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Noise propagation in quantitative positron emission tomography Palmer, Matthew Rex 1985

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NOISE PROPAGATION IN QUANTITATIVE POSITRON EMISSION TOMOGRAPHY By MATTHEW REX PALMER B.A.Sc, The University of British Columbia, 1981 A THESIS SUBMITTED IN PARTIAL FULFILLMENT O F T H E REQUIREMENTS FOR T H E DEGREE O F MASTER O F APPLIED SCIENCE in T H E F A C U L T Y OF G R A D U A T E STUDIES Electrical Engineering We accept this thesis as conforming to the required standard T H E UNIVERSITY O F BRITISH COLUMBIA July 1985 • Matthew Rex Palmer, 1985 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head o f my department or by h i s or her r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of / ^ / ^ / , ^ • / ^ A y ? / H * * r '^ 1 The U n i v e r s i t y of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date \J r — - y -DE-6 (3/81) ABSTRACT Image noise in Positron Emission Tomography (PET) is the result of statistical fluctuation in projection data. The variance properties of images obtained with the UBC/TRIUMF Phi 1 VI tomograph are studied by analytical methods, computer simulations, and phantom experiments. The PETT VI image reconstruction algorithm is described and analyzed for noise propagation properties. Procedures for estimating both point-wise (pixel) and region of interest (ROI) variances are developed: these include the effects of corrections for non-uniform sampling, detector efficiency variation, object self-attenuation and random coincidences. The analytical expression for image-plane variance is used in computer simulations to isolate the effects of the various data corrections: It is shown that the image precision is degraded due to non-uniform sampling of the projections. The RMS noise is found to be increased by 9% due to the wobble motion employed in PETT VI. Analytical predictions for both pixel and ROI variances are verified with phantom experiments. The average error between measured and predicted ROI variances due to noise in emission data for a set of seven regions placed on a 20 cm cylindrical phantom is 9.5%. Images showing variance distributions due to noise in emission data and due to noise in transmission data are produced from human subject brain scan data collected by the UBC/TRIUMF PET group. The maximum ratio of image variance due to noise in transmission data to that due to noise in emission data is calculated as 2.6 for a typical F D G study, and 0.082 for a typical fluorodopa study. Total RMS noise varies between 0.4% and 11.6% for a typical set of ROI's placed on mid-brain slices reconstructed from these data sets. Procedures are suggested for improving the statistical accuracy of quantitative PET measurements. ii Table of Contents Abstract " List of Tables v List of Figures vi Acknowledgements viii I. INTRODUCTION 1 II. THEORETICAL B A C K G R O U N D 1 10 2.1 Imaging Errors and Compensation 13 2.1.1 Projection Statistics 14 2.1.2 Photon attenuation 15 2.1.3 Compton Scatter IV 2.1.4 Random Coincidences 19 2.1.5 Detector Efficiency 21 2.2 Image Reconstruction 23 III. N O N - U N I F O R M SAMPLING 27 3.1 Rebinning 30 3.2 Noise Backprojecuon 31 3.3 Simulations and Calculations 34 IV. DATA CORRECTIONS A N D I M A G E - P L A N E VARIANCE 39 4.1 Noise Backprojection 42 4.2 Efficiency Correction 47 4.3 Random Coincidence Correction 49 4.4 Attenuation Correction 52 V. REGIONAL MEASUREMENTS 57 5.1 Image Covariance 58 5.2 ROI Variance - 6 2 VI. EXPERIMENTAL VERIFICATION 66 6.1 Phantom Measurements 67 iii 6.1.1 Transmission Scan Duration 71 6.2 Clinical Examples 73 VII. CONCLUSIONS 77 References 80 iv List of Tables I Average Variance Ratios • 37 II ROI variance 70 III Transmisson and Emission Variances and Slice Sums 75 IV F D G ROI Variances 76 V Fluorodopa ROI Variances 76 v List of Figures 1.1 PETT VI 3 1.2 Normal F D G image 4 1.3 Normal fluorodopa image 6 1.4 Normal water image 7 1.5 Comparison of PET, CT and MRJ 8 2.1 PETT VI scanning geometry 11 2.2 Ideal point-spread function (PSF) 12 2.3 Effect of Compton scattering 18 2.4 PETT VI efficiency characteristics 22 2.5 One-dimensional projection 24 3.1 Wobble motion 28 3.2 Wobble normalization function 29 3.3 Wobble; analytic variance 34 3.4 Wobble; statistical variance 35 3.5 Wobble; point sources 36 3.6 Sinograms • 37 4.1 Uniform phantom 45 4.2 Effect of efficiency correction 48 4.3 Effect of random coincidence correction 50 4.4 Effect of attenuation - uniform phantom 52 4.5 Effect of attenuation - point source, centred 53 4.6 Effect of attenuation - point source, 6 cm 54 4.7 Effect of attenuation - point source, 10 cm 55 5.1 Covariance weighting 59 5.2 Image-plane covariance function 60 5.3 ROI variance versus area - Emission noise 64 5.4 ROI variance versus area - Transmission noise 65 vi 6.1 Experimental data; statistical variance images 67 6.2 Statistical variance - emission noise 68 6.3, Statistical variance; transmission noise 69 6.4 ROI size and placement map '. 70 6.5 PET real data 73 6.6 Variance image - F D G 74 6.7 Variance image - fluorodopa 75 vii ACKNOWLEDGEMENTS I would like to thank my thesis supervisor Dr. Michael Beddoes and my associate supervisors Dr. Mats Bergstrom and Dr. Brian Pate for their generous support and advice throughout the course of this work. I would also like to thank the members of the UBC/TRIUMF PET Project; Dr. Michael Adam, Dr. Ronald Harrop, Dr. Wayne Martin, Dr. Joel Rogers, Dr. Tom Ruth, and Christine Sayer for their technical advice and helpful discussions. I gratefully acknowledge the technical collaboration of my collegues; James Clark, and Lawrence Panych. This work has been supported through the Natural Sciences and Engineering Research Council of Canada (grant number 67-3279) and the Medical Research Council of Canada (grant number SP-7). viii I - INTRODUCTION Positron emission tomography (PET) is a modem medical imaging technique used in the non-invasive study of physiological processes in the organs of an alert and comfortable human subject Quantitative imaging of injected radioactive tracers is possible for studying such body functions and parameters as blood flow, blood volume, and metabolic rates. A variety of brain and whole-body PET cameras have been built in the last decade capable of high sensitivity imaging with resolutions below 1 cm [10, 34]. In a typical PET study, a compound whose physiological characteristics in the body are to be monitored, is labelled with a positron emitting radioisotope. This involves the synthesis of the chemical compound in a laboratory and "labelling" the molecular structure with the radionuclide produced by an on-site cyclotron. The medically important radio-nuclides are n C , 1 3 N , 1 5 0 , 1 8 F, and 6 ! G a that decay by positron emission with half lives in the range of two minutes to two hours. Some typical brain imaging compounds are; lsO labelled water for blood flow studies, and 1 ! F labelled fluoro-deoxyglucose (FDG) for glucose metabolic rate studies. At the start of a PET brain study, the patient is injected intravenously with the tracer compound and his brain is placed in the field of view of the camera. A positron emitted in the field of view annihilates with an electron, to produce two 511 keV gamma rays which travel off in directions close to 180 degrees apart The PET camera consists of a set of one or more rings of crystal detectors which surround the volume to be imaged. The camera electronics are designed to count coincident gamma arrivals for all the possible detector channels (a channel is any pair of crystals). In the course of the study, histograms are generated for each detector ring plane (or "slice") with entries for each detector channel within the plane. The number of counts recorded in a detector channel is an estimate of the sum of activity along a ray connecting the two detectors. Histograms are partitioned into sets representing mutually parallel channels called projections or profiles. The mathematical techniques used to reconstruct the two dimensional activity distribution from the sets of one-dimensional projections is termed tomography, from the Greek for imaging in slices. The UBC/TRIUMF PETT VI camera [22], is a development of the Washington University at SL Louis, PETT VI [23,46,50]. The camera consists of 288 Cesium Fluoride (CsF) scintillation detector crystals arranged in four circular rings of 72 crystals each. A sixteen point wobbling motion is employed to increase the radial sampling density. Two photons, coincident in time, arrivingN at crystals within a ring or in adjacent rings are recorded by the detection electronics, and tallied in a C A M A C histogram memory. In this manner, coincidence data for seven slices (four "straight" slices and three "cross" slices) are tallied by the histogram memory. Machine control and monitoring for PETT VI is performed by a PDP-11 micro-computer which is attached to a VAX-11/750 minicomputer. Data is transferred from the C A M A C memory to the VAX computer which performs the image reconstruction. Images are displayed and analyzed on a Ramtek 9460 colour display system under the control of software that resides on the VAX. The tomograph collects data for seven cross-sectional slices simultaneously. The in-plane resolution is about 8 mm (high-resolution mode), the effective slice thickness varies from 9mm to 14mm, and the slice separation (centre to centre) is 14 mm. A typical scanning session involves a series of paired data collections, where the chair position is offset by 7 mm between scans in the pair. This provides fourteen interleaved sets of cross-sectional data obtained at 7 mm slice separations. Compared with computed x-ray tomography (CT) and magnetic resonance imaging (MRI or NMR - nuclear magnetic resonance), PET offers relatively poor spatial resolution and often low image statisitcs. In contrast however, PET provides almost unlimited potential for measuring isolated metabolic processes in the body with a sensitivity unrivaled for dealing with the microscopic quantities of material involved in neurotransmission. Both CT FIGURE 1.1 PETT VI: An external view of the UBC/TRIUMF PETT VI showing the head-holder, couch and tomograph gantry. and MRI scanning provide anatomical information with high contrast, and excellent spatial resolution (on the order of 1 to 2 mm) but little or no functional or physiological information. For example, pre- and post-mortem CT scans of a patient would be identical while the corresponding PET scans would vastly differ. PET has proven to be a powerful tool for research studies into brain and other organ function and in future may be exploited as a diagnostic tool. Efforts are being made to bring the cost of camera hardware down, as well as to provide simplified chemistry procedures for the production of scanning agents. FIGURE 1.2 Normal F D G image: Fourteen axial slices through the brain of a normal subject following injection of F D G . Medical researchers with the UBC/TRIUMF PET Program at present have three radio-tracer compounds available for human subject brain scanning. 2-fluoro-2-deoxy-glucose (FDG), fluorodopa, and 150-labelled water are routinely produced at the TRIUMF cyclotron facility. F D G in the blood behaves as normal blood sugar and labelled with " F it provides a means to measure the glucose metabolic rates in areas of the brain. Figure 1.2 shows a series of fourteen slices from a PET study of a normal subject following injection of F D G . Glucose is metabolized most strongly by the grey matter of the brain which is located in the cortex and deep brain nuclei. The combined ventricular fluid and white matter show up as dark (low metabolism) butterfly-shaped areas in the mid slices. Fluorodopa, is the chemical dopa (L-3,4-Dihydroxyphenylalanine) labelled with 1 8 F, and is metabolized by the brain to produce fluorodopamine. This neurotransmitter is active primarily in the basal ganglia, which are areas of the brain responsible for some aspects of movement. Figure 1.3 shows a series of fourteen slices from a PET study of a normal subject following injection of fluorodopa. The only significant activity level visible in these scans is in the basal ganglia - the two banana-shaped regions in mid-slices. The concentration of radio-active dopamine in the basal ganglia at some time after injection has been found to be lower than normal in subjects with Parkinson's desease or those who have been exposed to the neurotoxin MPTP. 150-labelled water is used in the study of cerebral blood flow. The relative radio-activity some time after injection is used as an indicator of the rate of blood perfusion to the various areas of the brain. Figure 1.4 shows a series of seven slices from a PET study of a normal subject following injection of l iO-water. Because of the short half-life of 1 5 0, water scans are usually performed in the low-resolution, high-efficiency scanning mode to improve image statistics. In this mode, the full surface areas of the detector crystals is exposed in order to improve the machine efficiency. This results in a resolution of about 15 mm, and consequently a smoother looking image. The real power of PET lies in its ability to obtain quantitative regional measurements of radionuclide concentration. The radioactivity distributions obtained by the PET camera can be converted to concentrations of metabolites of the compound under study by using a suitable metabolic model [39]. Whereas in CT or MRI, all that is desired is to obtain a visual contrast between types of tissue, or pathological and normal structures, a PET image is actually a data matrix whose values correspond to metabolic measurements. For this reason, special care is taken to correct the data for various imaging degradation effects. The errors involved in PET imaging are usually broken down into the two categories of statistical and systematic errors. Statistical errors, are those that arise from the FIGURE 1.3 Normal fluorodopa image: Fourteen axial slices through the brain of a normal subject following injection of fluorodopa. probablilistic nature of radioactive decay. The term "precision" refers to the effect of these statistical errors on images or regional measurements. The effects of statistical fluctuation on the image are usually referred to as noise. Systematic errors are those which are repeatable given an identical imaging situation. Some causes of systematic errors are patient movement, limited resolution of the detector crystals, and detector non-uniformity. The result of these errors are usually referred to as the image "accuracy". These two types of errors are somewhat complementary as we shall see. Accuracy can be improved in many cases where machine performance can be predicted but often at the expense of reduced precision. The precision of a regional measurement can be 7 FIGURE 1.4 Normal water image: Seven axial slices through the brain of a normal subject following injection of 150-water. improved, for example, by averaging over a larger area, but at the risk of reduced accuracy since the activity may not be constant over the larger region. Throughout the development of PET, a lot of effort has been made to improve the image accuracy. Corrections for many systematic degradations are standard in most PET cameras and will be discussed in Chapter 2. The main goal of this work is to study the image-plane variance effects of the main data corrections. This is achieved through a series of calculations performed on the computer that isolate the various corrections to obtain analytic predictions for image variance. These predictions are then verified by phantom experiments in which statistical 8 FIGURE 1.5 Comparison of PET, CT and MRI: Slices through the brain of the same subject with three imaging modalities. PET - F G D (a), MRI - Inversion Recovery (IR) sequence (b), MRI - Spin Echo (SE) sequence (c), and CT (d). variance measurments are made. In Chapter 2, the theory of imaging with PET is discussed, along with the systematic errors and how they are compensated. This is followed by a discussion of the methods of reconstruction from projections, with special emphasis on the filtered backprojection method that is employed in most PET imaging systems. Chapter 3 describes the analysis of the noise effects of detector wobble motion. This work has also been published elsewhere [38]. In Chapter 4, each data correction step involved in the reconstruction of PETT VI images is examined fox its effect on image variance. Isolating these effects with real data is often impossible and at best difficult. By simulating the effects individually on the computer, a better understanding of the various contributions to image variance is obtained. The image covariance is studied in Chapter 5 entitled Regional Measurements. The analysis of PET images usually involves averaging the tracer intensity data over as large an area as possible in order to reduce the effects of noise. The size of the region of interest (ROI) may vary from just a few pixels (picture elements) to a few hundred pixels, depending on the region of the brain under study. The noise between pixels in a PET image is correlated and the precision of ROI measurements cannot be simply computed from the variance image alone. Chapter 6 entitled Experimental Verification, presents the results of experiments performed on the UBC/TRIUMF PETT VI camera. These experiments were directed to verifying the accuracy of the simulations in predicting image noise and ROI variances. A description of the phantoms and scanning methods is included as well as an analysis of the results. Clinical examples are also presented; the computer program developed to implement the image and ROI variance calculations is applied to real data. Signal to noise ratios for typical measurements on F D G and Fluorodopa images are presented. D - THEORETICAL BACKGROUND The basis for positron emission tomography is the detection of positron annihilation events and accumulation of these in a series of one-dimensional projections. In most modern PET cameras, such as the UBC/TRIUMF PETT VI [22, 23, 46, 50], a number of scintilation crystals are placed in one or more rings (four for PETT VI) surrounding the patient and are coupled to photomultiplier tubes. The scanning geometry for the PETT VI camera is shown in figure 2.1. A gamma photon, produced from positron annihilation, enters the crystal, and interacts to produce visible light that is converted to an electrical pulse by the photomultiplier tube. When a coincidence is observed between pulses produced at two crystals on opposite sides of the field of view, the positron is assumed to have been annihilated somewhere along the line that intersects the two crystals. Coincidences are tallied separately for each detector channel (a pair of detectors forming a line that intersects the field of view) and in this way, ray-sums of activity along lines through the field of view are accumulated over time. In theory, it is possible to measure the difference in arrival times between the two photons of an annihilation event to determine the point along the flight path at which the gamma rays were produced. However, the practicalities of crystal response time and delays in the electronic pulse creation circuitry make this measurement difficult In machines with very fast crystals and electronics an estimate of the annihilation site can be made which can improve the signal to noise ratio. This approach is called time of flight PET (TOFPET) and has been exploited by Ter-Pogossian et. al. [47] as well as others. Most PET machines in use, however, do not make use of the time of flight information, and therefore the analysis in this work does not deal with it In an ideal camera, it follows from elementary geometrical considerations that the axial point-spread function (PSF) for crystal detectors with rectangular sufaces is triangularly shaped at the centre of the field of view, and rectangular at the crystal 11 F I G U R E 2.1 PETT VI scanning geometry: Two segments of a 72-detector circular ring. surface. This ideal PSF, shown in figure 2.2, is obtained by considering a point source of activity which radiates isotropically, placed at various locations with respect to two detector crystals. The crystals are assumed to be opaque to the incident radiation. The x-axis in the PSF graph represents the perpendicular distance from the line connecting the midpoints of the two detectors, and R(x) represents the relative count rate recorded by the detector pair. The PSF of a PET camera is a measure of the resolution. It is often expressed as the full width at half maximum (FWHM) of the PSF. The resolution for the ideal detector system then is D/2 at the centre of the Field of view, where D is the detector width. This factor, D/2, is often referred to as the intrinsic resolution of the camera, although in practice, it is only a First approximation. A number of factors contribute to the degradation of the ideal PSF, and the precise measurement of ray-sum activity. Gamma radiation is both attenuated and scattered by the body and its environment Coincidences are falsely detected by the camera due to FIGURE 2.2 Ideal point-spread function (PSF): The detector channel response to a point source located at the centre of the field of view (a) and next to a crystal surface (b). randomly conicident gamma ray arrivals from separate annihilation events. Furthermore, the integration time for ray-sum accumulation is limited and it is usually necessary to deal with projections that have relatively poor statistics. In the following sections these effects and the corrections that are done to compensate the data will be examined. 13 2.1 - Imaging Errors and Compensation The main sources of systematic errors are photon attenuation by the emitting object itself, Compton scattering of gamma radiation within the object, air, and detector crystals, the random coincidences of photons arising from separate annihilation events, and the variation in detector efficiency with angle of gamma-ray incidence. Also, in PET cameras that employ detector motion to increase the radial sampling density (such as the UBC/TRIUMF PETT VI), a non-uniform sampling pattern results. In most PET systems, an attempt is made to correct the data for these effects by compensation in the projections before the image is reconstructed. Two small, unavoidable effects that arise from imaging with positrons are worth noting. The positron travels a small distance (its "range") before annihilation. This is a distributed quantity whose average value depends on the isotope used for imaging. Derenzo [18] and Cho et al. [13] have measured positron ranges for medically important isotopes that range upward from 1 to 2 mm. While this effect is small compared with the 8 to 15 mm intrinsic resolutions of most PET cameras in use today, it does set the practical upper bound for the next generation of cameras as the resolution approaches 4 to 5 mm [45]. Derenzo et al. [20] recently reported resolution measurements on a high-resolution PET camera under development that depended on the isotope in accordance with its positron range. The second effect is the slight deviation from 180 degrees of the two gamma rays. This has been measured by Colombino et al. [17] to be a Gaussian distribution with F W H M equal to about 0 .4 ° . Typical crystal ring diameters for head scanning PET cameras are in the range of about 50 cm. This means that on the average the deviation from a straight line for gamma-rays emitted as a result of annihilation at the centre of the field of view is less than 1 mm 14 2.1.1 Projection Statistics It is well known that the number of counts recorded by each detector channel in a PET system obeys Poisson statistics and is uncorrelated between samples of projections and between different projections [25, 12, 1]. A Poisson distribution has the property that the mean and variance are equal. Let the expected value of ray-sum concentration be represented by c, then the signal power to noise power ratio is R p = cVc = c (2.1) The important result here is that the signal to noise power ratio can be made arbitrarily high by increasing the number of events recorded. Of course, in practice there are other factors to consider in determining the time during which events may be accumulated, such as the half-life of the nuclide involved, radiation dose, and some subtleties such as patient comfort 15 2.1.2 Photon attenuation Photons are attenuated in matter according to the formula; N = N 0 e ~ M X (2.2) where N 0 is the number of incident photons, N is the number after penetration of a distance x through an isotropic medium with a linear attenuation coefficient u [30]. For the soft tissue in the head, the linear attenuation coefficient is about 0.096 cm"1 [7]. For a distance of 15 cm, which is a representative chord length for a human head, the fraction of photons attenuated is about 75%. The problem of attenuation then is one that cannot be overlooked in order to achieve meaningful quantitative information from the data. The results of Huang et al. [30] suggest that this correction must be done as accurately as possible due to the sensitivity of the information to this effect There are several approaches to correcting data for object self-attenuation. One method is to scan the object, before administration of the tracer isotope, with an external source of positron annihilation gamma-radiation. The resulting transmission profiles can be used to correct the emission profiles by division of emission data by transmission data. This method is accurate, but increases the radiation dose to the patient, and adds significantly to the noise in the final image [30]. A second method of correcting for attenuation is to obtain a description of the object exterior boundary and calculate the chord lengths for all rays. The attenuation factor for each point in the projections can then be calculated by equation 2.2 above and applied to the emission data. The advantages of this method are that it requires no added radiation dose to the patient, it shortens the total scanning procedure, and adds no noise to the final image. Its drawbacks, however, arise from the irmacuracies in determining the object boundary and the fact that the attenuation coefficient, n varies slightly among 16 different tissues of the body, especially if bone or air is present As mentioned, errors in these methods may lead to significant errors in the final image. Some approaches that have been attempted along these lines are; fitting an elliptical skull outline to an uncorrected brain image and re-reconstructing with the calculated attenuation factors [6], automatic computer detection of edges in the projections [7, 29], and using CT images to obtain object boundaries. 17 2.1.3 Compton Scatter Compton scattering of gamma-rays between their production and detection causes degradation of the PET image in the form of a loss of resolution. Several researchers have studied the effects of scattered radiation on the shape of the PSF in PET systems [3, 21, 35]. Bergstrom [3] has stressed the importance of this correction to the accuracy of quantification in PET. He has demonstrated that a quantification error of about 20% can occur in certain geometries, if scatter compensation is not included in the PET imaging procedure. Compton scattering occurs in material when a gamma photon is deflected by an atom of the material, giving up some of its energy. The gamma rays produced during positron annihilation are at the 511 keV energy level and to a limited extent, the PET camera can reject scattered photons by energy discrimination at the PMT outputs. The effect of scatter on the PSF is to add long, low-intensity flanks at either side of the central peak. These can be seen clearly in figure 2.3 as linear in the logarithmic PSF graph, their slopes are dependent on position of the source of gamma-rays in the field of view. The measured scatter distribution also depends on the type of crystals used, size and shape of inter-crystal shielding septa, and the energy discrimination threshold setting [19]. For these reasons it is usually necessary to determine the scatter distribution experimentally for use in an empirical model for correction. The scatter correction methods Bergstrom et al. [3], and Egbert et al. [21] both involve experimental determination of that portion of the PSF due to scatter. This is done by obtaining projections for a point source located in a scattering medium, and separation of the scatter flanks from the central peak. Projections in a PET study are then corrected .for scatter by deconvolving them for the experimentally determined scatter functions. FIGURE 2.3 Effect of Compton scattering: The logarithmic scale shows scatter flanks as straight line segments on either side of the central PSF peak (from [4]). 19 2.1.4 Random Coincidences Random coincidences occur between photons generated by separate annihilation events in a PET study. The expected random coincidence count rate recorded by a detector channel is R = RjRj 2T (2.3) where Rj and Rj are the individual count rates recorded at detectors i and j, and T is the coincidence window time [27]. It is important to note that while the true coincidence rate is proportional to the intensity of activity in the field of view, the random coincidence rate is proportional to the square of the intensity. This is one of the factors limiting studies conducted at high counting rates. In some studies the random coincidence fraction is considerable and an accurate correction is essential (e.g. in studies of regional cerebral blood flow with ^O-labelled water which has a half life of about 2 minutes). There are at least two methods in use to correct for random coincidence contribution to measured activity. The method of Bergstrom et al. [5] involves measuring the individual detector count rates directly, then computing the expected random contribution by equation 2.3 above. This method is accurate but requires additional hardware and histogram memory space to obtain the additional information. In the PETT VI, individual detector counting rates are not monitored. The algorithm developed by Ficke et al. [50] is based on the method of Hoffman et al. [27] and assumes that the random coincidence contributions to all projection points are equal. The events recorded by detector pairs whose intersection lines fall just outside the field of view are averaged to yield a single estimate for the random and scatter fraction. The assumption is that all the activity outside the field of view is due to random coincidences and scattered radiation. This estimate is subtracted from each projection in the slice. This 20 approach is adequate for studies where the activity distribution in the head is localized centrally in the field of view but may cause problems otherwise. Recent studies by Bergstrom et al. [8] of the PETT VI camera have shown that detector wobble motion introduces complex radial and ring structure into the distribution of random coincidence events. Such effects will not however be considered here. 21 2.1.5 Detector Efficiency t | j The detector efficiency for scintillation crystals employed in PET cameras is dependent on the photon angle of incidence. In addition, the collimators used to control inter-crystal scatter in some designs, such as the PETT VI, cause the effective crystal surface area to be reduced with increasing angle away from a line normal to the detector face. For circular ring detector arrangements, the angle of incidence increases towards the edges of each projection. As well, multi-ring cameras (such as PETT VI) that use inter-ring coincidences to obtain between-ring (or cross-slice) data, will exhibit different efficiency characteristics for true and CTOSS slices. The PETT VI tomograph design incorporates a unique feature that allows the user to trade-off between resolution and sensitivity. In the high-resolution mode of the PETT VI . tungsten sheilding fingers are placed over the outer parts of each crystal surface to reduce the effective area of the crystal and narrow the shape of the PSF. This causes severe efficiency drop-off towards the edges of each projection. The correction for efficiency variation with position in the projection is straight forward. Correction factors are usually found experimentally for image slices in the various modes of the machine, and applied to the projection data. The experimentally determined efficiency curves for the UBC/TRIUMF PETT VI are shown in figure 2.4. For true slices in the high-resolution mode, with 100% efficiency arbitrarily set at the centre of the projection, the efficiency has dropped to about 20% at the edge of the projection. 20 40 60 B0~ 100 P o s i t i o n (b in number) 120 B. <u c 3 O > ~i5 *o (3 3 too lzo Position (bin number) FIGURE 2.4 PETT VI efficiency characteristics: High-resolution true slices (even projections) (a), and low-resolution true slices (even projections) (b) (from [4]). 23 2.2 - Image Reconstruction The problem of reconstructing a two-dimensional function from a set of one-dimensional projections has been studied extensively as it applies to the fields of radioastronomy, electron microscopy, and more recently, CT, MRI, and PET scanning. There are several methods available for reconstruction from projections; direct Fourier inversion [41], filtered backprojection [2,43] and algebraic reconstruction techniques (ART) [24, 49]. The field of PET has drawn heavily on the work in CT and most PET systems employ the filtered backprojection method. While there is renewed interest in the iterative ART methods due to its potential for better accuracy [44], these methods are slow and impractical at the present time. The following analysis applies to the filtered backprojection algorithm of Shepp and Logan [43] Consider the problem of reconstructing the two-dimensional activity distribution function, c(x,y), from a set of parallel projections. Ignoring all degradations, (i.e. randoms, scatter etc.) a projection at angle <p is given by P0(u) = SlcSlo, c(x,y) 5(u-x) dx dy (2.4) where x' = x cos <j> + y sin <p (2.5) and 6(u) is the Dirac delta-function. This projection-reconstruction geometry is illustrated in figure 2.5. While the symbol P^ is used, to emphasize the fact that each projection is a one-dimensional function, if we consider the set of projections at all possible angles <t>, we have a two-dimensional function, p(0,t), that represents an integral transformation of the original function c(x,y). This is called the Radon transform of c, named after the 24 FIGURE 2.5 One-dimensional projection: P^, is the projection at angle <t>, of the two-dimensional activity distribution, c(x,y). mathematician J. Radon who first studied its properties in 1917. The imaging procedure in PET, as well as in other medical imaging systems, is a sampling of the Radon transform at a discrete set of projection angles and a discrete number of points on each projection [40]. Taking the Fourier transform of both sides of equation 2.5, and comparing this with the two-dimensional Fourier transform of c(x,y), we obtain the Fourier slice theorem [41] which is expressed as SJco) = C(co,0) (2.6) where C(u,(f>) is the two-dimensional Fourier transform of c expressed in polar coordinates, and S^(w) is the one-dimensional Fourier transform of the projection, P^. The Fourier slice theorem expresses the equivalence of a one-dimensional Fourier transform of a parallel projection of the function c, and the two-dimensional Fourier transform of c evaluated along a line through the origin in the two-dimensional frequency plane. This suggests a conceptually simple method for recovering the function from a set of its projections; i.e. take individual one-dimensional Fourier transforms of available projections, interpolate from the set of radial lines to obtain the transform on a rectangular grid, finally take the inverse Fourier transform to obtain an approximation to c. This is in fact the direct Fourier inversion technique. The computation involved in accurate reconstructions from irregularly sampled, two-dimensional functions has been studied [16]. The problems associated with direct Fourier inversion arise mostly from the interpolations required which tend to be computationally intensive. Some comparisons between this and the filtered backprojection method have been made by Ortendahl et al. [37] for application in MRI scanning. By expressing the inverse Fourier transform of c in polar coordinates, and using the Fourier slice theorem, it can be shown that c(x,y) = /J Q0(xO d* (2.7) where x' is given by (2.5) above, and Q0(u) = rT^S 0(w)H(cj)] (2.8) where H(o>) = \co\ (2.9) The multiplication in the frequency domain represents a filtering process .with the filter H. The function is reconstructed by backprqjecting samples from filtered projections. The inverse Fourier transform of H doesn't exist in a strict sense. If we assume that the projection function is band-limited to cu0, then we can create a new filter in the frequency domain which is zero beyond a>0. This modified filter is called a "truncated ramp". In practice, the truncated ramp, or rectangular windowing results in ringing artifact in the reconstructed images. Chesler and Riederer [11] have shown that the ripple introduced in the reconstruction process is reduced when using a Hann window. |o>|[l + cos(nWcL>0)] |CJ| < u)0 H(w) = { (2.10) 0 otherwise In the practical imaging situation, we obtain m discrete parallel projections of a two-dimensional function, c(x,y). The filtered backprojection process can be expressed as m m c(x,y) = ( £ ) L Ci(u) h(x-u) du (2.11) j = l where h is the spatial domain representation of the Hann-weighted ramp filter, m is the number of projections, j is the projection index, and x' is given by x' = x cos0. + y sin<£. (2.12) ID - NON-UNIFORM SAMPLING In order to improve the sampling density, most PET systems employ some type of motion of the detector assembly during the scanning operation. Several methods have, been proposed and are in use, such as a wobble motion of the detector assembly by Bohm et al. [9] and Ter-Pogossian et al. [48], a dichotomic ring sampling scheme by Cho et al. [14, 15], and a "clam shell" motion by Huesman et al. [33]. One of the popular schemes, and the one adopted in the UBC/TRIUMF PETT VI is the wobble motion [22, 23]. The wobbling motion of the detectors in a PET camera is a circular movement of the entire detector assembly in which the assembly does not rotate, but its centre of cylindrical symmetry traces out a circular path whose centre coincides with the tomograph axis. In such a motion, each detector traces a circular path as shown in figure 3.1. Typically the wobble circle is divided into a number of arcs of equal length and detection events are tallied separately for each arc. Considering the set of parallel rays connecting the arc centres of opposing detectors we see that the spatial sampling pattern is non-uniform, with each sample representing an equal time interval, if the wobble rate is constant Due to the complexity involved in dealing with non-uniform sampling patterns, projection data are normally sorted into sets representing parallel rays, equally spaced on each projection axis. This process is commonly referred to as "re-binning". The re-binned projection data, although uniformly spaced, now represent varying collection times, and must be normalized by dividing the number of recorded events in each bin by the average time spent accumulating data for that bin. This average time is related to the density of sample lines through the projection axis. This is called the wobble normalization function, and denoted by w.(u) where subscript j is the projection index (or angle) and u is the FIGURE 3.1 Wobble motion: Detector trajectories and arc midpoint sample lines are shown. position along the projection axis. The discrete form of Wj(u) for a single projection, is denoted by w ,^ which is called the set of wobble normalization coefficients, where i is the bin index. The wobble normalization function has a periodic tendency but is not strictly periodic due to the fact that the density of sample lines through the projection axis increases towards the edge of the field of view. As a first approximation, we can consider it to be periodic, since the field of view size is small compared to the detector ring diameter (PETT VI; field of view is 25 cm, ring diameter is 60 cm). Typical wobble normalization functions for even and odd projections are shown in figure 3.2. The difference between even and odd projections is due to the way detector elements line up in alternate projections. Even projections have their centre bins line up with the top and bottom of wobble circles: Odd projections have centre bins line up with edges of adjacent wobble circles. 29 Wj(li) Wj(u) II • t 9 9 f • f f f • u u m * • • * ? u FIGURE 3.2 Wobble normalization function: Representing time spent in each bin for even projections (a) and odd projections (b). 30 3.1 - Rebinning Consider a uniformly-emitting, non-attenuating, non-scattering radiator being scanned by a camera with ideal efficiency characteristics using wobble motion sampling to improve the sampling density. The wobble normalization function, figure 3.2 describes the average time spent counting events as a function of position in the projection. While projection samples normalized by this function correctly reflect the ray-sum passing through each bin, the variance or noise component in the projection data is affected by the same wobble normalization function. To consider the effects on variance, the wobble normalization function is first constrained, by making its average over a wobble cycle equal to one. This will allow comparisons with uniformly sampled projections that contain the same number of events. Before normalization, the projection data follow a Poisson distribution and the effect of dividing by the wobble normalization function is to introduce a 1/Wj modulating term into the variance expression. Denoting the normalized ray-sum by Cj(u), and the expected projection value before normalization as Pj(u)> then o 2 c j(u) = Pj(u) / w.(u) (3.1) As before, u represents position on the projection axis, and subscript j is the projection index (corresponding to projection angle). 3.2 - Noise Backprojection Using equation 2.11, and the fact that projection samples are uncorrected, it can be shown that the variance in the image is _ m o2(x,y) = & Z S_aa'.(u) h2(x'-u) du (3.2) j = l 1 Here, o2c is the variance in the projection, h is the backprojection filter, m is the number of projections and x is given by equation 2.12. Alpert et al. [1] have suggested that, ignoring all corrections, the measured projection can serve as an estimate of the projection variance, which yields an estimate for the image variance function _ m <r2(x,y) = i^y I / _ „ Ci(u) h2(x-u) du (3.3) j = l J Now substituting (3.1) into equation 3.2, yields the image plane variance expression for a non-attenuating radiator sampled under wobble conditions: o2(x,y) = (^) JZ_ i/roBPj(u) /wj(u)h ,(^-u)du (3.4) This result is very powerful because it means that a variance "image" can be reconstructed with the same filtered backprojection algorithm by replacing the Harm-weighted ramp filter with its square. To compare the average variance in images sampled uniformly to that of images sampled under wobble conditions consider the variance in the projections. The linearity of the filtered backprojection algorithm allows us to do this, and using a squared filter has no effect on the linearity property, (c.f. equation 2.11 and equation 3.3). Consider the variance in the projections of a non-attenuating phantom, uniformly filled with positron emitting radioactivity, for both uniform sampling and wobble sampling conditions. Since the average wobble normalization coefficient has been made equal to one, variances can be compared on the basis of equal total number of recorded events in each case. The noise-free projections of a cylindrical phantom located on the tomograph axis are identical at all angles and are elliptical in form. To a first approximation, the projection can be considered to be made up of a series of straight line segments, with one segment spanning each wobble cycle. Since the normalization weights, Wj are symmetric about the centre of a wobble cycle, it can be easily shown that Z° p./w. « p Z D 1/w. (3.5) i = l 1 1 i = l 1 where p is the average modulated projection value across one wobble cycle, and is the number of bins in a wobble cycle. Therefore with this simplification, the ratio, R y , of average variance under wobble sampling conditions to that of uniform sampling conditions is given by R v - 1 /N b L° l/wj (3.6) Define the arithmetic mean over a wobble cycle as m A = 1/N b 2 ° wj = 1 (3.7) the harmonic mean as m H = N b / 2 D l/wj (3.8) and note that R y is simply the ratio of the two, or R v = m A / m H (3.9) 33 From the harmonic-arithmetic mean inequality [42], the harmonic mean is always less than or equal to the arithmetic mean, with equality holding if and only if all weights, Wj, are equal. We can therefore conclude that (given the earlier approximation) is always greater than unity, or that uniform sampling is always the most efficient scheme in terms of minimum average variance. To see when this approximation may not hold, consider a point source of activity in the centre of the field of view. Since the activity is highly localized, the significant weighting terms will be those at the centre of the wobble cycle for even projections, and those at the edge of the wobble cycle for odd projections. The average of the inverse of these two values may be less than unity, and if so, the average variance in the image will be lower than that for a uniformly sampled point source. This special case will not normally arise in PET studies. A complex radiator will normally make contributions to projections over several wobble cycles, and regions of interest will be chosen such that the average variance is increased. 34 FIGURE 3.3 Wobble; analytic variance: Variance images for uniformly sampled phantom (a), wobble sampled without offsetting alternate projection wobble centres (b), and wobble sampled with alternate projection wobble offset (c). 3.3 - Simulations and Calculations To observe the effects of non-uniform sampling in the image, noise-free projection data were computed, for a uniformly-radiating, non-attenuating, 24 cm diameter phantom; These projections were next multiplied by the wobble normalization function according to equation 3.1 above. In order to exaggerate the wobble variance effects, projections were also produced in which the wobble cycle centres were not offset in alternate projections. The variance images were reconstructed using a squared Harm-weighted ramp Filter. Figure 3.3 shows the resulting variance images for the uniform and wobble sampled phantom. FIGURE 3.4 Wobble; statistical variance: Mean (a) and variance (b) images from forty-two reconstructions of simulations employing wobble sampling. Using a Poisson noise generator, noise was added to computed projections of the same phantom as above. The variance of the noise was adjusted for each bin according to equation 3.1. Reconstructions were done for forty-two sets of projections in which the noise samples were statistically independent from bin to bin, and from projection to projection. Alternate wobble centres were not used in order to exaggerate the wobble effects. The point by point mean and variance images were computed and these results are shown in Figure 3.4. The concentric family of circles that shows up in the analytic variance image (figure 3.3 (b)) is clearly visible in the statistical variance image. FIGURE 3.5 Wobble; point sources: (a) uniformly sampled point source centrally located, (b) to (f) variance images for wobble sampled point source at progressive displacements from the centre. The next simulation, was performed to study the variance properties of a point source in varying positions. Noise-free projections for a point source at five different positions were computed, starting at the centre, and moving towards the periphery of the field of view. Again wobble cycle centres of alternate projections were not offset in order to exaggerate the wobble effects. The resulting analytical variance images are shown in figure 3.5. The effect in the variance image is to produce radial steaks converging at the point of activity and varying in number. To understand this effect it is helpful to employ a sinogram image. A sinogram is achieved by assembling projections to form a rectangular array in which position in 37 FIGURE 3.6 Sinograms: Sinograms of the four off-centre variance images in figure 3.5 (i.e. (c), (d), (e), and (0). Size (cm) 27 18 11 Rv (no offset) Rv (offset) 1.1853 1.1895 1.1887 1.1887 1.1874 1.1892 TABLE I Average Variance Ratios. projections is plotted horizontally, and projection angle is plotted vertically. Figure 3.6 contains sinograms of the four off-centre variance images of figure 3.5. It is clear from the sinogram representation that as the point of activity approaches the periphery of the 38 field of view, it sweeps a path covering a greater variation in position in the projections. The wobble normalization function can be observed in the sinogram of the variance image as . a modulating effect that is periodic in the horizontal, and vertically invariant The number of wobble cycles that are traversed by the point source in the sinogram corresponds to the number of radial streaks in the variance image. I To verify that the average variance is increased for wobble sampling conditions, the variance images of figure 3.3 were averaged over three square regions of interest The regions of interest had sizes of 27 cm, 18 cm, and 11 cm, on a side, and were centred on the images. The ratios of averages taken within the regions of interest between the wobble sampled phantoms and the uniformly sampled phantom were computed and the results are summarized in table I. Using the PETT VI wobble parameters, the theoretical value for R y as defined in equation 3.6 was computed to be 1.1889, which agrees very well with the measured average variance ratios. This appears to justify the simplification made in deriving R y . 39 IV - DATA CORRECTIONS AND IMAGE-PLANE VARIANCE The scanning protocol for the UBC/TRIUMF PETT VI involves three separate data collection procedures called normalization, transmission, and emission scans. In the normalization and transmission scans, a ring source containing 6 ! Ge is placed in the camera. The 6 8 Ge (with a half-life of a little less than one year) decays to 6 8Ga, a positron emitter, with a half-life of 68 minutes. In this way, a relatively stable source of external activity is maintained. The ring source has an internal dimameter of 27.5 cm which is large enough to encircle the patient's head and head support apparatus. Each of the three scanning procedures are done with the tomograph acquiring data while wobbling as described earlier. The data collected by each detector channel is then "rebinned" and normalized with the wobble normalization function described above. The resulting profiles or projections are nj(u), yj(u), and ej(u), for normalization, transmission, and emission respectively. Index j is the projection number, 1 < j < m (m is 72 in PETT VI), which represents angle, and u is the position in a projection. The normalization scan is done before the arrival of the patient, with no objects (other than air) in the field of view of the camera and with the camera in low-resolution mode (high sensitivity). The purpose of the normalization scan is to calculate a constant, N 0 , which represents the average ring source intensity. The primary purpose of the transmission scan is to measure the attenuation profiles for the patient It also has the effect of allowing correction for detector efficiency effects. This scan is done prior to administration of the radioactive tracer, with the patient in position, the ring source surrounding the head. The measured transmission profiles, yj(u) can be expressed as y j(u) = N„ S(u) F L j (u) Aj(u) (4.1) 40 where N 0 is the average source intensity, S is the ring source profile distribution, is the low-resolution efficiency curve, and A is the attenuation profile. The ring source profile distribution, S, and F ^ are functions that are determined experimentally, and do not change over time. The emission run is normally done in high-resolution mode, with the patient in position after the radioactive tracer injection has been made. The measured emission profiles, ej(u) can be expressed as e j(u) = Cj(u) Aj(u) F H j ( u ) + r W(u) (4.2) where Cj(u) is the ray-sum of activity for projection j, position u, FJ_J is the high-resolution efficiency curve, r is the random coincidence contribution, and W is a rectangular window that delineates the field of view. As mentioned previously, the random contributions are assumed to be constant for all projections and estimated by taking an average of counts recorded by detectors channels whose rays fall just outside the field of view. We can solve for Cj(u), the corrected projections, using the above three equations as Cj(u) = N.Mj(u) [ej(u) - rW(u)] / y j(u) (4.3) where M is given by M.(u) = S(u) F L j (u) / F H j (u ) (4.4) and is an empirically determined function that is unchanging over time. M is called the set of "ring factors". Finally, the corrected ray-sum of activity, Cj(u), is used in the filtered backprojection algorithm mentioned previously. The resulting image is thus corrected for random coincidences, detector efficiency characteristics, and gamma-ray attenuation. 4.1 - Noise Backprojection The assumption made in the equations for image plane variance in Chapter 3 are not strictly true for the corrected projections in the PETT VI reconstruction procedure. Since a single random estimate is subtracted from all projections the corrected projections are correlated. For the purpose of analyzing the noise propagation, c can be approximated by Cij(u) represents the portion of the corrected projection in which noise samples between bins and between projections are independent, and c2j(u) the dependent portions of the corrected projections. The "bar" notation has been used to indicate statistical expectation here and elsewhere, thus "y j(u) is the expected' value of >j(u). Using the linearity property of statistical expectation, equation 2.11 becomes Cj(u) - N„M j (u ) [e j (u ) /y j (u ) - rW(u)/yj(u)] (4.5) or C:(u) =* cM(u) - c2.(u) (4.6) Hc(x,y)} = (I) Z Fic.(u)} h(x'-u) du i = 1 J (4.7) Now introduce the notation; ACj(u) = c.(u) - HCJ(U)} (4.8) then we can express the image-plane variance as m „ _ m a2(x,y) = H /" 0 BAc.(u)h(x'-u)du • /! 0 0Ac k(v)h(s'-v)dv } (4.9) j — 1 -k — 1 where s' = x costf>k + y cos0k (4.10) Once again, the linearity properies of expectation are employed to obtain m m a> a> oJ(x,y) = (l)1 2 2 HAc,(u)Ack(v)}h(x-u)h(s-v) dudv (4.11) j = lk = l J Finally, this is expanded by substituting the separated form of Cj(u) m m o-2(x,y) = (Z)2 2_ 2_ HAClj(u)AcIk(v)}h(x-u)h(s'-v)dudv j — 1 k — 1 m m + 2 (I) 2 2_ i 2 _ i SZBSZC HAc1:J(u)Ac2k(v)}h(x'-u)h(s-v)dudv m m + (^)2 2 2 HAc2:(u)Ac2k(v)}h(x-u)h(s'-v)dudv (4.12) j = l k=l J Now these three additive terms can be simplified as follows: The middle term is identically zero, i.e. H AC l j(u)Ac 2 k(v) } = 0 (4.13) because the noise in the random estimate is independent of the noise in the emission and 44 transmission data. The first term in equation 4.12 is simplified by observing that 0 j^k , u * v H Ac,.(u)Aclk(v) } = { (4.14) ^2 cj(u) j = k, u = v where a 2 C ] j(u) - N0 2MYu)[a2 e j(u)/yYu) + a2yj(u)eJj(u)/y4j(u)] (4.15) and finally E Ac2 j(u)Ac2 k(v) } * No'a'jM'jMW'OO/y'jOi) (4.16) So equation 4.12 can be re-written as m _ <x2(x,y) - N„ J { ( l ) J E_ i /_ 0 B a' e j (u)M 1 j (u)/y 1 j (u)h 1 (x '-u)du m 1 1 1 «, + (s,)^ ; r . « ' 1 v i ( u ) M , i ( " ) / y 4 i ( u ) e J i ( u ) h 2 ( x ' - u ) d u m i = i °° yj v~'"" J V"' '' J V m _ + o\ [ ( ^ ^ . . M j ^ W C u J / y j ^ h C x ' - ^ d u ] 2 ] (4.17) In this equation, the variance effects of emission, transmission, and random corrections are represented as separate additive terms. The image-plane variance functions for each of these effects can be calculated separately by using simple modifications to the standard backprojection algorithm (equation 2.11): For emission and transmission effects, the Hann-weighted ramp filter, h, is replaced by its square. For random coincidence effects (third term in equation 4.17) the usual Hann-weighted ramp Filter is used, but the FIGURE 4.1 Uniform phantom: Image (a), and variance image (b) for a non-attenuating, uniformly-emitting, cylindrical, 20 cm phantom. Also shown, (c) and (d), are plots of lines through the centres of (a) and (b) respectively. resulting image is squared. Equation 4.17 forms the basis for the work in this chapter. Each correction is isolated to determine its effect on the image variance. With this analytical expression, variance computation is relatively straight forward using computer simulations. As a First example, all corrections were ignored and noise free, analytically-determined projections were produced on the computer for a non-attenuating, uniformly-emitting, cylidrical, 20 cm phantom. These projections were convolved with a Gaussian PSF of 8mm F W H M , which is a first approximation to the PSF of the PETT 46 VI [28]. Since no corrections are to be applied, the same projections serve as estimates of the projection variance (c.f. equation 3.3) and the image and variance image were reconstructed using the normal filtered backprojection algorithm, utilizing a Hann-weighted ramp filter, and squared Hann-weighted ramp filter respectively. The resulting images are shown in figure 4,1. While the phantom image has uniform intensity within its border, the variance image is bell-shaped, with a peak at the centre, and tails that extend beyond the phantom boundary. 4.2 - Efficiency Correction An efficiency correction is employed, as mentioned earlier, to correct the projection data for the drop in detector sensitivity with increasing angle of .incidence. For a non-attenuating, random-free object, sampled uniformly the transmission data expression, equation 4.1, becomes yj(u) = N 0 S(u) F L j(u) (4.18) and the emission data expression, equation 4.2 becomes e j(u) = c.(u) F H j (u) (4.19) Recall that FJJJ(U) and FLJ ( u) ^ ^ e ^Sh. low-resolution efficiency characteristics, and these experimentally determined curves were shown in figure 2.4. The image-plane variance expression, equation 4.17, in this case reduces to o2(*,y) = ( | I ) , Z_ i / ! ! 0 B a J e j (u) /P H j (u)h i (x ' -u)du (4.20) In the idealized case where sampling is uniform along projection axes, the emission profiles follow Poisson statistics, therfore aVu) = ¥ J ( U ) = ^ j ( u ) F H j ( u ) ( 4 2 1 ) and the image-plane variance expression becomes _ m c2(x,y) = (fj) 2! ;_ s , c i (u) /F H i (u)h J (x ' -u)du (4.22) j = l J 48 UBC/TRIUMF PET PROGRAM FIGURE 4.2 Effect of efficiency correction: Image shows the effect of the PETT VI high-resolution efficiency characteristics on noise distribution. Variance image (a) and plot along a central line (b). Noise-free, analytically- determined projections for a uniformly-sampled, uniformly-emitting, non-attenuating, random-free, 20 cm phantom were produced on the computer. These projections, Cj(u), were then divided by the high resolution efficiency function and backprojected using a squared, Hann-weighted ramp filter. The resulting variance images are shown in figure 4.2. These results can be compared' with those of figure 4.1 which represent image-plane variance for a machine with an ideal (flat) efficiency characteristic. The effect of the PETT VI high-resolution efficiency characteristics which exhibit severe efficiency fall-off towards the edge of projections, is to flatten-out the image-plane variance function. 4.3 - Random Coincidence Correction The procedure described by Ficke et al. [50] and mentioned earlier, is employed to estimate the approximate contribution of random conicidences. It is assumed that contributions are uniform in all projections and therefore a count average from detector channels outside the field of view is used as the estimate, and subtracted from each projection (c.f. equation 4.3). For non-attenuating objects sampled uniformly by a machine with ideal efficiency characteristics (FTT-(U) = 1), the emission data expression, equation 4.2, becomes e,(u) = c:(u) + rW(u) (4.23) and the transmission data expression, equation 4.1, becomes y.(u) = N„ S(u) (4.24) Projections, corrected for randoms are simply c,(u) = e,(u) - rW(u) (4.25) Therefore the image-plane variance expression, equation 4.17, reduces to (4.26) FIGURE 4.3 Effect of random coincidence correction: Contribution to image-plane variance due to random estimate subtraction. Variance image (a) and plot along a central line (b). Equation 4.26 is simply the variance image expression that was obtained earlier while ignoring all corrections, added to a term which is a squared image multiplied by the variance of the random estimate. The squared image is that which arrises from the reconstruction of "flat" projections of value 1 (inside the field of view). This squared image, representing the variance effects of the random correction, is shown in figure 4.3. It is clear from the figure that the contribution of this term to the image-plane variance is maximum at the edge of the field of view. The random estimate is obtained by averaging N r (N f = 432 for PETT VI) samples from detector channels located outside the field of view. The number of recorded 51 events follows Poisson statistics and therefore ° \ = T / N r ( 4- 2 7) The emission data also follow Poisson statistics, and therefore f J e j ( u ) = ejOl) = C . ( U ) + TW(u) (4.28) The image-plane variance expression now becomes m tf2(x,y) - (I)'Z J_Jc>)+rW(u)]h 2 (x--u)du j = l J + T / N [(I)Z ;* t tW(u)h(x'-u)du] 2 (4.29) j = l The variance images for a 20 cm phantom were reconstructed according to equation 4.29. These reconstructions assume ideal detector efficiency characteristics, and a non-attenuating, uniformly-emitting phantom. The variance image resulting from the second term in equation 4.29 is shown in figure 4.3 along with a plot of a central line through the image. The largest contribution due to random coincidence estimate subtraction to image variance occurs at the periphery of the field of view. FIGURE 4.4 Effect of attenuation - uniform phantom: Variance due to noise in emission profiles (a), and transmission profiles (b), and plots along a central lines (c) and (d) of (a) and (b) respectively. 4.4 - Attenuation Correction For the case of an attenuating radiator, sampled uniformly in the tomograph with ideal efficiency characteristics, and no random coincidences or scattered radiation, the expression for the emission profile data becomes e j(u) = Cj(u) Aj(u) (4.30) and the transmission profile data FIGURE 4.5 Effect of attenuation - point source, centred: Phantom outline and point source (a), variance due to noise in emission profiles (b), and transmission profiles (c). yj(u) = N 0 S(u) Aj(u) (4.31) If we further assume that the ring source distribution is flat (S(u) = l), then the variance in the image is given by m o2(x,y) - (J,) 1! ;_.a 2 ^/AMuJh^-uJdu j = l J J + ^ \ (£)2£_ i/!00^2yj(u)c2j(u)/A'.(u)h2(x-u) du (4.32) FIGURE 4.6 Effect of attenuation - point source, 6 cm: Phantom outline and point source (a), variance due to noise in emission profiles (b), and transmission profiles (c). Define a2 ie(x,y) = (|,)Jf_ ; . e , ff ' e ! J (u)/A l j (u)h i (x'-u)du (4.33) J and o\fry) =N^ 3 ( i ^ ^ r . a ' y j W j W / A ^ h V - ^ d u (4.34) so that FIGURE 4.7 Effect of attenuation - point source, 10 cm: Phantom outline and point source (a), variance due to noise in emission profiles (b), and transmission profiles (c). a2(x,y) - a2 i e(x,y) + a2iy(x,y) (4.35) The contributions to image-plane variance are thus separated into the variance due to noise in emission projections, a2je(x,y), and that due to noise in transmission projection data, o2jy(x,y). Now, for uniformly sampled projections, both emission and transmission data follow Poisson statistics therefore, these two terms become m _ o2 ie(x,y) = ( i ) 1 I_ i /!! 0 B C j (u) /A j (u )h I (x ' -u )du (4.36) and I 1 _ 111 o\y(*,y) = N-J, ( ^ ^ ^ ^ c Y ^ / A ^ h ^ x - ^ d u (4.37) Simulations were done on the computer to observe the variance characteristics for a uniformly-attenuating, scatter/random- free 20 cm phantom. In the first case, the activity was uniformly distributed, and in the second, a series of point sources were located at several radial positions within the phantom. In each case, the linear attenuation coefficient, M, was chosen to be 0.096 cm"1 which corresponds to that of soft tissue in the brain. The results of these simulations are shown in figures 4.4, 4.5, 4.6, and 4.7. The two terms contributing to image variance, a2ie(x,y) and a2jy(x,y) are illustrated separately in the figures. V - REGIONAL MEASUREMENTS Due to the nature of PET imaging, it is usually desirable to make regional estimates of activity obtained by averaging the image intensity over a region of interest (ROI). While the variance images obtained in the simulations of Chapter 4 are useful in studying noise propagation in a qualitative manner, they do not contain enough information to compute the variance of ROI measurments. This is due to the fact that while noise in projections is statistically independent, it becomes correlated through the reconstruction procedure. The Hann-weighted ramp filter, and the backprojection process itself produce an image which is highly correlated. The variance images cannot therefore be averaged to obtain ROI variances. This chapter explores the covariance properties of images reconstructed by the filtered backprojection procedure. As it turns out, a covariance description of reconstructed images, while illustrative, is an impractical tool in computing ROI variances. A method attributable to Huesman [32] is presented, and modified for use with the PETT VI reconstruction procedure. This method is computationally more efficient than calculating image covariance matrices. 5.1 - Image Covariance Equation 2.11 from chapter 2, describes image-plane activity concentration for the ideal imaging conditions (no attenuation, scattered radiation, or random coincidences, and ideal detector efficiencies). The image-plane concentration, c(x,y), is a function of the projection-space concentrations, Cj(u), and the backprojection filter, h(u), i.e. _ m c(x,y) = ( £ , ) ! j ^ c O O h t x ' - ^ d u j = l J where x' = x cos 0j + y sin 0j The covariance between points (x,y) and (s,t) is by definition, Covar[c(x,y),c(s,t)] = H [c(x,y)~ c(x,y)] [c(s,t>-c(s,t)]} (5.1) Since under these ideal imaging conditions, projection data values at different positions on the projection, and beween projections, are independent, it can be easily shown that Covar[c(x,y),c(s,t)] = ^^Z^^xx) h(s-u)h(x'-u)du (5.2) where s' = s cos 0j + t sin c6j (5.3) Making use of the fact that, h(u) = h(-u) (5.4) FIGURE 5.1 Covariance weighting: Hann-weighted ramp filter used as for backprojecting covariance images with respect to a fixed point (s,t). equation 5.2 can be expressed as _ m _ Covar[c(x,y),c(s,t)] = (lyL j"00g.(u,Oh(x'-u)du j = l J where g.(u,s) = a 2 .(u) h(u-s) 60 FIGURE 5.2 Image-plane covariance function: Points, (s,t), located at progressive displacements from the centre of a 20 cm phantom. Legend (a), arrows in (c) and (d) point to the phantom centre. For a fixed point (s,t), equation 5.5 is really just the filtered backprojection equation with the regular Hann-weighted ramp filter, and multiplied by a constant 7r/m. Keeping in mind that the point s' is the perpendicular projection of the point (s,t) onto the projection axis, then the function g represents the projection variance, a J H(u), windowed by the Hann-weighted ramp filter, centred at the point s'. This is illustrated in figure 5.1. 61 Equation 5.5 suggests an algorithm for producing covariance "images". That is; for a fixed point (s,t), the image-plane covariance function can be computed by windowing the variance profiles according to equation 5.6, then using the standard filtered backprojection algorithm. Figure 5.2 illustrates the results of applying this algorithm to points at various locations within a uniformly-emitting, non-attenuating, random/scatter-free, 20 cm cylindrical phantom. It is clear from the figures that the covariance function is complex, its shape and symmetry being dependent on location in the phantom. Furthermore, an algorithm that involves reconstructing a covariance "image" for each point in an ROI, would be impractical as a method for computing ROI variances. For this reason, an alternative method must be developed. 62 5.2 - ROI Variance As mentioned, a typical measurement obtained from a PET image involves integrating over a number of pixels, i.e. for a measurement a , a = 1/A J 7 R c(x,y) dxdy (5.7) a wh'ere R Q is the two-dimensional region of interest (ROI) with area A and c(x,y) is the concentration of activity in the image-plane. Substituting the expression for c(x,y), equation 2.11 from chapter 2, into this expression, we get a = 1/A ; / R (f.)I /" 0 Bc i(u)h(x'-u)du dxdy a j = 1 J where, as before, x' = x cos </>j + y sin <t>^ Now re-ananging this expression, yields m a = 1 / A a <m> 2_ i ; _ 0 , c j (u ) ; ; R a h(x-u)dxdy du (5.9) or a = 1 / A a c j ( u ) g a j ( u ) d u ( 5 - 1 0 ) where (5.8) g aj ( u> = /J"R h ( x ' - u ) d x d y ( 5- n) As pointed out by Huesman [32], this expression is very useful for two reasons. The first is that since gQj(u), is a function only of the area, and the backprojection filter, it can be calculated once and applied to many sets of projection data. This is useful in dynamic studies where fast calculation of a few ROI measurements is required. The other reason why this expression is powerful is because it simplifies the calculation of the ROI measurement variance, since g „ : ( u ) is a deterministic function. Using the linearity property of statistical expectation, and the same notation as in chapter 4, where A denotes deviation from the mean, then the variance in the ROI measurement is Recall that we can separate the projection data into two functions, Cij(u), and c2j(u), where the noise in Cij(u) is independent between positions on the projection axes, and between projections, and the noise in c2j(u), while independent of that in Cij(u), is a single value, i. e. Simplifying this equation in a similar manner as for 4.17 in chapter 4, we obtain (5.12) a = < 1 / A a ) ( m) [ ' - » Clj<u) &aj ( u ) d u + c ' j ( u ) &a j ( u ) d u ] (5.13) + o\ [ 2_ ; ! ' 0 0W(u)M j(u)/y j(u) g a j(u) du ]' } (5.14) 64 o o ibToQ 4 ^ 0 0 a T o O 12.00 18.00 20.00 24.00 A r e a ( p i x e l s ) FIGURE 5.3 ROI variance versus area - Emission noise: Circular ROI's centred at a point 5 cm from the centre of a 20 cm cylindrical phantom (squares) compared to similar ROI's placed on an uncorrelated image (circles). A program for calculating a2a has been implemented on the computer. To determine the effects of ROI area on ROI variance, computations were made for a 20 cm uniformly emitting, uniformly-attenuating (M = 0.096) cylindrical phantom. The efficiency characteristics for PETT VI were used and the random and scatter contributions were ignored. Circular ROI's with increasing diameter were placed at a radius of 5 cm. The variance terms due to emission noise and transmission noise were calculated separately. The results of this study are summarized in the graphs of figures 5.3 and 5.4 (squares). Also plotted (circles) is the theoretical variance versus area curve that would apply to ROI 65 °0J00 4T0O iToO 12.00 16.00 20.00 24.00 A r e a ( p i x e l s ) FIGURE 5.4 ROI variance versus area - Transmission noise: Circular ROI's centred at a point 5 cm from the centre of a 20 cm cylindrical phantom (squares) compared to similar ROI's placed on an uncorrected image (circles). measurements on an image containing uncorrected noise. The obvious effect of noise correlation through the filtered backprojection is to reduce the rate of ROI variance reduction as the ROI area is increased. For example, considering only the noise due to emission, to obtain an ROI variance which is approximately 20% of the central pixel variance, an ROI with an area of 11 pixels would be required on a PETT VI image. This is considerably larger than the 5 pixel area required for the same noise reduction on a theoretical image with uncorrected noise. VI - EXPERIMENTAL VERIFICATION In an effort to verify the analytical predictions of the previous chapters, a phantom experiment was designed and performed on the UBC/TRIUMF PETT VI tomograph. 1 8 F - F D G was added to water to create a uniform source of activity. The standard scan protocol was followed: Before the activity was placed in the phantom, normalization scans, followed by transmission scans were performed in low resolution mode. Finally emission scans were performed in high-resolution mode, following injection of approximately 1 mCi of 1 8 F into the phantom. With this low level of activity, the rate of random coincidences is negligible. The experiment made use of the standard 20 cm calibration phantom that is used routinely for machine checks. The goal of the experiment was to isolate, and statistically measure the image and ROI variances due to emission and transmission noise. FIGURE 6.1 Experimental data; statistical variance images: Images represent variance due to noise in emission data (a), and that due to noise in transmission data (b) for a 20 cm uniform phantom. 6.1 - Phantom Measurements Twelve sequential scans were performed in both transmission and emission modes. The emission scan duration was progressively increased to keep the total number of recorded events per slice (slice sum) constant as the 1 ! F decayed. The number of independent samples for each scanning mode was increased to forty-eight by considering the four true-slice data sets in each scan to be independent sets of samples from the same distribution. This is justified by the cylindrical geometry of the phantom and because the four detector rings behave identically and independently. Data for "cross" slices was o-. 20.00 .00 60.00 80.00 RADIUS (MM) 100.00 120.00 140.00 FIGURE 6.2 Statistical variance - emission noise: Plot of variance due to noise in emission data along a radial line in a 20 cm uniform phantom for statistical measurements (points) and analytical predictions (solid line). discarded because the efficiency of these differs from that of "straight" slices. Forty-eight images were reconstructed with the standard PETT VI software, from the forty-eight independent sets of emission data corrected for attenuation by profile averages from the forty-eight sets of transmission data. Since the same average set of transmission profiles was used in all emission reconstructions, fluctuations for any point in the set of images is due solely to noise in emission data. In a similar manner, forty-eight images were reconstructed from the profile averages of the emission data, each one corrected for attenuation by a different set of transmission data. Point by point statistical variance images were computed for the two classes of reconstructed images. These are shown in in figure 6.1 o o © 1TJ-, OJ o O-cu 69 o LU U o ZO <o' HO. > o o o. Ifl o o < = b V 0 0 2 0 . 0 0 40.00 60.00 80.00 100.00 l5o7oO U0 00 RADIUS (MM) FIGURE 6.3 Statistical variance; transmission noise: Plot of variance due to noise in transmission data along a radial line in a 20 cm uniform phantom for statistical measurements (points) and analytical predictions (solid line). The statistical variance images are themselves very noisy and are thus difficult to compare directly with analytical predictions. Averaging by . radius was performed on each variance image and plotted along with the analytical prediction for this phantom and each scanning mode. These results are shown in the graphs of figures 6.2 and 6.3. Statistical calculations and analytical predictions are in good agreement in both cases. Next ROI variance predictions were examined by placing several ROI's in the same position on the forty-eight independent reconstructed phantom images from each class. ROI arrangement is illustrated in the map of figure 6.4 and the results are summarized in table II. In both cases, the predicted ROI variance figures are in good agreement with those measured statistically. The average error between measured and predicted ROI variances is 9.5% for emission data and 18.5% for transmission data. FIGURE 6.4 ROI size and placement map: ROI numbers for calculating statistical ROI variances. ROI # diam. (mm) Emiss i o n V a r i a n c e T r a n s m i s s i o n V a r i a n c e p r e d i c t e d measured p r e d i c t e d measured 1 8.1 442.9 403.8 1065. 1130. 2 16.2 78.1 85.3 212. 208. 3 24.3 31.6 30.1 86.4 70.3 4 32.4 17.5 16.0 49.5 35.4 5 32.4 9.3 10.6 23.3 29.8 6 24.3 20.7 19.6 44.8 55.0 7 16.2 40.1 47.8 108. 133. TABLE n ROI variance. 71 6.1.1 Transmission Scan Duration The method of attenuation correction by direct measurement using a transmission source is in theory very accurate. One of the main disadvantages is that noise from the transmission profiles contaminate the attenuation-corrected profiles. Transmission noise can be reduced by increasing the number of counts recorded in the scan either by increasing the scan duration, or the strength of the source. This, however increases the radiation dose to the patient and either increases the total scanning time or poses potential rate problems for the tomograph. It is necessary to make a trade-off between these factors. The choice of emission scan duration is not usually as flexible, but is made on the basis of practical limitations involved in measuring dynamic metabolic processes. In addition, it is often desirable to add images from several short scans together in order to reduce the total noise. Since multiple emission scans are typically reconstructed with the same transmission data, the image noise due to transmission data is not independent and cannot be reduced in this manner. It is therefore necessary to keep the ratio of image variance due to noise in transmission data to image variance due to noise in emission data low. From equations 4.36 and 4.37 it can be shown that for a uniformly emitting, uniformly-attenuating phantom, the ratio,of image variance due to noise in transmission data, o2jv(x,y), to image variance due to noise in emission data, o2je(x,y), is given by RT(x,y) = (C/N,) q(x,y) (6.1) where C\ has been removed from the equation as the total number of counts recorded per emission slice, Nj is the total number of counts recorded per transmission slice, and q(x,y) is a function which depends only on phantom geometry and machine performance characteristics. From figures 6.2 and 6.3 it is clear that R-p is maximum towards the 72 centre of the phantom. In this experiment, approximately 320,000 counts were recorded per emission slice, and 1,100,000 counts recorded per transmission slice. Ry, in this case was, found to be 3.4. near the centre of the image. Therefore as a guide to determining transmission scan duration, to keep the maximum value of Ry below 1.0, approximately ten times the number of counts must be recorded in transmission mode as in emission mode. UBC/TRIUMF PET PROGRAM FIGURE 6.5 PET real data: F D G (a) and nuorodopa (b) PET slices selected to demonstrate variance calculations. Legends (c) and (d) correspond to ROI's placed on images (a) and (b) respectively. 6.2 - Clinical Examples As a final illustration of the application of the methods developed in this work, some variance images and ROI variances were computed for human subject data. Data was collected in each case by the UBC/TRIUMF PET research group using the PETT VI tomograph. Figure 6.5 shows the slices selected from F D G and fluorodopa PET studies overlaid by sets of ROI's typical for these studies. The first patient study involved injection of 4.2 mCi of F D G . This slice was reconstructed from data obtained over a 15 minute duration beginning 60 minutes after injection. The second patient study selected for FIGURE 6.6 Variance image - F D G : Variance due to noise in emission data (a) and variance due to noise in transmission data (b). illustration involved injection of 2.5 mCi of fluorodopa and a 10 minute data collection period, 150 minutes after injection. Variance images were produced on the computer and these are shown in figures 6.6 and 6.7. Both transmission variance images and emission variance images are shown separately for each case. The total variance images would be obtained by adding the two contributing terms. Since each image is scaled to fill the full brightness scale, they cannot be compared for absolute variance contribution. The ratios of transmission variance to emission variance, R T , and the ratios of total emission counts recorded in the slice to total transmission counts recorded in the 75 UBC/TRIUMF PET PROGRAM FIGURE 6.7 Variance image - fluorodopa: Variance due to noise in emission data (a) and variance due to noise in transmission data (b). Max. Var. Trans. Max. Var, Emission C l N l RC R T / R C Phantom 15040 4391 3.4 321505 1109354 0.29 11.7 Patient 62240 23850 2.6 1499743 9799625 0.15 17.3 (FDG) Patient 140 1710 .082 101524 8508910 .012 6.8 (Fdopa) TABLE IH Transmisson and Emission Variances and Slice Sums. 76 ROI # Mean Var.(Em.) V a r . ( T r . ) RMS N o i s e (%) 0 1393. 3263. 7033. 7.3 1 1505. 3486. 8425. 7.2 2 1563. 16. 32. 0.4 3 1412. 3068. 6547. 6.9 4 1402. 3394. 8311. 7.7 5 1839. 3494. 8497. 6.0 6 1678. 3404. 8551. 6.5 TABLE IV F D G ROI Variances. ROI # Mean Var.(Em.) V a r . ( T r . ) RMS N o i s e (%) 7 172. 223. 25. 9.1 8 179. 217. 24. 8.7 9 196. 223. 25. 8.0 A 190. 226. 27. 8.4 B 117. 170. . 13. 11.6 C 115. 159. 11. 11. 3 TABLE V Fluorodopa ROI Variances. slice for each of these studies and for the 20 cm phantom experiment are listed in table III. It can be seen that a rough linear relationship exists between the two ratios and that the comments made above concerning the choice of transmission scan duration for a 20 cm phantom can be extended to F D G and fluorodopa studies. It is interesting to note that in the case of the F D G study, image noise is made up mainly of transmission noise. Image noise in the fluorodopa image on the other hand, is largely the result of noisy emission data. The results of ROI calculated variances for the F D G and fluorodopa slices shown above are summarized tables IV and V. The percentage noise figure is calculated as the square-root of the total noise (due to both transmission and emission) divided by the average signal over the ROI. Noise varies between 0.4% and 11.6% with generally higher values from the fluorodopa data than from F D G . vn - CONCLUSIONS 77 A method for reconstructing analytical variance images from emission and transmission projection data in positron emission tomography has been described. This method involves computing profile variance functions and reconstructing images with a modified form of the filtered backprojection algorithm. A review of the theory of imaging with PET was given along with a description of the data corrections that are implemented as part of the UBC/TRIUMF PETT VI reconstruction procedure. The detector wobble motion and the subsequent profile normalization which are employed in PETT VI have been studied and their effects on image variance have been observed. Based on the assumption that projection data from uniform cylindrical phantoms can be approximated by straight line segments, it was shown that the most efficient sampling scheme is uniform sampling. In a trade-off against improved sampling density, non-uniform sampling due to wobble motion always increases the average variance in an image. The average variance increase due to wobbling for a cylindrical phantom has been computed to be about 19% using the PETT VI wobble parameters. This corresponds to an RMS increase of 9%. It was also shown that this is largely independent of location within the image plane, but dependent on the sum of the inverse of the wobble normalization coefficients. The increase in variance is a direct result of the increase in sampling density non-uniformity in the projections. A uniform non-attenuating cylindrical radiator and point sources were reconstructed using the analytical variance expression with wobble parameters taken from PETT VI. The large phantom variance image exhibits circular symmetry and radial ripples. The variance images of point sources exhibit radial streaking, with the number of streaks being dependent on the distance from the centre of the field of view. These variance patterns are the result of normalizing the projections with a function that is largely periodic. 78 The data correction steps for detector efficiency, random coincidences, and object self-attenuation were studied in Chapter 4. Each correction was isolated and its effect on the image-plane variance function illustrated. The sharp drop-off in efficiency with decreasing angle of incidence characteristic of PETT VI, has the effect of flattening the image-plane variance function for a uniformly-emitting, non-attenuating cylindrical phantom. The random correction as implemented in PETT VI, subtracts a constant value from all projections. This value is obtained from averaging 432 independent estimates of rate obtained outside the field of view. The resulting image-plane variance effect was illustrated and found to be maximum at the edge of the field of view. There are two aspects to the variance effects of an attenuating medium. The first is the effect of attenuation correction on the noise from emission profiles, and the second is the propagation of statistical errors if the correction is done with noisy, transmission data. Both of these aspects were illustrated with simulated data on the computer. The final step in a PET study usually involves ROI measurements being taken from the image data matrix. This involves averaging pixel values within an anatomical area outlined on the image and has the effect of reducing the variance of the measurement from that of the individual pixel values. Image-plane noise is correlated between pixels, complicating ROI variance measurements. A method for computing covariance images for points in an image was developed and demonstrated on a 20 cm, phantom. The above covariance computation proved to be impractical for computing ROI variances and therefore an alternate method was developed. The effect of increasing ROI area on measurement variance was explored using simulated data for a 20 cm phantom. These results were compared with those from a theoretical image containing additive, uncorrected noise. It was shown that for. a given reduction in ROI variance a substantially larger ROI is required on a PETT VI image compared with the theoretical image. A phantom experiment was designed and implemented to measure statistically the pixel and ROI variances. This involved multiple scans of a 20 cm phantom to obtain forty-eight independent emission and forty-eight independent transmission data sets. By reconstructing images from sample emission data sets corrected for attenuation by a common transmission data set, and images from a common emission data set corrected for attenuation by sample transmission data sets, the effects of noise from the two sources was separated. Point-by-point variance images were computed for both classes of reconstructed images and found to be in very good agreement with analytical predictions. ROI variances were also statistically measured by placing identical regions on multiple images. Once again, analytical and statistical ROI measurements were in close agreement Finally, the variance analysis procedure was applied to real PET data obtained from human subject F D G and fluorodopa scans. Mid-brain slices were chosen to illustrate the image variance distribution and ROI variances for typical PET measurements. It was demonstrated that most of the image noise in an F D G study is the result of noise in transmission profiles. 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