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

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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  •  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  / ^ / ^ / , ^•/  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  r—-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  •  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  •  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 ° . 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 (£) 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<£.  (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  • t  u m  9 9 f  •f  f f •  * ••* ?  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)  /_„  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  Z°  p./w.  i= l  where  p  1  « 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  L° l/wj  (3.6)  Define the arithmetic mean over a wobble cycle as  mA  =  1/Nb 2 °  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)  =  N„ S(u) F L j ( u ) Aj(u)  (4.1)  40  where N 0 is the average source intensity, S is the ring source profile 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) -  N„Mj(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  „ /" 0 B Ac.(u)h(x'-u)du • j—1  H  _ m  /! 0 0 Ac k (v)h(s'-v)dv } -k — 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 _ j— 1 k— 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  N„J {  m  1 1 1  (l)JE_i/_0Ba'ej(u)M1j(u)/y1j(u)h1(x'-u)du  «,  (s,)^ ;r.«'1vi(u)M,i(")/y4i(u)eJi(u)h2(x'-u)du m i = i °° 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)  =  ¥  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)  =  N„ S(u)  (4.24)  Projections, corrected for randoms are simply  c,(u)  =  e,(u) - rW(u)  (4.25)  Therefore the image-plane variance expression, equation 4.17, reduces to  (4.26)  FIGURE 4.3 Effect of random coincidence correction: Contribution to image-plane variance due to random estimate subtraction. Variance image (a) and plot along a central line (b).  Equation  4.26  is  simply  the  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  °\  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  + ^ \ (£) £_ /! ^ (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 (£,)! 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 „ : ( 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)  [  ' - »  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  °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 ©  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. Mid-brain slices were chosen to illustrate the image  variance  demonstrated  distribution and ROI variances  that most of the image  noise  for  typical PET measurements.  It was  in an F D G study is the result of noise in  transmission profiles. 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