 and

33, and operation "subtraction plus one, modulo one" and an "arithmetic sequence" (using

subtraction) (Marsaglia & Zarnan, 1987). The method passes al1 of the tests for random number

generators, thus it satisfies condition 1. The code is fast and uses minimal memory space, meeting

condition 2. This generator has two seed variables yielding 9.4 million different seeds (with O

< = Seedl C = 3 1328 and O < = Seed2 < = 30081) (Marsaglia & Zaman, 1987), tùlfilling conditions 3 and 4. Thus this generator is ideai for this paper. The program was modified to

produce a range of numbers (not restricting it to [O,l]). The program was also modified :O

produce one random number per call, as opposed to an array of random numbers.

2.3 Special Functions

2.3.1 Random SinKos Pair Generator

In this thesis, the need for the generation of random angles, and their corresponding sines

and cosines, is multi-fold. They are used in the conversion between coordinate systems,

generation of three dimensional random unity vectors and in the calculations associated with the

non-collinearity of the annihilation photons. There are a few different methods for generating

corresponding sines and cosines. Firstly, a random angle can be generated and the sine and cosine

of that angle would then be calculated. This method guarantees 100% correspondence between

the siw-cosine pair. However, either a very large look-up table or a Taylor series expansion with

nurnerous t e m is needed to yield the accuracy to the required number of decirnal places. Thus,

these solutions are costly in memory or computation tirne.

Ellen (1969) uses a superior solution which ignores the random angle and just generates

the sine-cosine pair. It requires two random numbers and it must satisfy the condition that these

random numbers lie within the unit disk. Specifically, for random numbers r, and r2:

As can be seen, this generates the sine and cosine pairs, which filfils the Pythagorean sinekosine

relation:

The uniformity of the angle a distribution was checked by taking the arcsin and the arccos of one

million sine-cosine randomly generated pairs, and placing the results in eveniy distributed bins.

For the a distribution of the arcsin(sin) combination, the mean number of ci per bin was 2000.0,

with a standard deviation of 42.24. For the a distribution of the arccos(cos) combination, the

mean number of a per bin was 2000.0, with a standard deviation of 42.00. These results show

that this method of sine-cosine pair generation was equivalent to taking the sine and cosines of

random angles uniformly distributed around the unit circle.

Another test was performed on this method of generation, to determine the accuracy of pair

correlation. Letting P, = arccos(cos a) and = arcsin(sin a), sinErr was defined as sin P, minus sin a, and cosErr was defined as cos R minus cos a. This test was done one million times, and the accuracy of each decimal place was determined. For the fast three decimal places, there is

one hundred percent correlation. The next three decimal places yield at least 98.9% correlation.

Correlation drops to just over 64% at the eighth decimal place and maintains this correlation for

the remaining decimal places. The accuracy to seven decimal places (89 %) shows that there is a

good correlation between the random generation of the sin-cosine pair. Therefore, this method

is acceptable; it is quick, not encurnbered with calculations, and its accuracy with its sin-cosine

pair correlation is acceptable for this thesis.

C-8

2.3.2 Quaciratic Equation Solver

The quadratic equation (see equation C-3) is very common in geometric applications

and an efficient algorithm is needed to accurately solve this equation. Jackson (1994) noted a few

problems with the conventional methods of solving the quadratic equation, namely that they are

unabie to handle floating point errors; specifically overfiow, underfiow and catasnophic

cancellation. Overtlow and underflow occur when the floating point exponent exceeds its

maximum and minimum respective limits. Catastrophic cancellation may occur when floating

point numben are summed and the f m l answer is rnuch smaller than the intermediate terms. The

larger intermediate tems canceled each other, however, before they canceled, they may have

"swallowed" up some smaller terms which are on the order of the h a 1 answer, but were not

included in the fmal answer. A simple example of this would be having two floating point

numbers, f, and f,, where f, > > f2 such that f, + & = f, . Thus f, + f, - f, = f, - f, = 0, and not fi which would be the desired result.

Catasuophic cancellation may occur if b' > > 4 1 ac ( , such that:

This can be avoided by calculating the second root (ie 5 = (-b - JbL-4ac)/2a) and by noting, from equation C-5, that r, = c I (a rd.

However, overflow and underflow may arise if b or a and c are near the floating point maximum

or minimum Limit, because bZ or ac may cause overfïow or undefflow respectively. Both problems can be avoided through factoring out the dominant term under the radical, such that:

Thus, the negative root of equadon C-6 is solved, and the positive root is calculated from

r l = c / (a r2).

The quadratic equation solveî was adapted from a computer lab (Jackson, 1994), where

it was tested with a variety of cases. Regular quadratic equations as well as quadratic equations

exhibithg boundary problems of catastrophic cancellation, overflow and undeflow were tested.

This solver successfully passed each test.

2.3.3 Fast Square Root

The traditional iterative methods for evaluating square roots are ofien too slow when a vast

quantity of square roots are to be evaluated (Lalonde & Dawson, 1990). As well, when a few

digits of accuracy are required, a faster approach would be preferable, because the traditional

methods (eg. sqrto function in most C libraries) retums a double precision result, even though a

single precision number is received (Lalonde & Dawson, 1990).

In binary, a floating point number consists of a mantissa and an exponent, in the form of

f mm.. .m x 2*"-', where m and e represent bits. In general, a floating point number is 32 bits

long. To get an extra bit of accuracy, floahg point numbers are expressed in a normaiized fom.

On Suns and PCs the form is *2? x l .m , whereas on the Vax, the normalized form is 12" '

&") x O. lm. Thus the exponent is normalized around a bias point to take into account positive and

negative. For an eight bit exponent, on the Vax the exponent bias is 128, whereas on the Sun and

PC the exponent bias is 127. Another difference between these formats is the storage of floating

point numbers on the computer. The Sun and PC store floating points as foiiows:

sqe,. . .e,e+22m2,. . .ml, ml,m,,. ..&, whereas the Vax stores its floating point as foilows:

m,,m ,,.. .m, .e,e,,m,q,. ..q6, where s = the sign bit. This storage information is criticai as the fast square root algonthm uses bitwise operations on the floating point nurnbers through an

irregular use of C pointers.

The square root of a positive floating point number can be considered as:

which is sirnply taking the square root of the mantissa and halving the exponent (Lalonde &

Dawson, 1990). A look-up table of stored mantissa square roots, used to accelerate the

caiculation, is the limiting precision factor. The larger the table, the greater the precision, but the

more mernory that is required. A bitwise shift right is the most efficient method for halving the

exponent (after the exponent bias has been removed). However, an odd exponent leaves a

remainder when divided by two. To avoid this remainder problem, the exponent is forced to be

even (qe,. . .e,O) with a corresponding quatemary mantissa v ,m, , . . .m, (Lalonde & Dawson, 1990). This quaternary mantissa is in the range [O. 1. .0.4). So the lookup table stores the square

root values of 2"xO. lm,m,, . . .m, (for the VAX). The fust st-bits (for n-bit precision) of the quatemary mantissa are used as the key to the

lookup table. The lookup table is an array of 2"+' (Lalonde & Dawson, 1990) n-bit bytes. The

array is initialized through calculating the square root of every possible quaternary mantissa up

to n-bit precision (ie q,m,rn,,... mm-,,), and storing the square root in the array at the place

designated by the corresponding quaternary mantissa. Upon calculating a square root, the

program will separate the exponent from the quaternary mantissa, shift-right the exponent bitwise,

and use the corresponding quaternary mantissa as the index of the array to fmd the new mantissa.

To minimize the program's start-up tirne, the loohp table was previously calculated and saved

to disk (in the appropriate VAX, PClSun format), so it needed only to be loaded, and not

calculated, upon prograrn initialization. There is a trade-off between accuracy (table size) and

speed (Ha, 1993); the algorithm was originally designed to be a high speed, low precision square root approximation.

The physicd limitations of the positron range (see section 3.5.2.2) requires the simulation

to have an accuracy of no greater than 0.05 percent error for the fsqrt algorithm. The accuracy

of die fsqrt algorithm was tested by generating one million random numbers in the range between

zero and one million. The fast square root was taken for each of these numbers, and the result

was squared and compared with the original. A percent error was calculated as follows:

1 Original Nmber -JasIJ~rigina~ Nurnber ' 1 % Error = x 100%

Original Num ber (C-8)

In the one million trials, only fifty four had a percent error greater than 0.05%; however, the

percent error of these fifty four trials was under 0.1 %. Therefore, the current settings of the fsqrt

algorithm are acceptable, and speed does not need to be compromised to increase accuracy.

Table C-1 shows the speed cornparison of the fast and regular algorithrns on the VAX, both

with and without the table load. As can be seen, in ten runs of one million trials of taking the

square root of a random number multiplied by the run counter, the fast algorithrn is superior to

the regular algorithm, although the table load takes a noticeable arnount of t h e . However, since

the table load will be done oniy at the initialization, the fast algondun is superior and was

therefore used in the prograrn.

Table C-1: Speed cornparison of fast and regular square root algorithms

2.3.4 Generate Random 3-D Unity Vectors

Throughout this thesis, there is a need to randomly generate a three dimensional unity

magnitude direction vector (ie vx2 + v,' f v: = 1). This vector may represent positron or photon directions. There are a few dif5erent methods to generate this vector.

First, the Cartesian coordinates-normalkation method. Three random numbers (0,1] are

generated, one for each direction. The vector's values are the normalized values of the three

Algorithm

fast, with table load

1

Mean Run Time (s)

2.5

fast

Standard Deviation (s)

0.5

28.1 0.35

random numbers (ie v, = rl * [r12 + r: + r:]-', V, = r2 * [r12 + r22 + $1-', v, = r, * [r: + r: + r:]*'). Method 2 is the spherical method. Choosing r = 1 (and hence, unity magnitude), with random 0 and a. However, instead of calculating the sines and cosines of 0 and $, it is more efficient to randomly generate two cosine-sine pairs (see section 2.3.1).

Thus the vector would be: v, = cosA sinB, v, = sinA sinB, and v, = cosB.

These methods were compared through ten runs of one million generations of random

vectors. The initiakation tirne for the fsqno table load (see section 2.3.3) was not included in

the tirne calculations because it will be loaded for the rest of the simulation, regardless of the unity

vector generation algorithm. As can be seen in Table C-2 , the spherical method of unity vector

generation is superior to the Cartesian coordinates-nonnalization method.

Table C-2: Com~arison of unitv vector ~eneration methods

II spherical 1 2 22.9 1 0.35 II

Method

Cartesian

2.3.5 Normal Distribution Sarnpling

The two parameters which classify a Gaussian distribution are the mean ( p ) and the

standard deviation (0). The normal probability density function is (Chatfield, 1983):

f(x) reaches a maximum when x = p. However, there is no closed form for the distribution

function. Thus, numericai rnethods are needed to solve the distribution integral (Chatfield, 1983).

Although available, a table of values inherently limits the uniforrnity of the distribution.

A superior method was found in Law and Kelton (1991). Provided that X-N(0,l) (ie

random variable X is normaily (Gaussian) distributed with p = O and o = l), the more general

Mean time (s)

166.4

Standard deviation (s)

O -49

normal distribution X ' - N ( ~ , ~ ' ) can be obtained by setting

x' = p o - X (C- 10)

Thus, a N(0,l) distribution is needed to obtain the desired inverse Gaussian distribution.

Law and Kelton (1991) suggested an algorithm, based on Box and Muller's method, which

produces a one-to-one correspondence between the random numbers used and the N(0,l) random

variate produced, which is beneficial for a Monte Car10 simulation. The method is to generate

U,, Uz U(0,l) and set X, = cos(2xUJJ(-2 1n(U, ))A = sin(2xUJJ(-2 1n(U,)). This, upon substitution into equation C-IO, will yield two X's. This method was then modified by using the

aforementioned random sinekosine generator (see section C-2.3.1) instead of calculating the sine

and cosine of a random nurnber. The potential problem with this method occurs when U, and U2

are not truly U(O, 1). This happens when usbg a random nurnber generator whose output depends

on the previous random number produced (see section C-2.2) (Law & Kelton, 1991). However,

the generator used in this project does not have this problem, and, according to Law & Kelton

(199 l), this method is accurate.

An important parameter of the normal probability density hinction is Full Width Half Max

(FWHM), which is defmed as the width of the curve, 6, at half of the peak height. FWHM occurs

when x = 6/2 + p and f(x) = % f ( x h . Equation C-Il shows the relationship between the

standard deviation and the FWHM.

2.4 Coordinate System

The simulator uses the Cartesian coordinate system for the majority of calculations. This

system was selected becauso inherently it has the simplest methods for ail vector operations for

vectors which do not intersect the origin. Other coordinate systems, such as cylindrical and

spherical, are used for specific routines when they prove more efficient.

3 .O Source Simulator Module

3.1 introduction

The source simulator module consists of four components. The input to this module is a

userdefmed source to be simulated as well as the parameters for the simulation itself. The user-

defined source is comprised of multiple sub-sources. in the Source Simulator, positrons are

uniformly distributed within each sub-source. This uniform distribution is, in effect, assuming

perfect rnixing of the sub-source at the beginning of the scan. Although this assumption removes

the ability to sirnulate tracer time curves within a specific sub-source, it is sufficient for the

purposes of this thesis, to compare camera performance for various camera configurations.

Carnera performance is calculated by scanning uniformly distributed phantorns (Karp et al., 199 1)'

which are mixed just before the scan begins.

This module produces a photon information file which contains a surnmary of the

simulation parameters as well as a listing of contents of the generated photon data files. One

photon data file is created for each sub-source that was simulated. These files contain information

about each annihilation photon, including the location of the photon's intersection with the outer

sub-source. The outer sub-source is the boundary between the Source Simulator module and the

Camera SImulator module (see 4.0).

The user-defrned source is constructed in the Create-Source sub-module (section 3.2). The

user-defined simulation parameters are invoduced in the Source-Simulation-Configuration sub-

module (section 3.3). The material dependent attenuation coefficients are generated in the XGAM

sub-moduIe (section 3.4). The simulation itseif is carried out in the Source-Simulation sub-module

(section3.5) .

3.2 Create Source

The purpose of this sub-module is to defme the source distribution. Every source is

comprised of a) an outer cylinder filled with air, and b) huer sub-sources. The outer cylinder is

used as the dividing line between the Source and the Camera Simulator Modules. If the

annihilation photon was not photoelecrricaliy absorbed inside the source, then it must intersect this

outer cylinder. If the intersection point is on either of the two ends of the cylinder, then the

annihilation photon would miss the camera ring. However, if the intersection takes place on the

axial body of the cylinder, then the annihilation photon will potentially hit the camera ring. Thus,

the outer cylinder is used to filter out photons which are guaranteed to miss the camera ring.

Inner sub-sources may be comprised of spheres, cylinders, or rectangular prisms, al1 of

various sizes, locations and materials. When assembled, sub-sources may be distinct or imbedded

within each other, but complex geometric functions were designed to ensure that no two sources

would intersect. This was done to avoid the problem of generating material-dependent interaction

coefficients for the intersection region of two sub-sources comprised of different materials. The

problem of assigning a generated positron to a specific sub-source is also avoided with the

imposition of non-intersecting sub-sources. Each sub-source may be comprised of any material

provided that the density and the chernical composition of the material is known, so chat the

attenuation coefficients can be generated through XGAM (section 3.4).

3.3 Source Simulator Configuration

The purpose of this sub-module is to create the source sirnulator configuration file. The

configuration file contains the user-defmed simulation specific parameters. The primary parameter

is the number of positrons to generate for each sub-source. In an effort to maxirnize the number

of relevant annihilation photons saved in the photon history data me, the user defmes limits on the

outer cylinder - if a photon intersects the outer cylinder within the limits, it is saved, otherwise it is treated as a miss. Finaiiy, the contiguration sub-module checks to see if the sub-sources have

already been sirnulated. If so, then the user has the option to redo the simulation or continue the

simulation from where it was left off.

In order to avoid enormous photon data files, and in order to reduce the Source Simulator's

simulation tirne, there is an option for octant simulation. Octant simulation generates positrons

in only the x > O, y > 0, z > O octant. In the Camera Simulator, each positron is pseudo- generated eight times - the filst time the positron's annihilation photons are loaded from the data Ne. The other seven times the annihilation photons' location and direction are reflected into the

other seven octants, as shown in table C, C-3, 3:

Table C-3: Octant Reflections

Where "O" represents the original location and direction and "Rn represents the reflected location

and direction. Octant simulation reduces the simulation time and photon data file by a factor of

eight. However, octant simulation is valid only when the source is symmetrical about al1 eight

octants - the cylinder, sphere, or rectangular pnsms must be centred about the origin.

Octant

x

Y

z

3.4 XGAM

The material dependent attenuation coefficients are generated through XGAM, a National

Institute of Standards and Technology's X-ray and gamma-ray attenuation coefficients and cross

sections database. XGAM takes the chemical form of the material and produces a data file with the

photoelectric absorption coefficient and the total attenuation coefficient for energy levels between

16 keV and 511 keV, in 5 keV increments. The XGAM data is presented in a specific (ie density

independent) format: cmt/g. As the source simulator uses a density dependent format, XGAM~MAT

converts the XGAM data file into a density dependent format of mm-'.

3.5 Source Simulator

3.5.1 Introduction

The Source Simulator sub-module is event based. Each event is comprised of two parts:

positron (section 3.5.2) and photon (section 3.5.3). The positron section includes positron

generation, range and annihilation. The photon section includes photon generation, which includes

non-collinearity, and photon path, which tracks the photon motion through the source. This sub-

module generates events for each sub-source. The number of events generated is dependent on

the source simulator configuration Ne. However, the event is not considered successful if b o t .

1

O

O

O

2

R

O

O

3

O

R

O

4

R

R

O

5

O

O

R

6

R

O

R

7

O

R

R

8

R

R

R

of the resulting annihilation photons fail to intersect along the outer cylinder, between the b i t s

set in the configuration sub-module (section 3.3). The outputs of the source shulator are the

photon information and data history files. The format of these photon files is discussed in section

3.5.4.

If the simulation is a continuation of a previous simulation, then it will start where the

previous simulation stopped. Since the simulation is dependent on the random number generator,

conthuity of the simulation is ensured by starting the randorn number generator at the same point

it stopped. The sirnplest implementation method is counting the number of randoms produced,

and reproduce those numbers. Practically, a simulation of a small sub-source requires billions of

random numbers. Generating billions of random numbers before begiming the simulation will

take the, and will waste system resources. This effect is rninimued by reinitializing the random

number generator with a new seed every million random numbers. One million random numbers

was selected because it is small enough to be generated very quickly, and large enough that the

random seed initialization is infrequently called.

3.5.2 Positron

3.5.2.1 Positron Generation

The instantaneous d g assumption (see section 3.1) forces each potential location of

positron generation to be equiprobable within a sub-source. There are two methods for

equiprobable positron generation: outer rejection, and random within boundary. The frst method

randornly samples the uniform distribution to get x, y, and z