NOISE PROPAGATION IN POSITRON QUANTITATIVE EMISSION T O M O G R A P H Y By MATTHEW REX PALMER B.A.Sc, The University of British Columbia, 1981 A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T 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 O F 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 â€¢ 1985 Matthew Rex Palmer, 1985 In p r e s e n t i n g this thesis r e q u i r e m e n t s f o r an of British it freely available agree that Columbia, Library s h a l l make for reference and study. I f o r extensive copying of department or by h i s or be her g r a n t e d by s h a l l not the be of this / ^ / ^ / , ^â€¢/ The U n i v e r s i t y o f B r i t i s h 1956 Main Mall V a n c o u v e r , Canada V6T 1Y3 Date DE-6 (3/81) \J râ€”-y- ^ this A y ? Columbia / H ** r '^1 thesis my It is thesis a l l o w e d w i t h o u t my permission. Department o f further head o f representatives. copying or p u b l i c a t i o n f i n a n c i a l gain University the p u r p o s e s may for the the I agree that permission that f u l f i l m e n t of advanced degree at for scholarly understood in partial written ABSTRACT Image fluctuation noise in described Phi 1 and and Positron projection UBC/TRIUMF simulations, in data. VI The experiments. for noise Tomography variance tomograph phantom analyzed Emission are The (PET) is properties studied PETT propagation of by VI result of images obtained analytical methods, image properties. the reconstruction Procedures for statistical with the computer algorithm estimating is 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 U B C / T R I U M F 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 FDG 0.4% study, and 0.082 for a typical fluorodopa study. Total RMS noise varies between 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 measurements. ii accuracy of quantitative PET Table of Contents Abstract " List of Tables v List of Figures vi Acknowledgements viii I. INTRODUCTION 1 II. T H E O R E T I C A L B A C K G R O U N D 10 1 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 III. N O N - U N I F O R M 23 SAMPLING 27 3.1 Rebinning 30 3.2 Noise Backprojecuon 31 3.3 Simulations and Calculations 34 IV. D A T A CORRECTIONS A N D IMAGE-PLANE VARIANCE 39 4.1 Noise Backprojection 42 4.2 Efficiency Correction 47 4.3 Random Coincidence Correction 49 4.4 Attenuation Correction 52 V. R E G I O N A L M E A S U R E M E N T S 57 5.1 Image Covariance 58 5.2 ROI Variance - VI. E X P E R I M E N T A L VERIFICATION 6 2 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 â€¢ 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 6 image 1.4 Normal water image 7 1.5 Comparison of PET, C T and MRJ 8 2.1 PETT VI scanning geometry 11 2.2 Ideal point-spread 12 function (PSF) 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 â€¢ 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 - 68 emission noise 6.3, Statistical variance; transmission noise 69 6.4 ROI size and placement map '. 6.5 PET real data 70 73 6.6 Variance image - FDG 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 U B C / T R I U M F 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 Research through the Natural Sciences and Engineering 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 are to be synthesis with monitored, is labelled with a positron emitting radioisotope. This involves the of the chemical compound in a laboratory and "labelling" the molecular structure the radionuclide radio-nuclides are n C, produced 1 3 N, 1 5 by 0 , 1 8 F , and an 6! on-site cyclotron. are; ls for glucose metabolic rate studies. labelled water for blood flow studies, and At The medically important G a that decay by positron emission with half lives in the range of two minutes to two hours. Some O in the body 1 ! typical brain imaging compounds F labelled fluoro-deoxyglucose the start of a PET brain study, the patient is injected (FDG) 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 histograms channels are (a channel generated for is each any pair detector of ring crystals). plane (or In the "slice") course of with entries the study, for each detector channel within the plane. The number of counts recorded in a detector channel is an are estimate of the sum of activity along a ray connecting the two detectors. partitioned into sets representing mutually parallel channels called Histograms projections or profiles. The mathematical distribution techniques used to reconstruct from the sets of one-dimensional projections the two dimensional activity 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 S L Louis, PETT VI [23,46,50]. The camera consists of 288 Cesium Fluoride (CsF) scintillation detector sixteen crystals arranged in four circular rings of 72 crystals point wobbling motion is employed to increase photons, coincident in time, arrivingN at crystals the radial within for seven slices (four density. Two a ring or in adjacent rings are recorded by the detection electronics, and tallied in a C A M A C manner, coincidence data sampling each. A "straight" histogram memory. In this slices and three "cross" slices) are tallied by the histogram memory. Machine micro-computer control which from the C A M A C and is monitoring attached to a for PETT VI is performed by VAX-11/750 minicomputer. Data a is PDP-11 transferred memory to the V A X 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 V A X . The in-plane varies typical tomograph resolution from is 9mm to scanning collects about data for seven 8 mm (high-resolution 14mm, and the slice session cross-sectional involves a series separation of paired mode), simultaneously. The the effective (centre data slices to centre) collections, slice thickness is 14 mm. A 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 (MRI or N M R - tomography (CT) and magnetic nuclear magnetic resonance), PET offers resonance imaging 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 C T FIGURE 1.1 PETT VI: An external view of the UBC/TRIUMF head-holder, couch and tomograph gantry. PETT VI showing the 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 exploited studies into brain and other organ as a diagnostic tool. Efforts are being function and in future may made to bring the cost of camera hardware down, as well as to provide simplified chemistry procedures of scanning agents. be for the production 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 radio-tracer researchers with compounds 2-fluoro-2-deoxy-glucose the UBC/TRIUMF available for PET Program human (FDG), fluorodopa, and 15 subject 0-labelled at present have brain three scanning. water are routinely produced at the T R I U M F 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 subject injection following of a series of fourteen F D G . Glucose is slices from a PET study metabolized matter of the brain which is located in the cortex and deep ventricular fluid and white areas in the mid slices. matter show up as dark (low most strongly of a normal by the grey brain nuclei. The combined metabolism) butterfly-shaped Fluorodopa, is the chemical dopa (L-3,4-Dihydroxyphenylalanine) labelled with 18 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 concentration of radio-active has been found to be - the two banana-shaped dopamine in the basal ganglia lower than normal in subjects with regions at in mid-slices. some time after Parkinson's The injection desease or those who have been exposed to the neurotoxin MPTP. 15 0-labelled radio-activity some perfusion the from short to water time is used after various in the injection areas of the is study used of as cerebral an brain. Figure indicator 1.4 shows a PET study of a normal subject following injection of half-life high-efficiency of 15 0, water scans are usually scanning mode to improve image blood li flow. of a the The rate relative of blood series of seven slices O-water. Because of the performed in the low-resolution, 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 can compound under study camera by using be a converted suitable to concentrations metabolic model of [39]. metabolites of the Whereas in C T or MRI, all that is desired is to obtain a visual contrast between types of tissue, or pathological and normal structures, a PET image measurements. For imaging degradation The errors this is actually reason, a special data matrix care is whose values taken to correspond correct the to metabolic data for various effects. involved in PET imaging are categories of statistical and systematic errors. Statistical usually broken down into the two 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 on the errors image on images or are usually regional referred to measurements. The effects as noise. Systematic of statistical errors are those fluctuation 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 0-water. 15 improved, accuracy for since development Corrections of example, by the activity PET, a averaging may lot of for many systematic over a larger not be constant effort has degradations area, over the been made are standard but at larger to the of reduced region. Throughout the improve in most risk the image accuracy. 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 variance. These isolate the various predictions are corrections then verified to obtain by phantom analytic predictions experiments in which for image 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 is often impossible and at best difficult. By simulating the effects with real data 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 an area as interest possible (ROI) usually involves averaging the tracer intensity data over as in order to reduce the effects may vary from just a few large of noise. The size of the region of 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 performed verifying 6 entitled on the Experimental UBC/TRIUMF PETT Verification, presents VI camera. These the results experiments of experiments were directed to 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 U B C / T R I U M F 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. enters the crystal, and interacts pulse by the photomultiplier produced at two crystals to have been A photon, produced from positron annihilation, to produce visible light that is converted to an electrical tube. on opposite annihilated somewhere Coincidences are tallied separately line that intersects gamma When a coincidence is observed between pulses sides of the field of view, the positron is assumed along the line that intersects the two crystals. for each detector channel (a pair of detectors forming a 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 the gamma delays in rays the determine the point along the flight path at which were produced. However, the practicalities electronic pulse creation circuitry make of crystal this response time and 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 axial point-spread triangularly shaped function at the (PSF) centre for of the crystal field geometrical detectors of with view, and considerations rectangular rectangular at that the sufaces is the crystal 11 FIGURE 2.1 PETT VI scanning geometry: surface. This ideal PSF, shown in figure Two segments of a 72-detector circular ring. 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 as the full is a measure of the resolution. It is often expressed 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. camera, This factor, D/2, is often referred to as the intrinsic resolution of the although in practice, it is only a First approximation. A number precise measurement of factors contribute to the degradation of the ideal PSF, and the 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 with projections accumulation is limited and it is usually necessary that have relatively poor statistics. In the following sections and the corrections that are done to compensate the data will be examined. to deal these effects 13 2.1 - Imaging Errors and Compensation The main sources of systematic errors are photon attenuation by the emitting object itself, the Compton scattering random coincidences of gamma radiation within the object, air, and detector of photons arising from separate annihilation events, crystals, and the variation in detector efficiency with angle of gamma-ray incidence. Also, in PET cameras that sampling employ detector UBC/TRIUMF an attempt PETT is made motion to increase the radial density VI), a non-uniform sampling pattern results. to correct the data for these arise from effects (such In most by as the PET systems, compensation in the projections before the image is reconstructed. Two small, unavoidable noting. The positron travels effects a small that distance (its imaging with "range") before positrons are worth 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] PET camera under recently reported resolution measurements development that depended on the isotope in on a high-resolution accordance with its positron range. The second effect is the slight deviation from 180 degrees of the two gamma rays. This has FWHM been measured by Colombino et al. [17] to be a Gaussian distribution with equal to about 0 . 4 Â° . 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 of view is less than 1 mm emitted as a result of annihilation at the centre of the field 14 2.1.1 Projection Statistics It is well known that the number of counts recorded by each detector channel in a PET and system obeys Poisson statistics and is uncorrelated between samples of projections between different projections [25, 12, 1]. A the mean and Poisson distribution has the property that 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 The p = cVc = c (2.1) 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 as the half-life patient comfort of the nuclide involved, radiation dose, and be accumulated, some subtleties such such as 15 2.1.2 Photon attenuation Photons are attenuated in matter according to the formula; N = N0 e ~ (2.2) M X where N 0 is the number of incident photons, N distance is the number after penetration x through an isotropic medium with a linear attenuation coefficient u of a [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 fraction of photons attenuated cannot data. be overlooked The results of in is representative chord length for a human head, the is about 75%. The problem of attenuation then is one that order Huang a to et achieve al. [30] meaningful suggest quantitative that this information correction must from the be done as accurately as possible due to the sensitivity of the information to this effect There are several approaches method is to scan the object, before to correcting data for object self-attenuation. administration of the tracer isotope, with an external source of positron annihilation gamma-radiation. The resulting transmission profiles used to correct the This method is emission profiles accurate, but by division of increases the significantly to the noise in the final image A object second exterior method of correcting for boundary and One calculate the emission radiation dose data to can be by transmission the patient, and data. adds [30]. attenuation chord is lengths to obtain a for all description of the 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 these methods have been uncorrected may attempted lead to along brain image and significant these errors lines are; re-reconstructing As mentioned, errors in in the final image. fitting with an elliptical the calculated Some skull approaches outline attenuation to factors that an [6], automatic computer detection of edges in the projections [7, 29], and using C T images to obtain object boundaries. 17 2.1.3 Compton Scatter Compton degradation of scattering the of gamma-rays PET image in the between form of a their production loss of and resolution. detection Several causes researchers have studied the effects of scattered radiation on the shape of the PSF in PET systems [3, 21, 35]. Bergstrom of quantification can occur in [3] has stressed the importance in PET. He has certain geometries, demonstrated if scatter that of this correction to the a quantification compensation is not accuracy error of about included in 20% the PET imaging procedure. Compton atom scattering of the material, effect can reject at the 511 scattered photons of scatter on the PSF is central peak. These in material giving up some of positron annihilation are camera occurs when a gamma its energy. keV energy by to add can be seen clearly energy photon is The gamma deflected by an rays produced during level and to a limited extent, the PET discrimination long, low-intensity in figure 2.3 as at flanks linear the P M T outputs. The at either side of the in the logarithmic PSF graph, their slopes are dependent of view. The measured scatter distribution also depends on the type of crystals used, size and shape of inter-crystal [19]. For these reasons on position of the source of gamma-rays in the field shielding septa, and the energy it is usually necessary to discrimination threshold determine the scatter setting distribution experimentally for use in an empirical model for correction. The scatter correction methods involve experimental et al. [3], and Egbert et al. [21] both determination of that portion of the PSF due to scatter. This is done by obtaining projections of the scatter flanks Bergstrom for a point source located in a scattering medium, and separation 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 events in a coincidences PET study. occur The between expected photons random generated coincidence by separate count rate annihilation 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]. rate is proportional to the It is important to note that while the true coincidence 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 contribution to measured the individual contribution methods in use to correct activity. The method of Bergstrom detector by two count equation 2.3 rates above. directly, This then method for et al. [5] computing is random accurate coincidence involves the but measuring expected requires random additional hardware and histogram memory space to obtain the additional information. In the PETT algorithm developed and assumes that VI, individual detector by Ficke et al. [50] counting rates are not monitored. The is based on the method of Hoffman et al. [27] 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 assumption averaged to yield a single estimate for the random and scatter fraction. The 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 centrally in the field for of studies where the view Bergstrom et al. [8] of the introduces complex radial and but PETT ring may activity cause VI camera structure distribution problems have into the in the otherwise. shown that distribution events. Such effects will not however be considered here. head Recent detector of is localized studies by wobble motion random coincidence 21 2.1.5 Detector Efficiency t | j The dependent detector efficiency for scintillation crystals employed in PET cameras on the photon angle of incidence. In addition, the collimators used inter-crystal scatter in some designs, such as the PETT VI, cause the is to control effective crystal surface area to be reduced with increasing angle away from a line normal to the detector face. For circular ring detector arrangements, edges well, of inter-ring each projection. coincidences efficiency characteristics The PETT As to obtain the angle of incidence increases towards the multi-ring between-ring cameras (or (such as PETT cross-slice) data, will VI) that exhibit different for true and CTOSS slices. 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 VI . tungsten sheilding the reduce the effective severe efficiency The use fingers are placed over area of the crystal and outer parts of each narrow the shape of the crystal PETT surface to PSF. This causes drop-off towards the edges of each projection. 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 efficiency curves for the U B C / T R I U M F in the high-resolution mode, with data. The experimentally determined PETT VI are shown in figure 2.4. For true slices 100% efficiency arbitrarily set at the centre projection, the efficiency has dropped to about 20% at the edge of the projection. of the 20 40 Position 60 B0~ 100 ( b i n number) 120 B. <u c 3 O > ~i5 *o (3 3 Position (bin number) too lzo 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 one-dimensional projections has a two-dimensional been studied extensively function from as it applies to the radioastronomy, electron microscopy, and more recently, CT, MRI, and PET a set of fields of scanning. There are several methods available for reconstruction from projections; direct Fourier inversion [41], filtered backprojection The field of PET algebraic reconstruction techniques (ART) has drawn heavily on the work in CT the filtered backprojection methods due [2,43] and impractical at the present time. The Consider the most PET systems employ method. While there is renewed interest in the iterative to its potential for better algorithm of Shepp and and [24, 49]. accuracy [44], these methods are slow ART and following analysis applies to the filtered backprojection Logan [43] 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 SlcSlo, c(x,y) P (u) = x' = x cos <j> 0 5(u-x) dx dy (2.4) where and + y sin <p (2.5) 6(u) is the Dirac delta-function. This projection-reconstruction geometry is illustrated in figure 2.5. While the symbol P^ one-dimensional function, if we we is used, to emphasize the fact that each projection is a consider the set of projections at all possible angles <t>, 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 coordinates, and S^(w) two-dimensional of a parallel theorem expresses the projection transform of c evaluated of the its projections; projections, i.e. interpolate rectangular of c expressed in equivalence polar function c, of a one-dimensional Fourier and the two-dimensional along a line through the origin in the two-dimensional plane. This suggests a conceptually simple of transform is the one-dimensional Fourier transform of the projection, P^. The Fourier slice transform Fourier take from individual the set frequency method for recovering the function from a set one-dimensional of grid, finally take the inverse Fourier radial lines Fourier to Fourier transform transforms obtain the of available transform on a to obtain an approximation to c. This is in fact the direct Fourier inversion technique. The computation two-dimensional Fourier functions inversion computationally involved has arise been mostly intensive. in from accurate studied the Some comparisons reconstructions [16]. The interpolations between this from problems required and the irregularly associated which filtered sampled, with tend direct to be 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 Q(xO 0 where x' is given by (2.5) Q 0 (u) where = d* (2.7) above, and rT^S 0 (w)H(cj)] (2.8) 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 in the practice, the truncated ramp, or rectangular reconstructed images. Chesler and Riederer windowing results in ringing artifact [11] have shown that the ripple introduced in the reconstruction process is reduced when using a Hann window. H(w) In = { |CJ| < u)0 |o>|[l + cos(nWcL> )] 0 0 the practical (2.10) otherwise imaging situation, we obtain m discrete parallel projections of a two-dimensional function, c(x,y). The filtered backprojection process can be expressed as c(x,y) = m (Â£) L j= l where h is the spatial m Ci (u) h(x-u) domain representation du (2.11) 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<Â£. (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 al. [9] [14, and Ter-Pogossian 15], schemes, and a "clam et al. [48], a dichotomic ring sampling scheme shell" and the one adopted motion by Huesman in the U B C / T R I U M F et al. [33]. by Bohm et by Cho et al. One of the popular 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 Typically the detection events motion, each wobble are circle is detector divided tallied separately connecting the arc centres traces into for of opposing a a each detectors is non-uniform, with each sample representing circular number arc. path of as arcs shown of Considering the we see in figure 3.1. length and of parallel rays equal set that the spatial sampling pattern an equal time interval, if the wobble rate is constant Due to the complexity involved in projection data are normally sorted dealing with into sets representing non-uniform sampling parallel rays, each projection axis. This process is commonly referred to as patterns, equally spaced on "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 denoted by w^, which is called the set of wobble normalization coefficients, projection, is where i is the bin index. The periodic wobble due to normalization the fact that function the density has of a periodic sample tendency lines through but is not strictly the projection axis increases towards the edge of the field of view. As a first approximation, we can consider it to be diameter periodic, since (PETT normalization difference the field of view size is VI; field of functions for view is even and 25 cm, ring odd between even and odd projections in alternate projections. small compared diameter projections are is to the detector 60 cm). Typical shown in figure ring wobble 3.2. The is due to the way detector elements line up 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 â€¢ t u m 9 9 f â€¢f f f â€¢ * â€¢â€¢* ? u 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 time spent counting events as a function of position samples normalized by this function correctly reflect the variance or noise component in the projection describes the average in the projection. While projection the ray-sum passing through each bin, data is affected by the same wobble normalization function. To consider constrained, by comparisons with Before the making effects its on average over uniformly sampled normalization, the variance, projection a the wobble wobble cycle projections data follow that a normalization equal to function one. is This will contain the same number of Poisson distribution and the first allow events. 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) As before, = Pj(u) / w.(u) u represents position on the projection index (corresponding to projection angle). (3.1) axis, and subscript j is the projection 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 _ & o2(x,y) = Here, o is 2 c the variance number of projections that, m Z j= l ignoring all S_ a'.(u) a h2(x'-u) du (3.2) 1 in the projection, h is the backprojection filter, and x is given by equation 2.12. Alpert et al. [1] corrections, the measured projection can serve as an m is the have suggested estimate of the projection variance, which yields an estimate for the image variance function _ m i^y I j= l <r2(x,y) = Now substituting (3.1) /_â€ž Ci (u) h 2 (x-u) du (3.3) J into equation 3.2, yields the image plane variance expression for a non-attenuating radiator sampled under wobble conditions: o2(x,y) This result is reconstructed = (^) J Z_ /ro B Pj(u) / w j(u)h , (^-u)du very with (3.4) i powerful the because same filtered it means that backprojection a variance algorithm "image" by can replacing be 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 variance on the in the linearity property, projections of a (c.f. equation non-attenuating 2.11 and equation phantom, uniformly 3.3). Consider the filled with positron emitting radioactivity, for both uniform sampling and wobble sampling conditions. Since the average wobble normalization coefficient the basis spanning of recorded to one, events variances in each can be projections of a cylindrical phantom located on the tomograph axis are identical be number equal noise-free to total made on considered equal been compared at all angles and are of has case. The elliptical in form. To a first approximation, the projection can be made up of a series of straight line segments, with one segment each wobble cycle. Since the normalization weights, Wj are symmetric about the centre of a wobble cycle, it can be easily shown that ZÂ° p./w. i= l where p 1 Â« p Z D i= l 1 1/w. (3.5) 1 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 Rv - 1/Nb LÂ° l/wj (3.6) Define the arithmetic mean over a wobble cycle as mA = 1/Nb 2 Â° wj = 1 (3.7) the harmonic mean as mH = and note that R y Rv = N b / 2 D l/wj (3.8) is simply the ratio of the two, or mA / mH (3.9) 33 From the harmonic-arithmetic mean inequality than or equal to the arithmetic mean, with Wj, are We can therefore equal. always greater [42], equality conclude that the harmonic mean is always less holding if and only if all weights, (given the earlier approximation) is 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 equation 3.1 were next multiplied by the wobble normalization function according to above. In order also produced in which to exaggerate the wobble variance the wobble cycle centres were not offset The variance images were reconstructed using a squared effects, projections in alternate were projections. Harm-weighted ramp Filter. Figure 3.3 shows the resulting variance images for the uniform and wobble sampled phantom. F I G U R E 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 projection. Alternate wobble centres were not used bin to bin, and from in order projection to 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 source in next simulation, was performed varying positions. Noise-free to study projections for the a variance point properties source at of a point 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 figure 3.5. The effect effects. The resulting analytical variance images are shown in 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). S i z e (cm) 27 Rv (no o f f s e t ) R (offset) v 18 1.1853 1.1895 11 1.1887 1.1887 1.1874 1.1892 T A B L E I Average Variance Ratios. projections is plotted contains sinograms horizontally, and projection of the four off-centre the sinogram representation angle is plotted vertically. Figure 3.6 variance images of figure 3.5. It is clear from 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 number of effect wobble that cycles is periodic that are in the traversed horizontal, and by the point vertically source invariant The 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 wobble sampled phantoms of averages taken within and the uniformly sampled results are summarized in table I. Using the PETT value well for R y as the regions of interest phantom were between the computed and the VI wobble parameters, the theoretical defined in equation 3.6 was computed to be 1.1889, which agrees very with the measured made in deriving R y . average variance ratios. This appears to justify the simplification 39 IV - DATA CORRECTIONS AND IMAGE-PLANE VARIANCE The scanning protocol for the U B C / T R I U M F PETT VI involves three separate data collection procedures normalization camera. The and 68 called normalization, transmission Ge (with a scans, half-life a transmission, ring of a source and emission containing 6! Ge little less than one year) scans. In the placed in the is decays to 68 Ga, 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 "rebinned" and earlier. The data normalized with resulting profiles and emission described or projections the are respectively. Index j wobble nj(u), collected by each detector channel is then normalization function described yj(u), above. The and ej(u), for normalization, transmission, is the projection number, 1 < j < m (m is 72 in PETT VI), which represents angle, and u is the position in a projection. (other The normalization scan is done before than of air) low-resolution in mode the field (high view sensitivity). The the arrival of the patient, with no objects of the camera purpose of the and with the normalization camera scan in 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 This scan is position, the It also has the effect of allowing correction for detector efficiency effects. done prior to administration of the ring source radioactive surrounding the head. The measured tracer, with transmission the patient in profiles, yj(u) can be expressed as yj (u) = Nâ€ž 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 the low-resolution profile efficiency distribution, S, and curve, and F ^ are A is functions the that attenuation are profile. determined distribution, The ring experimentally, is source and do not change over time. The position emission after the run is normally done in high-resolution radioactive tracer injection has been mode, with the patient in made. The measured emission profiles, ej(u) can be expressed as ej where (u) = is Cj(u) high-resolution rectangular Cj (u) the Aj(u) F H j ( u ) ray-sum efficiency of curve, + r W(u) activity r is the for (4.2) projection random j, coincidence position u, FJ_J is contribution, and W is the a 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)] / yj (u) (4.3) where M is given by M.(u) = and is S(u) F L j ( u ) / F H j ( u ) an empirically determined set of "ring factors". (4.4) function that is unchanging over time. M is called the Finally, backprojection the corrected algorithm ray-sum mentioned of previously. activity, Cj(u), The resulting is used image is random coincidences, detector efficiency characteristics, and gamma-ray in thus the filtered corrected attenuation. for 4.1 - Noise Backprojection The not assumption made in the equations for image plane variance in Chapter 3 are strictly true for the corrected projections Since a single random estimate is subtracted in the PETT VI reconstruction procedure. from all projections the corrected projections are correlated. For the purpose of analyzing the noise propagation, c can be approximated by Cj(u) - Nâ€žMj(u)[ej(u)/yj(u) - rW(u)/yj(u)] (4.5) or C:(u) =* cM(u) - Cij(u) c2.(u) (4.6) represents the portion of the corrected projection in which bins and between projections corrected projections. The here and elsewhere, thus are "bar" independent, and notation has Now = been used samples between dependent portions to indicate statistical of the expectation "y j(u) is the expected' value of >j(u). Using the linearity property of statistical Hc(x,y)} c2j(u) the noise (I) Z i=1 expectation, equation 2.11 becomes Fic.(u)} h(x'-u) du (4.7) J introduce the notation; ACj(u) = c.(u) - HCJ(U)} then we can express the image-plane variance as (4.8) a2(x,y) = m â€ž /" 0 B Ac.(u)h(x'-u)du â€¢ jâ€”1 H _ m /! 0 0 Ac k (v)h(s'-v)dv } -k â€” 1 (4.9) where s' = x costf>k + y cos0 k (4.10) Once again, the linearity properies of expectation are employed to obtain oJ(x,y) = m m 2 2 j = lk = l (l) 1 a> a> HAc,(u)Ac k (v)}h(x-u)h(s-v) dudv J Finally, this is expanded by substituting the separated o-2(x,y) = Now these m m (Z) 2_ 2 _ jâ€” 1 kâ€” 1 2 (I) 2 + (^) three 2 m m 2_ 2 _ i i SZBSZC m m 2 2 j=l k=l additive terms form of Cj(u) HA C l j (u)Ac I k (v)}h(x-u)h(s'-v)dudv 2 + (4.11) can HAc1:J(u)Ac2k(v)}h(x'-u)h(s-v)dudv HAc2:(u)Ac2k(v)}h(x-u)h(s'-v)dudv (4.12) J be simplified as follows: The middle term is identically zero, i.e. H A C 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 H 0 j^k, u * v Ac,.(u)Ac lk (v) } = { ^ 2 c j(u) j = k, u = v (4.14) where a 2 C ] j (u) - N 0 2 MYu)[a 2 e j (u)/yYu) + a2yj(u)eJj(u)/y4j(u)] (4.15) and finally E No'a'jM'jMW'OO/y'jOi) Ac 2 j (u)Ac 2 k (v) } * (4.16) So equation 4.12 can be re-written as m <x2(x,y) - + + In this Nâ€žJ { m 1 1 1 (l)JE_i/_0Ba'ej(u)M1j(u)/y1j(u)h1(x'-u)du Â«, (s,)^ ;r.Â«'1vi(u)M,i(")/y4i(u)eJi(u)h2(x'-u)du m i = i Â°Â° yj v ~'"" J " ' '' J V V m _ o\ [ ( ^ ^ . . M j ^ W C u J / y j ^ h C x ' - ^ d u ] 2 ] equation, corrections are represented for _ the variance effects of emission, transmission, as separate additive terms. The image-plane each of these effects can be calculated separately (4.17) and random variance functions by using simple modifications to the standard backprojection algorithm (equation 2.11): For emission and transmission effects, the Hann-weighted (third term ramp filter, h, is replaced in equation 4.17) the usual by its square. For random coincidence effects 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 isolated to 4.17 determine forms its the effect basis on for the the image work in this variance. chapter. Each this analytical With correction is expression, variance computation is relatively straight forward using computer simulations. As a First example, analytically-determined projections uniformly-emitting, Gaussian cylidrical, 20 PSF of 8mm F W H M , all corrections were ignored were produced on the computer cm phantom. These projections and for were a noise free, non-attenuating, convolved with a 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 reconstructed variance (c.f. equation using the normal filtered ramp filter, and squared 3.3) and the image and variance image were backprojection algorithm, utilizing a Hann-weighted 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, the phantom boundary. with a peak at the centre, and tails that extend beyond 4.2 - Efficiency Correction An data efficiency correction is employed, as mentioned earlier, to correct the projection for the drop non-attenuating, in detector random-free sensitivity object, with sampled increasing uniformly the angle of .incidence. transmission data For a expression, equation 4.1, becomes yj(u) and N 0 S(u) F L j ( u ) (4.18) the emission data expression, equation 4.2 becomes ej Recall and = (u) that these = c.(u) F H j ( u ) (4.19) and F L J ( u ) ^ FJJJ(U) experimentally ^ determined e ^Sh. curves low-resolution efficiency were shown in figure 2.4. The characteristics, image-plane variance expression, equation 4.17, in this case reduces to o (*,y) = ( | ) Z _ / ! ! a ( u ) / P ( u ) h ( x ' - u ) d u 2 In the I idealized case , i where 0B J ej sampling is (4.20) i Hj uniform along projection axes, the emission profiles follow Poisson statistics, therfore a and V u) = the image-plane c2(x,y) = Â¥ J(U) = ^j ( u ) F Hj(u) variance expression ( 4 2 1 ) becomes _ m (fj) 2 ! ;_s,ci(u)/FHi(u)hJ(x'-u)du J j= l (4.22) 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 uniformly-emitting, computer. These non-attenuating, projections, function and variance images are figure 4.1 efficiency which backprojected which shown represent characteristic. exhibit Cj(u), using projections random-free, were a in figure then squared, 4.2. image-plane 20 for cm phantom divided by the Hann-weighted These results variance a for can a were high ramp be fall-off the image-plane variance function. towards the produced resolution the efficiency resulting compared' with those of machine filter. on The The effect of the PETT VI high-resolution severe efficiency uniformly-sampled, with an efficiency edge of projections, is ideal (flat) characteristics to flatten-out 4.3 - Random Coincidence Correction The to procedure described by Ficke et al. [50] and mentioned earlier, is employed estimate the approximate contributions are uniform channels outside the contribution of in all projections field of view is used random conicidences. and therefore as the a estimate, It is assumed count average from and subtracted that detector from each projection (c.f. equation 4.3). For non-attenuating objects sampled characteristics (FTT-(U) = e,(u) and = c:(u) uniformly by a machine with ideal efficiency 1), the emission data expression, equation 4.2, becomes + rW(u) (4.23) the transmission data expression, equation 4.1, becomes y.(u) = Nâ€ž 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 while ignoring all corrections, added variance of "flat" projections expression that to a term which is a squared the variance of the random estimate. reconstruction image of image, representing the variance effects The squared value image 1 (inside was image is that which the field of obtained earlier multiplied by arrises from the view). This squared 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 Nr (Nf = 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 Â°\ The T J e j (u) = N r (4 -27) ejOl) = C.( )+ U TW(u) (4.28) image-plane variance expression now becomes tf2(x,y) - + The 4.29. / emission data also follow Poisson statistics, and therefore f The = These m (I)'Z J_Jc>)+rW(u)]h2(x--u)du J j= l T/N [(I)Z ;* t t W(u)h(x'-u)du] 2 j= l (4.29) variance images for a 20 cm phantom were reconstructed according to equation 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 image variance occurs at the periphery of the field of view. estimate subtraction to 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 ej (u) = Cj (u) Aj(u) and the transmission profile data (4.30) 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 o (x,y) 2 m (J,) ! ; _ . a ^ / A M u J h ^ - u J d u j=l 1 2 J J + ^ \ (Â£) Â£_ /! ^ (u)c (u)/A'.(u)h (x-u) du 2 2 i 00 2 2 yj j (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 a 2 i e (x,y) = (|,) J f_ ; . e , f f ' e ! J ( u ) / A l j ( u ) h i ( x ' - u ) d u J (4.33) and o\fry) so that =^ N 3 (i^^r.a'yjWjW/A^hV-^du (4.34) 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) - The noise a 2 i e (x,y) contributions to + a 2 i y (x,y) image-plane in emission projections, variance a 2 j e (x,y), and (4.35) are thus that separated due to noise into the variance due to 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 o 2 i e (x,y) and = m _ (i)1I_ /!! Cj(u)/A (u)hI(x'-u)du i 0B j (4.36) I 1 o\ (*,y) y Simulations were 111 (4.37) N done uniformly-attenuating, was uniformly _ = -J, ( ^ ^ ^ ^ c Y ^ / A ^ h ^ x - ^ d u on the computer scatter/random- free to observe the variance characteristics 20 cm phantom. In the first for a case, the activity 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 two terms contributing to image in the figures. are shown in figures 4.4, 4.5, 4.6, and 4.7. The variance, a 2 i e (x,y) and a 2 j y (x,y) are illustrated separately V - REGIONAL MEASUREMENTS Due to the nature of estimates of activity obtained PET imaging, by averaging it is usually the image desirable to make regional 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 impractical attributable Huesman [32] is procedure. This method to reconstruction image covariance matrices. an tool presented, is and in computing ROI modified computationally for more use variances. with efficient the than A method PETT VI calculating 5.1 - Image Covariance Equation 2.11 from chapter 2, describes ideal imaging conditions (no attenuation, ideal detector efficiencies). projection-space image-plane scattered The image-plane radiation, concentration, activity concentration for the or random c(x,y), is a coincidences, and function of the concentrations, Cj(u), and the backprojection filter, h(u), i.e. _ m (Â£,)! j^cOOhtx'-^du J j= l c(x,y) = x' = x cos 0j + where 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 Covar[c(x,y),c(s,t)] _ m _ = (lyL j" g.(u,Oh(x'-u)du j= l where g.(u,s) = a 2 .(u) h(u-s) 00 J 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 equation a fixed point with the regular (s,t), equation 5.5 is really just the filtered backprojection 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) the projection axis, then the function g represents the projection variance, onto 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 a the fixed point (s,t), the variance backprojection profiles image-plane according algorithm. Figure to 5.2 covariance equation illustrates function can be 5.6, the then results for computed by windowing using of "images". That is; the applying points at various locations within a uniformly-emitting, non-attenuating, standard this filtered algorithm to random/scatter-free, 20 cm cylindrical phantom. It is clear from the figures that the covariance function is complex, its shape and symmetry involves being dependent reconstructing impractical as a a method method must be developed. on location in the covariance for "image" computing ROI phantom. Furthermore, an for each variances. point For in this an algorithm ROI, reason, an would that be alternative 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 wh'ere R Q the = 1/A J7 R a c(x,y) dxdy (5.7) is the two-dimensional region of interest (ROI) with area concentration of activity in the image-plane. Substituting the A and c(x,y) is expression for c(x,y), equation 2.11 from chapter 2, into this expression, we get a = 1/A ; / a (f.)I /" 0 B c i (u)h(x'-u)du dxdy J j= 1 + y sin <t>^ R (5.8) where, as before, x' = x cos </>j Now re-ananging this expression, yields a = 1 / A a a = 1 / A a m <m> 2 _ i ; _ 0 , c j ( u ) ; ; R a h ( x - u ) d x d y du (5.9) or c j(u) g aj(u) d u ( 5 -10) where gaj(u> = /J"R h (x'-u) d x d y (5-n) As The first filter, it useful The pointed out by Huesman [32], is that can be since gQj(u), calculated is once a and this expression function only applied to of many is very useful the area, and sets of for two reasons. the backprojection projection data. in dynamic studies where fast calculation of a few ROI measurements This is is required. other reason why this expression is powerful is because it simplifies the calculation of the ROI measurement variance, since g â€ž : ( u ) is a deterministic function. Using the linearity property chapter 4, where A denotes of statistical deviation from expectation, and the same notation as in the mean, then the variance in the ROI measurement is (5.12) Recall that where the between we can noise in projections, separate is Cij(u) and the the projection independent noise data between in c2j(u), into two positions functions, Cij(u), on the projection while independent of that and c2j(u), axes, and in Cij(u), is a single value, i. e. a = <1/Aa)(m) [ ' - Â» Cl j<u) & a j ( u ) d u + c ' j ( u ) &aj(u) d u ] (5.13) Simplifying this equation in a similar manner as for 4.17 in chapter 4, we obtain + o\ [ 2_ ;!' 0 0 W(u)M j (u)/y j (u) g a j (u) du ]' } (5.14) 64 o o 4 ^ 0 0 a T o O ibToQ 12.00 Area 18.00 20.00 24.00 (pixels) 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 a has 2 a 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 characteristics for ignored. Circular PETT VI were used ROI's with increasing (M = and 0.096) cylindrical the diameter random and were placed phantom. The efficiency scatter at a contributions radius were of 5 cm. The variance terms due to emission noise and transmission noise were calculated separately. The results of this study plotted (circles) is are summarized in the graphs of figures 5.3 and 5.4 (squares). Also the theoretical variance versus area curve that would apply to ROI 65 Â°0J00 4T0O iToO 12.00 16.00 Area (pixels) 20.00 24.00 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. reduce The obvious effect the rate of ROI variance considering only approximately be required the of noise noise due correlation reduction as to emission, through the to the filtered ROI area obtain backprojection is increased. an ROI For variance is to example, which is 20% of the central pixel variance, an ROI with an area of 11 pixels would on a PETT VI image. This is considerably larger than the 5 pixel required for the same noise reduction on a theoretical image with uncorrected noise. area VI - EXPERIMENTAL VERIFICATION In an effort to verify the analytical predictions of the previous chapters, a phantom experiment 1 8 F-FDG was designed and performed on the was added to water to create a uniform UBC/TRIUMF source PETT VI tomograph. 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 18 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 The emission recorded scan events the four duration per independent samples true-slice scans were performed in both transmission slice was progressively (slice for each data increased sum) constant scanning mode as the to 1 ! was increased keep F and emission the decayed. total modes. number of The number of to forty-eight by considering 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 80.00 100.00 RADIUS (MM) 140.00 120.00 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 the forty-eight independent sets of emission data corrected PETT VI software, from 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 averages of the emission data, each one were reconstructed from the profile corrected for attenuation by a different set of transmission data. Point by point statistical variance images were computed for the classes of reconstructed images. These are shown in in figure 6.1 two 69 o o Â© 1TJ-, OJ o cu O- o LU Uo ZO <o' HO. > o o o. Ifl o o < = bV0020.00 40.00 60.00 RADIUS 80.00 l5o7oO 100.00 U0 00 (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 compare statistical directly with variance images are analytical themselves very noisy and are predictions. Averaging by . radius was thus difficult to 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 calculations and analytical predictions are in good agreement and 6.3. Statistical 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 table both those II. In measured illustrated in the map cases, the predicted statistically. The of figure 6.4 and the results are ROI variance figures are average error between in good measured variances is 9.5% for emission data and 18.5% for transmission data. and summarized in agreement with predicted ROI FIGURE 6.4 ROI size and placement map: ROI numbers for calculating statistical ROI variances. ROI # 1 2 3 4 5 6 7 diam. (mm) 8.1 16.2 24.3 32.4 32.4 24.3 16.2 Emission Variance p r e d i c t e d measured 442.9 78.1 31.6 17.5 9.3 20.7 40.1 Transmission Variance p r e d i c t e d measured 403.8 85.3 30.1 16.0 10.6 19.6 47.8 TABLE n ROI variance. 1065. 212. 86.4 49.5 23.3 44.8 108. 1130. 208. 70.3 35.4 29.8 55.0 133. 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 duration, or the strength of the source. This, however increases patient and either increases the total the tomograph. It is necessary basis of practical increasing the scan the radiation dose to the scanning time or poses potential rate problems for to make a trade-off between these factors. The choice of emission scan the by limitations duration is not usually as involved in measuring flexible, but is made on dynamic metabolic processes. In addition, it is often desirable to add images from several short scans together in order to reduce the total noise. same transmission data, Since multiple emission scans are the image noise due to transmission cannot be reduced in this manner. It is therefore variance due to noise typically reconstructed in transmission data 4.36 it necessary to image with the data is not independent and to keep the ratio of image variance due to noise shown that for a in emission data low. From equations uniformly-attenuating and 4.37 phantom, the can ratio,of be image variance due to uniformly noise emitting, in transmission data, o 2 j v (x,y), to image variance due to noise in emission data, o 2 j e (x,y), is given by R T (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 characteristics. From figures depends 6.2 and only 6.3 on phantom geometry it is clear that and machine performance 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 duration, centre to keep of the the image. Therefore maximum value of as a Ry guide to below 1.0, determining transmission approximately ten number of counts must be recorded in transmission mode as in emission mode. times scan the 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 tomograph. overlaid by Figure sets case by 6.5 of the shows ROI's UBC/TRIUMF the slices typical for selected these PET research from studies. group F D G and The first using the PETT VI fluorodopa PET studies 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 6.6 and 6.7. separately Both transmission variance images and emission variance shown in figures images are shown 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 be compared for absolute The total are ratios emission of counts brightness scale, they cannot variance contribution. transmission variance recorded in the slice to emission to total variance, transmission RT, and counts the ratios of recorded in the 75 UBC/TRIUMF PET PROGRAM FIGURE 6.7 Variance image - fluorodopa: Variance due to noise in emission data and variance due to noise in transmission data (b). Max. V a r . Max. Var, Trans. Emission C l N l R C (a) R T / R C Phantom 15040 4391 3.4 321505 1109354 0.29 11.7 Patient (FDG) Patient (Fdopa) 62240 23850 2.6 1499743 9799625 0.15 17.3 140 1710 101524 8508910 .012 6.8 .082 T A B L E IH Transmisson and Emission Variances and Slice Sums. 76 ROI # 0 1 2 3 4 5 6 Mean Var.(Em.) Var.(Tr.) 1393. 1505. 1563. 1412. 1402. 1839. 1678. 3263. 3486. 16. 3068. 3394. 3494. 3404. 7033. 8425. 32. 6547. 8311. 8497. 8551. RMS N o i s e (%) 7.3 7.2 0.4 6.9 7.7 6.0 6.5 TABLE IV F D G ROI Variances. RMS N o i s e (%) 9.1 8.7 8.0 8.4 11.6 11. 3 25. 24. 25. 27. 13. 11. 223. 217. 223. 226. 170. . 159. 172. 179. 196. 190. 117. 115. 7 8 9 A B C Var.(Tr.) Var.(Em.) Mean ROI # 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 cm phantom can above be concerning the choice of transmission extended to F D G and fluorodopa studies. scan duration for a 20 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 square-root of the total noise (due to both noise figure is calculated as the 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 . 77 vn - CONCLUSIONS A method transmission method projection involves modified for reconstructing data in computing analytical positron profile variance emission variance images tomography functions and from has been reconstructing emission and described. This images with a 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 U B C / T R I U M F The detector employed in PETT wobble PETT VI reconstruction procedure. motion and the subsequent be approximated sampling scheme non-uniform normalization which are VI have been studied and their effects on image variance have been observed. Based on the assumption that projection data can profile is by straight uniform line segments, sampling. In a it from uniform was trade-off shown against sampling due to wobble motion always increases cylindrical that the phantoms most efficient improved sampling density, 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. RMS This corresponds to an 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 using the analytical variance expression with wobble parameters large phantom variance images of point image sources dependent on the distance were reconstructed taken from P E T T VI. The exhibits circular symmetry and radial ripples. The variance exhibit radial streaking, with the number of streaks from the centre of the field of view. These variance are the result of normalizing the projections with a function that is largely periodic. being patterns 78 The data self-attenuation correction steps for detector efficiency, random coincidences, and object 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 from rate random correction as all projections. obtained implemented This value is obtained outside the field of in PETT VI, subtracts from averaging view. The resulting 432 a constant value independent estimates of 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 from that of the complicating and has individual ROI variance the effect pixel values. measurements. of reducing the variance of the Image-plane A method noise for is correlated measurement between pixels, computing covariance images for points in an image was developed and demonstrated on a 20 cm, phantom. The above covariance variances and therefore area These on measurement results uncorrected substantially were noise. larger computation an alternate variance was compared It was proved to be impractical for method was developed. The effect explored using simulated with those shown that from for. a ROI is required on a PETT a data theoretical given VI image computing ROI of increasing ROI for a 20 cm phantom. image reduction in containing ROI compared with additive, variance a the theoretical image. A pixel phantom and experiment ROI variances. forty-eight independent reconstructing images was designed This involved emission and from sample and implemented to measure multiple scans of a forty-eight emission sets the 20 cm phantom to obtain independent data statistically transmission corrected for data sets. By attenuation by a common transmission data set, and images from a common emission data set corrected for attenuation was by sample separated. transmission Point-by-point data variance sets, the effects of noise were computed images reconstructed images and found to be in very good agreement ROI variances were also statistically measured by placing the variance analysis procedure was for applied two both sources classes of with analytical predictions. identical images. Once again, analytical and statistical ROI measurements Finally, from the regions on multiple were in close agreement to real PET data obtained from human subject F D G and fluorodopa scans. 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Noise propagation in quantitative positron emission tomography Palmer, Matthew Rex 1985
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Title | Noise propagation in quantitative positron emission tomography |
Creator |
Palmer, Matthew Rex |
Publisher | University of British Columbia |
Date Issued | 1985 |
Description | 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 PETT 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 FDG 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. |
Subject |
Positrons - Emission |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-05-28 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0096299 |
URI | http://hdl.handle.net/2429/25131 |
Degree |
Master of Applied Science - MASc |
Program |
Electrical and Computer Engineering |
Affiliation |
Applied Science, Faculty of Electrical and Computer Engineering, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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