APPLICATIONS OF GAMMA RAY TOMOGRAPHY TO
NON—DESTRUCTIVE TESTING OF INDUSTRIAL MATERIALS
A THESIS SUBMITTED FOR THE AWARD OF THE
DEGREE OF DOCTOR OF PHILOSOPHY TO THE
UNIVERSITY OF LONDON
BY
MOHAMED M. ENNAMI
REACTOR CENTRE
IMPERIAL COLLEGE OF SCIENCE, TECHNOLOGY AND MEDICINE
SEPTEMBER 1988. This thesis is dedicated to my wife Mariam, my son Haroun and
to our parents 1
ABSTRACT
This study investigates the Possibility of applying computerised gamma—ray tomography in non—destructive testing o-f industrial materials in transmission and emission modes.
Measurements -for tomography using gamma—rays and the operation of radiation detectors are based on photon interactions with matter. The theory governing these interactions is discussed.
The theory o-f the mathematical reconstruction of a two dimensional distribution from its projections is shown? and
reconstruction techniques and applications are reviewed.
The principles of gamma—ray detection and measurements are
presented in order that the physical significance of the data
recorded can be assessed. Detection characteristics pertinent
to imaging applications were measured for the horizontal
dipstick Ge
importance of these characteristics in determining the
suitability of this detector for imaging applications is
di scussed.
A prototype scanning rig was designed for transmission and
eiii i ssi on tamour aph y , T’.,; r i o wiii-, ‘ * • used with phantoms t o
determine contrast and spatial resolution.
The usefulness of line scans and contrast measurements in
quantitative analysis is shown.
The effect of scattering on image quality is discussed
briefly in terms of the quality of the data used for
reconstruction? and scatter subtraction is used for all
reconstructions. The effect of attenuation in Single Photon Emission
Computed Tomography is studied and the available analytical
attenuation correction techniques were briefly discussed. The
use of multi—energy scanning as a viable industrial
tomographic technique was experimentally investigated. The
application of multi—energy scanning as an experimental
attenuation correction method was suggested and also verified
by means of scanning an ai umi ni urn phantom containing six vials
of europium nitrate solutions at four different Y~ray
energies. The reconstructed data was then used to derive a
semi —empirical function for attenuation correction.
The results obtained from all the scanning experiments
illustrate the possibility of using the tomographic system
developed here at the Reactor Centre for non—destructive
scanning of industrial samples and nuclear waste packages in
both transmission and emission modes. They also show the use
of multi—energy scanning as both a tomographic technique and
an attenuation correction method. ACKNOWLEDGEMENT
For his enthusiasm, advise, guidance and constant encouragement throughout the period of this research t would
like to sincerely thank my supervisor Dr- Desmond MacMahon-
I would like to thank Drs- Peter Gray and Bill Glauert -for their help regarding the scanning and image display software.
I would like to also thank the Reactor and workshop staff for their help with collimators, Phantoms an^, source preparations
I am most grateful to friends and collegues in the Reactor
Centre for helpful discussions and providing a pleasant envi recent - CONTENTS
PAGE NO.
ABSTRACT i
ACKNOWLDEGEMENT i i i
CONTENTS i v
CHAPTER Is Introduction 1
CHAPTER 2: Interaction ofPhotons with Matter 5
2.1 Introduction 5
2.2 The Photo—electric Effect 5
2.3 Gamma Ray Scattering B
2.3.1 Coherent(Thompson) Scattering by 8
a Free Electron
2.3.2 Incoherent Scattering by a Free electron 10
2.3.3 Scattering from a Bound Electron 12
2.4 Pair Production 17
2.5 Total Attenuation Coefficients 18
2.6 Attenuation Coefficients in
Elemental Analysis 21
CHAPTER 3: The Principles of Gamma Ray Detection and
Measurements and Applications 26
3.1 Introduction 26
3.2 General Properties of Radiation Detectors 26
3.2.1 Generalised Operation of Detectors 26
3.2.2 Pulse Height Analysis 28
3-2.3 Detector Efficiency, Energy Resolution
and Dead Time 29
Gamma Ray Spectroscopy V
3.3.2 Theoretically Predicted Pulse Height
Distribution 33
3.3.3 Practical Spectroscopy with
Semi-conductors 3B
3.4 Application of Gamma Ray Spectroscopy
in Tomography 41
3.4.1 Emmission and Transmission Tomography 41
3.4.2 Determination o-f Source Depth by
Scatter to Peak Measurements 49
CHAPTER 4: The Theory of ReconstructiveComputerised
Tomography 53
4.1 Introduction 53
4.2 Non—Reconstruct!ve Tomography 53
4.3 Reconstruct!ve Tomography 54
4.3.1 Statement of the Problem 57
4.3.2 ReconstructionTechniques 59
4.3.2.1 The Analytical Recontruction
Technniques 60
4.3.2.2 The Fourier TransformMethod 61
4.3.2.3 Back Projection 64
4-3.2.4 Filtered Back Projection 67
4.3.3 Iterative Methods 73
4-3.3-1 Type of Corrections 77
4.3.3.2 Weighting Factors 78
4.4 Discussion 80
CHAPTER 5: The Scanning System 83
5.1 Introduction 83
5.2 The Scanning Rig 85 vi
5.3 The Stepping Motors and Stepper
Motor Interfaces 87
5.4 Detector and Source Colli mation 90
5.5 Detector and Counting Electronics 93
5.6 Experimental Control and Data
Acquisition Program 93
5.7 Image Reconstruction and Display 97
5.8 Conclusion 99
CHAPTER 6: System Characterisation 101
6.1 Introduction 101
6.2 Sensitivity of Imaging Systems 101
6.2.1 Detector Homogeniety and
Size Determination 102
6.2.2 Detector Efficiency and Resolution 104 6.3 Spatial Resolution and Sensitivitya* 108 6.3.1 Definition of the Point Spr^jfi Function 110
6.3.2 Geometrical Analysis of PSF of a
Single Bore Collimator 112
6.3.3 Modulation transfer Function 116
6.3.4 Sensitivity and Resolution 119
6.4 Discussion and Results 121
CHAPTER 7: Experiments in Tomography 140
7.1 Introduction 140
7.2 Scanning Geometry 140
7.2.1 Transmission Geometry 140
7.2.2 Emission Geometry 141
7.3 Tomography Experiments 143
7.3.1 Scanning Summaries 143
7.3.2 Scanning Details 144 vi i
7.4 Reconstructed Images and Analysis 148
7.4.1 Contrast In image Analysis 148
7.4.2 Analysis of Images Obtained in
Tomography Experiments 151
7.5 Discussion and Conclusion 156
CHAPTER 8: liul ti —Energ y Scanning and The problem of
Attenuation in SPECT 172
8.1 Introduction 172
8.2 The Effects of Attenuation On
Reconstruction 172
8.2.1 Attenuation Problems for Positron ECT 173
8.2.2 Attenuation Problem for Single
Photon ECT 175
8.2.3 Example of Single Photon
Attenuation Effect 177
8.3 Attenuation Compensation 179
8.4 The Multi—Energy Scanning Experiment 180
8.4.1 Source and Phantom Preparation 180
8.4.2 Scanning Details 181
8.4.3 Reconstructed Images and Analysis 183
8.5’ The Use of Multi—Energy Scanning for
Attenuation Correction in SPECT 187
8.6 Discussion and Conclusion 189
CHAPTER 9: Conclusions 210
9.1 Recommendations for Further Work 214
REFERENCES 216 1
CHAPTER ONE
INTRODUCTION
The -fundamental aim of tomographic techniques is to produce an image of a slice through an object which is free from interference effects from the underlying and overlying planes- For many years focal plane tomography in which the underlying and overlying layers are blurred whilst the slice of interest is kept in focus was employed by (BOC—1921,
ANG—1968) before computerised tomography was introduced.
However, the complete elimination of the effect from neighbouring layers is not possible using this non—reconstructive technique.
The basic principle of computerised tomography is that the internal structure of an object can be reconstructed from the infinite set of all possible projections of the object- The mathematical theory governing this principle, that is, a two or three dimensional object can be reconstructed from the infinite set of its projections, was proved by the Austrian mathematician ,J. Radon (RAD—1917) working on gravitational theory many years before the first commercial computerised tomography scanner was introduced (HOU-1973).
The first application of mathematical reconstruction was
in radioastronomy in 1956 (BRA-1956) which was followed in
several other fields including optics and electron microscopy
(ROW—1969, De ROS-1968). However it is its medical application
that has had the greatest impact and provided impetus for
widespread research (HOU—1973; SHE-1974; BUD—1974; DUB—1977).
The measurement of projections by means of the detection
of a probe which may originate within, be transmitted through 2
dp stimulated within the object provides the basic data -for reconstruction. A projection is therefore either a measure of properties of the materials comprising the object which governs the transmission or stimulation of the probe or is the concentration of the probe itself.
The number of types of probes used in imaging is large and continues to increase, including: nuclear magnetic resonance
(SHE—1980, CRO—1982), electrical impedance (PRI-1979), thermal microwaves (SCH—1979), X—ray fluoroscopy (PAT—1980, BAI—1979) and pions (WIL—1980).
However, the probes that have been most widely applied in both me'dica! and non—medical fields, and have therefore been the subject of a very large number of research studies, are x — and y— rays. X— and Y-rays have been extensively employed in transmission and emission tomography systems since the very start of medical imaging (KUH—1963; PHE—1977; WIL—1979;
MUE—1976; HOF-1981; KUH-1976; HOU-1973) . The application of
Compton scattered photons has been studied (CLA—1069;
STOK-1981; HAR—1982; BAL-1986).
Although the mathematical reconstruction of images was first applied in non—medical fields (RAFD—1917; BRA—1956; De
ROS—1968), since then the majority of research studies and hence the major advancements have been with respect to medical imaging applications. However, more recently as the technique of computerised become more widely appreciated the range of applications in non—medical fields particularly in non-destructive testing, has expanded, (KRU—1981;
SANDERS0N-1979; SCH-3 980 HOP-1981; GIL-1982; SAN-1983; SPY-1984). 3
In imaging systems where a gamma ray is the probe, as in this study, the detection and measurement of photons -form the basis o-f the imaging process. It is therefore important that the fundamental principles of radiation detectors are studied particularly the characteristics of detectors pertaining to imaging such as efficiency, resolution, homogeneity, scatter contribution to peak areas and quality of data used in reconstruction. In addition to the detector properties the scanning system must be studied from the point of view of coilimation, spatial resolution, line and point spread function, modulation transfer function, system sensitivity and
image resolution.
The application of computerised tomography to non—destructive testing has received a great deal of attention at the Reactor Centre and forms the basis of this study. It is proposed that tomographic techniques could be routinely used to examine various industrial samples such as radioactive waste packages and others in both transmission b-nd emission modes using gamma rays.
Since the attenuation coefficients of the material comprising industrial samples are normally higher than those
of medical imaging the energy of radiation used must be also be higher. The combined effects of high gamma ray energies and high atomic numbers of samples is to produce a large number of
scattered photons within the object, therefore, the precise
selection of data is particularly important in industrial tomography in order to minimise the number of scattered photons recorded by the detector and hence optimise the
quality of the data used in the re t ; .i • r uc t i on process. 4
Because of the large variation of attenuation coefficients i.e. large variation in density and atomic numbers in industrial samples the use of multi—energy scanning as a tomographic technique is studied and the possibility of using this multi—energy scanning as an effective experimental technique for photon attenuation compensation is also studied. 5
CHAPTER TWO
INTERACTION OF PHOTONS WITH MATTER
2.1 Introducti on
When photons propagate through matter, they interact with the medium in a number o-f ways dictated by the physical properties of the material as well as the photon energy. Fano et—al (FAN. 1959) have summarized the possible modes of interactions. Each interaction process can be considered either as an absorption or a scattering event. In the former, the photon disappears as its energy is converted into the kinetic energy of some atomic particle. In the latter, the gamma ray is deflected away from the primary beam of gamma rays. In the energy range of practical usefulness (0.01 MeV —
2MeV) especially for tomography, there are three generally accepted important interactions. These are photoelectric effect, Compton scattering and Rayleigh scattering. A significant amount of Rayleigh scattering generally occurs within the the photopeak window usually set for data collection in all tomographic scans. Recently tomographic images using Rayleigh scatter photons have been published
(HAR. 1985).
2.2 The Photo-Electric Effect
In the photoelectric absorption process, an incoming photon undergoes an interaction with a bound electron and the photon energy is completely transferred to the electron which is ejected -from its bound shell. The energy of the photoelectron, Ee, is gi ven by 6
Ee = EY " Eb 2.1 where Ey is the incident photon energy and is the binding energy o-f the ejected electron. The interaction also creates an ionised atom with a vacancy in one of its bound shells.
This vacancy is -filled by the capture o-f a -free electron -from the medium or by rearrangement of electrons in other shells and hence characteristic X—rays may be generated or, less frequently, the emission of Auger electrons may occur.
Photoelectrons may be ejected from any K,L,M .... shells
but a free electron can not absorb a photon and becomes a
photelectron since a third body, the nucleus, is necessary for
conserving momentum. Therefore the probability of
photoelectric absorption increases rapidly with the tightness
of the binding of the electrons so that at higher energies
greater than the binding energies of K and L shells the
absorption due to outer shells is negligible(e.g. BOX of
photoelectric absorption takes place in the K—shel1 when
energetically possible). Furthermore, for a given shell, the
interaction is largest for photon energies just above the
ionisation potentials for the shell and falls off rapidely
with decreasing energy because the photon energy then becomes
smaller than the binding energy of some of the electrons and
therefore the number of electrons for which the process is
energetically possible is suddenly reduced.
Calculation of the dependence of photoelectric absorption
cross—secti on, photon energy Ey have been reported by Heitler (HEI—1953) and Evans (EVA-1955) but no single analytical expression exists 7 which is valid over all ranges of Ey and Z, however, an approximation is n m <7 a Z / E m2/atom 2.2 pe v the value o-f the exponent m depends on both Ey and Z. At energies less than 0.1 MeV the cross-section is complicated by the absorption edges where thecross-section shows discontinuous jumps. At energies greater than 0.1 MeV m varies between 3 and 3.5 and is greater -for lower energies. For a given energy range m is higher -for low Z and decreases with increasing Z The values of the exponent n varies between 4 and 5 and depends on Ey? in the range 0. 1 to 3 MeV n i s found to i ncrease from 4.0 to 4.6 as Ey increases(EVA—1955) The linear attenuation coefficient is given by Hpe = where N is the number of atoms per cubic metre. The energy absorption coefficient appe, for the photoelectric effect is equivalent to the linear attenuation coefficient since the remaining fraction of the photon energy not converted into kinetic energy of the photoelectron represents the exitation energy in the residual atom and is given up as characteristic X—rays or as Auger electrons. These are both absorbed within a distance comparable with the range of photoelectrons so that the effective energy absorption is represented by ppe, i . e. a ^pe ~ l*pp m 2. 4 B 2.3 Gamma Ray Scattering Apart -from the type o-f photon interaction resulting -from total absorption of the in-coming photon, the other major form of interaction is the scattering of the photon by the potential centered on a particle. This particle could be a free electron , a bound electron or the nucleus of the atom. In the energy range of interest, the types of scattering of any significance are those involving a free electron or a bound electron . The interaction involving the nucleus requires high energy photons energy the probability is very small. Both classical and quantum theories have been employed to explain the behaviour of gamma rays when scattered by a free electron. The binding of electrons within the atom complicates the derivation of the theory to explain the behaviour of photons scattered from them. 2.3.1 Coherent (Thompson) Scattering by a Free Electron When a photon undergoes this kind of interaction with a free electron, the emerging photons essentially have the same frequency as the in—coming photons. The theory behind this has been well documented by Thompson. In his study each electron is considered to respond only to the force generated by the electric vector of the electromagnetic wave (the photons), the electron then oscillates at the same frequency as the incident radiation and emits scattered radiation at the same energy. The magnetic vector component of this magnetic wave is assumed to have little or no effect on the electron. This type of interaction is termed Thompson scatter ing. 9 Thompson, in his classical model of photons scattering ■from a stationary electron, made a number o-f assumptions that is to limit the applicability of his result. These ares (i) the atomic number o-f the scattering medium must be small so that the electrons could be considered -free; (ii) the wave length o-f the radiation must be shorter than the characteristic size o-f the atom m) so that the interaction is with the electrons rather than with the whole atom. The implications o-f these conditions are that the incident energy must be greater than 10 keV. The upper limit o-f the photon energy is set by the requirement -for its energy to be small compared to the rest mass o-f the electron, in other words, the incident photon energy must not be relativistic so that the magnetic component of the radiation could be considered negligible compared to the electric vector component. Thus the energy range in which the theory is applicable is quite narrow and its usefulness falls off rapidly with increasing atomic number of the material. This is the result of the need for low binding energies in the electrons to ensure the validity of the freeelectron assumption. The requirement is summarized in the inequality statement Eb « Ey « 513 keV where Eb is the binding energy of the electron, and Ey the primary photon energy. The value of the cross-section for this type of interaction had been theoretically calculated to be cJu t L 6. 66x1 O'-2 9 (m2) by Thompson. This multiplied by the number of electrons per unit area of target material, represents the fraction of the incident photons which will be scattered in 10 all directions (4 *t> by Thompson scattering- The differential cross—section which is the -fraction of the intensity scattered into a differential solid angle dft by a single electron around a scattering angle 8, is given by the relation d the quantity in the brackets in equation 2.5 is generally referred to as the classical electron radius. 2.3.2 Incoherent Scattering by a Free Electron The incoherent scattering of photons by a free electron is otherwise known as Compton scattering. The theory for Thompson scattering ceases to hold as the photon energy approaches 511 keV. In this case the momentum of the incident photon hv0/c could not be neglected and the law of conservation of momentum must be obeyed by the whole system. With the exception of the case of the zero scattering angle? the scattered photon always leaves in a direction which is not parallel to the initial direction of the primary photon, thus the scattered photon always has a smaller energy than the incident photon, the rest of the energy being carried away by the electron. Compton derived a theory for incoherently scattered photons by a stationary tree electron by applying the principles of conservation of energy and momentum? a relation between the primary and scattered photon energy is given as 1 + 11 where Ey being the scattered photon energy- The remaining energy carried away by the electron is generally relativistic and therefore it is necessary to derive a relativistic theory for scattering. Using Dirac's relativistic theory of an electron, Klein and Nishina d (m2/electron)/steradian 2.7 where re = (e2 / me2) is the classical electron radius and ey . me2 2 8 At low energies where K « 1, the K1ein—Nishina equation reduces to Thompson classical formula 2.5 _d_ d Q rTh i , r e ' 1 :os'~ as required. Integration of equation 2.7 over all energies yields the total Klein— Nishina cross—section. 'ft KN r f d 2.9 1 I ds> 0 0 12 where dSi = sinB dB dot 2.10 and a is the azimuthal angle such that 0 < cx < 2*t. Thus i+k: 2(1+K) 1n (1+2K) •> 0, KN = 2f( . K2 . 1+2K K In(1+2K) 1+3K m2/electron 2.11 2K (1+2K)2 This multiplied by the number o-f electrons per target area (scattering medium) gives the -fraction of the incident photons that is scattered in all directions. The values o-f have been tabulated -for energies up to 3 MeV using equation 2.11 (HUB-1969). 2.3.3 Scattering -from Bound Electrons In most applications the effect of bound electrons on the calculated scattering cross—section has been either disregarded or approximated by combining them with Rayleigh scattering (see below). However, it has been shown by Jackson and Hawkes (JAC—1981) that for low Z materials, binding effects are important for photon energies less than 50 keV. , while for medium Z materials like calcium the effect of binding could not be neglected even for energies as high as 300 keV. The deviation of the ratio of the differential cross-section per atom to atomic number E (da Klein-Nishi na cross-section (de The assumption made in the K3ein—Nishina and Thompson equations are that the eJprtrr.ni is free and also stationary. 13 Jauch and Rohr lick (JAU—1955) has improved on this -for the case where the electron is tree but in motion. The binding correction has been treated, taking into account not only the K—shell, but all the atomic electrons. This generally involves applying a multiplicative -factor to these approximated cross—secti ons. < Cross Section tBarns/Electron) Figure 2.1 Variation of the K1ein—Nishina cross—secti on and of the incoherent cross-section ^^incoh/j with photon energy p When the wave length of the incident photon 1 s comparable with the atomic radius, the interaction i s usual 1y with i hr-- whole atom and not with an individual electron. These 1 4 electrons will oscillate in phase and the photons emitted by these oscillating electrons will also be in phase. Since the amplitude o-f these emitted (scattered) photons addsi the intensity also adds. The total energy will be equal to the energy o-f the incident photon. This type o-f coherent scattering is usually referred to as Rayleigh scattering. Coherent scattering from a bound electron implies no change in the internal energies of the electrons and that the atom as whole remains in its ground state after the interaction. Using a non—relativistic plane wave approximation the differential cross—section for scattering from atoms is given by (VIE—1966). d d o-Th rcoh e dfl dfl 2. 12 Where F0(q> is the multiplicative factor correcting for the case of bound electrons and is called the form factor. The square of this form factor,F^ (q), is the probability that the Z electrons of the atom take up a recoil momentum, q, without absorbing any energy. If the i nc: i des < i: photon momentum is F-Kj and the scattering photon momentum is fvK.f , then q~= K~ - K ~ ~ 2. 13 is the momentum transfer. For coherent scattering since I Kj ! = I K.f I = E/hc f-q = 2 (E/mc2) (me) sin(8/2> 2. 14 the form factor is defined by lq.r Z e ?0 (r ) d V F 0 (q) I 2. 15 15 where ?o(r) is the electron density distribution. The density distribution is defined such that F0 TjC- ■16 as q — » 0 as q — > m thus the atomic cross-section for coherent scattering is proportional to Z2 at small momentum transfer. Under the situation where the incident photon has a wave length shorter than the atomic dimension, then the interaction is with the bound atomic electrons. This form of interaction is called inelastic scattering and results in excited atomic electrons leaving the whole atom in an excited state. The excitation could be into another but higher bound level, or if the energy transferred is sufficient, into the continuum. The incoherent cross-section is the sum of inelastic cross-sections to all possible final states. Veigele et—al (VEX—1966) using a plane wave impulse approximation have written the inelastic cross-section as rKN dfl -inc - dQ Z. S(q) 2. 17 where S(q) is the incoherent scattering function, and q is the momentum transfer defined by *.q = 2Ksin&6 -CL l+(Ka+2K) sin%83^ C 1+2K sin203_i3- 2.18 The scattering factor Z S(q) has the limiting behaviour such that Z S(q) ---» 0 Z S(q) ---» Z 2. 19 as -» o as 4 and 16 a s q ------» CD Thus, at high energies, the incoherent cross—section is proportional to Z, the atomic number o-f the scattering material. This factor corrects the K1ein—Nishina equation for the bound effect on the electron and it represents the probability that an atom is raised to an excited state or even ionised as a result of momentum transferred to it due to a photon collision. Calculations of S(q) and F0(q) depend on knowledge of the atomic wave functions. They can be done analytically for hydrogen and, for other atoms, in various approximations based on the Thomas—Fermi, Hartree—Fock, or other models. The procedures that could be used for these calculations have been adequat.1 y reviewed by Jackson and Hawkes (JAC—1981). In a thin foil of thickness dx the number of electrons/m2 is NZdx where N is the number of atoms / m3 and Z is the atomic number (e/atom>. If an incident beam of monochromatic photons contains n photons then the number of photons removed from the beam is given by —dn / n = N Z dx c1nc 2.21 then the number of unscattered photon, n', transmitted through a thickness x is given by n = n e-Mr- x 2 . 22 where Mc = N Z <7C m 1 2.23 and i Lf*fi •»?a1 i or * <: oe-f f i c i e n t _-. Tt n . could also be written an 17 Fc a^c + s^c 2-24 where apc and 5MC are the Compton absorption and scattering coefficients respectively. The significant of this distinction between scattering and absorption coefficients is particularly important in Compton scattering since only the absorption coefficients represents the detectable effects of the interaction process which are caused by Compton electrons, and the detection of scattered photons depends on further interactions. The energy absorbed per unit volume of an absorber as a result of Compton interactions is equal to I.anc where I is the incident photon i ntensi t y. 2. 4 Pair Production In the pair production process the photon disappears and a positron—electron pair is produced. For the eventto occur, the photn energy must be greater than the total rest mass of the pair, that is, 2m0c2 = 1.022 MeV. This proces can only occur in the Coulomb field of the nucleus which is required for consevation of momentum. The total energy of electron positron pair is equal to the photon energy Ey with the excess energy above 2m0c2 going as kinetic energy (KE) shared by the electron and positron, i.e. K.E = E y — 2itij)C2 2.25 There is no single expression to describe the pair production cross-section but its magnitude is proportional to Z2 of the absorber and the natural logarithm of the incident photon energy,E Y i i « e . 1 8 o-pp tx Z2 Zn Ey m2/nucleus 2.26 The linear attenuation coefficient for pair production is simply the product of Ppp « N Z2 Zn Ey 2.26 In pair production only a portion of the photon energy appears as kinetic energy of the electron—positron pair so that the absorption coefficient, appp, can be written as a^pp = ppp £1-2 m0c2 / Ey 3 2. 27 and the remaining 2m0c2 of the photon energy is given up when positron annihilates with an electron after being slowed down by ionising and radiative collisions. Therefore the total pair production cross—section can be written a^PP = a^pp + b ^p p 2 . 28 where PP 5 the pair production scatter coefficient given by s ^p p — Pp p 2m 0 C 2 / Ey 2 . 29 However S^PP is generally negligibly small at low energi es (<3 MeO) due to the dominance of Compton effect and at hi gher energies because 2m0c2 / Ey becomes smal1 . Therefore it can be assumed that a^pp ~ ^pp 2 . 30 » vJ Total Attenuation Coefficients The probability that a photon will interact when passing through matter is given by the attenuation coefficient. The linear attenuation coefficient p ^ i m 1) is the interaction probability per unit length which is related to the atomic 19 cro55-5Bction , = cN = Where N is the number of atoms per m3 , Nq is Avogadro's number, ? is the density and A is the atomic mass. The mass attenuation coefficients * :-m *\e = ^m- ? 2.33 The interaction of gamma rays with matter is characterised by the fact that each photon interacts individually in single events. Therefore the number of photons removed from a beam by either scattering or absorption, dl, is proportional to the thickness of the material dx and the number of incident photons I, i.e. dl = -pg I dx 2.34 For a homogeneous material and monochromatic gamma rays, Hjg will be constant and integration of equation 2.34 gives I = I0 expC— x) 2.35 where I is the number of photons remaining from the incident beam containing I0 photons after traversing a thickness x of the absorber. Therefore a collimated gamma ray beam of a particular energy passing through a medium follows an exponential law of attenuation. Due to the fact that the interaction processes are i ndependent of each other the total linear attenuation coef f i r i ent taking into account al i p»ossi bl e contr i but i ng 20 processes is VJl (total ) = ^(l)+^^(2)+...... +^(N) 2.36 Therefore if a collimated beam of monochromatic photons of initial intensity I0 passes through a medium of thickness x the intensity of the primary photons that undergo no interaction of any kind is I = Io e>:p <-vz (total) ■*> 2-37 where the total attenuation coefficient is the sum of the coefficients of all possible interaction processes and for the energy range .01 to 10 MeV is given by (total ) = + ^pe + **c + ^pp ** 2.38 where pr , npe, pc and ppp are the total linear coefficients for Rayleigh scattering, photoelectric absorption, Compton scattering and pair production respectively. Similarly, the intensity of the primary photons not absorbed along the path x within a medium is Ia = lo exp(total)*M* 2.o9 where ^a(total) the energy absorption coefficient. The relative magnitudes of the interaction processes in equation 2.38 for water and lead are shown in figures 2.2 (a) and (b) and figure 2.3 illustrates the relative importance of Compton scattering, photoelectric absorption and pair production as a function of atomic number and photon energy. Three areas are defined on the plot within which each of these processes predominate and the lines defining the regions represent the energies where the coefficients of the neighboring areas are equal. It is clear -from these two graphs 21 that the photoelectric effect predominates for high Z and low photon energy, pair production for high Z and high photon energy and Compton scattering for the entire domain of intermediate Z and photon energy. The primary attenuation of gamma rays in chemical compounds or mixtures of elements is assumed to depend on the sum of the cross sections of all the atoms in the mixture and be independent of the chemical state since chemical bonds are only of the order of few eV and hence have no significant effect on the photoelectric, Compton or pair production interactions. Therefore the total mass attenuation coefficient, pjvj, can be calculated from **„ = [ wi 2.40 i where Wj is the weighting factor for element i with mass attenuation coefficient (pm>j and is given by wi = mi / M 2.41 where m^ is the mass of the i element and M is the t Lri dl mass of the mixture. 2.6 Attenuation Coefficients in Elemental Analysis A method of elemental analysis based on the measurement of differences in mass attenuation coefficients has been proposed by Kouris and Spyrou (KOIJ—1978) , and applied (SAN—1982) . The mass attenuation coefficient of a mixture, provided that the effect of chemical binding is neglected, is given by equation 2.40. 22 -figure 2.2a linear attenuation coefficients for photons in water (taken from EVANS—1955) f i gu.r e 2 .2b linear attenuation coefficients for photons in lead (taken from EVANS—1955) 2 3 •figure 2.3 showing the regions where the three main interactions dominate (taken from EVANS-1955) If an additional element (n+1) is added to the mixture then the new total mass attenuation coefficient Hjvj' is given by 2.42 •V = wn+l where r wn+l + L. Wj = 1 2.43 Therefore the change in mass attenuation coefficients, assuming that there is no change in density, is MM' “ f = ------2.44 and hence the minimum detectable mass fraction, wn+i , becomes 2 4 f*M w . = ------. f 2.45 n+l . and can be -found by measuring the change in mass attenuation coefficient after the addition of an extra element. ^ n ^ n + l » PM then wn+l ~ ^M • * 7<»1m)n+l 2.46 and if wn+i ” 2.47 This theory can be extended and applied to tomography where linear attenuation coefficients are measured (KOU—1981). The raysum is defined P = In ( I0 / I ) 2.48 and if the photons are monoenergetic and the medium is homogeneous PL = PL L 2.49 where L is the path length. If an element is now added with linear attenuation coefficients (n^)c and is represented by an equivalent length £, then pc = P|_ (L - £) +(Pjg>c £ 2.50 and the change in the raysum due to the addition is f = < Pc - P[_ > / PL 2.51 and the minimum detectable length, £ /L, is £/L = pl .f / < and si mi I ar 1 y £/L PL - i 7 (P£>r for (Pjg^ 2 5 and £/L ~ — f -for << hj_ 2-54 Minimum detectable -fractions -for length and mass of various elements are not as low as for other techniques. However, the advantage of the theory lies in its application in fields such as tomography where the necessary data already exist. 26 CHAPTER THREE THE PRINCIPLES OF GAMMA RADIATION DETECTION MEASUREMENT AND APPLICATION 3.1 Introducti on The detection and measurment o-f radiation -forms the basis in all imaging systems where ionising radiation is the probe used to acquire the data for reconstruction- An understanding of the fundamental mechanisms underlying the operation of radiation detectors and the physical assessment of the data they produce is therefore essential in any radiation imaging stud y. 3.2 General Properties of Radiation Detectors 3.2.1 Generalised operation of detectors The fundamental mechanism underlying the operation of all radiation detectors is the dissipation of the energy of the particle within a suitable medium with a net result that an electric charge is produced. This charge is normally collected by the application of an electric field so that a current flows through the detector for a time tc after the particle has interacted. The time taken for complete charge collection varies with detector material and depends on the mobility of the charge carriers and the average distance travelled to the electrodes e-g. in ion chambers the collection time is typically few milliseconds whereas in semiconductors it is a few nanoseconds. (KNO—1979) A current is produced for every quantum of radiation that interacts and as the inter aria or) r ai f* increases then the 2 7 situation arises when the current flowing within the detector is due to more than one interaction. The magnitude and duration of each current depends on the type of interaction. A distinction can be made between two modes of operation of detector systems. In one mode of operation the average d.c. current produced by the detector is measured and this is known as the current mode. The recorded current is the time averaged effect of many interactions within the detctor and therefore individual events are not distinguishable. Detectors operating in the current mode are normally applied to the general field of radiation monitoring? such as radiation dosimetry. However? for the majority of applications? detectors are more usefully employed operating in the pulse mode. In this type of operation the output from the detector is a series of signals? or pulses? corresponding to each quantum of radiation that has interacted within the detecting medium. There are two main advantages of the pulse mode operation. Firstly? the amplitude of each pulse depends on the amount of charge generated due to the individual interaction of which it is a result and is an important means of analysing the type and frequency of interactions taken place within the detector? hence providing information on tfie incident radiation. The second advantage is that a greater sensitivity can be achieved since in current mode there is a minimum detectable current which is equivalent to a certain interaction rate whereas in pulse mode single interactions can be detected limited only by the local background radiation level. 2 0 Pulse height analysi s The pulses produced in a detector due to radiation interactions will not be all of the same amplitude. The variation may be due to -fluctuations in the detector response or to differences in radiation energy. It is pulse height analysis with respect to incident radiation energy that -forms the basis of most detection systems and the result is normally displayed as the di-f-ferenti al pulse height distribution. The di f-f erenti al distribution is a plot of dN/dH, where N is the number of events, versus the height H, where H is the magnitude of the pulse (volts), and is normally known as the spectrum. The pulse amplitude scale on the abscissa ranges -From zero to a maximum value which is larger than the amplitude of the largest recorded pulse, H|vj, and the ordinate, dN/dH, has units of inverse amplitude, volts-1? therefore the number of pulses with amplitude between two limits,H1 and Ho? can be found by integrating over the region, and is given by H- dN dH 3. 1 dH H so that the total number of pulses in a spectrum is dN r dH c> J dH o Since pulse height is proportional to the energy dissipated in the detector by the interaction of a quantum of radiation the pulse height spectrum ! r>di cates information 2 9 about the incident radiation. A peak in the distribution, Hp in -figure 3.1, indicates an energy about which a larger number o-f particles of similar energy exist and vice versa -for a low point or valley H|_- Analysis o-f the di-f-f erenti al pulse height spectrum must always involve the use o-f areas since the quantity dN/dH is a physically meaningless parameter until multiplied by the abscissa H. 3.2.3 Detector eficiency. energy resolution and dead time In theory an output pulse should b e produced -for every interaction that takes place within t h e active volume o-f the detector. However, whilst this is possible with charged particles, the more penetrating radiations such as gamma rays or neutrons must undergo signi-tic ani interactions in order to 3 0 produce a detectable pulse and can travel large distances between interactions and detectors are therefore rarely lOOX effi cient- Efficiencies are catagorised in two ways* firstly, by the nature of the event that is recorded. The total efficiency is that calculated using the total number of pulses recorded by the detector and includes all types of interactions that have taken place within the detector. The peak efficiency, however, only includes interactions that have resulted in the particle depositing all its energy within the detector and corresponds to the area under the peak in the spectrum. Secondly, efficiecies are subdivided into intrinsic and absolute. Absolute efficiency is defined as ______number of pulses recorded______a“5 number of radiation quanta emitted by the source and is therefore not only dependent on the detection properties of the detector but also on the geometry of the experiment. The intrinsic efficiency is defined as _ number of pulses recorded _ „ inr number of quanta incident on the detector “ and is therefore independent of geometry. The most useful and commonly reported efficiency is the intrinsic peak efficiency since there is no geometrical dependence. Also the unquantified effects of scattering from neighbouring objects and s p u r i o u s noise have more influence on the total number of pulses in a spectrum than on the peak area thus making pea}-: efficiencies more reliable. In rad iatio n spec t r r.«sc. up* y uu im portant property of a d e t e c t or is the? ability t o distinguish between radiation 31 energies that are slighty different. This parameter is called energy resolution and is determined from the differential distribution produced by a detector in response to a monoenergetic source. Energy resolution is measured in terms of the width of the full energy peak and is normally quoted as the full width at half maximum (FWHM). The smaller this value is the better the resolution of the detector. If the detector is calibrated then each pulse height H is equivalent to a certain energy E and the FWHM can be expressed in key. An alternative definition is given by FWHM ______^ R = ---□------x loo/: 3.5 Hp The width of the full energy peak arises from fluctuations in the size of the pulses produced b y the quanta of the same energy. The sources of these fluctuations include random noise within the detector and associated instrumentation, drift in operating conditions of the detector and statistical noise due to the discrete nature of the signal. The first two sources are specific to the detection system and are therefore, in that sense, under the some control. However, the third source will always be present, irrespective of how perfect the system is, and in a wide range of detection systems is the dominant and hence the limiting source. When a charge Q is created within a detector it consists of a discrete number of charge carriers- the number of which is random and fluctuate? from event to event. Assuming that the formation of each charge carrier is a Poisson process and that N carriers are created then the standard deviation will b e / M. If there is no other s o u r c e of fluctuation then the differentia! response function '-nil hr haussi an with a 3 2 standard deviation 2-35 RP = 2.35 K / N / KN = 2.35 / / N 3.6 The good resolution ot semi conductors arise From the tact that the number ot charge carriers produced per unit ot deposited energy is high. However, experimental measurement have shown that resolution better than those predicted by Rp can be achieved which led to the derivation ot the Fano tactor. This tactor is detined as F= observed variance i n hi / Poisson predicted variance 3.7 and since the variance is equal to Rs = 2.35 K / N •/ F / KN = 2.35 / (F/N) 3.8 The value ot the Fano tactor depends on the type ot detector and generally F<<1 tor semiconductors. It the incident tlux ot radiation on a detector is high then a situation may arise that the time between pulses is so short that individual pulses cannot be resolved. The minimum time between pulses, such that each one is recorded, is known as the dead time, T, and it the pulse rate is very high then the recorded number ot pulses may be very much less than the actual number and should theretore be corrected. the exact torm ot this correction depends on whether the system response is paral y?able or' not. In the paral ysable case it the time between two pulses is less than the dead time then the second pulse is not recorded and the dead time ot the system is Further ext ended h y v. i her ot nr e, it the pulse rale is very high the system may become "paralysed" since a period of r is added to the total dead time tor everyunrecorded event and the total dead period is unlimited. However* systems can generally be assumed to be nonparalysable in which case the system is dead -for a period equal to the dead time after each event and events that occur within that period are neither detected nor contribute to the dead time. Therefore the total dead time of the detector is fixed by f and the recorded count rate m and the true incident flux n can be calculated from n = m / (1 — m *r> 3.9 3.3 Gamma Ray Spectroscopy 3.3.1 Introduction The detection of gamma rays is dependent on the photon undergoing an interaction in which it transfers energy to an electron which can subsequently be detected. Some of the interactions that photons may undergo are described in detail in chapter 2; those that are important in gamma ray spectroscopy are the photoelectric effect* Compton and Rayleigh scattering and pair production. The secondary electrons produced in photon interactions must be fully absorbed within the detector for spectroscopy of gamma radi ati on. 3.3.2 1heoretical 1y predicted pulse height di stri buti ons The predicted response of a detector primarily depends on its size with respect to the mean free path of the secondary gamma rays produced in trie interactions of the original r ad 3 at inn. I)i le t o the short r ringe of F'l s Irons in most solid 3 4 materials (a tew mm -for e's ot upto a tew MeV energy) it can be assumed that there is complete absorption ot electrons produced in photon interactions. It is usetul to examine the expected response ot the two extreme cases ot small and large detectors when considering the response ot real detectors which generally lie between the two extremes. A small detector is one whose dimensions are small compared with the mean tree path ot the secondary gamma rays (generally this applies to detector dimensions ot ten to twenty millimeters). There are two sources ot secondary gamma rays; tirstly Compton scattered primary photons and secondly, annihilation photons produced when the positrons created in pair production annihilate. Figure 3.2 illustrates the predicted response spectrum tor gamma ray energies ot several MeV. Since the detector dimensions all compare with the mean tree path ot the secondary photons only single Compton interactions are assumed to occur, theretore the ratio ot the photopeak area to the Compton continuum area is the same as the ratio ot the respective cross-sections, Similarly, the annihilation radiation escapes and only the electron and positron energies are deposited resulting in the double escape peak, with energy Ey 2moc2. For energies below 2m0c2 where pair production is impossible the spectrum will be almost identical to tigure 3.2 but wi t ft out the double escape peak. 3 5 Figure 3.2 Predicted response -for Ey up to several MeV in small detector I-f the detector dimensions a r e large compared with the mean -free path o-f the secondary gamma rays and (-for ideal case) the radiation source is in the centre of the detecting volume? then the secondary photons will interact within the detector and none would escape ( in practice this will require unrealistic detector dimensions o - f hundreds of millimeters) (KNO—1979). If the energy of the primary photons is greater than 2mgC^ then secondary gamma r a y s may b e produced in both Compton scattering events and annihi j ation photons from pair production. These secondary p h o t o n s may then be Compton scattered again until eventually pfini oel ectric absorption takes pi ace and the total energy o-f t h e original photon has been deposited within the det ec tor , the t i m e between t h e 36 absoption oT the Compton electron and the subsequent absorption o-f the scattered photon is very short compared with the response time o-f the detector and similarly the time between the absorption o-f the el ectron-posi tron pair and annihilation photons is also very short- This results in the -fact that the pulse produced in the detector will be the sum o-f the electron energies produced in all interactions and a single Tull energy peak is produced (figure 3.3). F-igure 3-3 Predicted response TorE^up to several MeV Tor large detectors The response ot real detectors combines the properties of these two extreme cases and i s further romp 1 i cated by other T actor s i e. g - the sur r oi tndi ng mat e*r i ai s (figure 3.4). 3 7 Energy(E) Figure 3.4 Predicted response -for a detector of intermediate dimensions The secondary photons produced in Compton scattering may be -further scattered and not undergo photoelectric absorption , giving rise to a distortion in the Compton continuum and having energies up to Ey. The ratio of the photopeak area to Compton continuum will be higher than the ratio of the cross section due to the contribution of multiple events to the photopeak. The annihilation photons from pair production may either escape or be absorbed giving rise to single or double escape peaks (of energies Ey—m0c3 or Ey—2m0c2). Secondary radiation produced in the material surrounding the source—detector system contributes to the spectrum. Characteristic X-rays may he generated following photoelectric absorption i u the surrounding mat er i h which may be detected 3 8 when they are of su-f-ficienti y high energy to reach the active detector volume. Primary photons may undergo Compton scattering in the surrounding material and -for all scattering angles greater than about to photons of almost identical energy, the combination of which results in the backscatter peak. Annihilation photons generated in the surrounding material may also be detected. The spectra obtained with semiconductors are similar to the predicted response functions due to their excellent energy resoluti on. Practical spectroscopy with semiconductors In designing a spectroscopy system careful consideration mu.st be given to the geometry of the experiment and particular care must be taken with respect to the surrounding materials since they can contribute features to the spectrum which yield no Lisefui information and may interfere with features of interest. The production of characteristic X—rays is the most serious problem. The energy of characteristic X—rays increases with atomic number and the higher the energy of the X—rays the higher the probability of their detection. Therefore, it is desirable to keep the atomic number of the surrounding material as low as possible. However, the most useful and widely used collimator and shielding materials are those with high Z, particularly lead. A solution to this is to line the lead with a low Z material to absorb the characteristic X—rays from the lead and whose own X—rays have very low energy s^o that the possibility of detecting them is very smai1. 3 9 Correct collimation and detector shielding are important, particular1y for reproducibility, to eliminate the effect of other structures in the experimental system such as support stands and the source holders. A detector collimator should extend back along the length of the detector completely shielding the active volume. Peak area measurement is the most common parameter in spectroscop y. By comparing the peak area of a sample containing unknown activity to a sample of known activity of the same energy and under the same counting geometry it is possible to determine the unknown activity. This method of measuring activity is used extensively in activation analysis. Attenuation measurements are normally carried out by measuring relative peak areas; and in emission tomography and in transmission tomography using isotopic sources the area within a window set approximate! y around the* peak for each raysum is the basis of the data that are used in the reconstruction. In the analysis of spectra for peak areas two general classes of methods are used. The first of which is the digital calculation method which involves the adding up of raw data contained in the photopeak and subracting the underlying background continuum; no assumption is made on the shape of the peak or the continuum on either side, only the position of the (usually linear) base-line. A number of methods have been proposed to implement this calculation each defining a somewhat different photopeak area. Among the most popular are t h e Coveli method mi ni—computer 5 . The -former method de-fines a calibrated -fraction o-f the peak which is assumed to bear a constant relationship to the total area contained in the peak. The TPA method relies on linearly extrapolating the continuum under the peak i -figure 3.5, thus defining the integrated region to be enclosed within the limits of fwhm) (corresponding of a Gaussian peak area). The precision obtainable with these and other digital integration method has been studied by Baedecker(BAE—1971), who found that truncation of the peak boundries in combination with a base line approximation similar to that used by the TPA method, lead to improvemed precision in peak area determination over both the TPA and Coveil methods. A similar result was obtained by Robertson et—ai (ROB—1975) in a comparis. on of Nal The second general class of net peak area computation is the fitting method in which an analytical function, usually Gaussian form reflecting the statistical nature of the system noise and of charge collection process, is fitted to the observed data. First, by iterative minimisation of the square of the residuals between data and the function requires the power Df a computer in ail but the most trivial context. Several authors have reviewed the choice of analytical function EMcNel les and Camnel 1 (McfM—1975); Takeda (TAR—1976)j and also the relative merits of digital and fitting methods distorted by the addition of e>: ponent i al tails. Tailing is particularly severe on the low energy side of the peak, attributable to incomplete charge collection in the detector, whilst high energy tailing due to pulse pile—up may become appreciable at high count rates. This function forms the basis of SAMPO gamma ray spectrum analysis code written by Routti and Prussin (ROU—1969) which is now the most widely used of its kind. A) Ot + Cg 2 Figure 3.5 Principles of spectral analysis. 3.4 Applications of Gamma Ray Spectroscopy in Tomography 3.4.1 Emi ssi on and Transrrn ssion_ T omograph y The effect of scatter in transmission and emission imaging have been until r e c e n t ] y ignored throughout the development of 4 2 radiation imaging techniques. Gamma ray spectroscopy is rarely carried out with detectors incorporated into imaging systems. In systems where spectroscopic data are available* mostly in radionuclide emmission studies* the discriminator window is ■frequently set subjectively around the photopeak with no true spectroscopic analysis. However* more commonly, spectroscopic data are unavailable in emission systems and the data are accepted -from within a di scr i mi nator window of width equal to a fixed percentage of the photopeak energy centered on the peak. In transmission tomography systems the detectors are normally operated in current mode and hence spectroscopic information is lost and the total number of integrated counts is recorded and forms the data for reconstruction. Scattered photons are the result of both coherent (mainly Rayleigh) scattering and incoherent (Compton) scattering. Table 3.1 taken from Sanders (SAN—i9S2) shows the ratio of the incoherent scattering cross-sections to coherent scattering calculated from the tabulated data of Storm and Isreal (STO—1970) as a function of the incident gamma ray energy. The relative scattering contribution from coherent scattering increases with atomic number and decreases with energy and is considerable for low gamma ray energies and high atomic number. Furthermore, it is obviously impossible to reduce the amount of coherently scattered phot on a accepted by a detector by means of ener gy discrimination. 43 Table 3.1 Rat i o of i ncoher ent scat ter i rig cross section to cotier ent scatter i ng cross sect i on ( a5 (Compton) / However ? the number of incoherent! y scattered photons accepted b y a detector can be geometrically minimised by the u s e of collimators and shielding and also by means of energy discrimination in the pulse height spectrum. The efficiency of energy discrimination depends on the energy of the incident photon and the the energy resolution of the detector. Table 3.2 shows the minimum energy of Compton scattered photons (corresponding to ISO0 scattering angle) as a function of the incident gamma ray energy. It is clear from these results that at low incident gamma ray energies, the energy transferred to the recoi1 electron is small that the energy of the scattered photon differs only slightly from that of the incident photon. Therefore- as table 3.3 summarises? the reduction in the number of Compton scattered photons in the photopeak by means of trie detector energy resolution varies with energy.The efficient discrimination against Compton scattered photons is important since Compton scatter i ng is the predominate interaction process for energies above 100 keV and up to sever a 3 MeV for most mater iais. 44 Table 3, Mi ni mum energy of Compton scattered photons corresponding to 180 0 scattering angle and the percentage of the primary photon energy transferred to the recoil electron as a function of the incident gamma ray energy. incident photon minimum energy of 7. of primary energy energy scattered photon transferred to keV photon keV recoil electron 20 IB. 55 7.2 50 41.82 16.4 100 71.87 28. 1 200 112.18 43.9 300 137.98 54.0 400 155.91 61.0 500 169.10 66.2 1000 203.50 79.6 Table 3.3 Fraction of Compton scattered photons accepted by a detector as a -function of incident photon energy and detector energy resolution taken -from (BRI—1972} incident Ev Detector resolution (keV) keV 1 3 10 30 0.30 0. 99 1.0 H v O 50 0. 072 0.35 ■ 100 0.0066 0.051 0. 32 200 0.00054 0.0046 0.045 400 0.00004 0.00039 0.0042 Scattered photons do not contribute useful information to an image and in fact reduce its contrast and increase the noise level leading to overall image degradation (BEC—1969, HOF—1971). Therefore it is desirable to reduce the amount of scatter accepted by the detector in radiation imaging. Studies on how to reduce the ef f e* t. of scattered photons in emission and transmi ssi on i magi ng carri ed out by several o authors resolution i.e. employing a de.'t E?ct or with higher energy resoluti on. Figure 3-6 illustrates the geometry -for the transmission imaging situation where a uniform slab o-f thickness L is irradiated by a collimated beam of monoenergetic photons of medium plane Figure 3.6 Transmission geometry in which a uniform slab is irradiated with a thin beam of monoenergetic photons. 7he differential cross-section dc£-"r'vdft for Compton scattering is given by equation 2.7 as (1-cosB) 21 d o rKN < e > 1_ L i+K(1—cos(8) J - i-tecs1-8+ — - t-ttz------— I dft 2 L 1+K(1—cos0)J where r^ = ec-/m0c2 is the cla? ssif.. al electron radius and K= Ey/m0c2- For 1 ow photon energies ,E:Y, K becomes very small and the expression is reduced to dct;KN/dfi Z r l-+-cosr'0 3 3- 10 T h e + I i tx of unscat 1 e r e h phot - through a volume Adz situated 46 in the slab at a distance z t rorn the exit face is I (z ) = I0 expC — py (L—z ) 1 3. 1 1 where py is the total linear attenuation coetticient and I0 is the incident photon -flux. It ne is the electron density ot the material then the number ot electrons in Adz is neAdz and the total number ot photons scattered per unit solid angle is I The solid angle subtended by a detector element ot area d2r located a distance r trom the axis ot the photon beam is dQ = d2r . cos36 /(s+z)2 3.13 where 8 = arctan (r/ (z+s) ) Theretore, the number ot photons scattered into the solid angle subtended by the detecting element d2r by Adz is given by the product ot the two equations 3.12 and 3.13. However the scattered photons may be attenuated in the distance zsecB bet ore leaving the slab so that the total number ot scattered photons reaching the detecting element, h=i(r) is Lp h.5(r)=1e r§ne Adz I0 j expt-py(L-z>3 expt—py zsec83 0 CDS3 9 l+cos26) dz 3. 14 < s+z ) 11 the distance trom the detector to the scattering medium,s, i s greater than or equal to L then cos8 varies little with z and 8-8' - arctan Hr/ (s+}£L> 3 3.15 and (s+z)2 - (s+}£l ) 2 3.16 so the expression now becomes h5 (r)= # r 2 rit.3 A T f,exp (-- p jL_) cos3B* The total detected photon density in the beam is h < r > = hs (r) + hu (r ) 3.18 where hu (r) is the unscattered component given by hu (r) = I0 exp(-hjL) = I(L) 3.19 assuming that r lies within the beam. In -figure 3.7 hs (r) (in the units of % rg neAI 0exp ( — hjL) is plotted against r for several scattering medium thicknesses L and distances to the detector s. Two important conclusions should be noted from these results. Firstly, that scatter gives rise to broad structurel ess wings around the point image and that whilst the amplitude of this spread is low for a single thin beam the accumulative effect becomes significant as the number of thin beams comprising a real beam Df finite width increases. Secondly, from equations 3-17 and 3.18 it is clear that the relative importance of h5 (r> can be reduced by increasing the distance front the scatter medium to the detector,s, since this reduces r;s-r } by a factor l/(s+ /$_> 2 but has no effect on hu (r) when tne primary beam is well colli mated. Until now the number of s c a t t e r e d photons detected has been discussed i n terms of their relative rejection and acceptance either geometrically or by means of a lower level pulse height d i scri mi mator. However , an alternative approach is the subtraction of scatter by means of setting two single channel analyser windows, one over the full energy peak or a portion of it and the other i n t h e scattered photon region of the spectrum (normaij v defined in terms of the angle of llompt nn sc ai t or i n g cor t u- po : =.-i ■ -.c c : * n e photon energies), the 48 data is corrected tor scatter by subtracting the data in the scatter window -from the data in the the full energy peak. Another method is to use a multi channel analyser and collect the region ot interest which covers the entire -full energy peak , evaluate the peak area and the scattering contribution to the peak. This spectroscopically determined data is then used -for image reconstruction. The advantage of this method is the use of spectroscapical1y determined data. The main disadvantage of this is that the time spent collecting data is hi gh and also takes a lot of memory. This method of acquiring data is the one used for all the experiments carried o u t i n this work. Results of the effect of scatter subtraction on the image quality are reported by Beck (BEC--1972) and indicate - 02-1 3 0 0 mm slab thickness ------distance to detector, s=500mm • 01- PSF of hs(r) 500 radius (mm) 300mm distance to detector slab thickness, L = 100mm 500 radius(mm) Figure 3./ Plots of PSF of hB (r) for various slab thickness 2 and detector distances (in units of 1/2 r0neAI0 exp (-p-fL) ) taken from SANDERS (SAN--19S2) 4 9 that the scatter subtracted data gives sharper image contrast than the uncorrected data. Experimental con-firmation o-f the theoretical advantages of reducing the scattering component of the reconstruct!on data in emission imaging has been provided by Dresser (DRE—1972), Hoffer (HOF—1971) and Sanders (SAN—1982) . Hoffer compared images of phantoms collected with Nal(TU) and Ge(Li) detectors, thus selecting different scattering components by means of the different energy resolution of the detectors. The images reconstructed from Ge(Li) detector data in all cases, especially when a muitienergetic spectrum is used, indicate better definition and contrast than those reconstructed from data obtained by a Nal(l£) detector. Dresser achieved scattering component reduction by means of both energy resolution and geometrical selection with the use of collimation. Sanders compared images obtained by Nal (Be—wi ndow) , CdTe, CsF and BGO < detectors thus achieving scatter reduction by means of varying energy resolution and also by means of energy discrimination, setting different energy windows around the peak,!.e. using FWHM and FWTM areas for reconstruct!on. 3.4.2 Determination of source depth by scatter to peak ratio measurements The Tf)ajOY''A problem in single photon emission tomography is that of attenuation. It is impossible to make quantitative measurements without taking into account the amount by which the emitted photons attenuate within the object. This involves accurate measurement, or at least, excellent approximation of 5 0 the linear attenuation coef-f icient and the depth o-f the activity. In emission computed tomography the quantity o-f interest is the distribution o-f activity inside a sample and thus the raysum is proportional to the entire detector signal while in transmission tomography the quantity o-f interest is the distribution of attenuation coefficient and so the raysum is only proportional to the logarithm of the detector signal. There are many methods of attenuation compensation for single photon ECT. These include: 1— Constant Attenuation, 2— Least Squares for variable and constant attenuation, 3— modification of projection data before reconstruction which include a— Geometric Mean modification i.e. the use of the square root of the product of two opposing projections as a projection data, b— Hyperbolic Sine correction, c— Average the Minimum and Maximum Exponential Factor and d— Average of Exponential Factor, 4— Iterative Convolution Method for variable and constant attenuation. All of these methods are well explained in literature (HER—1979). Therefore,the scatter to peak ratio (SPR) method of activity depth determination employed in -Scintigraphy is suitable. However, this requires spectroscopic data which is not readily available in all commercial scanners. A simple expression (WAT—1972) can be derived assuming small angle scattering events only. In figure 3.8 the number of scattered photons ds produced by a thickness dx is ds=Ip£ dx 3.20 where I is the source intensity at x and is given by I = I0 exp ( —p-j (Ey) * x) 1 5 1 and hq is the linear Compton scattering coefficient and py(Ey> is the total linear absorption coefficients of the material for the primary photon energy Ey Figure 3.8 Representation of small angle scattering geometry for a point source in a scattering medium Therefore the number of scattered photons from dx reaching the surface of the medium is ds exp<-py(E5). d>i 3.22 where py(E5) is the linear total absorption coefficient for the scattered photon energy Es. If only photons scattered though small angle, 8, are accepted by the detector then the difference in energy between the primary photon and the scattered photon is small and it can be assumed that py(Ey> - Py(Es >. Therefore equation 3.22 becomes 5 2 I0 exp( —py d) dx 3.23 Integrating along x gives the total number of scattered photons for a source depth di d S^. = J I0 exp <-py d) hq dx = I0 exp (— py d) Hq d 3.24 0 Therefore the number of scattered photons recorded by the detector is S'= 8S Sy 3.25 where 0B is the solid angle subtended at the detector by the scattered photons. The number of full energy peak photons recorded, p, is P = 8p I0 exp <-py d) 3.26 where 0p is the solid angle subtended at the detector by the source. The SPR is therefore I0 exp(-py d) d pq 85 S/P = ------3.27 I0 exp<-py d) 8p In the case of a well collimated or small detector 85 is small and can be approximated to 8S = 0p. Therefore the SPR for a point source varies linearly with depth SPR = pq d 3.28 criurfy 5 3 chapter four THE THEORY QF RECONSTRUCT IVP COMPUTERISED TOMOGRAPHY 4.1 Introduction The problem of image reconstruction Prom projections has arisen independently in a number o-f scientific* medical* and technical fields. These include finding the internal structure of the solar corona* the radio-brightness of a portion of the sky , electron microscopy , the distribution of radionuclides indicating the physiological functioning of the human body and the dynamic behaviour of the beating heart of a patient. The solution that all these imaging problems have in common is the same mathematical -foundation (HER—1979,EAV—1979) - Of all these applications , probably the greatest impact on the world at large-at least from the public awareness point of view, has been in diagnostic medicine = Medical diagnosis using X-ray radiography is limited by blurring effects caused by the collapse of a three dimensional object onto a two-dimensional plane. The ability to view a body section or a layer clearly without interference from other regions, has long been a goal of medical radiology. Attempts at this goal could be divided into two major groups , these are the conventional non—reconstructive and the reconstructive tomography, 4.2 Mon-Reconstructi ve tomography The earliest attempt at tomographic imaging using X—ray transmission was by Bncage in 1921 (P.R(3—1974) . His technique, general 1 y referred to as longitudinal or focal plane 5 4 tomography, involved movi ng the X—ray tube and the -film at compatible speed in opposing directions. This technique is illustrated in figure 4.1. While the X—rays pass through many layers of the body, the relative movement ensures that only the desired layer is maintained in focus while other layers are blurred out. The method has also been applied in radioisotope imaging (ANG—1969,CASS—1969,MIR—1970). Here the source of radiation is a gamma radionuclide intravenously injected into the patient. The tomographic effect results from the use of focusing collimators combined with various types of patient and detector motions. The latest in non—reconstructive tomography is Compton scattering tomography. All these techniques with the exception of Compton tomography (BAL—1986) have the disadvatage of the presence of blurred unwanted planes which tend to reduce the quality and hence the diagnostic value of the image. .3 Reconstructive Tomography In reconstructive tomography a narrow beam of gamma or X-rays traverses the plane of interest so that all the unwanted planes are excluded (fig, 4.2). If this beam is fine enough, it could then be assumed that the plane being imaged is of a "negligible " thickness and hence two dimensional. Images of cross-sections of the human body are produced from data obtained by measuring the attenuation of X—rays along a large number of lines through the cross-section. Apart from the application in radioisotope imaging (emission tomography), image reconsfrut iion has been applied to other diagnostic imaging technique*-, using probes other tharr Do X or gamma rays.These include alpha radiography (CROW-1975) ,proton radiography (COR-1976) » neutron tomography (KUS—1985), ultrasound (6RE—1975) and nuclear magnetic resonance (LAU-1975). X -ray tube position i x _ ray tube position i i figure 4.1 Conventional Tomography- points on the plane x are always in focus whi 1 e pen rd on the other planes such as p and q are smeared onto rSi -s k erent ?»pi rits over the film 56 -figure? 4.2 Typical tomographic motions include successive linear stepping -followed by angular stepping over 180 degrees -for transmission and usually 360 degrees -for emission scans. 57 4.3. 1 Statement of the ker: on struct i on Pr obi em The mathematical problem involved in image reconstruction ■from projection is as follows, There is an unknown ,two dimensional, distribution of some physical Parameter- A -finite number of line integrals of this parameter can be estimated -from physical measurments. We wish to estimate (reconstruct) the original distribution. If as in figure 4.3 we denote the plane under investigation by ^artesian coordinates (x,y). the contribution of each point towards the detected signal is the density function f (x,y). In X—ray CT case the density distribution is that of X—ray linear attenuation coefficient of tissues in the the body while for' radioisotope imaging the f(x,y) represents isotope density « The ray ^aths are more conveniently described in a set of orthogonal ccurdinatejf Each point along the ray is now specified using polar coordinates (r , $> (figure 4.3). Where is the angle of the ray with the y—axis and r is its distance from the origin. The coordinate s represents the path lengt/i' along the ray. The integral of f < x ? y) along the ray t r , f P \ r - ® ) = j t- i x , y; d s 4.1 < r , Tor X-ray imaging or in gener a J transmission tomography the raysum is proper ti on-ii 1 o the logarithm of the detector si gnal since for monoener get i c photons, the transmitted beam inf ensif v l s 58 -figure 4-3 Coordinate system. Positions within the object is specified by an (x ? y> coordinate. While the rays are indicated by the rotated I M (>:,y) ds ) I0 *>:p (- I 4.2 where I0 is the incident beam intensity. If we identify \i in eq. 4.2 with f in eq. 4.1 then it is easy to see that P = — Zn ( I / I0 ) 4.3 For radioisotope imaging or generally emission tomography, p is proportional to the total detector signal provided that the effect of attenuation could be neglected. A complete set of raysums at a particular angle is called a projection in that direction. The coordinate r of a ray that passes through a given point (x,y) at an angle r = x cos 4.4 as can be seen from figure 4.3 Ideally f(x,y> is a continuous two-dimensional function and an infinite number of projections are required for reconstruction. In practice, however, fCx,y> is calculated at a finite number of points from a finite number of projections. For data collecting purposes the object is usually limited to a circular domain of diameter d say, and if the image is reconstructed at points arranged rectangularly with spacing w , then there are n = d / w 4.5 points along the principle diameter. This is the number of raysums contained in a projection. Each square cell with spacing w is called a pixel, short for picture element. 4.3.2 Reconstruction techniques Many numerical procedures tor the estimation Dt the 60 density function f (x,y) -from the measured projection data have been applied to parallel beam geometry. These could be put generally under two headings which are the analvtic reconstruction and the i.ler^JLive reconstruction methods. Furthermore the analytic technique could be subdivided into two basic -forms , the -first one is known as the two-dimensional Fourier transform method and the second one is the filtered back projectionmethod having as its seed germ the earlier method of simple back projection. The simple back projection was used in the early experiments. Though it is the simplest to implement , it produces reconstructions with substantial artifacts so restricting its use to merely a demonstration technique and a stepping stone to a more accurate method refered to as filtered back projection. There exist three basic iterative techniques , classified according to the sequence in '--ihich the corrections are made and incorporated during the iteration. These are , the iterative least square technique (ILST), the simultaneous iterative reconstruction technique (SIRT) •> and the algebraic reconstruct!on technique (ART). '1.3.2. 1 The Analytic Reconstruc ti on Technique Analytic, reconstruction is based on direct solutions of the image equation (a) the image may he ret onst r itc. i ed or; an arr ay of points with spat i rig w * whr-r 61 w = 1 / 2 km 4.6 (c> Fourier trans-forms may be replaced by a discrete Fourier series. Etand-1 i mi t i ng is the only approximation that is necessary in analytic image reconstruction. While all analytic methods are theoretically equivalent? the various forms of solutions lead to significant differences in implementation. The two methods described here are the Fourier transform method and the Filtered Back Projection method. Their use depends on the available hardware and computer time. 4.3. 2.2 The Fourier Transform Method The problem posed is to find the best estimate of a set of density functions f (x,y) from a set of projection data p(r? Fourier transforms of the distribution. One can find the values of the Fourier transforms on a rectangular grid by interpolation? then take the inverse Fourier transform to obtain the distribution as shown in figure 4.4. The various interpolation schemes had been exhaustively discussed in the literature (HER—1974 ? etc.) The starting point -for the derivation of the Fourier method is the representation of the density function as a two—ai mensional i ntegrai CD CD f ? y > F (1 exp (2f(i (x y Ky) dK>; dKy 4.7 -CD W i I | ‘ f i f' f ! hf-:. u-i ■* i -sent od as a superposition 62 erf sinusoidal waves. This parameters K>; and Ky are the wave numbers in the x and y directions. Taking the inverse transform of equation 4.7 gives the Fourier coefficients F < Kv , Kv ) (D (D CO F(K>; ,K y) = | J f (x * y > expC -2*(i — CD — CD if the < x , y > axes i s rotated b y a n angle $ to a new axes (r,s) (figure 4.3) where K figure 4.4: A geometrical representation of the central sections theorem illustrating the requirement for interpolati on. = arc tan ( Kv / i 4. v then using rotati nna Ml * 63 Kx = K cos Ky = K sin f 4.10 where K = Kv, + K y K = ! K ! = / ( Kg +~K^~T 4.11 so that F (Kx ,Ky5 ~ [ j f (x,y) p>: p L-2di K (x cos?+ ysin Now because both (x,y) and (r,s) coordinates are Cartesian we can define dx dy = ds dr 4. 13 and since from equation 4.4 we have r cos© + v sin? then (• F ( P. y ) = J exp <-2?ii. Kr > ds dr 4. 14 Ky J J The inner integral of equat i on 4» 14 coni d be identified wi th the ray-projection p (r , F In prose what this equation means is that each Fourier coefficient* or wave-amp ] i t tide , of the density function is equal to a correspond!ng Fourier coefficient of the projection taken at the same angle as the Fourier wave. Equation 4.15 can be used f or i mage reconstruct i on * as fed 1 nwr . f 3 f si t aff f he cue di mem:-i' M>rii ( our l er transform of 6 4 the projection , -followed by interpolation to provide a two dimensional array o-f Fourier coe-f f i c 1 ent s (see -figure 4-4) and -finally , the inverse two dimensional transform- The requirement for interpolation arises because the Fourier coefficients obtained from the projection do not fall on a rectangular matrix , as required for the inverse two dimensional transform. Though it is possible to obtain exact interpolation using a sampling theorem (BRA—1965), the large amount of computer time involved has made it prohibitive for most applications- A number of alternative approaches have been developed (CRO—1970, MER—1973, THD-1974) - One of these, kown as linear interpolation, is widely used- One important advantage of the Fourier method is the ‘ ^ possibility of processing the data projection by projection, opening up a possibility of on-line processing using dedicated mini computers. However, until the advent of the fast Fourier transform technique the r econstr lict i on time was dominated by the two dimensional Four 1 er transformation procedure. This has led to the introduction of an equally accurate but less time—c.onsuming method termed "filtered back projection" . This is an off—shoot of the early approximate method of reconstruction known as the back-projection method. 4.3.2.3 Bac k Pr o j ec t i on This was the ear 1 i est attempt at reconstructive tomography. It is the easiest to implement without a need for a computer or complex mathematics Reconstruction is per f or rued b y smear i ng each, prof i \ e bac k across the image plane. hue- the m agm t ube of ea« h ^avsnm is applied to all p f < i r • r -• t f;at ma * e ■ ip t h e v a v „ T hi ' pr or e s s may be described 6 5 mathematically as m C'(K ,y) = [ P ( r j , tp j ) Aq> 4. 16 j = 1 where $ j is the jsil projection anole , A is the sum of all raysums passing through the point. From the procedure discribed above it is obvious that back projection can not produce an acr~ * i r =*T reconstruction. This is due to the fact that each raysum is applied not only to points of high density - but to all points along the ray. This results in appearance of a st ar sr 11f art around a point of high density. Four profiles obtained from a circular object are illustrated in figure 4 =-5. It could be observed that a point o outside the object will receive positive corstributi ons from projections A and B during back projection. If enough projections are taken , the effect is to blend each individual ray of the star together creating a general fogging of the background. This may result in a loss of subtle differences that. may exist in the bac. kground - 6 6 / Fig. 4.5 Simple back projection showing f our prot i les o-f a circular object. Note that the cell shown outside the object receives positive contributions -from profiles A and B causing the famous star artifacts It can be shown that there exists a relationship between the approximate reconstruction produced by back—projection and the true image (BA«—1971> * If we wr i to equation 3.16 in its integral form p (r - one can obtain- by substitution into equation 4.17 it ® exp L 2dti k (xcos with the Integra! mu .1 tin.! i eb and divided by !K I so that the 6 7 equation now assumes the form o+ a two-dimensional Fourier integral in polar coordinates. It we take the two dimensional Fourier transform of equation 4. 18 we have P(K, The implication of equation 4.19 is that the back projected image is the same as the original image except that the Fourier coefficients of the true image are divided by the magnitude of the spatial frequency. 4.3.2.4 Filtered Back—Projection As shown in the previous section (equation 4.18)? simple back projection could be made to work by proper modification or filtering of trie projection before back projection- Thus if the profiles shown in figure 4 . 5 are modified as illustrated in figure 4.6 by including negative components? contribution by a point outside the original object could be made if proper filtering is applied. in figure 4.6? filtered profiles are back projected to reconstruct the original object. It could be seen that though the point o outside trie object receives positive contributions f r on* filtered profiles A and B? these are effectively cancelled by the negative contributions of profiles C and D, With many projections in use? this could eliminate the star artifacts (HER-1980 ? BRQ—1976 ? CHO—1974) . There exists three forms of filtered back projection classified according *o the ^ re r»- -tillering employed. These are 68 C F‘ Fig, 4.6 Filtered Bnckprojectian.Filtered profiles(p*) are back projected to reconstruct the original circular object.Note that a point outside the circular object receives positive contribution -from A and B and of -f setting negative contributions from C and L>, thereby eliminating the star artifact (BRO-1976). a) The Fourier Filtering This could be obtained by considering the Fourier integral of T < x- y) (equat i on 4.IS) q m r f- i y} ex p i 2-xu K r - *p ) 1 i K i df( d? 4.20 0 — a> in polar coordinates. Noting that from equation 4.19 F(Ky-K v) = P(K,?) 4.21 then we can write Tf p * (r , ©) d $ 4.22 0 with p * (r i ) I K ! p (K. i e-f p f Kr I d*: 4.2 6 9 In practice, equation 4.22 is replaced by its digitized -form, (equation 4.24), m -f ( ;>* « v ) = VL (r 4.24 j = i where m is the projection number and is the angular interval between each projection. Equation 4.24 could be compared to equation 4.16 o-f the simple back projection method except that back projected profiles p* used in this case have been -filtered (equation 4.23) before back projection. The change in p to -form p* is illustrated in -figure 4.6. The effect of filtering is to increase the frequency components of the profile in accordance with their wave number producing a diphasic function with average value of zero (BRO—1976) . The procedure for implementation of this filtering method is to first take the Fourier transiorm of a projection, multiply each coefficient by the magriitu.de of the spatial frequencies iKl, backproject onto the image plane using interpolation and then repeat for successive projections. b) Radon Filtering Equation 4.23 is in effect the inverse Fourier transform of the product of two functions namely P(K, iKl. There is a well known convolution theorem which sta tes that the Fourier transfor m of a product is equal to the jconvolution of the individual Fourier transforms (BRA—1965). Note the Fourier transform of p < r , '» is P (K, transform of l K ! =—1 ^ (1 1 B--1 9 5 S * BkiJ-- 1 976) . Thus applying t he r-envoi ut i an theorem we could write equation 4.23, using equal i on 4 .2s i as 70 m -1 P (r , q>) (r , ) = dr 4. 25 '*T‘ (r-r ' )2 — CD The integral which contains a quadratic singularity, must be evaluated by parts (LIG-195S, BRD--1976)- yielding 5p (r ' , ® > / dr P* (r , which could be evaluated using the Cauchy—Reiman method i.e. as a principal part integral. Equation 4.26 describes the -filtering process as a single convolution i ntegral in which the derivatives of p are added together, weighted by the inverse distance -from the point at which the filtered value is desired. Questions of ac cur ac v arise when equation 4.26 is implemented digitally , such as a) handling of singularity at r=r , b) approximating the derivative and c) replacing the i ntegr al by a summation. An-swers to these questions are not r eadi1y found in literature, although it is of interest that Beattie (BEA— 1 975> per t ormecl a summation by parts during his implementation and arrived at the convolution method described in the next section. Equation 4.22 and 4.26 were first derived by Radon in 1917 for solving gr avs tati onal equations. In recent years the method has been r ec appl i eo to i he c« y ? mag. ny by (.or mat: 1973 and by Beattie 71 1975. c) Con v'olut i on filterin g The divergence o-f equation 4.25 is caused by the factor !K! in equation 4.23 whose Fourier transform contains a square term in the denominator. The divergence can be avoided if iKl is replaced by a function equal to IK I for iKl 4 Km , but zero for f K! >- Km. This replacement will not affect equation 4.23 because P(K,S>) is zero for lKf>Km , thanks to the assumption of band—1imiting we had to make for digitization of equation 4.1 to be valid. ft f (>; ? y) = p(r? * F (f (r) ) d 9 4.27 where F ( f < r > ) = I K I and Kf 4 Kmm where Km is t h e maximum frequency r u t — off. We have by the inverse Fourier transform in the rotated axis fig. 4.3 Km f(r) = | IK I expt—2fiiK r ] dK -Km ! -m e x p ( —Z f t i K r ) dK f <-! xp—2‘fti K r> ) dK -■■P• • m m K c o s (2ft K r > d k Integrating by parts- we have 7 2 Km 2K sin(25(K r) f (r> = (2*(K r) dK 2*fr r fr 1 s i n k-m = , 2Km sin(2rtKmr) 2 i t r 2«gr g <=ds (2*k V> | sin (25fKmr) = 2Km + ■ (—2 s i n 2 <5fKmr)) 2-rtr r f (r) = 2Km2 sine (2Kmr) - Km2 sine2 (Kmr) 4.2B where sine x = sin(*rx)/ *fx i-f we now replace IK! in equation 4.23 by its cut—oft version, we obtain (D l r it(r— r ) *T2 (r—r ) 2 —© . dr 4.29 using the convolution theorem. The result obtained tor the filter function has been derived in various forms (BRA—1967). Various approximations have been used (RAM—1971,BRA—1967,SHE—1974) . The first term in the bracket is the well known sine function which has the effect of removing frequency components greater than Km in a convolution integral (BRA—1965). This is because the Fourier transform of 2Km sine(2Kmr) is a tophat function of unit height out to frequency Km and it is assumed (band limiting assumption) that p(r,«p> contains no frequencies in excess of Km (BRD—1976). The implication of this is that p(r,$) is left untouched after convolution with this term, and we have 73 sin2 L*fK P * One of the implications of band limiting is that the integral in equation 4.30 could be replaced by a summation, with poi nts spaced at intervals w=l/2Km (BRA—1956). In addition sin2(4TKm(r—r )) = f * r r , odd multiple o-f w 1,1 l 0 r—r even For computer implementation, equation 4.30 is written in the -form m P (r-j ) P* (r j , with the summation taken over all j -for which i— j is odd. This method o-f modifying the projections by convolving them with a function prior to back projection has found favour in most applications , especially with the first generation type of scanners where data collection time predominates. The main reason being that the reconstruction could be done projection by projection thereby saving computer core space. The advantage of this becomes more apparent if the scanner is micro-processor based. This is the method used for most reconstructions in this thesis. 4.3.3 Iterative methods The other major method of image reconstruction from projections is the iterative reconstruction technique. This refers to all the methods which re]y on the successive appr o>: i mat n on of the image parameters using arbitrary initial 74 cell densities in an attempt to match the measured ray projections. the process is repeated until the calculated projections agree with the measured ones within a desired accuracy. In order to implement this technique, the object is ■first approx i mated by an array of size n x n of uniform density f1(x,y). Each projection is broken into strips, as against the idealised rays of equation 4.1 , of width equal to the dimension of the cell (figure 4.7). The raysums are then calculated according to the contributions from each pixel intersected by the ray. i-e. for the j*-*"1 ray N pj = [ wij n 4-32 i = 1 where Wi j is the weighting factor that represents the contribution of the i^*~* cell to the jraysum, Pj, note that most weighting factors are zeroes, since only a small number of cells contribute to a given ray. Equation 4.32 represents an array of M equations in N unknowns, with the density f^ as the variables. This equation could, in principle, be sol ved by inverting the matrix Wj j i . e. , M fi = i This procedure has proved impractical (BRO—1974), this is because it is not usually possible to have enough projections to provide N independent equations (i.e. M N ), there will 75 be many possible solutions. More-over inconsistency in the data due to noise render the solution so obtained unrepresentative o-f the image. Another problem often encountered in this procedure is that o-f an unmanagebl y large matrix. One method devised to overcome these di-f-f iculties is to adjust the density values -F j iteratively until the calculated projections agree with the measured ones. The procedure is as -follows, a uniform density distribution is first assumed and raysums are calculated for these, the density of each pixel is then adjusted to compensate for the difference between calculated and measured raysums. When this is done for all projections the first iteration jth ray Figure 4.7 The n x n image reconstruction matrix used in the iterative reconstruction technique. The beam has a finite width, w. is complete. t he procedure is then repeated until the desired accuracy in t d] -f i er F-nce het.weein measured and calcuiated projections 1 S achieved. thi- procedure could be 76 mathematically expressed as M fj* = + E ^ i j £ 4.34 j = l where fj^—* , f are the density before and after the iteration and Afij^ is the correction applied to the i^il pixel from the j^Jl ray- At the end of the first iteration one arrives at a solution which is essentially what is obtained for the case of simple back projection if the starting density was zero throughout the image field. Because this result is not adequate, a correction factor is introduced into the succeeding iterations. There exists three basic iterative techniques, classified according to the sequence in which these corrections are made and incorporated during the iteration. These are, the iterative least square technique ( ILST ) , the simultaneous iterative reconstruction technique ( SIRT ), and the algebraic reconstruction technique (ART). The simplest of these methods is the ILST . All projections are calculated at the begining of the iteration and corrections applied simultaneously to all pixels. Because each pixel is recorrected for each ray passing through it the result is an oscillation about the correct solution. To arrest this behaviour, a damping factor can be applied to all corrections so as to produce the best least square fit after each iteration. In SIRT, sometimes referred to as point-by-point correction method, each iteration begins with a partici ular 77 point and corrections are applied to all raysums that contribute to this point. The procedure is repeated for every other point while taking all the previous corrections into account in the present iteration. Each point, pixel, receives a correction in proportion to its current density. The most widely used procedure , however, is the algebraic reconstruction technique ( ART >. This is illustrated -for the simple case of a -four pixel object. At the begining of each iteration, one raysum is calculated and corrections are applied to points that contribute to the ray. This is repeated for each raysum in a projection and -for each projection with the previous corrections embodied in the succeeding cal cl? 1 at i ons (HOU-1973,KUH-1973). 4.3.3.1 Types of Correction Suppose, during the iteration, that a calculated raysum Pjc is compared with the measured value pj , and an error is found, given by A P j = p j - p jc 4.35 corrections Afjj^' are to be calculated so that, if applied to all cells constituting the it!], ray, the calculated raysum will be increased by Apj. (Application to cells is actual 1y performed with I LSI and ART ; wi i L SIR! only one point at a I i me' i s r. or r ec. t e d - > T wc* common* used ways to distribute the 78 correction among the pixels are described here. a) Additive Correction. Each pixel receives a correction in proportion to its weight Wj -j , N A fi = Wij A pj / £ Wi j2 4.36 The denominator is a normalizing -factor to ensure that the total change in the raysum, it all pixels are corrected , is Apj. b) Multiplicative Correction. Each cell receives a correction in proportion to its present assigned density f^a , tia Ap F. _ Afi j J 4. 37 .c again the normalization is such that the total ray correction is Ap .. Using equation 4.30 and rearrange we obtain J + i- pj tta + A -f±j£ = ------4.38 Pjc which says that the corrected density is equal to the -former density multiplied b y the ratio o - f the measured and calculated ra/sums. 4.3.75.2 Weighting Factors The weighting -factors Wj j are usually precalculated and stored -for use during reconstrnetion. Because of the large number of values, the table may be compressed and approximations used. Theoretical! y, w1 5 is equal to the actual area of intersection between the i *^ cell and the j^l ray as shown in -f i gure 4 r / „ fit ;ict c h o i t e s ar e aval i abl e , however , as 79 1 1 9 t 13 5 7 a) 6 n ►B 0 0 2 2 , * * t t 0 o 1 1 o 1 1 1 1 1-0 9-0 11/2 9/2 l 4. 4. rr .J » >_J 4.5 [ 5.5 4.5 — > 10 * 5.5 4. 5 5.5 4.5 H 10 —» 10 10 10 10.2 9* B r- ! 6.5 i 5.5 ] <— 12-10 6. 6 1 5.4 1 ji +—12/10 1 t. ,... 4,5 ) -v— ;--i — 1 c> 4.4 3.6 4--B / 1 U 7-10 13- i 0 7/1 0 13/9.e -> y j *r h 7 4.0 7.2 ! 1 ! 6 5.8 2.5 1 2 (b) (c) Figure 4.8 Numerical illu s tr a tio n of ite r a tiv e reconstruction ART. a) Original object and six raysums b) Additive correction, starting from a value of zero. A correction is first applied to the two vertical rays by subtracting the calculated values (O and 0 )from the measured values \li and 9> and d istribu tin g the difference between the two pixels. ihis^ process is repeated for the horizontal and diagonal rays, Thus completing the first iteration. In this example, the reconstruct i or> is perfect after one iteration. c) Multiplicative correction. The starting values <1 and 1) are m ultiplied by the* ra tio of the measured vertical raysum <9 and 11) to the calculated values <7 and 2). This process is repeated for the horizontal anu diagonal rays. In this exampi e, tur t tier i t er at i one ar r- r egm r r»i f 80 follows: a) 1 or 0 , according to whether the centre of the pixel lies within the ray. This choice is the simplest to implement, and reduces the calculaiion time considerably, but is generally considered to be less accurate. b) The distance -from the centre of the pixel to the centre of the ray. this choice divides the contribution o-f a cell between the two adjacent rays in a linear -fashion. c) As a function of this distance. This choice was used by Hounsfield in the original EMI scanner (HOU—1972). 4.4 Di scussi on Generally speaking there exists no set criteria for choosing a reconstruction algorithm, ft number of authors have compared the different methods (HER—1973,RAM—1971,SWE-1972 * SMI- i973) , however, the conclusions are specific to the article being examined and an overall comparison is yet to be reported. The suitability of a particular reconstruction technique to a given situation is normal 1y expressed in terms of speed- accuracy, image quality and its response to incomplete data. The relative importance of each of these depend on the appli cati on. When complete data are available, that is the projections are obtained at fine angular steps, it is generally accepted that the analytical method is the algorithm of choice. This is because, in this case, there exists little need for interpo]ati on which accounts for most of the time used during all the ana] ytiral r e( oust r tu i i on method,-.. However, projection data ran not he obtained for ai J aiipif". net ween 0 and iX. fhe 81 back projection integral has to be replaced with a sum. The question arises as to the number of projections required for a circular domain of diameter d if the image is to be correctly reconstructed for frequencies up to the cut-off maximum Km - Brae ewe 3 1 arid Ri dd"; f-:RA- • 1 97 1 ) suggested that if a good reconstruction is to be obtained, since the total number of density values to be estimated i s \.N==f\n2/ 4), it is clear that the minimum number of projections needed to fully determine the densities is 11 y\ c r . - f - w / /I and furthermore, the projections should be equally spaced from 9=0 to resulting image is said to be over-determined and represents an average of the redundant information* The effect of this is to reduce noise and minimize interpolation requirements. If a fewer number of projections are avail able for reconstruction, the image is said to be under determined. The behaviour of analytic and iterative techniques are fundamentally different. While the analytic methods assume the missing projections are identical to the available ones, iterative methods on the other hand generally converge to the smoothest image obtainable. Another difference is that under-determined data results in increased reconstruction time when using analytic methods since the time spent on interpolation for intermediate points predominates. On the other hand, tor iterative methods, the requirement for time is reduced these considerations, iterative reronstructions may become the method o-f choice i n higni y under-determined sitnations. 82 In practice, reconstructi oris are never exact. With analytical techniques, the two limiting factors are band-1imitin9 and interpolation. The density distribution f(x,y) contains frequency components in excess of Km , the maximum we can reconstruct. Removing these high frequencies causes a ripple artifact especially near sharp edges, these ripples can be damped considerably while retaining an acceptable image by changing the convolving function so that instead of a sharp cut-off frequency at Km, they are rolled off gently according to F(f(r)) = IK I [ 1 + cos CrfK'/ Km ) j IK'I 4 Km 4.39 this is termed the Hanning window and is wel 1 documented in the literature (BRA— 1967,RAM— 1971,SHE—19/4). It is clear from the discussion that all the r econstr uc.t i o n techniques wor k well depend! ng on the practical situation. For complete sampling the analytic method is better but when sampling i s coarse a n d circular symmetry could not be assumed, it is often accurate and less time consuming to use the iterative techniques. 8 CHAPTER FIVE THE SCANNING SYS I EM Introdu.cti on With the medical C-T scanner- the projection data -for the patient is obtained by rotating the source and detector assembly around the patient, This is mechanically cumbersome to implement and partly accounts tor the high cost ot commercial CT scanners. For mobile inanimate objects this is unnecessary. The simple instrument tor computed transverse section tomography ot inanimate objects is a rectilinear scanner which moves the sample back and -forth, across the radiation beam while keeping the source and detector stationary. This is repeated after rotating the object through a small angle until sufficient angular data are collected tor reconstruct! on, as p'fO^neer ed by Kuhl and Edwards (KUH—1963) . The scanner used tor the tomography work in this thesis is a rectilinear gamma ray scanner developed tor transmission and emission studies. The scanning system, figure tut consists ot the following components: 1— Scanning rig, 2- Stepping motors and stepper motor interface, 3- Detector and source collimators, 4- Detector and counting electronics-. 5— Experimental control and data acquisition and 6— Image reconstrnetion and display In Single Photon Emission tie collimated whi ie in transmission tomography the source as uie t s as f t jo del ec to r i,-- 1o he " f - * i 2J * i -m.;t - cm ? to ‘ define. r • — the • • r a y •. o:1 irnated • jv , L < 1 d s t e c t u f Pre-antoiiner 84 Source collimator '•graphics-! CALCOM GDC ; tier min pi! I. Plotter. K'n'j yr»* im a o ’ ^rj *~ffi liiVj 'J 'r icJlru toeCLt’ OfnCS 8 5 path and to reduce the scattered radiation. 5.2 The Scanning Ri9 The scanning rig, figure 5.2, is a Uni slide assembly made by lime__and Rreci^ijen__Ltd* It comprises the following three stages: 1— A motorised linear slider housed in a base section which hard and black anodised aluminium with a usable length of ^lm. The slider is 150mm in length and is made of high grade aluminium with a motorised leadscrew of 1.27mm pitch and '''5mm lead giving a resolution of 'v.01mm at 400rev/s. Two safety • nii t t c: h(~s are mount t-' ms t»I. j > ends 1 hi * scanner base. 2— A motorised and manual eievatory stage to lift a 40kg sample a distance of 150mm and fitted with micro switch safety limits. The eievatory stage could be used for scanning vertically a segment of the sample to locate regions of higher density but it is mainly used to adjust the height of the sample so the detector source combination is looking at the slice of interest. 3— A motorised precision built rotary stage which has a table diameter of 120mm and a central aperture of 20mm. The large stepping motor and general construction allows loads up to 40kg to be used. Precise movement and extreme accuracy are achieved by the high specification main bearing and thrust race. A smooth 90:1 ratio worm and wheel assembly transmits the drive to the rotary table top which has engraved graduation of 1 degree intervals and up to 360 degrees. Every step of the motor provides a standard resolution of 0.005 denr ees> ■» repeatability nf i.i.oi degrees and accuracy of + U.OV 86 F i g u r e Photographs s h o w i n g ( t o p ) t h e scanning system? stepper mot or c o n t r o 1 d e t e c t o r and the source stand < b o t t o m ) the scanning rig B7 degrees- The rotary stage is made o-f high grade aluminium alloy, stainless steel and phosphorous bronze and weighs 3.O k g . 5-3 The Stepping Motors and Stepper Motor Interfaces A stepping motor can be described as a rotating machine in which the motor/shaft moves through discrete angular steps in response to a current pulse applied to the motor winding giving an incremental rather than continuous motion as in most other motors- The number of steps executed by the motor is equal to the number of command pulses. This characteristic makes the stepping motor highly compatible with digital electronics- Stepping motors are connected directly to the leadscrew giving a direct linear movement for each step of the motor. Speed is controlled by the rate at which pulses are applied to the motor. This then enables accurate control of position and speed as the required angular movement can be expressed in terms of a precise number of steps. Stepping motor performance is a function of many parameters involving system loads and driving techniques, as well as motor characteristics. The motor drive can be a resistance limited unipolar or bipolar type depending on the speed, torque and movement precision required- Unipolar drives are commonly used for reasons of simplicity and require only a single polarity power supply, however, bipolar drives compared to unipolar drives on the same motor, will produce more torque at low speeds. The present system uses a unipolar dr i ve. 8 8 The stepper motor interface card IF1 permits up to three stepper motors to be controlled via an RS232C serial link. The controller may be a simple nonintel1igent terminal, a mi croprocessor, a programable logic controller or a main-frame computer, in -fact any suitable device with an RS232C port. In our scanning system the controller is an IBM—PC. Information is transmitted to the interface in a form of a series of commands giving the required speed, direction, distance etc. From this information the interface generates a ramped pulse wave form suitable for feeding directly to the clock input of a stepper drive. The diagram shown in figure 5.3 shows a typical three axis system, including the interface, stepper drives and power supplies. The interface motherboard incorporates an RS232 socket for direct connection from the controller. The IF1 receives and sends data using an RS232C port connection on the controller. The RS232C specification defines all the essential details of the communications link, like signal levels, connector type, pin allocations and the general data format. However there are numerous variations possible within the basic framework, and both interface and controller must be configured to operate in the same way. This method of communication is referred to as "asynchronous serial". The data is transmitted one character at a time, a character comprising a number of bits sent along the same piece of wire one after the other. Start and stop bits mark the begining and end of each character. The time interval between each bit of t V\9 character dp H ned 0 c\urfa FIG.(5.3) IF 1 STEPPER MOTOR INTERFACE 9 0 start and stop bits, can be transmitted at any time with no reference to any timing wave -form, hence the description "asynchronous" . 5.4 Detector and Source Collimation Good detector collimation is essential in both emission and transmission tomography when using most common detectors which are not position sensitive. Therefore, the Ge(Li) detector incorporated into our imaging system was well collimated -for all the experiment. It is important that the detector is -firmly positioned since the scattering due to surrounding materials may otherwise vary (section 3.3.2). The detector-col1imator is held in position by a steel collar to the steel frame which incorporates the detector and the scanner so the entire assembly is kept in a rigidly fixed position. The collimator used in all the tomographic experiments was a single bore lead collimator of the jacket type which extends back to cover the length of the crystal. Figure 5.4 shows a cross—sect i onal view of this type of collimator with the dimensions used. Mathematical analysis of the collimator properties and experimental studies of the collimator response functions is given in chapter 6. The collimator aperture is either 1mm or 2mm in diameter and the position of the aperture is 8mm off centre of the collimator to avoid centering the gamma ray beam on the central dead region (p—core) of the detector crystal (chapter 6>. In transmission tomography good collimation of the detector is pointless withoui equally good collimation of the K ------» 95mm 30mm Figure 'd H typical single bore collimator cross-section with the dimensions of the collimator used 9 2 Figure S.Sshowing a schematic representation of the source collimator 9 3 source- The source collimator, -figure 5-5, is made -from a hollow lead cylinder of '''32mm thickness- A 1mm aperture was drilled into the cylinder forming the collimation. A solid aluminium cylinder is inserted inside the lead cylinder, the source is then placed up against the aperture and on top of the aluminium cylinder. 5.5 Detector and Counting Electronics The chain of the electronic counting and control system is illustrated in figure (5.1). The counting chain starts in the Ge(Li) detector which is an open ended coaxial with a horizontal dipstick. A power supply is used to provide high voltage for the detector. The current pulses originating from the detector are passed through a linear amplifier where amplification and pulse shaping is carried out. The pulses are then passed through an ADCAM —multichannel buffer and a multiplexer. The spectrum is then collected and displayed on an IBM—PC with an MCA emulator program. The general control of the whole system is centered on the IBM—PC which also controls the stepping motors interface card IF1 via an RS232C serial port. The characteristics of the detection system are descibed in detail in chapter six. 5.6 Experimental Control and Data Acquisition Program The starting point of the developement of a control program for a tomographic scanning system is to decide the required motion sequence. This involves linear stepping motion followed by successive angular indexing for normal tomographic scanning. Other important points to be considered include the rp^iijul inn o-» the virions motion? wroth determine the minimum 9 4 number o-f pulses to be sent to each motor and the timing parameter of the interface- The pulse length shoud be a minimum of 40ps while the time between pulses must not be less than 770ps. This frequency is dependent on the inertial load applied to the system- A fast pulse could cause stalling of the motor and a subsequent loss of steps. The control program is written in a series of modules or small programs linked together in a batch file called CYCLE1. All the modules are written in either BASIC or Fortran languages. These modules are: 1- CREATE.BAS 2- SYSF’REP. BAS C 3- SPPREP.EXE Y 4- SYSCYCLE.BAS C L L 5- START ACQ.EXE 0 E 6- STORE.EXE □ 1 P 7- STSPEC.EXE B 8- SPECAS.EXE A 9- PEAK22.EXE T i) Before entering the batch file CYCLE1.BAT the user has to execute the program CREATE.BAS. This program creates all the sets of coordinate points to be used by the scanner. These points are calculated from the size and diameter of the object to scanned, the number of raysums and the number of projections needed to fully reconstruct the image. The coordinate points are given as (X,Y-Z) where X is the coordinate representing the translation movement in front of the detector. Y is the coordinate representing the elevation of the scanner i.e the movement of the scanner vertically to locus the scanning position on the layer (region) o-f interest. In general cases, 9 5 looking at a single slice, the Y coordinate is kept constant. Both X and Y coordinates are given in mm. The coordinate Z is given indegrees and represents the angular increment between projections. The coordinate (0,0,0) is set such that the detector source assembly is ■focussed on the centrepoint o-f the slice of the object the scanner is lookingat. These coordinate points are stored in a data file. ii) Having created the input data the user can now run the batch fileCYCLE i. BAT as CYCLE! DATA TIME ,the parameter DATA is the file containing all the coordinate points and the parameter TIME is the live time per raysum which is chosen depending on the source strength and the required image resolution- The first thing that CYCLE1 does is to copy the DATA file to the T0M3EQ file and then executes the system preparation program SYSPREP which initialises the tomography unit and prepares the DATA file. This program will ask the user to turn on the tomography unit and then it will open the communication, setting the baud rate at 300, the number of data bits per character to 7 and the parity to even. iii) CYCLE1 executes the program SPPREP which prepares a random access file for spectral data. This program will ask the operator to enter the initial and final channel numbers, these two channels should enclose the region of the gamma—ray spectrum which contains the photopeak and a similar number of channels on either side of the peak to be used -for scatter subtraction as discussed in Chapter T hr ee * 9 6 Once the previous programs have been executed, CYCLEI starts a 1 OOP which contains four programs dealing wi th movement of the scanner to the next position, data acquisition and data storage. CYCLEi will continue looping until all the data -for all the coordinate points have been col1ected. i v) The -first module to be executed in the loop is the system cycle program called SYSCYCLE which determines the next position and moves the tomography unit. This program also sets up XLOW = —450.0mm- XHIGH = 450.0mm, YLDW = — 100.0mm, YHIGH = 100.0mm, ZL0W = 0.0 and ZHIGH = 360.0. The maximum scan distance is im so that XLOW and XHIGH are set to avoid getting in contact with the safety switches on either side of the scanner bed. This also sets the XCALIB = 80.0, YCALIB = 1000.0, ZCALIB = 100.0, XSPEED = 2000, YSPEEE) = iOOO and ZSPEED = 5000 rev/s. The calibration values are conversion ratios between motor revolutions and 1mm but for the Z it is conversion from revolutions to 1 degree. This program also calculates the relative positions i.e. xrelative = new position — old pc?s:.. t on and i -f ;; re 1 a ' 0 irh?- -r xdirect ion =,, + " el sc-- xdi recti on = " —" and similarly for the others. The next two modules to be executed in the loop are the STARTACQ which takes into account the MCA number, segment number and live time and the STORE program which takes into account, in addition to the mca number and the segment number, the channel file T0MCHN. The last program io be executed in the loop is the SlSfTT which stores arid writes out the spectral data to a 9 7 random access -file SPEC. At the end of the previous step CYCLE1 goes back to the begining of the loop and repeats the same procedure until it reaches the end of TCJMSEQ. v) Program SPEC-AS converts the region of interest into ASCII format and writes out the mca number, segment number, date, initial channel, final channel and for every coordinate (X,Y,Z> it writes out the time, real time, live time and counts for every channel to a file called TOMOUT. PEAK22 is a Fortran program that will calculate the peak area and the scatter area and it will write the output to a file called TOMRES to be used later for reconstructi on. 5.7 Image Reconstruction and Display The mathematics of image reconstruction from projections for different reconstruction techniques have been discussed in chapter 4. The reconstruction technique used is the Filtered Back Projection method with convolution filtering see section 4.3.3.^- A computer program REC1.BAS was written to work on the IBM—PC. The advantage of this method of image reconstruction is also discussed in chapter 4. The reconstructed data (density values) represent the attenuation coefficient in each pixel in the case of transmission tomography or the activity distribution in the case of emission tomography. Images obtained are first displayed on the IBM—PC using simple BASIC programs written by the author. This is done in two way5»0) values of the pixels are displayed as intensity levels- these intensity levels are presented as shades of 9 8 gray. ( i o The intensity levels could also be displayed in colours, but on the IBM—PC we could only get a choice of four colours if we are using a high resolution (number of screen pixels) and a total of eight colours if low resolution is used. Displaying the images on the IBM—PC is done only as a quick look option but not for image analysis. Most images presented in this work are surface plots which are perspective or isometric projections of a surface produced from a matrix of gridded data which is supplied through a local file called TAPE60. SURFACE is a graphic package available to use interactively on the Imperial College Computer Centre (RAB—19S4) . Each matrix or block data is formatted into a given number of rows and columns which correspond to the X and Y directions of the grid, respectively. The surface plot is constructed from line plots in either one or both grid directions. A line is plotted in one of the directions X or Y followed by the next line in that direction, parallel to the first, and so on, until all lines have been drawn in one direction. A number of different surface views can be produced from a given block of data, but data from more than one block can not be used to produce a single view. Each view presented causes the previous one to be erased so that only one view appears on the screen at a given time. The entire data set could be used for plotting and it could also be partitioned. SURFACE generates a representation of surfaces, as outlined above- from gridrieo f*J ovation values and also gives freedom to manipulate the data and pictures. SURFACE is roi troll ed from a gr aphi c 1 er mi nal . By entering the commands 9 9 at the terminal during a SURFACE run, the user can adjust the viewing angle and perspective distance, and change picture magnification. Other commands enable the user to partition and interpolate the data, eliminate the hidden lines, and clip the height of the pictures. A hard copy command is incorporated into SURFACE to allow microfilms from a 35mm camera and Benson r» 1 o fit. £-.••'? •• i? • tp».t o !' t-' obtained. MATMAP is another graphic package used (RAB—1986). This package generates the picture in the same way as SURFACE and has similar routines to it but it also allows the user to draw tonal maps with a maximum of 10 shades selected by the minimum and maximum values in the data. The tonal effect is produced by the combination of various characters. 5.8 Conclusion A single detector gamma ray scanner has been developed that could be centered around an IBM—PC micro—computer. the scanner could carry a maximum load o-f 40 kg and scan an object of IQO 0mm diameter. The control program has the capability of collecting spectroscopical data for both emission and transmission studies and could also be used to vertically scan a small region of an object, this is done by the use of the elevatory stage which also gives us the capability of scanning many planes of the same object in succession. The reconstruction program is written in BASIC to run on the IBM—F'C and also has the capability of handling both emission and transmission reconstructions, A display program on the microcomputer has been developed but -further improvement of the package is envisaged. 1 he main display packages used are 100 SURFACE and MATMAP which are available on the college mainframe computer and can be used interactively through any graphic terminal. These two packages provide good display opti ons. 101 CHAPTER SIX SYSTEM CHARACTERISATION 6.1 Introduction The scanner is an assemblage of different elements whose combined characteristics determine how accurate an image produced represents the original object- These elements are the gamma ray source, the detector, the source and detector collimators, the scanning mechanism, the reconstruction algorithm and the display system. The quality of an image is then determined by a number of physical factors which include, attenuation of photons, uniformity of response with depth, detection sensitivity, physical sampling of the field of view (linear and angular sampling) and elimination of non-linearity due to unwanted Compton and multiple scattering events- Whilst most of these factors are common to various types of tomography, the effect of photon attenuation and non-linearity produced by the acceptance of unwanted scattered photons are of particular interest in Emission Tomography- Their effects will be investigated later- The other factors affecting image quality could be studied under two headings: firstly, detector efficiency and energy resolution, and secondly, detector—col1imator spatial resolution. 6.2 Sensitivity of imaging detectors The sensitivity of a tomographic system is important as the noise introduced by the reconstruction is approximately an inverse square function of the total detected events (BUD-1977). The acquisition of statistically significant counts per raysum of an image' may take minutes in our type of 102 single detector scanner. Much work is directed at present to increasing detector ef-f iciency and the total detector area exposed to the object (KAV-1978, 0RT-19B0, CHU-1978, ENT-1973, BAL-19B6). Preliminary experiments were carried out to determine: (i) the homogeneity of the detector i.e. the variation of sensitivity across the face of the detector along a diameter; (ii) the crystal size (diameter and volume); (iii) the intrinsic efficiency as a function of energy; and (iv) the energy resolution. 6.2.1 Detector Homogeneity and Size Determination Before finally setting up the detector— source—scanner arrangement, the scanner was used for scanning the surface of the detector. 0.37 MBq 137Cs, 133Ba, 60Co point sources from the Radiochemical Centre, Amersham were collimated to 1 mm, with the surface of the collimator 5 mm from the surface of the detector. The full energy peak counts were taken at 1 mm steps over a length greater than the physical size of the detector s outer casing. An allowance of 10 mm was given on either side of the crystal housing to allow for total scan of the detector. The same procedure was followed for the length of the detector. Figure 6.1 below shows the dependence of spatial resolution (resolving distance) on the collimator dimensions where is the length of the collimator, D^ is the collimator bore diameter and z is the distance between the collimator face and the surface of the detector. From the symmetry between the two triangles ABC and ADE we have 1 0 Figure 6. 1 Dependance o-f spatial resolution on collimator dimensions R / 6 (z) = R = •for our measurements we have Lb=30mm, Db=lmm, z=5mm R = 0.67 mm thus the resolving distance is less than 1mm. The graphs shown in -Figures 6.2* 6.3 and 6.4 show the response of the Ge(Li) detector, from the graphs the width of the detector was determined to be 32.5mm ± 0.2mm, the three graphs also indicate the homogenei t y of the detector. In f i gure 6.2 which is the scan using *33Ba (356 keV) the homogeneity is well defined where one can see a clear dip i n the centre of the detector, where the count rate drops down almost to the background level. This dip is not unexpected due to the fabrication of the crystal. The coaxial nature of the semi-conductor crystal leaves a central dead layer called a p—core which is not responsive to radiation passing through it. The p-core is where the electric field is generated and it is very important since charge collection is carried out 104 under the influence of the electric field. The width of the p-core is estimated to be 4.0 ± .2mm. The depth of the p—core is constant (manufactured) and does not depend on energy. However as can be seen from figure 6.4, when higher energies like &0Co- 1333 keV are used to scan the detector, the result does not show the existence of the dead layer, this is because of higher scattering at higher energies. Thus to determine the crystal and the p-core diameter and depth, it is important to use low energy photons. If the p-core extends from top to bottom, the detector is called true or open ended coaxial and if it does not, it is called closed ended coaxial. Figure 6.5 which is a scan along the side of the detector using a i37Cs 662 keV point source, shows that the response is homogeneous and the p-core extends from top to bottom i.e. it is a true coaxial and the length of the crystal was found to be 32.0 ±0.2mm, and thus the crystal is nearly a right circular cylinder. If the dead layer in the crystal is parallel to the face of the detector, then it is called a planar detector. The presence of the p—core (dead layer) is a fact that has to be noted in any application of this detector to tomography, since collimators focussed on what is considered to be the most efficient region, by some workers, results in a considerably reduced efficiency ; therefore collimators have to be focussed on the region of highest efficiency (away from the centre of the crystal). 6.2.2 Detector Efficiency and Resolution The photo—peak efficiency of the detector was measured using a number of standard point. sources witn certified 1 0 5 nominal activities (Radiochemical Centre, Amersham). The various sources, 24iAm, 133Ba, 137Cs, &0Co, 22Na, 5*Mn and 57co were chosen for their simple pulse height distribution to cover the energy range of interest, namely between 80 keV and 1.33 MeV. Two sets of sources of nominal activities 0.37 MBg and 0.037 MBq were used. The high activity sources were used for calibrating the detector at large detector to source distances while at smaller separations the weaker sources were used. This was done to reduce the effect of pulse pile—up in the spectrum obtained. In order to minimize the effect of scattering from the environment, the source holder used for reproducible geometry was made of low density, low atomic number plastic material. Each pulse height distribution was acquired into 2000 channels using an IBM—Ortec system which comprises of the following units as can be seen from figure 6.6. 1) EG? 2) multiplexer unit number 476—4 3) ADCAM 918 (multi-channel buffer) 4) an 8000 channel MCA emulator divided into four sectors, and 5> an IBM—PC—XT Figure 6.6 Block diagram of 1 he detection system. 106 In determining the photo—peak area o-f the energy distribution of each spectrum, the full width at the tenth maximum ( FWTM ) of the photo—peak height is taken. The number of counts in the photopeak was substituted in the following formula to determine the intrinsic efficiency. (ADAM—1970) w II < M e = nd / n0 ------> QJ (nd / n0) *v_1 6.2 where Or = (1 / 2) C 1 - 2 / ( Z2 + r2 )* 3 is the relative solid angle. rijj = detection rate i.e., photopeak counts n0 = absolute gamma ray emission rate of the source r = radius of the crystal and Z = distance from the source to the front of the crystal i.e. Z = z + z0 where z0 is the distance from the crystal to the surface of the can. The ratio n,-j/n0 is the absolute efficiency. n0 was calculated using half-life and branching ratio values and energies taken from Lorentz (LOR—1983) . the distance Z used was from the crystal surface rather than from the front surface of the can. The distance z0 defined above was obtained by fitting a straight line to the inverse square root of the detector response (C—^ ) along its axis. The value of z0 was obtained from the intercept on the x-axis when C—^ was plotted against Z and found to be " 21mm. Figure 6.7 shows the detector casing and crystal dimensions. The efficiency as a function of energy was obtained for the detector by fitting a curve to the measured efficiency points. The fitting function used is the six-parameter function developed by P. W. GRAY and A. AHMAD (GRA-1984)who have studied different classes of Ge(Li) detector efficiency functions and have agreed that the class 107 of -functions defined by 4 (E,p)=Cp1+P2£n(E)+p3£n2 (E) +p4^n3 (E) +ps,6n5 (E) +p&£n7 (E) 3/E 6.3 is best for the energy range (80-1850 keV). In the equation E is the energy in MeV. The minimisation routine MINUIT developed by CERN (JAM—1978) was employed to obtain the six fitting parameters, these parameters and their errors are listed in table 6.1. The fitted efficiency curves are shown in figures 6.8 and 6.9 and they show that the maximum efficiency is for photons of energy about 100 keV. Table 6.1 showing the values of the parameters in equation 6.3 and their errors parameter val ue error P 1 10.96 . 14 CO 1 P2 • . 3 P 3 -9. 4 1.2 P4 -6. 2 1. 0 P5 . 76 . 14 P& -. 038 .008 P-CORE "32.5mm ~74mrn .aluminium casing ~32mm 21mm^ 6e(U) Crystal 76mm Figure 6.7 the dimensions of the detector crystal and casing 1 0 8 Another property of the detector to be considered in imaging is the energy resolution- Good energy resolution is important because the contribution of scattered photons to the photopeak region, due to both coherent (Rayleigh) and incoherent (Compton) scattering, lead to image degradation (DEC—1969). Though it is possible to discriminate energetical1y against Compton scattering, its extent depends on the resolution of the detector since only small angle scattering events are accepted- However, if the spectrum of interest is of a multi-energetic type, as is the case of neutron activated samples (Neutron Induced Gamma Ray Emission Tomography) the scattering contribution from peaks of higher energies could prove to be very important. The resolution of the system as measured by the FWHM of the photopeak as well as the FWTM were measured for different energies. A function of the form fwhm The same minimization routine used earlier was used here to determine the values of the parameters- Listing of these parameters are available in tables 6.2a,b. The variation of the fwhm, the fwtm and their ratio with energy are shown in figures 6.10 -- 6-13. The ratio of the fwtm to the fwhm as a function of energy should be constant and for a good Gaussian it should be ~ 1.8. 6.3 Spatial Resolution and Sensitivity The spatial resolution obtainable by an imaging system depends on the following design parameters: 1 0 9 Table 6.2 showing the parameters of equation 6-4 a) for FWHM parameter val ue error Pi 2. 50 . 0 9 P2 1.18 . 17 P3 -3. 1 1. 0 b) for FWTM parameter val ue error Pi 1. 43 . 10 P2 -2.39 . 0 4 P 3 . 30 . 0 4 1) incident beam size which is a function the source collimator's diameter and length* 2) detector collimator size and 3) linear data sampling. Phelps et al- (PHE—1977) have shown, using the sampling theorem, that if the detector response is Gaussian in shape, then for the optimum signal to noise ratio, the FWHM image resolution should be less than or equal to the FWHM inherent spatial resolution of the detector. However, the spatial resolution of the detector is determined by the type of the collimator used. In our experiments, only one type of collimator was used, which is a bore-hole type of either 1mm or 2mm diameter and 30mm length. The collimator is of the jacket type with an extension to cover the crystal length of the detector. Figure 6.14 is a typical cross-sect i on of this kind of collimator. Both detector and source collimators are presented in chapter 5. 110 imm 15mm 3 0 mm Figure 6-14 typical single bore collimator cross-section with the dimensions o-F the collimator used. A -few -fundamental experiments were carried out using this collimator- Three di-f-ferent line scans were taken at three different energies- The sources used were point sources from Amersham ,namely 133Ba , 136Cs and &0Co . The scanning program described in chapter 5 was used for high positioning accuracy. Each count, within the photo—peak was taken for 100 seconds at a 1mm intervals. The response curves for the line scans using i33Ba and 137Cs, point spead functions, shown in figures 6-15 and 6.16 were obtained and the resolution of the detecting system was calculated from the FWHM of these point spread functions. 6.3.1 Definition of the Point Spread Function — P 5 F — In order to calculate the point spread function of a 111 scanner, we should state -first what we mean by the PSF. The PSF is defined as the instantaneous count rate of the detector as a function of the position of a point source- There are two difficulties with this definition. The first is that the term "instantaneous count rate" must be understood in a statistical average sense. If the same measurement is repeated several times, or an ensemble of identical systems is considered, then the count rate of concern is ATTj- / At, where ^Tc is the mean number of photons detected in a vanishingly small time At- In practical life one does not have an ensemble of systems nor does he have the time to repeat the measurements many times- A single measurement over a very small At will probably yield only zero or 1 detected photon and give practically no information about the mean rate. Some sort of averaging is clearly needed. This problem is solved by the use of a ratemeter The second difficulty with defining the PSF as the instantaneous count rate is that it has the dimensions of counts per unit time while the point spread function should have the dimensions of counts per unit area. In a scanner the conversion from counts / time to counts / area, involves two mechanical parameters; i) the linear scan speed vs and ii) the number of scan lines per unit length nj> (in the direction perpendicular to the rapid scan). Dimensional analysis then shows that the number of counts / area is the instantaneous count rate x and have width of 1 / n^ and v^ is assumed to be constant (BAR-1981) . 1 12 6-3-2 Geometrical Analysis of the P5F o-f a Single Bore Collimator Consider the geometry shown in -figure 6-17 below * where a collimator o-f constant bore diameter Dj-, and a bore length is viewing a point source a distance z from the -face o-f the collimator- Consider the case , z = 0, where the source point is virtually in contact with the face of the collimator- Then, assuming there is no penetration of gamma rays through the walls of the collimator, the count rate must be zero when the point source lies outside the disk region of diameter directly in front of the bore- Furthermore, since will normally be much larger than , the collection solid angle will , to a good approximation, be independent of the location of the point within the disk region- Finally,note that the scanner will have a magnification of one since the count rate will be maximum when the collimator is positioned directly over the point source. To image a second point source located a distance L away , the collimator must be moved by L. With these simple observations, an expression for the PSF of a single bore collimator can now be written down (BAR—1981) PSFsb (z=0) = psb (r;z=0) = const, circ <2r / Dfcj) 6.5 The vector r* represents the two dimensional position of the point on the plane z=Q, the collimator is assumed to be centered on r* = 0- To fix the constant in equation 6.5 , consider a point source emitting K photons per second- Its emission density function is K.6(F*) and the instantaneous count rate within the disk region is K. ft / 4*r, where Q is the solid angle subtended by the detector- The PSF must satisfy 113 scon direct' on L \ L Figure 6-17 Geometry -for calculating the PSF of s b c. Since the solid angle in this case is *fD§ /4L§ , the PSF is given by 6.7 Note that psb has the dimensions of time per area* so that we can directly convolve psb with f function describing the two dimensional source density. To determine the general behaviour of psb we need to distinguish three regions in the source plane- The first , which is called the umbra region is the disk shaped region of diameter Db directly in line with the collimator bore. The PSF in the umbra region is approximately constant. The second region is termed p e n u m b r a . It is the region between the umbra and a circle of diameter Db»(1+(2Z/Lb)) as shown in figure 6-17. In this region part of the detector is obscured by the collimator, reducing the collection solid angle and hence the PSF. In the third region outside the circle of diameter Db. (l+(2z/l_b)), all of the detector is obscured and the PSF- is zero (neglecting penetration of gamma rays through the collimator). From these considerations one can sketch the PSF for any z. In figure 6.18 the image of two well resolved point sources is shown. The sloping lines in the penumbra region are drawn straight but more careful analysis shows that they have slight curvature. -L count role Figure 6.18 image of two points with a single—bore col1i mator Delector Detector Figure 6.19 showing the equivalence between a sbc and a pair of circular apertures. 1 1 5 Further analytical treatment with the aid o-f figure 6.18 shows that the PSF for any value of z is . nj> C P5b The integrand in equation 6.8 is a product of a circ functi on of diameter Dfc, with a second circ function of di ameter D^/a- Since the parameter a is less than one, the second circ function is larger than the first. The interpretation of this result is that the larger circ function represents the lower aperture as projected from the source point into the upper aperture. The PSF is proportional to the area of over lap between the two circ functions, as illustrated in figure 6.20, or the portion of the detector that can be seen from the source location. Figure 6.20 a diagram to help in interpreting eg.6.8 Examine equation 6.8 in the z— > 0, or equivalently, a— > O and b— *1 , then the first circ function is much smaller than the second and may be treated as a delta function: 116 circ (2r ' ' /D > = < D? / 4 > S 6.8 with the result P . The behaviour of p5^ (r*;z) for non zero values of z is illustrated in figure 6.21. Figure 6.21 the PSF of sbc for various values of Z. 6.3.3 Modulation Transfer Function A more general way of characterizing the imaging system is by determining the modulation transfer function MTF. An ideal imaging system is one which produces the image of an object under examination without loss of information content. 1 1 7 Methods based on the use of spatial sine waves are commonly used in the practical measurements of the MTF. Other types of methods involve Fourier transformation of the measured point spread functions and the use of coherent optical systems. Detailed descriptions and extensive' bibliographies can be found in the review by Dainty (DAI—1971) and in the theses by Ozek (OZE-1973). The sine wave method assumes a test object of known one dimensional physical quantity distribution E(x), of the form E(x) = a + b cos (2-rtfx) 6.12 where f denotes the spatial frequency and b/a the modulation. This distribution could be closely approximated by moving a point source at a constant stepping interval across the surface of a collimated detector. The wave form representation of the object is shown in figure 6.22. Through the use of this method, covering an appropriate range of frequencies, an evaluation of a system performance should be possible from a comparis; on of the characteristics of the recorded line spread function with those of the test object. The ratio of the amplitude of the response of a system at a given frequency to the maximum amplitude at any frequency is a measure of the relative response of the system. When such ratios are plotted against frequency values, there results a relationship which expresses the system response as a function of the spatial frequency in the test. Such a relationship is called a modulation transfer function, MTF, denoted in a normalized form as M(f) by the equation M(f) = Mi / MD 6.13 where 1 1 8 Mi ^maxi E-f i and ^maxi + ^mlni ^ m a x o ^fo M0 = ------6. 14 ^maxo + ^mino are the modulation in the image and the object respectively and Cmax = Maximum amplitude Cmin = Minimum amplitude and = amplitude at any -frequency In the experiment to determine the MTF of our detecting system, the response -function obtained in section 6-3 was used. The terms are illustrated in figure 6-22 a,b showing the physical object and its corresponding image i-e- the PSF of the object and the PSF of the image. Since the modulation in the object is unity the modulation transfer was obtained simply by calculating the modulation in the response of the system for the varying frequencies. The MTF as a function of spatial frequency obtained with this method is shown in figure 6.23. A ma ? % a 'max a di Figure 6.22 a ) object contr ast h ) image contrast 1 1 9 Another method involves the Fourier transformation of PSF, equation 6-8 which results in the MTF being given by 1psb z > 1 2J± lim MTFsb = I 2J ± (*tDbp) /rtbb? I 6.16 z--->0 which only depends on the bore diameter Db. 6-3.4 Sensitivity and Resolution A method normally used for measuring the sensitivity of a scanner is the planar sensitivity S. This is defined as the count rate obtained when the scanner is viewing a planar source of activity 1 (iiCiVcm2. Such a source is described by the emission function f (r*) = constant- The value of this constant is fixed if we assume that each nuclear disintegration produces exactly one emitted gamma ray with an energy suitable for detection by the scanner- Then, since 1pCi corresponds to 3.7 x 10* dis./sec.,we have f(r*)= 3-7 x 10* emitted photons/cm2 sec 6.17 Different isotopes give different values for the numerical constant in equation 6.17 since, in general, the number of usable gamma rays per disintegration is different from one. The count rate is given in terms of the PSF by count rate = (v5/n£> psb Since f Ssb = —^ (3.7x10*) p5b 3.7x10* 2r ' ' 2 I ar* ' '+br*l‘ Ssb=- d2. d2r circ ci re 6.20 4ff(Lb+z)2 » fl) The integral over r* in equation 6.20 involves only the second circ function and thus could be integrated with the results 3.7x10* *Db *sb d2r '' circ(2r''/ Db ) 6.21 4* (Lb + z)2 4|^2 CD 2 2 . 4 3.7x10* 3. 7x10 * 4tD. counts/sec b b □ S5b =' 2 „ ,2 6.22 4* (Lb + z)* A i t 64 Lb nCi/cm2 This is independent of z and the scan parameters n^ and vs. To put these expressions for sensitivity into proper perspective, it is useful to restate it in terms of the collimator resolution distance defined as the FWHM of the PSF. For a single-bore collimator, the resolution distance 6sb(z) can be estimated from the approximate sketches in figure 6.18, where the curvature of the PSF in the penumbra region is neglected. We find 6sb(z)~ Db (1 + z/ Lb> 6.23 and if we regard Db as the variable parameter and hold z and Lb fixed, we have Ssb o: 6sb 6-24 121 Thus a two-fold improvement in resolution (reduction in 6sb> must come at the expense of a sixteenfold decrease in sensitivity* and even this drastic behaviour understates the problem when noise is consdered. It has been shown that the signal to noise ratio (SNR) is determined by the number of detected photons per resolution element,Ng. For a given source* we have Nfi S 5 [j. Ssb ®- $5b0 6.25 thus to keep the SNR constant when 6S^ is reduced by a factor of 2* therefore, means either the source activity or the total scan time must be increased by a factor of 64- Figure 6-24 shows the resolution distance as a function of z, the source to collimator distance, for D^, = 1mm and 2mm and Lj-, = 30mm. 6.4 Discussion and Results A number of system characteristics of great importance in tomographic imaging has been studied in this chapter for the tomographic system used- These include the variation of sensitivity across the detector surface* the determination of crystal size, diameter and volume, determination of the p—core diameter and depth, distribution of efficiency as a function of energy, variation of resolution ,fwhm* and fwtm with energy, the spatial resolution of the detector collimator system, analysis of the point spread function for the single bore collimator, variation of the point spread function with distance, modulation transfer function of the system and the system sensitivity as a function of the spatial resolution- 122 The -first part o-f this chapter described the determination of homogeneity across the detector surface i.e. the variation of the detector response along a diameter, and the determination of the crystal size and shape- Figure 6-2 is a scan across the detector face using i33Ba 356 keV line. This scan was carried out with the source collimated to 1mm and the collimator detector distance was ~ 5mm. The reason for the source being collimated and the separation being only 5mm is that we are considered© to be looking at a point ( Figures 6.3 and 6.4 show line scans using i37Cs, 662 keV, and &0Co ,1333 keV, while it could easily be seen that the crystal dimensions have not changed, the response along the detector face has changed i.e. the variations expected are not as clear as in the lower energy case. This is because of the higher probabi1ities of higher energies being scattered and also the collimator efficiency is reduced i.e. more photons penetrate the collimator surface than at lower energies. The &0Co scan as shown in figure 6.4 shows almost no variation. Looking back at figure 6.2 we see that there is a dip in the centre of the crystal. This is the dead layer known as the p—core. The diameter of the crystal and p-core were determined from the fwhm of the detector response to be 32.5 ± 0.2mm and 4.,0±.2mm 123 respectively- To determine the length o-f the crystal and the depth o-f the p—core* a scan along the side of the detector was taken, the response was uniform as can be seen from figure 6.5 and thus the p—core extends from top to bottom i.e. the crystal is a true or open ended coaxial. The length of the crystal was measured from the fwhm of this response and found to be ~ 32mm. This result and that of the diameter were used to calculate the crystal active volume and is found to be ^ 29500mm3. Fi gure 6.7 shows the relative dimensions of the crystal and the p—core in the detector casing. The second part of this chapter dealt with the variation of intrinsic efficiency with photon energy and the variation of the fwhm and the fwtm with energy. The photo—peak count rates were obtained using a set of standard point sources. The efficiency was calculated using equation 6.2. The efficiency values and their corresponding energies were then fitted to the polynomial of equation 6.3, the parameters p a— p& and their errors were found using the minimization routine MINUIT and they are listed in table 6.1. Figures 6.8 and 6.9 show the fitted efficiency as a function of energy and also show the experimental points and how good the fit is. Another important detector characteristic is the energy resolution as measured by the fwhm of the photo—peak. The fwhm was obtained for several energies.These points were then fitted to the empirical formula in equation 6.4, the parameters of which were then determined using the same minimisation routine. The values of these parameters are listed in table 6.2a. The fwhm is shown to increase with energy due to multiple scattering increase and then reaches 1 2 4 almost a constant (asymptotic) value, that is why the empirical -formula was chosen. As well as the variation of the fwhm with energy, the variation of the fwtm with energy was also studied. The same empirical formula was applied but noticeably the fwtm increased a little faster with energy than the fwhm. The parameters for the fwtm are shown in table 6.2b. We generally assume the detector response to be Gaussian in shape and for a good Gaussian the ratio of the fwtm to the fwhm is constatnt and is ^ 1.8, however for our detector system this ratio is ^ 1.9. Figures 6.10 — 6.13 show the variation of the fwhm, fwtm and their ratio with energy and also show how good our empirical formula fitted the experimental values. The third part of this chapter dealt with the collimator detector response functions. The collimator used in most of this work was a single bore collimator of the jacket type as illustrated in figure 6.14. The bore—hole diameter was either 1mm or 2mm positioned 6mm off the axis of the collimator to get the maximum efficiency.A line scan across the collimated detector was taken using i33Ba ,356 keV, to study the point spread function of the system, the fwhm of the PSF was measured to be 'u3mm at a source—col 1 imator distance '''70mm- The graph of this point spread function is presented in figure 6.15. A similar line scan was taken using i37Cs, the fwhm of the point spread function for this scan,figure 6.16, was again ^Smm but the fwtm and the ratio of the maximum to the minimum were much worse than that of the 133Ba because in the 133Ba scan the collimator efficiency was much higher due to low penetration probability. Because 137Cs was the main source 1 2 5 used in the later work, the 137Cs line scan was later used to determine the modulation transfer -function of the collimator detector system- The geometrical and analytical analysis of the PSF for a single bore collimator was used to obtain an analytical expression for the PSF at any value of z, equation 6 .8 . The modulation transfer function was studied using two methods; one of them using a sine wave method which assumes a test object of known one—dimensional quantity distribution, equation 6.12, and this is approximated to a point source positioned at a constant Z and moving at constant stepping intervals across the surface of a collimated detector. Using this method the MTF of the detection system was calculated and shown in figure 6.23. An analytical expression for the MTF of the system was derived by the Fourier transformation of the PSF of equation 6.8, this expression which gives the MTF, equation 6.15, for a single bore collimator as a function of Z, L j-j and D^. Finally an analytical expression for the sensitivity in terms of the PSF, collimator dimensions, was derived - equation 6.22—. This expression shows basically the dependance of sensitivity on the collimator length and bore diameter. The resolution as a function of Z, and was also estimated. The variation of resolution with Z for Dj-, =lmm and 2mm and L|3= 3 0 m m is shown in figure 6.24 and this dependance is linear. COUNTS «10’ -40-00 0.00 ACROSS -4.00 DET. IN o.oo MM 4.00 24.00 28 loo .o .N o o o o CM o o n o o o o° • » 1-1 / O 5500 . cn CD CD & cn o o o o o o o CO cn o CO o o o 7 1 2 7 5 cn detector response ro N> o o o V A- — cn o o o o — cn o o o o o UMEN025 UMEN025 12.07.*18. 87/06/10. AFNM I 48356 Figure 6.3 Variation of sensitivity using Cs 13/ PLOT COMPLETE. PENS USED UMEN025 09.19.19. 87/06/08. ACYJ I 48020 iue . Vrain f estvt uig Co—60 using Sensitivity of Variation Figure 6.4 4 1 2 9 i UMEN025 16.58.54. 87/07/09. ACKD 51695 cTl rtc ID H 5000. o U1 O ’ o OI< o” I rt-3" rt-di cn ‘ id 0 - o o~ ida 3 rt- 0 1 nID ■*» ^ ' rt- CD o - 0 ID rt 3 o~ n *•in i •< r+M- u>' in < o - rt- Hi rt o~ f' ■< c l c lb to* i in •- ro ' «• o <~+ o - 3 3 Q 10 (0 o ~ 5° n - n rt- CD in 3- CD 1 1 id Co Q o - w in CD Nj M. r> 'D o “ a Q O - ID Co - Co - Coc o - <~+ • _ S- o «QCD . CO 1 . - Con o - Q. 6o CD o ~ "N O - ro ' o - • _ ^r o cd . Q. U> ■ CD o - «-+ • - CD o m =32m fwhm o <-+ o -b. ‘ "D a - o ~ . cn ' o - o o oCD -' o~ PLOT COMPLETE. PENS USED - M UMEN025 15.09.44. 87/06/22. ABJY I 49730 iue . Te aito o dtco efcec as efficiency detector of variation The 6.8 Figure fnto o energy of function a energy energy rtei/o PLOT COMPLETE PENS USED - 49734 AAI0 efficiency 17.16.04. 17.16.04. 87/06/22. o o o UMEN025 Figure 6.9 The variation of detector efficiency as PLOLCOMPLETE. _PENS USED - a function of energy 1/ 1 3 2 F W H M ke o — r\3 cu cn cn Energy Me v. UMEN025 UMEN025 12 43.26. 87/11/06. APAI I 65016 Vor!athr. if r W H M Figure 6.10 Variation of FWHM with Energy j 1 3 3 r. UMEN025 12.47.37. 87/11/06. APAI 65017 f-WHM keV o ± ,__J UMEN025 15.29.35. 87/11/06. APT I 3 65031 69510 .00 2 0 1 3 5 UMEN025 UMEN025 18.28.55. 88/01/05. ADDH Figure 6.13 variation of the ratio of FWTM to FWHM with energy PLOT COMPLETE. PENS USED ---- POINT SPREAD.FUNCTION USING BA133 Figure 6-15 Point Spread Function using Ba-133 iue61 Pit ped ucin sn Cs-137 using Function Spread Point 6.16 Figure letector response ie cn sn cs!37 using scon line CDro PLOT COMPLETE. PENS USED PLOT COMPLETE PENS USED 1 4 0 CHAPTER 7 EXPERIMENTS IN TOMOGRAPHY 7-1 Introduction The detector collimator system characterized in chapter 6 was incorporated into the scanning rig described in chapter 5. The tomography system was then tested by scanning different phantoms and samples in both transmission and emission modes. A selection of images obtained are discussed qualitatively and in some cases the reconstructed data are quantitatively displayed by means of line scans and analysed usi ng the concept of contrast and the measured full width at half max i mum ( FWHM ) of the reconstructed samples in the 1 ine scans. Since one of the main objectives of this work is to develop a gamma ray scanner for industrial applications! most of the samples tested (imaged) in transmission tomography were metal samples. 7.2 Scanning Geometry 7.2.1 Transmission Tomography The geometry for transmission tomography is illustrated in figure 7.1, the detector and source collimators are described in chapter 5. It is important to note that from figure 7.1 the volume of acceptance for the detector is much larger than the narrow beam defining the raysum thus allowing a large number of scattered photons to reach the detector not only from other regions of the object being scanned but also from background sources (see the path ABC— figure 7.1). The volume of acceptance is the volume bounded by the two cones defined by the two collimator apertures and the separation distance between the two collimators. The angle of acceptance 0 depends only on the detector collimator aperture and length and for our collimator which has an aperture (Lb) of 30mm it is 0 = 2 tan-1 t (Db/2> / 7.2.2 Emission Tomography The geometry for emission tomography is very simi 1 ar to that of transmissi on tomography except that the source and source collimator are removed, the source is, in fact, replaced with the radioactive sample or embedded inside the object at which the scanner is looking. Figure 7.2 shows the scanning geometry used for emission tomography and again as in the transmission case the amount of photons accepted i.e., the volume of acceptance is larger than the narrow beam defining the raysum which in turn defines the pixel area of the image. As can be seen from figure 7.2 the number of photons recorded by the detector for a given raysum has contributions not only from the volume of acceptance but also from scattered radiation originating outside the volume of acceptance. In both emission and transmission tomography the volume of acceptance could, in fact, be decreased by reducing the collimator aperture, increasing the collimator length and/or reducing the sample collimator separation. This will lead to improving the spatial resolution of the scanning system and thus obtaining a well defined image of high contrast. 1 4 2 DETECTOR COLLIMATOR SOURCE COLLIMATOR figure 7.1 geometry for transmission tomography detector collimator defining the raysum j volume of acceptance figure 7.2 geometry for emission tomography 143 7.3 Tomography Experiments 7.3.1 Scanning Summaries The tomography experiments carried out were in two parts; one part was to study and test the system performance and capability of producing images of an acceptable contrast and spatial resolution. These experiments include in both transmission and emission tomography modes: 1— Scanning of a cross-section of a hollow alumimium cylinder. a) Coarse scan using only 13 projections where the projection is a complete profile of the object at a given angle (see figure 4.2) , and b) Fine s c a n using a reconstruction matrix of 49*49 pixels out of 31 projections, i.e. each projection is divided into 49 linear transverse steps or raysums. 2— Scanning of a solid steel cylinder (rod), 3— Scanning of an aluminium phantom containing lead and brass rods, 4— Emission tomogram of a point source, and 5— Emission tomogram of phantom containing three liquid caesium samples. The second part of the tomography experiments was done totally in emission mode and carried out using an aluminium phantom containing six vials of f (152EuN03)2■6H20 > europium nitrate solution of different diameters. This experiment was carried out to study the effect of energy on reconstruction, where the phantom was scanned and reconstructed at four different energies in an attempt to use the multi—energy scanning to compensate for photon attenuation. This part is the topic o* the next chapter. 144 7.3.2 Scanning Details i) In the first experiment, the sample studied was a hollow cylinder. This cylinder has an external diameter of 88mm and a wall thickness of *''19mm. This s c an was carried out twice. The first was a coarse scan and has the following details: 1— detector collimator diameter = 1mm 2— detector collimator length = 30mm 3— source collimator diameter = 1mm 4— source collimator length = 35mm 5— number of projections = 13 6— maximum scan distance = 104mm 7— number of steps = 13 8— step length = 8mm and 9— counting time per raysum = 200 seconds The detector and source collimator dimensions are the same in all transmission tomography experiments and the source used in all these was a i37Cs source with an approximate activity of llOMBq, a diameter of 5mm and a length of 7mm. The maximum scan distance was chosen to be greater than the actual cylinder diameter to e n s u r e that the entire cross—section of the cylinder is scanned. The image obtained is displayed in figure 7.3 and it shows the distribution of attenuation within the cross sction. This experiment was repeated with the number of projections being increased to 31 and the number of raysums per projection to 49 i.e. the number of transverse linear steps of the sample between the detector and source assembly for one angular increment is 49 and the number of 145 angular increments is 31. The step length was reduced to 2mm and the counting time was also reduced to 100 seconds per raysum. This reduction in counting time is done in order to limit the experimental time and is considered enough to give a reasonable statistical error of in the raysum counts. The image was then reconstructed on a square matrix o-f 49*49 pixels. This particular scan was done to see the -finer details of the density of the cylinder and to see if it is possible to pick up any small density variations due to cracks in the cylinder. The resultant image shows that in order to obtain the finer structure of the cylinder and to detect any possiple cracks lower energies and smaller collimator diameter must be used, i.e. the spatial resolution mu st be improved. The image obtained is displayed in figure 7.4 and a line scan along the image diameter is displayed in figure 7.5. ii) The second experiment reported in this thesis dealt with the scanning of a solid steel cylinder. The scanning parameters for this experiment were: 1— number of projections = 31 2— number of steps = 41 3— step length = 1mm 4— maximum scan distance = 41mm 5— the steel diameter = 15mm and 6— counting time per raysum = 100 seconds. The image of the attenuation coefficients in the cylinder is displayed in figures 7.6 and in 7.7 where the maximum and minimum pixel values are clipped to constant values in order to reduce image noise and increase the contrast- 146 iii) The scanning of an aluminium phantom containing lead and brass rods was carried out to obtain an image of different density distribution and to see the system response to different elements within the matrix i.e. the capabilities of the system to recognise small diameter objects in the matrix. The scanning parameters ares 1— number of projections = 31 2— number of steps = 31 3— step length = 2mm 4— maximum scan distance = 62mm 5— phantom diameter = 50mm 6— lead rod diameter = 9mm 7— brass rod diameters are 7 and 3mm 8— counting time per raysum = 200 sec. The images obtained are displayed in figures 7.8 and 7.9. In figure 7.8 the background pixels i.e. the pixels outside of the actual aluminium phantom with reconstructed values similar to that of the air are all clipped to a constant minimum value while in figure 7.9 the reconstructed values are all smoothed out to give a continuous image free of the statistical noise but on the expense of lower image contrast. The images show the z existence of the lead rod and the existence of the larger diameter brass rod but not the smaller one because the spatial resolution in this experiment was approximately 3mm which is similar to the diameter of the second brass rod. 147 iv) After carrying out several experiments in the transmission mode, the system was then tested -for tomography in emission mode. The -first experiment to be carried out in emission mode was the simple experiment of producing an image of a point source. This was also done to check the system resolution in emission mode. In this experiment, a i33Ba point source of nominal activity of 37kBg was scanned with the following parameters: 1- number of projections 2- source diameter = 1mm 3- number of steps = 17 4— step length = 1mm 5— total scan distance = 17mm 6— counting time per raysum = 200 sec., and 7— the energy used for scanning = 356 keV The image of the point source obtained is shown in figure 7-10 and a line scan across the image is shown in figure 7.11. This line scan shows that the spatial resolution of the image is approximately 3mm. v) After the point source experiment, the system was tested by the use of a polythene phantom containing three liquid caesium vials of different diameters. The schematic representation of the liquid 137Cs assembly is shown in figure 7.12. The activity concentration of 37kBq/ml is the same in all vials but the volume of the activity in the cross—section the detector is looking at is different since this is defined by the volume of acceptance (figure 7.2) and the vial diameter. The scanning parametrs are: 148 1— number o-f projections = 61 2— number o-f steps = 63 3— step length = 2mm 4— maximum scan distance = 126mm 5- vials diameters 12, 8, 5mm 6— total activity in pCi in each vial is 30, 10, 8 7— the volume of activity in ml in each vial is 3, 1 and . 8 8— the polythene container diameter is 100 mm and 9— counting time per raysum = 100 seconds. The images of the liquid caesium assembly obtained are shown in figures 7.13 and 7.14. 7.4 Reconstructed Images and Analysis. 7.4.1 Contrast in image analysis Image analysis is fundamental to the field of imaging since the purpose of obtaining an image, in any application, is to accurately identify features in the original object by means of reconstruct! ng the response of a probe of that object. The information content of an image is high and hence the process of evaluating and comparing images is complex. Subjective visual inspection of images can give reasonable qualitative results due to the excellent pattern recognition , ability of the human brain. However, the brain is incapable of 1 detecting small differences quantitatively and hence this important aspect of the information content of an image, which forms the basis for objective analysis, is lost in the human visual evaluation process. 1 4 9 The principles of describing an information transfer system in terms of its response to sinusoidal input function was first applied in electronic engineering to analyse communication systems and later in the fields of optics and photography to analyse imaging systems in terms of their spatial frequency response. The introduction of the transfer function analysis techniques in the field of radiography in order to quantitatively evaluate images and imaging systems began in the early 1960's (ROS—1964). Since then both theoretical and practical aspects of its application in medical imaging have been extensively developed (ROS—1968, ROS—1969, MET—1979). The ultimate aim of the transfer function analysis is to characterise systems in order that the output resulting from a known input can be predicted and also to enable inter—comparisons of systems. Firstly the generalised system is defined which incorporates every component involved in the production of an image from the original object, for example, in gamma ray emission tomography, this includes the effects on the primary photons of scattering and attenuation within the object, the actual imaging system and the reconstruction process. Thequantitative analysis of the input and output functions of the system forms the basis for transfer function analysis techniques. In imaging systems various types of input functions are employed, for example, in emission imaging the response of the system to point and line sources, i.e. the point and the line spread functions, are frequently determined and utilised in the characterisation of systems, (section 1 5 0 6-3). However, a more general approach is the quantitative analysis of simple objects and their corresponding images produced by systems using the concept of modulation (section 6.3) and contrast. The definition of the modulation can be applied to any situation where the distribution pattern is repetitive su ch as sinusoidal distributions. In the generalised form the modulation is defined as the amplitude of a distribution divided by the average value. Therefore, as can be seen from figure 6.22, the modulation m of a cyclic distribution is given by 6.14 C - C . max min m = ------r + r max min The concept of contrast is a generalised form of modulation, where—asmodulation applies to cyclic distribution, contrast can apply to any type of distribution and the contrast of one region in a distribution with respect to another can be given by C1 - C2 C = ------7.1 C1 + C2 where C ± and C2 are the intensities of the two regions of interest. The ratio of the contrast of the image to that of the object can be used to evaluate and compare imaging processes, i.e. C.l contrast ratio 7.2 C“o 151 where Cj and C0 are the contrasts of the images and the object respectively. 7.4.2 Analysis and Results of Images Obtained in Tomography Experiments The reconstructed images for the tomography scans descibed in section 7.3 are divided into two groups: (i) those in transmission mode and (ii) those in emission mode. Since most samples scanned in transmission tomography were metal objects and phantoms containing metal objects, the contrast of the metal samples and objects inside phantoms were calculated from equation 7.1 using the linear attenuation coefficients, i.e. the contrast of an object is given by - + (p ) A obj Z surr Since some of the objects were scanned in air, i.e- the surrounding medium is air, the object contrasts are considered equal to 1 , but for the objects which were scanned in an aluminium phantom the contrast is (p > - ( p > Z obj Z al C = ------7.3 Quantitative measurements can be obtained from the contrast and contrast ratio concepts. For example? since the contrast is a function of attenuation, if the contrasts of several elements are measured in a similar matrix and these values are plotted as a function of (p^) and some relation is found between the contrast and the linear attenuation coefficients, then this relation could be used to determine unknown materials in a matrix by simply measuring the contrast of the region of interest (ROI) of the image that contains the unknown material. An attempt to produce such a relation from a few points is shown in figure 7.15. For experiment 2 the object contrast is 1 but the image contrast for the steel rod is .82 ± .08. In this experiment the contrast was measured from the average maximum and the average minimum pixel values and as in the previous experiment CR is taken equal to CD ^ 0.82. In the experiment of the aluminium phantom containing lead and brass rods, (section 7.3.2, iii), the object contrasts for the ROI containig the lead rod was calculated from equation 7.3 and found to be 0.71 and for the ROI containing the brass rod CQ ^ 0.5. But the image contrast was estimated using equation 7.3 for both ROI and found to be 0.5 and 0.3 respectively and the CR for ROI 1 and 2 and found to be 0.7 and 0.6 respectively. The two image cont^ixsts for the brass and lead rods were also determined with respect to air and found to be 0 .8 8 and 0.93 respectively. Table 7.1 below summarises the contrast ratio results of the transmission tomography experiments and these are the same values used to obtain the relation of figure 7.15. 15 contrast element ratio mm * A1 .0 2 1 0.73 Fe -057 0.82 Brass . 064 0 .8 8 Lead . 125 0. 93 Table 7-1 The contrast ratios of four elements scanned in transmission tomography The concept of the FWHM could be used to obtain some quantitative information about the object dimensions from the line scans across the images, for example, in figure 7.5 the inner diameter and wall thickness of the cylinder were estimated from the FWHM of the line scan to be 47 and 18mm respectively and this compares very well with the actual dimensions of 50 and 19mm. Thus from the point of view of transmission tomography the scanning system was fully tested for metal (heavy) samples using primarily the energy of 662 keV of the i37Cs transmission source- In addition to qualitative information obtained by visual inspection of the image, quantitative information could also be obtained from the image through the use of FWHM of the line scan and the contrast ratio concepts as illustrated above. The choice of the transmission source depends primarily on the samples being scanned. Thus if a hydrogeneous material or low Z elements are being scanned as in the case of medical imaging the source used should be one of low energy so the attenuation inside the sample is significant and could be 154 measured. But for industerial samples of high Z elements and or large dimensions higher energy sources should be used to allow the radiation to penetrate the sample and reach the detector even though ideally low energies are pre-ferable because of high attenuation values and thus small variations in densities could be detected. Hence an adequate energy should be used to allow the photons to penetrate the samples and at the same time lose a considerable part of their intensities. In the emission imaging mode two basic experiments were carried out to test the scanning system. The details of these scans are mentioned in section 7.3.2. Considering the image of the point source in figures 7.10 and 7.11 the spatial resolution of the scanning system was determined from the FWHM of the line scan across the image to be "“3mm. Since the matrix containing the sample (point source) is an air matrix and there is no activity in the matrix other than that of the sample the object contrast was considered unity. Ideally if the matrix is an active one as in the case of a radioactive waste package then the object contrast which is now considered as the contrast of a ROI inside the matrix should be measured relative to a reference point or ROI in the same matrix. The reference ROI could be an additional object introduced into the matrix with its contrast being measured relative to air or a ROI already existing in the matrix with some properties such as high intensity, well defined dimensions etc. If this is considered then the contrast is calculated in the same way as before using equation 7 . 1 with the intensity being replaced by the activity per unit length 155 or the activity per step length. In our emission experiments the intensities are considered as Bq/mm. Thus for the point source the image contrast was measured to be 0.78 ± .08 and hence the CR is set equal to 0.78 + .08. The second experiment carried out in emission mode was the scanning of the polythene container with three liquid caesium vials. The scanning parameters are discribed in the previous section as well as the amount of activity in each vial. From the volume of activity and the vial dimensions the activities per unit length i . e. kBq/mm were calculated to be approx i matel y 42, 18.5, and 7.2 for the 1 2 , 8 , and 5mm diameter vials respectively. The contrast for each vial was then calculated from equation 7.1 relative to the smallest (5mm diameter) vial which is considered to have an object and image contrasts of 0 relative to itself (Mac—1988) . Thus the object contrasts for the 12 and 8mm vials are 0.71 and 0.44 respectively. The image contrasts then calculated from the average maximum pixel values for three ROIs containing the three vials and found to be 0.32 and 0.16 for the vials of 12 and 8 diameters respectively. The contrast ratio CR was then calculated and found to be 0.45 and 0.37 respectively. Table 7.2 below summarises the contrast results of this experiment. vi al aver, pixel object i mage contrast di am. value contrast contrast ratio 12 mm 145 ± 5 0. 71 0.32 0.45 8 mm 104 + 3 0. 44 0.16 0.37 5mm* 75 ± 2 0 . 0 * 0 . 0 * undefined Table 7.2 The contrasts of the three vial liquid Cs assembly <* relative to itself) 156 The concept of the FWHIi of the line scan across the image could be used in the emission mode as it is in the transmission mode. Figure 7.16 shows three line scans along the diameters of the three caesium vials. The full width at half maximum of the line scans were measured and found to be approximately 16, 14 and 10mm for the 12, 8 and 5mm diameter vials respectively. The reason for the discrepancy between the actual and the estimated diameters is mainly because of smoothing the data before plotting it and partly because of lower resolution for lower object diameters. 7.5 Discussion and Conclusion It can be seen from the above analysis of images obtained in both transmission and emission tomography techniques that the scanning system developed here for industrial use is capable of producing images of good contrast and spatial resolution for samples of different dimensions and phantoms containing different elements. The use of the concepts of image contrast and full width at half maximum of line scans through images are illustrated and quantitative information is obtained from the images through the use of these concepts. One of the main problems in both modes of reconstruct i ve tomography is the quality of photons recorded in the photopeak. Considering the geometries for transmission and emission tomgraphies (figures 7.1 and 7.2) it can be seen that the volume of acceptance is much larger than the narrow beam defining the raysum. Scattered photons originating from within and outside the volume of interaction reach the detector. These scattered photons result in image degradation and there 157 are di-f-ferent techniques -for scatter subtraction- In our experiments all data recorded in the photo—peaks and used -for image reconstruction are corrected -for scattering by means o-f scatter subtraction. This is done by collecting the counts in a region of interest containing the photopeak of the energy of interest- This region of interest is then evaluated for peak area and scatter is subtracted by means of evaluating the areas of two regions of interest on both sides of the photo—peak containing similar number of channels to those of the peak- This scatter area is then removed from the total events recorded in the peak region and thus obtaing a scatter free area- (section 3-4-1). This scatter corrected peak area forms the basis for data used in reconstruction. The images obtained from the transmission measurements represent the distribution of attenuation coefficients inside a cross-section of the object being scanned. The distribution of attenuation coefficients in industrial samples can vary considerably since the materials found in such a matrix could range from low Z (atomic number) or hydrogeneous materials to heavy elements and or high Z numbers. The transmission source is chosen in such a way to get the most information out of the image. Low energies are selected to get small variations in attenuation coefficients of low Z low density materials while high energies are used for the high density material to allow radiation to penetrate the sample and be detected. Our choice of the i37Cs transmission source is due to the intermediate energy it possess* the high gamma—ray emission probability and the availability of the source in a high activity. 15B In addition to the scattering problem in both tomography modes emission tomography suffers from another major problem* that is of attenuation. The photons are attenuated along the lines defined by the distance from the points of emission to the points of detection. This problem will be discussed in the next chapter. The results obtained from the emission tomography experiments in this chapter were not corrected for attenuation for the reasons of the energy used being high and the attenuating medium is very light i.e. attenuation is negli gi ble. The transmission and emission tomography results obtained and analysed in the previous section show that the scanning system can be used effectively for industrial application. Figure 7-3 Smoothed image o-f the hollow aluminum cylinder on a matrix o-f 49x49 pixels iue75 Variationof attenuation Figure inthe7-5 cylinder pixel value * 1 0 0 0 162 IUE 7-7 FIGURE H MNMM S LPE O HGE VLE HN N A IN THAN VALUE HIGHER R TO CLIPPED IS MINIMUM THE I I ro FIGURE 7.8 TRANSMISSION TOMOGRAM OF THE AL. PHAN. CONTAINING LEAD AND BRASS RODS 164 EMISSION TOMOGRRM OF R POINT SOURCE ERCH PIXEL IS 1 MM*1 MM PLOT COMPLETE. COMPLETE. PLOT USED PENS .0 .0 line scon across the Image of a point source with a fwhm of approx. 3mm .0 .0 • 0 • 0 .0 • 0 .0 .0 .0 -8 0 -7.0 -6.0 -5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0 6-0 7.0 8.0 distance across the Image mm Figure 7.11 Line scan across the image o-f the point source 167 Figure 7.12. a schematic diagram of the three liquid Cs assembly ( 63 , 1 ) FIGURE7*/3 RN IMAGE OF THE THREE LIQUID CS137 VOLUME SOURCES WITH THE MAX AND MIN CLIPPED TO CONSTANT VALUES VIEWING ANGLE IS (75 75) FIBURE7WTHE three CS137 VOLUME SOURCES WITH THE HIDDEN L IN E S E L I MI NR TI ED PI OT POMPI FTF OT PI POMPI PTMF IIFFH i l.OO-i o 0.90- 0.80- cr? uod o iu j 0.70 - 0.60- 0.50 -|---- r 1------1------1------1------1------1------r------0.000 0.050 0.100 CKI50 linear at ten. coeff. fiaure ( 7. is ) variation of contrast ratio as a function of linear attenuationcoeff. flour pixel 16 he ln s across te mae te he liquid three the f o age Im the s s o r c a s n a sc line Three rials / / ■ I ■ 172 CHAPTER 8 Multi-Energy Scanning and the Problem o-f Attenuation in 5PECT 8 .1 Introduction One o-f the main problems in single photon emission computed tomography (SPECT) is the problem o-f attenuation- In transmission computed tomography (TCT) the quantity of interest is the attenuation while in SPECT the quantity of interest is the distribution of activity inside the sample. Since the detection of the distributed photons inside the sample is governed by the attenuation inside the material the projection data used for reconstruction in SPECT are functions of both integrals of activity and attenuation distributions and hence they are proportional to the entire detector signal. Thus in order for SPECT to be used quantitatively photon attenuation must be considered. This chapter is dealing mainly with the attenuation problem in SPECT and the application of multi—energy scanning as an experimental technique for attenuation correction. 8.2 The Effects of Attenuation on Reconstructions For gamma ray transmission imaging the transmitted intensity I(r, rewritten from equation 4.2 is r r T 1 ICr, F (x y) 6 (r+x si n F t x 7 y) on the ray defined by the equation r+ x s in < p — ycos«p =0 (figure 8.1). The projection p(r, p(r,f) = —Zn fl(r,9 ) / I0 (r»?)} JJ p(x,y) 8 (r+xsin9-yco5 Thus we see that photon attenuation is the parameter measured in TCT. In contrast, ECT seeks the distribution of source intensities wherein some a priori information of photon attenuation is neccessary. A simple relation such as 8.2, the basis for TCT, cannot be used to represent the line integrals of source intensities from within an object or a region of interest. For accurate reconstruction of emission data attenuation coefficients must be determined by TCT or estimated by the assumption of a constant coefficient. This a priori information about the attenuation coefficients can be used to modify the projection data or can be used as weighting in the iterative reconstruction techniques. 8.2.1 Attenuation Problem for Positron ECT The strategies for attenuation compensation are relatively simple for positron ECT compared to single photon ECT. For positron annihilation coincidence imaging the projection Py y *1"’^* is defined by the integral equation PYY(r,q>) = exp JJ ? y) S (r+xsin?-ycos?)dxdy J . ? (x , y) fi (r+xsin Therefore, each datum is the line integral of the positron concentration distribution multiplied by an exponential attenuation factor determined from the line integral of 174 Fi gure 8.1 Parallel beam geometry -for data collected at projection angle detector Figure 8 The projection data PY attenuation coef f i ci ents over the total ray path. These data are easily modified for attenuation giving the corrected projection data (BUD-1979). p(r,q>) = exp £ JJ n (x i y) 6 (r+xsi n = exP £ JJ 9 ,(x y) 6 (r+xsin 8.2.2 Attenuation Problem for Single Photon ECT The attenuation compensation needed for single photon emisssion tomography is not a simple multiplicative correction of observed data as is the case for positron ECT. The projection PY(r, PY (r » V) ?(x,y) a(x,y,r, detector a(x,y,r, A single photon projection is the summation of isotope concentration at points (x,y) modified by an exponential e~z where z is the line integral of attenuation coefficients from the point (x,y) to the detector. Thus the single photon emission reconstruction problem is more difficult than either transmission or positron emission tomography. The influence of the term a(x,y,r, The effect of attenuation can be seen from figure 8.3 (taken from T.F. tfuddinger et-ai* (BUD-1979) where a disk of 1/6 Figure 8.3 The e-f-fect o-f attenuation on a transverse section reconstructed without attenuation compared -for various attenuation coe-f-ficients 1979) 177 230mm diameter and a constant attenuation is used to illustrate the effect of transverse section reconstruct i on without compensation for attenuation- Here reconstructed results for Y~rays of few MeV (p-0.05 cm-i) are compared to those of Y-rays of 511 keV Y—ray emitters compensation for attenuation must be employed. 6.2.3 Example of Single Photon Attenuation Effect As an example, let us calculate the projection Py(r, using parallel beam geometry for a disk containing a homogeneous emitter concentration 9 and a constant attenuation coefficients p. The functions p(x,y) and p(x,y) of equation 8.5 are defined by 2 2 o C , x * + y r ^ P < >: » y) = 8 . 6 { n otherwi se and P , + y- * rfc ji (x , y) = 8 . 7 { 0 , otherwise substituting these experessions in equation 8.5 we see that the problem is reduced to determining the factor e— ^ where £ is the distance between the element (x,y) and the boundary of the disk (figure 8.4). Thus for «p=0 the weight function a (x , y , r , just —x+/(rjj— r2). 178 P Figure 8.4 The case o-f circular disk with constant attenuation and constant emitter concentration, the -factor a(x,y,r,0) reduces to e~H£ where Z is the line segment between the point Figure 8-5 Projection values for a disk with attenuation 15 cm- 1 and a constant concentration 1 7 9 Thus Tor a disk because o-f circular symmetry equation 8-5 becomes +/ P y ( r ) = — £ 1— exp <—2p /(r§ - r2) j 8.9 This is the case -for constant attenuation coefficients p. To compare this to the case where there is no signi-ficant attenuation Py(r) = 2C /(r§ - r2) 8.10 These two functions 8.9 and 8.10 are shown for p=0.015mm— i, and r0=2 OOmm in figure 8.5. 8 .3 Attenuation Compensation The problem posed by the attenuation of distributed sources of photons in human organs, waste packages and other industrial materials is presented in the previous section. There are many methods used for attenuation correction most of which are analytical and there are some empirical techniques based on phantom measurements and theoretical modeling (BUD—1979). The attenuation compensation methods include: 1— Implementation of Convolution and Fourier space methods for constant attenuation (TRE-1978). 2— Least—squares methods for constant and variable attenuation. The attenuation effect in single photon imaging can be compensated for by using attenuation coefficients, which can be determined from a transmission experiment, or may be assumed constant over a particular region. Using least squares methods, the attenuation correction schemes assume 180 that the projection data -for the transverse section are the summation o-f pixel concentrations attenuated by a factor that is a function of the attenuation between the pixel and the edge of the object (BUD—1979). 3— Modification of projection data prior to conventional convolution or iterative methods (TAN—1983) for constant and spatially varying attenuations (ZEE—1985). There are various methods which compensate for attenuation by correcting the projection data before reconstructing. The reconstruction algorithm then assumes that the data represent the true unattenuated data and thus does not require spatial weighting to compensate for internal absorption. In most cases the corrected data are only approximations to the true projection data. These modification include: a— Geometric means b- Hyperbolic sine correction (BUD-1977a,b, 1974? SOR—1974) c— Average minimum and maximum exponential factors (KAY—1975) d— Average of exponential factors. 4— Iterative convolution for variable and constant attenuation (WAL—1976) 8.4 The Multi—Energy Scanning Experiment 8.4.1 Source and Phantom Preparation The source used for this experiment is a multi—energy i52ELl source. This source was prepared by irradiating ~ lg of europium nitrate crystals at one of the core tubes of the irradiation facilities of the 100KW CONSORT II Reactor with a thermal neutron flux of approximately 1012 n/cm2 /s. Since the europium nitrate contains about 51% i53Eu and 47% 15 1 Eu, there 1B1 are basically three neutron interactions: 1- 15 1 Eu(n,Y)152Eu , wi th -5900b and = 13.4 years 2 — i51Eu(n,Y)1 52Eum , wi th °"th ~3 --0 0 b and = 9.3 hours, and 15 3 Eu(n,Y)154Eu , wi th ath“ 312b and = 8.5 years. where <^th thermal cross—sect i on and is given in barns where b= 10—28 m2. Both hal-f —lives and thermal cross-sections are taken from (MUG—1964). As can be seen from the three above interactions that when this sample is irradiated most of the immediate activity comes from the second interaction because of the short half-life of the i52Eum compared to that of i52Eu. Since the cross-sections for both interactions are similar the ratio of their activities is proportional to the ratio of the half-lives. The irradiated sample was then allowed to decay for about one week until almost all the activity of the metastable isotope is decayed away. The sample was then dissolved in approximately 15ml of a weak nitric acid giving a concentration of -9.62MBq/ml. An aluminium phantom containing six polythene vials of different diameters was then used. These vials were then positioned in prepared places in the aluminium phantom. The amount of activity in each vial is different. Figure 8 . 6 shows a schematic representation of the phantom containing the vials and also shows the actual dimensions of the vials and the amount of activity in each vial. 8.4.2 Scanning Details With the europium phantom positioned on the gamma ray scanner described in chapter 5? a full scan was carried out. Because of the large amount of data to be collected the scanning procedure was done i n steps. Each step consists of 1B2 \ polythene int. volume tubes diam of solution 15mm 5.0ml B 12mm 3.4ml 8mm 2*1.0. ml D 5mm 2*0.65ml total solution ~11.7ml containing total activity of ~ 1/5 % Activity concentration Figure 8. £ the aluminium phantom containing six vials with Eu.(NOJ*6H 0 3 X 2 152 (Europium nitrate solution) with Eu 183 six to seven projections with each projection having 45 raysums,i.e. 270 or 315 spectra were collected for each step then each spectrum is analyzed and the peak areas for four different regions of interest were evaluated and scatter subtracted (section 3.4.1). These ROIs contain the four peaks of the four energies to be used for reconstruct!on. Figure 8.7 shows the multi—energy spectrum of a standard 152Eu point source taken using the same detection apparatus. The scanning parameters for this experiments are: 1- number of projections = 37 2- number of raysums = 45 3- step length = 3mm 4- maximum scan distance = 135mm 5- detector collimator length = 30mm 6- detector collimator bore diameter = 2mm 7- phantom diameter = 100mm 8- vial diameters in mm =15, 12, 2*8, 2*5 9- volume of activity in each vial in ml = 5, 3.4, 2*1., 2*. 65 10- energies used for scanning in keV = 121.78, 344.3, 778.9, 964. and 11- counting time per raysum = 200seconds. 8.4.3 Reconstructed Images and Analysis After col1ecti ng the data, the images were then reconstructed. Four images were obtained for the four different energi es. These images are displayed in figures 8.8-12, and looking at them qualitatively at different nri entati ons one can see c1 ear1y the existance of the six vi al s in all images hut the lower energy ones show better vial 'a(m )h Gmarv vcrm f u5 a maue i a eL) detector GefLi) a in measured Eu153 of as svectrum F'La.(8mf)The Gamma-rav Counts 6 hne no. Channel 185 definition, better contrast and less image noise simply because of better photon statistics, better detection efficiency and also the collimation efficiency for scattering is better. The larger diameter vials have better contrasts than the small diameter ones for the same energies because of the large amount of activity per mm and thus larger detector response. For each energy three line scans were taken across each image, they are: a— line scan along projection 13 shows the distribution of activity inside the aluminium phantom due to one of the 5mm diameter vials. b— line scan along projection 22 shows the activity distribution due to both 15 and 12mm diameter vials. c— line scan along projection 28 shows the distributions of activities due to one of the two Smm diameter vials. These line scans are shown in figures 3.13—16 for the energies of 121.78, 344.3 and 778.9 keV respectively. From the images obtained, with the help of the above mentioned line scans at different energies, the image contrast and the contrast ratios were calculated using equation 7.1 (section 7.4). The object and the image contrasts were calculated relative to the vial of largest diameter i.e. 15mm since it contains the highest activity per unit length which in turn has its contrast taken as 1 relative to air under the assumption that the activity in the air is negligible. Table 8.1 below summarises the object and image contrasts for the four different diameter vials at the four different energies used. 106 From these line scans the -full width at halt maximum (FWHM) ot the tour vials were estimated and found to be for the 121-78 keV image (figure 8.13) 16, 11, 9-6 and 6-6 mm for the 15, 12, 8 and 5mm diameter vials repectivel y. Thus it can be seen that these values of the FWHM are in excellent agreement with the actual diameters of the vials. The FWHM was also measured at different energies to see if there is any variation in the spatial resolution of the system as a function of energy- But as can be seen from figure 8-17, which shows the same line scan through the vials of 15 and 12 mm diameters at the four different energies used, there is no significant variation in the spatial resolution i-e. the collimation efficiency is good for all energies used. If, however, very high energy < >lMeV) were to be used then the collimation efficiency would be reduced considerably and the spatial resolution would become experimentally dependent on energy, though theoretically the spatial resolution as measured from the line spread function (see section 6-3) depends only on the object collimator separation and collimator parameters, i.e. the bore diameter and length- When scanning samples of very high densities and high atomic numbers the use of high gamma—ray energies is essential- The collimator thickness should then be increased correspondingly, so that the effec f. of hi gh energy photons penetrating the shield of the collimator would be reduced. 187 Table G. 1 The contrast analysis of the Eu—phantom ** . Energy vi al Act. pixel object i mage contrast 15 1718 67 1.0* 1.0* 1.0 12 1090 54 0.22 0.11 0.5 121.78 8 481 34 0.56 0.33 0.58 5 189 32 0.80 0.35 0.44 15 1718 48 1.0* 1.0* 1.0 12 1090 38 0.22 0. 12 0. 53 344.3 8 481 25 0. 56 0.32 0.56 5 189 93 0.80 0.35 0.44 15 1718 12 1.0* 1.0* 1.0 12 1090 9 0.22 0. 14 0. 65 778.9 « 8 481 5. 5 0.56 0.37 O O' Ch 5 189 4.9 0.80 0.42 0.53 15 1718 10. 7 1.0* 1.0* 1.0 12 1090 7.6 0. 22 0. 17 0.77 964. 8 481 4 9 0. 5 6 0.48 0.86 5 189 3.2 0. 80 0.58 0. 73 * relative to the air ** the contrast is taken as the absolute value of eq. 7-1 8.5 The Use of Multi—Energy Scanning for Attenuation Correction in SPECT From the reconstructed data of figure 8.17 which shows the same line scan through the images of the phantom at the four different energies it could be seen that there is a very large difference in the pixel values. This difference is due to the following parameters: efficiency, gamma ray intensity and attenuation. To use the multr-ener ny scanning for attenaution 188 correction the other two parameters must be eliminated. This is done by adjusting the pixel values to 10071 gamma—ray detection efficiency and 10071 gamma—ray emission probability thus eliminating the dependence on the these two parameters i.e. the total number of gamma ray photons for the four energies are the same, and hence any discrepency in the adjusted pixel value is only due to attenuation. If the attenuation for the different energies is the same then the adjusted pixel values should have the same constant value for all energies but that is not the case. Table 8.2 below shows the adjusted pixel values as a function of energy. These pixel values are taken as the average maximum reconstructed value for the centre vial of the 15mm diameter. The gamma ray intensities are taken from Nuclear Decay Data for Radionuclides used as Calibration Standards (LOR—1983). Figure 8.18 shows the variation of the adjusted pixel values (APV) with energy and also show the unadjusted pixel values (bottom curve). Detector Emi ssi on Energy Pixel Adjusted Intri nsi c probabi1i ty keV Efficiency I Y * Value Val ue 121.78 ^ 8071 28. 4 68 299.6 344.3 ^ 307. 26.6 45 558 778.9 ^ 157. 13 12 617.2 964 ^ 11 14.6 10.5 652.8 Table 8.2 summarises the adjustment of pixel values for the 10071 detection efficiency and 10071 emission probability. The adjusted pixel values increase with energy because of the reduction in attenuation but as can be seen from figure 8.18 they reach an asymptotic value. This asymptotic constant 189 value is the maximum pixel value that could be obtained in the case where the attenuation is almost zero. The difference between this maximum value and the adjusted pixel value at any energy is then used -for attenaution correction. The adjusted pixel values (APV) as a -function o-f energy shown in -figure 8.18 are -fitted empirically to a -function of the form: APV (E) = a + — | 8.11 where E is the energy in MeV and the constants a and b have the fitted values of 691.54 ± 36.4 and —47.4 ± 5.1 respectively. The correction due to attenuation as a function of energy is shown in figure 8.19. This correction is obtained by subtracting the adjusted pixel value at any energy from the maximum value, i.e.: CORR(E)= a - (APV(E)) = -b/E 8.12 The fitting function goes to zero at the approximate energy of 70 keV and thus this correction function only applies for E>70 keV. This is only because of the choice of the fitting function, other classes of functions will have discontinuities at other energies. Figure 8-19 shows that the correction is most needed at lower energies where the attenuation is most severe. There are however other classes of functions more suitable for this kind of distribution, namely exponential functions with three or more parameters, but the limited number of energies used in this experiment makes it difficult for such functions to be fitted. 8.6 Discussion and Conclusion From the above analysis of the photon attenuation problem us SF’F-.n it is very clear that if a truly quatitative 190 reconstruction of gamma ray emitters is to be carried out photon attenuation must be considered- Examples of photon attenuation effects were studied and different methods of attenuation compensation in Single Photon Emission Computed Tomography were briefly discussed. The application of multi—energy scanning was studied by carrying out an experiment using a 152Eu source- This source was prepared at the Reactor Centre irradiation facilities. An aluminium phantom was then prepared to hold six vials of different diameters containing europium nitrate solutions. Four images of the phantom were then obtained and analysed using both the concepts of contrast and full width at half maximum of line scans through the images. From the image analysis the useability of the technique as a tomography method is obvious especially if the sample scanned contains a variety of metals of different atomic numbers such as industrial waste packages so the lower energies could be used for the less dense or low Z areas and the higher energies used for denser and high Z materials in the same matrix. The useability of the multi energy scanning technique as a method of attenuation compensation was studied by eliminating all other parameters that affect the pixel value so the only effect remaining is that of attenuation and then using these adjusted pixel values to derive an attenuation correction formula. The attenuation correction formula given in equation 8.12 shows the correction is mainly needed for low energies is scanned then this low energy concept, is shifted more toward the high energy side* i.e. the low energy phenomenon is 191 basically a -function o-f the material being tested. UMEN025 16.45.54. 88/06/28. ABIZ I 82897 PLOT COMPLETE. PENS USED •figure 8.9b RN EMISSION TOMOORRM OF THE EU-PHRNTOM RT THE ENERGT OF 344 KEV THE VIEWING RNOLE 15 ' 607.60 1 DEO . i 1V6 e r u g i f 9c c .9 B THE EMISSION IMAGE OF THE EU-PHAN. RT 344 KEV i WITH THE RRYSUM5 OUTSIDE THE PHRN. BEING CUT-OFF RGIRIN THE VIEWING RNOLE IS '60160* cn cn 201 AND Y- ROTATION IS 60 DEGREES COMPLETE PENS USED PLOT 202 Mwg ig~.~e>7. uMewsge ue. "ee^oe/e?. iue81 Acontour imageofthe europium Figurephantom showing8.12 thesix vials r — a n a S 9 4 6 3 7 HO t.-i- HO pixel value CN o pixel value in N o - igre 15 .i- a i gur e pixel value ine unsmoothed unsmoothed ine itne crs te mg I mm In Image the accross distance line scans through through scans the 344keV mage N o o pixel value 2 0 7 - 66.0 - 54.0 - 42.0 - 30.0 - 18.0 - 6.0 6.0 18.0 30.0 42.0 54.0 66 . CN o co i pixel value 200 0 - - - 300 600 •- 400. - 500- - 800 .-i 700 •- 100-- Figure8- Thevariation ot pixel values IBwith energy o. - . o.o aito o te eosrce pxl au wt energy with value pixel reconstructed the of Variation / X i rD nry MeVr Energy 0.5 - » i - — U ) UA ■ ndutd ie value pixel unadjusted dutd ie value pixel adjusted 1 .0 « » bottom top 1.5 correction value 0 h - 10 o.o -figure 8-19The variation of attenuation correction asa I I ------1------1 functonof energy ------. 1.0 0.5 1 Energy Me V Me Energy ------1 ------1------1------1 ------1 ------1------1 ------ l.5 i 210 CHAPTER 9 CONCLUSIONS The design, construction and testing ofa gamma—ray tomographic scanner incorporating a data acquisition and a display system has been completed. The scanning rig has been employed in both transmission and emission modes of tomography to illustrate its capability and versatility. The rig was designed not only -for testing and developing tomographic techniques but also for carrying out industrial applications of tomography. The rig was built of a hard anodised steel with the capability of scanning an object of lm diameter and weighing up to 40kg with a high degree of precision and positioning accuracy. The characteristics of each component of the scanner's detecting system has been studied in order to determine its response to input signals. The variation of sensitivity across the cc—axial Ge(Li) semi-conductor detector used in all the experiments was studied and found to have a dip, p—core, in the centre of the crystal. This dip was found to be energy dependent. This energy dependance was only because of scattering. This behaviour of the Be(Li) detector is not unexpected as it has been shown by ftdesammi , Nic; ol&u and Bal ogt*n (ADE—1983, NIC—1933, BAL—1986) that it occurs because of the mode of fabrication of the co—axial semi-conductor detectors. The effect of this greatly reduced efficiency in the centre of the crystal must therefore be taken into account in the construction of collimators used with this type of detector in tomographic experiments. Our hor p-hol e collimator was designed to use the most 21 1 efficient part of the cr /stal . The resolution of this collimator detector system depends on the bore diameter and length and also on the object collimator separation. This was found to be appro;-: i mat el y 3mm at the centre of rotation of the scanning table for the scanning geometry used. The spatial resolution as defined by the full width at half maximum of the Point Spread Function was studied as a function of energy and found to be energy dependent. This dependance was only because of the collimator efficiency being reduced due to higher scattering at higher Y-ray energies. The variation of spatial resolution as a function of separation, distance between the detector collimator and the source, was studied analytically and found to increase with increased separation. The best analytically achievable spatial resolution is that of the collimator diameter, however, experimentally this is hard to achieve. The best experimentally achievable resolution is about 2 or 3 times the step length if the latter is set equal to the collimator bore diameter. From the Point Spread Function the Modulation Transfer Function of the system was obtained. The detector intrinsic efficiency and its energy dependance was studied by using standard point sources. The variation of which was found to fit best a logarithmic class of functions (GRA—1985). The variation of the full width at half maximum with energy was found to increase and reach an asymptotic constant value at high energies. The energy dependance of full width at tenth maximum was found to follow the same pattern as that of the full width at half maximum. However, the ratio of the full width ai tenth maximum to that 212 at halt maximum was almost constant and approximately 1.9 which compares very well with 1.8 expected tor a good Gaussian di stri buti on. The scanner and data acquisition system is controlled via an RS232C port by an IBM—PC. The data is collected tor all coordinate points specified by the object translation and rotation. The number of the coordinate points is defined as the number of projections (angular increments) multiplied by the number of linear steps (translations). This is predetermined and represents the total number of raysums needed to reconstruct the image. Since one of the most important problems in both emission and transmission tomography is that of scattering the data collected and used for image reconstruction must be scatter free. The scatter correction is achieved in two ways: either by geometrical means (i.e. collimation) or by subtraction of scatter data collected within the photo—peak region (i.e. energy discrimination). The concept of line scans through the image is utilised for quantitative analysis of the images. The full width at half maximum of these line scans was used to determine the object dimensions in the images. These dimensions are then compared to the actual object diameters. Line scans make the representation of the spatial distribution of image density easier and point out the exact locations of regions of higher and lower intensities. The concepts of image contrast and modulation were utilised. Modulation is mainly for cyclic distributions while contrast is a general representation o-f image quality and is 213 used -for all kinds of distributions. In all the transmission and emission tomography experiments carried out in this work both the concepts of full width at half maximum of line scans and contrast were used to determine the image resolution, the dimensions of objects within the image, the distribution of attenuation coefficients and the distribution of activities. Contrast ratios were used to indicate intensity relations between different regions of interest and a method of their use in determining unknown objects in matrices is reported. From all the results seen in chapter 7 it is very clear that the scanning system is capable of carrying out tomographic experiments on industrial samples of large diameters and comprising heavy elements. The main problem in single photon emission tomography is that of attenuation. Chapter S dealt mainly with the problem of attenuation and the possibility of developing an experimental attenuation correction technique. The use of multi—energy scanning was studied in part as another application of tomography and its usefulness was established by reconstructing a europium phantom at different energies. The images were then analysed and compared. It is also useful when imaging neutron activated biological or industrial samples and radio-active waste packages. These type of samples generally contain large number of isotopes and several Y~ray energies. To image the distribution of different isotopes in a matrix the multi—energy scanning is essential. In the case of transmission tomography the multi—energy scanning is also recommended especially when scanning large packages of large variation in attenuation coefficients. 214 The other use of multi—energy scanning was to develop an experimental attenuation correction technique. This is done by means o-f adjusting the reconstructed pixel values of the images obtained from the europium phantom at four different energies. The pixel values were adjusted for the variation in detector efficiency and the variation in gamma—ray emission probabilities of the energies used. Thus the effect of the number of counts per pixel due to detection efficiency and emission probability is eliminated. The only remaining problem is that of attenuation. The adjusted pixel values were then plotted and found to increase almost exponentially with increasing energy and reach an asymptotic constant maximum value. This asymptotic maximum value represents the situation were the attenuation is effectively zero. The difference between this maximum pixel value and the adjusted pixel values at any energy is then used as an effective attenuation correction technique for single photon emission tomography. 9.1 Recommendations for Further Work The existing tomographic and data acquisition system is controlled by an IBM—PC in conjunction with an ADCAM and a Multiplexer unit. The latter units is in the systemt to enable several users to carry out experiments simultaneously. However? the tomography experiments required sole use of the ADCAM system for long periods of time? so it would be very 'J valuable to have an independent data acquisition system for tomography. The data acquisition technique which involves the multi-channel buffer system and allows one to collect and 215 store a great number of spectra -for peak analysis is not essential in tomography and should in -fact be replaced with 2 or 3 single channel analysers. This will result in a drastic reduction of the amount of data recorded, hence leaving a large part of the computer disk space free for scanning larger matrices. Since the spatial resolution in the image depends on the collimator geometry and source detector separation the detector must be allowed to move as close as possible to the object being scanned. In the case of transmission tomography the allignment of the source and detector collimators is very critical. Thus the source stand must be mounted on top of a manual or an automated lift so precise allignment using a laser beam would be carried out. The display system employed uses main frame computer packages. This should be replaced with a micro-computer package thus having a fully automated scanner that could be operated totally independent of a main frame computer. This is very useful1 if the scanner is to be transportable. From the tomography point of view a great deal of work has been carried out to study the different imaging parameters and scanning techniques. The study of multi-energy scanning was carried out both as a tomographic technique and as a method for attenuation correction. The usefulness of m ulti—energy scanning is reported but for this technique to be fully utilised and for its capability ten be understood much more work is needed. 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