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Investigation of problems created in SPECT images by artifacts in the attenuation map and correction… Chang, Zheng 2002

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Investigation of problems created in SPECT images by artifacts in the attenuation map and correction methods by Z H E N G C H A N G B . S c , Shandong University, 2000 A THESIS S U B M I T T E D I N P A R T I A L F U L F I L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F S C I E N C E in T H E F A C U L T Y O F G R A D U A T E S T U D I E S D E P A R T M E N T O F P H Y S I C S We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A August 2002 © Zheng Chang, 2002 In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that conying or publication of this thesis for financial gain shall not be allowed without/my written permission. Department of Physics & Astronomy The University of British Columbia 6224 Agricultural Road Vancouver, B . C . , Canada V 6 T 1Z1 Date: Abstract This thesis consists of five chapters. Chapter 1 presents the basic ideas of SPECT imaging. This chapter also explains the phenomenon of attenuation and shows how the attenuation effect affects SPECT imaging. Chapter 2 reviews various attenuation estimation methods and attenuation compensation methods. The advantages and disadvantages of these methods are discussed in this chapter. Moreover, this chapter discusses three artifacts such as missing counts, truncation and cross-talk effect, and their corrections. Additionally, this chapter presents three map reconstruction methods: FBP, OSEM and OSTGS. OSTGS has been tested by using computer simulations described in Chapter 3. Chapter 3 presents transmission simulations performed analytically. The analytical transmission simulations can be used to simulate simultaneous transmission /emission data acquisition by combining Monte Carlo SLMSET emission simulations. Different reconstruction methods have been tested in this computer simulation and the thorax phantom experiment described in Chapter 4. The results of these tests indicate that the cross-talk correction performance achieved by using OSTGS is better than that achieved by using conventional subtraction cross-talk correction. Finally, a summary is presented in Chapter 5 in which three topics for future work are being discussed. ii Table of Contents Abstract i i Table of Contents i i i List of Figures vi List of Tables x i i Acknowledgement x i i i 1 Introduction 1 2 Attenuation Correction Methods 10 2.1. Review of Different Attenuation Estimation Methods 10 2.1.1 Estimation of Attenuation Distribution through the Transmission Measurements 10 2.1.2 Estimation of Attenuation Distribution without the Transmission Measurements 23 2.2 Review and Discussion of Different Attenuation Compensation Methods . 26 2.2.1 The Chang Algorithm 27 2.2.2 Ordered-Subset Version of the M L E M Algorithm ( O S E M ) 28 2.3. Review of Artifacts and Corrections 29 2.3.1 Review of Artifacts 30 2.3.2 Review of Artifact Corrections 37 3 Simulation Experiments 42 3.1. Emission Simulation 44 3.1.1. M C A T Phantom 44 i i i 3.1.2. Camera System . . . 45 3.1.3. Data Acquisition 46 3.2. Transmission Simulations 48 3.2.1. Monte Carlo Transmission Simulation 49 3.2.2. Analytical Transmission Simulation 52 3.3. Creation of "experimental" data 89 3.4. Data Analysis Method 89 3.5 Results and Discussion - Simulation Experiments 91 3.5.1. Reconstruction of Attenuation Maps without Emission Cross-talk Effect . 91 3.5.2. Reconstruction of attenuation maps with emission cross-talk effect . 98 3.5.3. Reconstruction of Emission Images with A C 103 4 Phantom Experiments 108 4.1. The Brief Description of the Phantom Experiments 108 4.2. Results and Discussion - Phantom Experiments 109 4.2.1. Reconstruction of Attenuation Maps 109 4.2.2. Reconstruction of Emission Images 112 5 Conclusion and Future Work 121 5.1. Review of the Work 121 5.2. Future Work 122 5.2.1. Ful ly three dimensional attenuation reconstruction with collimator blurring correction 123 5.2.2. Truncation Correction 123 iv 5.2.3. Cross-talk Estimation 124 Bibl iography 125 v List of Figures Figure 1.1: Illustration of geometry of theoretical S P E C T imaging system [1] . . . . 2 Figure 1.2: Right anterior oblique (RAO) , anterior (ANT) , left anterior oblique ( L A O ) , and left lateral ( L L A T ) theoretical projections of a point source in the liver of the M C A T phantom. The projections are consistent in the size and shape. [1] 3 Figure 1.3: Illustration of impact of attenuation on S P E C T imaging. Photon A is recorded as a count, and photon B is scattered such that photon B is not detected. The result is a decrease in the counts in the projection data. [1] 4 Figure 1.4: Right anterior oblique (RAO) , anterior (ANT) , left anterior oblique ( L A O ) , and left lateral ( L L A T ) attenuated projections of a point source in the liver of the M C A T phantom. The size and shape of the projections vary with angle position. [1] . . . . 4 Figure 2.1: Different configurations for transmission imaging on a S P E C T system [1]. . 13 Figure 2.2: The geometry of a transmission scan 31 Figure 2.3: Theoretical transmission profile data with Poisson noise effect . . . . 31 Figure 2.4: Truncation due to limited size of F O V 32 Figure 2.5: Truncation due to improper position of patient 33 Figure 2.6: Truncation due to misalignment of detector and transmission source. . . 34 Figure 2.7: Truncation in the axial direction (thorax phantom from [41]) 34 Figure 2.8: Schematic presentationof the energy spectrum of the simultaneous transmission/emission acquisition. Spectrum of photons scattered in detector (Nal) is not shown since the number of these photons is small compared to the number of photons scattered in the object 36 Figure 2.9: Schematic of the photon spectrum in the four energy windows 39 Figure 3.1: A , Activi ty distribution in transverse slice of M C A T phantom. B , Distribution of attenuating medium in transverse slice of M C A T phantom 43 Figure 3.2: The cross section of a parallel collimator 46 vi Figure 3.3: Emission data of the projection (at the projection angle of 34° with respect to the x-axis in Figure 1.1) acquired in four different energy windows 48 Figure 3.4: Attenuation distributions in the M C A T phantom 50 Figure 3.5: A , A transverse slice of sheet transmission source (rotating around M C A T phantom). B , Transmission sinogram of one slice over 360° 51 Figure 3.6: The geometry of transmission scan 54 Figure 3.7: The transmission scan geometry of two collimated linear transmission source. 56 Figure 3.8: One projection of two collimated linear transmission scan 56 Figure 3.9: A . A profile in the selected data zone. B . Gaussian function fit to the projection of the first collimated linear source. C. Gaussian function fit to the projection of the second collimated linear source 58 Figure 3.10: A . The profile obtained by summing selected data into one profile. B . Gaussian function fit to the projection of the first collimated linear source. C. Gaussian function fit to the projection of the second collimated linear source 59 Figure 3.11: One collimated line transmission source scan geometry in simulation. . 60 Figure 3.12: A theoretical projection of one collimated linear transmission source from simulation 61 Figure 3.13: A . 2 D H S gaussian function; B . 2D H S gaussian function in surf image format;C. 2D H R E S gaussian function; D . 2D H R E S gaussian function in surf image format 62 Figure 3.14: multiplication of HS gaussian function and H R E S gaussian function (the " H S - H R E S gaussian"). A . 2D H S - H R E S gaussian; B . 2D H S - H R E S gaussian in surf image format 62 Figure 3.15: The convolution of theoretical projection and H S - H R E S gaussian function. A , 2D theoretical projection. B , 2D H S - H R E S gaussian in surf image format. C , The projection with blurring effect 63 Figure 3.16: A , One line through the blurring projection. B , F W H M of simulated H S -H R E S collimator from line profile study 64 vi i Figure 3.17: A , Transmission scan geometry when the cylinder is placed at the center. B , Transmission scan geometry when the cylinder is placed closer to the source. . . . 65 Figure 3.18: A , One-rod transmission geometry when the rod was placed at 20 cm to the detector. B , Transmission projection of one rod transmission can. C, The profile of the one-rod transmission scan 67 Figure 3.19: A , One-rod transmission geometry when the rod was placed at 58 cm to the detector. B , Transmission projection of one-rod transmission can. C , The profile of one-rod transmission scan 68 Figure 3.20: Comparing the above two line profiles of one-rod transmission scans. . 69 Figure 3.21: A , Two-rod transmission geometry when the rods were placed at the position of 16 cm to the detector. B , Transmission projection of two-rod transmission can. C , The profile of two-rod transmission scan 70 Figure 3.22: A , Two-rod transmission geometry when the rods were placed at 50 cm to the detector. B , Transmission projection of two-rod transmission can. C , The profile of the two-rod transmission scan 71 Figure 3.23: Comparing the above two line profile studies of two rods transmission . 72 Figure 3.24:Transmission scan without any object in the F O V 73 Figure 3.25transmission projections without thorax phantom recorded in the different energy windows 74 Figure 3.26: Transmission scan with non-activity thorax phantom at the center of the F O V 75 Figure 3.27:. Transmission projections with a non-activity thorax phantom recorded in the different energy windows 76 Figure 3.28:A. 2D projection of the Uniform Blank; B . The profile of the 2D projection. 78 Figure 3.29: A . 2 D projection of the Profile Blank; B . The profile of the 2D projection . 79 Figure 3.30: The parallel strips of the center bin at the different projections. A . The parallel strip of the center bin at the first projection. B . The parallel strip of the center bin at the 15th projection. C . The parallel strip of the center bin at the 25th projection. . 80 vi i i Figure 3.31: M C A T attenuation phantom related to 100 keV 80 Figure 3.32: The ideal transmission data obtained by using the Uniform Blank. A . 2D transmission projection. B . The profile of 2D transmission projection 81 Figure 3.33: The ideal transmission data obtained by using the Profile Blank. A . 2D transmission projection. B . The profile of 2D transmission projection 81 Figure 3.34: A , The 2D Uniform Blank projection with blurring effect. B , 2D transmission projection with blurring effect. C , Their profiles of 2D blurring transmission and blank projections 84 Figure 3.35: A , The Profile Blank projection with blurring effect. B , 2D transmission projection with blurring effect. C , Their profiles of 2D blurring transmission and blank projections 85 Figure 3.36: A , The 2D Uniform Blank projection with blurring effect and Poisson noise. B , The related 2D transmission projection. C , Their profiles of 2D blurring transmission and blank projections 87 Figure 3.37: A , The 2D Profile Blank projection with blurring effect and Poisson noise. B , The related 2D transmission projection. C , Their line profiles of 2D blurring transmission and blank projections 88 Figure 3.38: A , The 2D Uniform Blank projection with blurring effect. Poisson noise and integer count effect. B , The related 2D transmission projection. C , Their profiles of 2D blurring transmission and blank projections 92 Figure 3.39: A , The reconstructed F B P (A) and O S E M (B) attenuation map was obtained by employing the uniform blank. The strength of the uniform blank is 500 counts/pixel. 94 Figure 3.40: A , The reconstructed F B P (A) and O S E M (B) attenuation map was obtained by the profile blank. The maximum strength of the profile blank is 500 counts/pixel . . 94 Figure 3.41: A , The reconstructed F B P (A) and O S E M (B) attenuation map was obtained by employing the uniform blank. The strength of the uniform blank is 2000 counts/pixel. 95 Figure 3.42: A , The reconstructed F B P (A) and O S E M (B) attenuation map was obtained by employing the profile blank. The maximum strength of the profile blank is 2000 counts/pixel 95 ix Figure 3.43: A , The reconstructed F B P (A) and O S E M (B) attenuation map was obtained by employing the uniform blank. The strength of the uniform blank is 10000 counts/pixel 96 Figure 3.44: A , The reconstructed F B P (A) and O S E M (B) attenuation map was obtained by employing the profile blank. The strength of the uniform blank is 10000 counts/pixel. 96 Figure 3.45: The first "experimental" transmission data was simulated by adding emission cross-talk and transmission data which was obtained by employing profile blank with the max strength of 500 counts/pixel 99 Figure 3.46: The second "experimental" transmission data was simulated by adding emission cross-talk and transmission data which was obtained by employing profile blank with the max strength of 2000 counts/pixel 100 Figure 3.47: The reconstructed attenuation maps by using F B P without cross-talk correction. A , The attenuation map obtained by using the first "experimental" data. B , The attenuation map obtained by using the second "experimental" data. . . . . . 101 Figure 3.48: The reconstructed attenuation maps by using F B P with subtraction cross-talk correction. A , The attenuation map obtained by using the first "experimental" data. B , The attenuation map obtained by using the second "experimental" data 101 Figure 3.49: The reconstructed attenuation maps by using O S E M with subtraction cross-talk correction. A , The attenuation map obtained by using the first "experimental" data. B , The attenuation map obtained by using the second "experimental" data 102 Figure 3.50: The reconstructed attenuation maps by using O S T G S . A , The attenuation map obtained by using the first "experimental" data. B , The attenuation map obtained by using the second "experimental" data 102 Figure 3.51 When profile blank with the max strength of 500 counts/pixel was employed, O S E M emission images were obtained with different attenuation corrections. . . . 105 Figure 3.52 When profile blank with the max strength of 2000 counts/pixel was employed, O S E M emission images were obtained with different attenuation corrections. 106 Figure 4.1: The reconstructed attenuation maps of the No.8 and No. 12 transverse slices by using F B P (A), O S E M (B), F B P S (C), O S E M S (D) and O S T G S (E) I l l x Figure 4.2: The No.63 cross sections of the cardiac vertical long axis (A), the No. 12 cross sections of the cardiac horizontal long axis (B), the No.74 cross sections of the cardiac short axis (C) and profiles of the No.74 cross section of the cardiac short axis (D) of the R_OSEM_wi thou t_AC 114 Figure 4.3: The No.63 cross sections of the cardiac vertical long axis (A), the No. 12 cross sections of the cardiac horizontal long axis (B), the No.74 cross sections of the cardiac short axis (C) and profiles of the No.74 cross section of the cardiac short axis (D) of the R _ O S E M _ w i t h _ A C _ F B P . 115 Figure 4.4: The No.63 cross sections of the cardiac vertical long axis (A), the No. 12 cross sections of the cardiac horizontal long axis (B), the No.74 cross sections of the cardiac short axis (C) and profiles of the No.74 cross section of the cardiac short axis (D) of the R _ O S E M _ w i t h _ A C _ O S E M 116 Figure 4.5: The No.63 cross sections of the cardiac vertical long axis (A), the No. 12 cross sections of the cardiac horizontal long axis (B), the No.74 cross sections of the cardiac short axis (C) and profiles of the No.74 cross section of the cardiac short axis (D) of the R _ O S E M _ w i t h _ A C _ F B P S 117 Figure 4.6: The No.63 cross sections of the cardiac vertical long axis (A), the No. 12 cross sections of the cardiac horizontal long axis (B), the No.74 cross sections of the cardiac short axis (C) and profiles of the No.74 cross section of the cardiac short axis (D) of the R _ O S E M _ w i t h _ A C _ O S E M S 118 Figure 4.7: The No.63 cross sections of the cardiac vertical long axis (A), the No. 12 cross sections of the cardiac horizontal long axis (B), the No.74 cross sections of the cardiac short axis (C) and profiles of the No.74 cross section of the cardiac short axis (D) of the R O S E M with A C O S T G S 119 xi List of Tables Table 3.1: The activity distribution used in simulation experiments 45 Table 3.2:The four different energy windows used in emission simulation 47 Table 3.3: Attenuation index number table 51 Table 3.4: F W H M s of gaussian functions describing different slices in the selected data zone 59 Table 3.5: F W H M of gaussian function obtained by summing data into one profile. . 60 Table 3.6: Comparing the F W H M s from transmission simulation and from experimentally acquired transmission data 64 Table 3.7: The Mean Ratios of two different experiments 77 Table 3.8: The relative standard deviations of the reconstructed attenuation maps in several different simulated situations 97 Table 3.9: The relative standard deviations of the reconstructed attenuation maps by using different reconstruction methods in two different "experimental" situations. . 103 Table 3.10: The relative standard deviations of the reconstructed O S E M emission images by using different attenuation correction methods 107 xi i Acknowledgement I would like to thank my supervisor, Dr .Anna Celler, for all her help and guidance throughout this project. She gave me all of her help and support. She always encouraged me when I needed encouragement. I feel grateful for all she did for me. What I learned from is not only how to be a scientist but also how to be a great person. It is my pleasure to have such a great person as my supervisor. M y thanks also to all of the members of M I R G . I have enjoyed working with you all. I would like to thank Eric Evandervoort and Kat Dixon for their many helpful discussions and suggestions. M y special thanks to Stephen Blinder for his help, his discussion and his interest in this project. Lately, I must express my gratitude to my parents and girl friend for their continual encouragement and support. M y final thanks for them, my deepest love. x i i i Chapter 1 Introduction 1 1 Introduction Single photon emission computed tomography (SPECT) is an important technique of nuclear medicine. SPECT is used for the imaging of spatial distribution of radioactive tracer inside the patients. In SPECT imaging, data acquisition is measured as illustrated in Figure 1.1. The figure shows a single-headed gamma camera measuring a source distribution f (x,y) at a rotation angle 0 with respect to the x-axis. The detector is equipped with a parallel-hole collimator. The parallel-hole collimator allows only photons emitted from f (x, y) paralleling the collimator septa to pass through to the detector (usually made of Nal(Tl)), then the photons are recorded as counts. The data acquired by the detector at A point source in liver Figure 1.1: Illustration of geometry of theoretical SPECT imaging system [1]. Chapter 1 Introduction 2 one angle position are called one projection. The object in Figure 1.1 is a transverse slice of the three-dimensional (3D) mathematical cardiac-torso ( M C A T ) phantom with a T c 9 9 m distribution. [1] The projection space can be represented in either Cartesian coordinates or in polar coordinates [2]. In Cartesian coordinates, the (x, y) coordinate system can be represented as a new coordinate system (t, s) rotated counter-clockwise by the angle 9. Then t and s can be written in terms of x, y and the rotation angle 9 as: t = x cosQ +y sind s = -x sinQ +y cosd (1.1) The ray sum (p (9, t')) is the line integral over f (t,s) with respect to s for t = t' as: oo oo oo oo oo P@,t')= \fe(t',s)ds= J jfe(t,s)S(t-t')dtds= ^fg(x,y)S(xcos0 + ysin0-t')dxdy (1.2) - c o — o o — o o — o o oo where 8 is the delta function, and t is expressed in terms of x, y, and 0. The function p (9, t') is the Radon transform of f (x,y) in a theoretical imaging case[2]. B y rotating the camera around the patient, a set of projections can be measured from different projection angles. The original source distribution, from which emission photons are emitted, can be estimated from this set of projections [1]. Chapter I Introduction 3 LAO 12000 1 0 S O O a ooo 6 000 •C 4 000 -1 2 000 H 30 40 P ro je c t i o n Bin Figure 1.2: Right anterior oblique (RAO), anterior (ANT), left anterior oblique (LAO), and left lateral (LLAT) theoretical projections of a point source in the liver of the MCAT phantom. The projections are consistent in the size and shape. [1] Figure 1.2 shows the ideal projections of a point source in the liver of the M C A T phantom at projection angles of 0° (left lateral), 45° (left-anterior oblique), 90° (anterior), and 135° (right-anterior oblique) with respect to the x-axis in Figure 1.1. The black point within the liver and the one on the x-axis in Figure 1.1 indicates where the point source is located. The location of the point source is shown as the black point within the liver and on the x-axis in Figure 1.1. In theory, the projections are of the same size and shape i f there is no attenuation and other factors. The positions of the projections vary, depending on the projection angles. The projections form a matrix called the sinogram i f they are represented in the space (t,9). Chapter 1 Introduction 4 In reality, the quality and quantitative accuracy of S P E C T image is affected by many factors such as attenuation, Compton scatter. Amongst those factors, attenuation effect is of the most gravity. Figure 1.3: Illustration of impact of attenuation on SPECT imaging. Photon A is recorded as a count, and photon B is scattered such that photon B is not detected. The result is a decrease in the counts in the projection data. [1] A s illustrated in Figure 1.3, both photon A and photon B are emitted from the body in the direction to the detector. Photon A travels through the body and is recorded as a count in the projection; photon B is scattered so that it w i l l be lost from the projection. Thus the attenuated projections (pA(6,t)) wi l l contain fewer counts than expected in theory. Figure 1.4 shows the attenuated projections of the point source of Figure 1.3. Chapter 1 Introduction 5 12000 P r o j e c t i o n B i n Figure 1.4: Right anterior oblique (RAO), anterior (ANT), left anterior oblique (LAO), and left lateral (LLAT) attenuated projections of a point source in the liver of the MCAT phantom. The size and shape of the projections vary with angle position. [1] A s shown in Figure 1.4, the size and shape of the attenuated projections vary with angle position. This means that attenuation effect depends on the thickness and nature of material through which photons travel. The extent of attenuation can be expressed mathematically by the transmitted fraction ("TF"). The transmitted fraction at angle 0 (TF (t', s', 0)) is the ratio of the transmitted emission photons from location (t', s') at angle 0 to the emission photons from location (t', s') at the same angle. The transmitted fraction's formula is given by: Chapter I Introduction 6 TF(t',s',e)= N ' ( t ' ' S ' ' e ) =exp(-\ju(t',s)ds) (1-Ne(t',S',0) J where Nt( t' ,s' ,6) is the number of photons from location (t', s1) that wi l l be transmitted through a patient at angle 0. Ne(t',s',6) is the number of photons emitted from location (t', s') at angle 0. ju(t,s) is the distribution of linear attenuation coefficients. Equation 1.3 is accurate provided that a photon, which undergoes interaction, is no longer counted as a member of the beam. This condition is referred to as the "narrow beam geometry" condition [3,4]. Also, Equation 1.3 is accurate only for a mono-energetic photon beam. Since different types of tissue have different attenuation coefficients, the attenuation effects obtained in the cases where photons are passing through muscle, lung and bone respectively, are different. This leads to the inference that T F varies even the distance between the location of source and the detector remains the same. Moreover, different patients have different attenuation distribution. For example, it was reported in [5] for cardiac imaging that the attenuation coefficient of an individual patient might have a 40% standard deviation with respect to the mean in a group of patients investigated. Therefore, the spatial distribution of attenuation coefficients (attenuation map or estimate of p.(t,s)) is patient-specific and the attenuation map regarding a particular patient is Chapter 1 Introduction 7 necessary for correcting the attenuation effect. With this patient-specific attenuation map, we can recover the loss of counts due to attenuation effect. Currently there are two general ways to obtain the attenuation map. One is to estimate the attenuation distribution using additional transmission measurements. This approach includes the following transmission measure methods: 1) Importing and registering attenuation maps from another modality. For example, attenuation maps can be obtained by using conventional x-ray C T . 2) Measuring transmission data, which is necessary for estimating the attenuation maps, with the gamma camera used in emission imaging. Attenuation maps can be obtained by using sinogram (and system matrix i f iterative methods are employed). The system matrix represents the contributions that each small volume in space (herein referred to as pixels) makes to each camera bin. For example, the element of system matrix l,j represents the contribution that pixel j makes to camera bin i . A s indicated in Equation 1.4 and 1.5, the sinogram (S (t,0)) can be calculated by using a blank scan (No (t,6)) that is measured without a body and a transmission scan (N (t,0)) that is measured with a body. Chapter 1 Introduction 8 0",<9)exp(- ^ju(t',s)ds) (1.4) where \x is the distribution of linear attenuation coefficients in the body. S(t',0) = log(No(t',0)/N(t',0)) (1.5) The approach of estimating the attenuation distribution using additional transmission measurements is advantageous in terms of the accuracy of the attenuation distribution achieved. This approach also has disadvantages that include additional hardware and more acquisition time (if transmission scan is not performed simultaneously with emission scan) [6]. The other approach of estimating attenuation maps is to estimate the attenuation distribution from the S P E C T data alone. Theoretically this approach can efficiently achieve the emission distribution, while it does not require additional hardware. Some studies reported that non-uniform attenuation correction in S P E C T imaging has been achieved successfully without using transmission measurements [7]. However, the methods performing accurate reconstructions without the knowledge of attenuation distribution have not been established successfully. In summary, accurate quantitative emission image requires accurate attenuation correction ( "AC") . In order to perform A C correctly, an accurate, patient-specific attenuation map is required. As discussed previously, while many different methods have Chapter I Introduction 9 been investigated to determine the attenuation map, they still have some practical limitations and disadvantages. Also , the quality of the reconstructed attenuation map is impacted by problems such as truncation and cross-talk effect. N o doubt, the inaccuracies of the reconstructed attenuation map decrease our ability to detect diseases. For example, in heart perfusion studies, the inaccuracies of the reconstructed attenuation map can artificially enhance the extra-cardiac activity, which confounds our ability to detect the defect of heart. For this reason, the investigation of problems created in S P E C T was performed in this work. Also , this work presents a new cross-talk correction method, which is tested by computer simulations and phantom experiments. Chapter 2 Attenuation Correction Methods 10 2 Attenuation Correction Methods 2 .1 . Review of Different Attenuation Estimation Methods Attenuation correction research in S P E C T imaging is very important in the area of nuclear medicine. To determine an accurate, patient-specific attenuation map is crucial to perform A C . As mentioned previously, currently two strategies have been employed for obtaining attenuation maps for A C : One is to estimate the attenuation distribution using additional transmission measurements. The other way is to estimate the attenuation distribution based solely on the S P E C T data [1]. 2.1.1 Estimation of Attenuation Distribution through the Transmission Measurements This approach allows people to obtain the information about attenuation distribution by using a Conventional X-ray C T or with a transmission scan by using a gamma camera. Chapter 2 Attenuation Correction Methods 11 2.1.1.1 Attenuation Distribution Obtained with a Conventional X-ray CT High-resolution attenuation coefficient images from another modality can be imported, and registered with the patient's S P E C T data as attenuation maps [8]. Besides their use for attenuation maps, these images can also provide anatomical contexts for the emission distributions. Practically, there are two problems associated with this approach. First, since attenuation coefficient varies with energy, the attenuation coefficient in the high-resolution images wi l l require scaling to the values appropriate for the energy of the emission photon [9]. Second, due to the constraint on health care costs, it is factually difficult for all patients have such high-resolution attenuation coefficient images available. A t the 1999 Society of Nuclear Medicine Annual Meeting, one instrumentation company showed a combined S P E C T and C T system that would image a patient on a shared imaging table. However, usually it is very difficult to accurately align the attenuation images from C T and emission images from S P E C T because C T images and S P E C T images are obtained on different machines at different times. The kind of difficulties in the alignment between the two different systems (CT and S P E C T ) may limit the application of the combined system. This system is also not desirable due to the high cost of manufacturing. Chapter 2 Attenuation Correction Methods 12 2.1.1.2 Attenuation Distribution Obtained with a Transmission Scan by Using a Gamma Camera To overcome inaccuracies in alignment between the attenuation maps from C T and emission images from S P E C T systems, the usual way in S P E C T is to perform a transmission scan and reconstruct the attenuation map from the transmission data measured by using a gamma camera. The transmission scan would be performed either sequentially or simultaneously with an emission scan. Simultaneous mode provides higher efficiency than sequential mode since simultaneous mode allows us to obtain a transmission scan and an emission scan on the same machine at the same time. Different kinds of S P E C T systems have been designed and manufactured so that transmission and emission data can be required simultaneously. Consequently, the reconstructed attenuation maps can be used for attenuation correction during the emission reconstruction. For example, in a three-head detector S P E C T system, the three detectors with fan-beam collimators are arranged in a triangle set. One detector acquires transmission and emission data, while the other two detectors acquire only emission data [10]. A line transmission source is located at the focal line of the detector acquiring transmission and emission data simultaneously. But the major disadvantage of this configuration is the contamination (cross-talk) of transmission caused by emission radiation or the contamination of emission caused by transmission radiation. Chapter 2 Attenuation Correction Methods 13 Several source/detector configurations for a S P E C T transmission scan have been investigated. As shown in Figure2.1, these configurations include: (1) Uniform sheet source opposite to a parallel beam collimator (2) Several fixed position collimated line source opposite to a parallel beam collimator (3) A scanning line source opposite to a parallel beam collimator (4) A single fixed line source opposite to a symmetric fan beam collimator (5) Offset line source opposite to an asymmetric fan beam collimator (6) A medium-energy scanning-point source opposite to a parallel beam 1 1 1 1 A . Sheet S o u r c e 1 1 i i P j i i i P i m j H i i j ^ ^ 'iiV y y V • y y i 'l 1' 1 3 , Multiple Line Source irrnn'TrTOTiinni • f M111 i i 111 n II i i Milan ii m m i N V D. Converg ing Collimatiion E. Asymmetr i c Fan B e a m R Septa l Penetration of Parallel Coll i mat ion Figure 2.1: Different configurations for transmission imaging on a SPECT system. [1] Chapter 2 Attenuation Correction Methods 14 1) Uniform sheet source opposite to a parallel beam collimator: The first configuration shown in Figure 2.1 A is the configuration of a sheet transmission source opposite to a parallel-hole collimator on the camera head. This configuration has been investigated for a number of years [11-13]. It has both advantages and disadvantages. The advantages of this configuration are as follows: • It provides a transmission source, which fully irradiates the camera head opposed to it. • The transmission source needs no motion beyond the one provided by the rotation of the gantry. This configuration has the following disadvantage: • It requires a hot source and a heavy collimator for the purpose of obtaining the "narrow beam geometry" [14]. Chapter 2 Attenuation Correction Methods 15 2) Several fixed position collimated line source opposite to a parallel beam collimator: The second configuration illustrated in Figure 2. I B is the configuration of the multiple line-source array [15]. In this configuration, the transmission irradiation comes from a series of collimated line sources. The line sources are parallel to the axis of rotation of the camera. The line sources' activities and spacing_are tailored to provide a greater irradiation near the center of the field of view (FOV) , where a patient has greater attenuation. A n optimum transmission activity profile is one that yields constant signal-to-noise ratios (SNR) across the whole patient. In this configuration, a proper profile can be obtained by using pairs of line sources. The pairs of line sources are placed symmetrically around the center and each pair is stronger than its neighboring pair that is more far away from the center. The advantages of this configuration are as follows: • It allows the activity distribution in the system to be optimized to the attenuation distribution in the body. • The multiple line source array provides full irradiation across the F O V of the parallel-hole collimator employed for emission imaging. • This configuration can decrease cost because not all of the line sources have to be replaced. Two new line sources are inserted into the center of the array. The Chapter 2 Attenuation Correction Methods 16 rest of the lines are moved outward by one position, and the weakest pair is removed. The configuration has the following disadvantage: • The transmission profiles are resulted from the overlapping irradiation of the individual lines. The irradiation of the individual lines varies with the distance between the source and the detector. Thus the blank data obtained in one experiment is applicable in other experiments only i f the distance between the source and the detector remains the same in those experiments. A new blank scan has to be done i f the distance between the source and the detector changes [1]. This makes the process more time consuming and more costly. 3) A scanning line source opposite parallel to a beam collimator: A s shown in Figure 2.1C, a single line source is moved along the detector in the direction perpendicular or parallel to the axis of rotation of the camera. The camera has a moving electric window to detect only the transmission photons. During each S P E C T projection, the whole area of the detector is scanned by the line source. The advantages of this configuration are as follows: • The entire field of view of the detector is irradiated. Chapter 2 Attenuation Correction Methods 17 • The configuration significantly reduces the amount of scattered transmission photons because the electronic window allows the detector to record only the photons detected in a narrow region opposed to the line source in the transmission data. • For the same reason, the scanning line source configuration also decreases the contamination (cross talk) of transmission data caused by emission radiation and contamination of emission caused by transmission radiation. With the moving electronic window, the detector records only the photons detected in the region not opposed to the transmission source in the emission data. [1] The disadvantages of this configuration are as follows: • This configuration requires additional electric and mechanical hardware that are designed to move the line and to employ electronic windowing technique. • Synchronization between mechanically motion of the source with the electronic windowing is necessary when the electronic windowing is employed to acquire the transmission photons. • Activity in the scanning line cannot be too high. Otherwise there is a problem of dead time when there is no attenuation medium between the source and detector. On the other hand, the source cannot be too weak. Otherwise there would not be enough counts in transmission data to reconstruct good quality attenuation maps when a highly attenuating patient is measured. This leads to the inference that the line sources have to be replaced often at the expense of high cost. Chapter 2 Attenuation Correction Methods 18 However, this method is currently the dominant configuration offered commercially, since its advantages overwhelm its disadvantages. 4) A single fixed line source opposite to a symmetric fan beam collimator: Figure 2. I D illustrates the use of a fan-beam collimator with a line transmission source at its focal distance [10,16,17]. The advantages of this configuration are as follows: • "Narrow beam geometry" is obtained since the collimator easily removes scatter. • Compared with a parallel collimator, a convergent collimators provide a better spatial resolution and better sensitivity for small objects such as the heart • Only one line source needs to be handled, shielded, or mounted on the system [1]. The disadvantages of this configuration_are as follows: • The truncation of the F O V caused by this configuration is more than that of parallel collimation on the same camera head. • The configuration does not use electronic windowing technique, which can assist in correction of cross talk between emission and transmission data. Chapter 2 Attenuation Correction Methods 19 • The source must be kept at the convergence location of the collimation. • The emission data are acquired by fan-beam collimators. The use of fan-beam collimators for estimation of attenuation maps results in possible truncation at some projection angles. There is no doubt that truncation presents a serious problem for the A C . Three approaches can be used to reduce the truncation artifact and to achieve attenuation map: (1) The first approach is to employ the long focal length (110-115cm) fan-beam collimator and a fast sequential transmission/emission acquisition protocol. Given sequential transmission and emission imaging, there is no need to correct contamination of emission data by transmission radiation. Also , data acquired in the transmission window during emission imaging can be used to estimate the contamination of the transmission data by the emission photons [16]. A disadvantage of this method is that it may be more difficult to ensure registration of the attenuation map with S P E C T data. (2) The second approach is the "Extrapolation of the truncated transmission data under clinically relevant count density conditions" [18]. The method includes four steps. First, the patient contour can be obtained by processing the scatter and photo peak emission data. Second, the contour is filled with attenuation coefficient of soft tissue. Then the created contour image is reprojected. Finally, the truncated projections can be extrapolated with the contour image projections. In this way, the quality of Transmission Chapter 2 Attenuation Correction Methods 20 Computed Tomography (TCT) image can be improved and truncation artifacts remarkably can be decreased. [18] (3) The third approach is to employ an asymmetric fan-beam collimator instead of a symmetric fan-beam collimator. In this approach, truncation can be eliminated, or at least dramatically reduced provided that data is acquired over 360 degree. [19-22] 5) Offset line source opposite to an asymmetric fan beam collimator: As shown in Figure 2. IE , the use of an asymmetric collimator results in the truncation of data on one side of the patient, instead of both sides. Conjugate views wi l l f i l l in the region truncated by rotating the camera with the collimator 360° around the patient. The advantages of this configuration are as follows: • Due to low collimation of the source, the converging fan beam geometry has high efficiency. • Attenuation maps obtained from a line source with a fan beam collimator have good spatial resolution. • A converging collimator reduces the scatter fraction in transmission data. This configuration has the following disadvantage: Chapter 2 Attenuation Correction Methods 21 • The line source has to be at a fixed distance from the detector during the measurements. 6) A medium-energy scanning-point source opposite a parallel beam: The problems arising from the process in which converging collimators acquire the emission data can be overcome as illustrated in Figure 2. IF . Because the photons from a medium-energy scanning-point source can penetrate the septa of the collimator, a*n asymmetric fan-beam transmission projection is generated through a parallel-hole collimator [23]. With the configurations discussed above, we are able to measure transmission data and blank data. Thus we are able to calculate sinograms by using Equation 1.5. With the sinograms, attenuation maps can be reconstructed by employing F B P (filtered backprojection) or O S E M (ordered-subset expectation maximization). Currently attenuation maps can be obtained using transmission/emission simultaneous acquisition protocol. While the simultaneous acquisition of emission and transmission data in S P E C T reduces the problems of the registration between emission and transmission data, it does result in contamination of transmission data caused by emission irradiation. The contamination is the cross talk between emission and transmission energy windows, which are used to record the transmission and emission photons respectively. For example, during the S P E C T imaging (with T c 9 9 m used as Chapter 2 Attenuation Correction Methods 22 emission source and G d used as transmission source), emission and transmission data are measured simultaneously. Because the energy of the transmission photons is different energy from the energy of the emission photons, the transmission photons and the emission photons can be recorded in different energy window by employing multi-energy window technique. However, since some T c 9 9 m photons down scattered into the energy window used to record G d photons, the cross-talk wi l l cause the contamination of the transmission data. It is necessary to do cross-talk correction before the transmission data is used to reconstruct attenuation maps. In general, a conventional subtraction method is used to correct cross-talk effect. When down scatter is estimated, it can be subtracted from transmission data before we reconstruct attenuation distribution by using transmission and blank data. However, this subtraction method wi l l decrease the S N R , and consequently wi l l incur inaccuracies in the reconstructed attenuation map. In this work, we propose an extension of Lange's maximum-likelihood transmission gradient ( " M L T G " ) algorithm [24] to correct cross-talk effect instead of the conventional subtraction method. The M L T G algorithm's formula is given as [25]: Ylc.expi-YW.h) (2.1) Chapter 2 Attenuation Correction Methods 23 where jU is the distribution of linear attenuation coefficients, c is blank data, T is transmission data, and ltj is the element of system matrix representing contribution that pixel j makes to camera bin i. The maximum-likelihood transmission gradient with scatter ( " M L T G S " ) is modified version of the M L T G and down scatter is incorporated in the formula. The formula of the M L T G S is given as [25]: where s is cross-talk estimate, and Zy is the element of system matrix representing contribution that pixel j makes to camera bin i. The M L T G - S is a better approach than conventional subtraction method in cross-talk correction since it incorporates down scatter factor in the formula and thus doesn't increase noise effect. Also , the M L T G S can stabilize the attenuation map in reconstruction process. Consequently, the M L T G S has a potential to offer a better attenuation map [25]. ^(c.expf-^MjlyJlu) c,exp(-^Mjlu) (2.2) I ciexp(-YJP"jhj ) + s> ) Chapter 2 Attenuation Correction Methods 24 2.1.2 Estimation of Attenuation Distribution without the Transmission Measurements While we can theoretically obtain accurate attenuation distribution by measuring transmission data, the method has several practical limitations as follows [26]: 1) The additional hardware is needed. For example, long-lived transmission sources are required and they should be carefully shielded. It also increases the quality control requirements. 2) These additional components w i l l raise the running costs of a camera. Especially, the transmission source is required to be replaced periodically. 3) The transmission scan may increase scanning time i f it is not performed simultaneously with an emission scan. More recently, there has been strong interest in the methods, which can achieve good attenuation distribution without the transmission measurements. Theoretical methods performing accurate reconstructions without a priori knowledge of attenuation distribution have not been found. However, some successful studies on attenuation correction without transmission measurements have been reported [7]. Moreover, studies showed that a complete set of transmission measurements may represent "overkill"[26]. For example, i f the region of interest (ROT) is restricted to some Chapter 2 Attenuation Correction Methods 25 small areas (such as the heart and liver), only small partial set of attenuation distribution has to be used in a standard iterative algorithm incorporating attenuation information. For the reasons stated above, there are many methods to reduce the need for transmission measurements. Several suggested algorithms can completely avoid transmission measurements in some applications. These algorithms can be divided into two classes: The first class of the algorithm applies segmentation algorithms to emission data or uncorrected emission images. The algorithm is used to locate regions of approximately constant attenuation, and then predefined attenuations coefficients can be assigned to these regions [27]. It is mainly applied in attenuation correction in P E T , not S P E C T . The second class of the algorithms attempts to obtain the information on the attenuation distribution directly from emission data without transmission data. There are three different types considered in the latter class of algorithms. 1) The first type is the simultaneous computation of attenuation map and the emission activity, both of which are directly obtained from the emission data. This can be done by iterative inversion of the forward mathematical model suggested by Censorin 1979[28], Mangos [29], and Bronnikov(with an approximate linear model)[30]. The major drawback of these methods is that the emission and attenuation distributions are inextricably linked in the imaging equations [31]. Chapter 2 Attenuation Correction Methods 26 2) The second type is based on the consistency conditions theory [31]. The approach was proposed by Natterer [32] and further explored by Welch [33] and Moore [34]. The consistency conditions of attenuated Radon transform provides a link between the transmission data and attenuated emission data. The relationship can be exploited to estimate the transmission data. However, the non-linear systems describing the consistency conditions are highly under determined. It is therefore necessary to input some a priori information on the unknown attenuation map [31]. For example, a new method of this type is Maximum A Posteriori ( M A P ) Reconstruction where the objective function is a combination of the likelihood and an a priori probability. The latter uses a Gibbs prior distribution to encourage local smoothness and a multi-modal distribution for attenuation coefficients [27]. 3) The third type is based on the recent development in mathematics of non-linear inverse ill-posed problems: For both activity and attenuation distributions unknown, the attenuated Radon Transform is a non-linear operator. Tikhonov regularization would be a standard way to recovery activity and attenuation distributions [26]. 2.2 Review and Discussion of Different Attenuation Compensation Methods During the last decade, we not only have greatly improved the ability to estimate attenuation distribution, but also have the ability to perform attenuation correction (AC) Chapter 2 Attenuation Correction Methods 27 once the attenuation distribution is known. Due to the great developments in computing power and improvements in the algorithms used in attenuation correction, it became practical to perform complicated computations. A n example is the development of ordered-subset or block iterative algorithms for the use of maximum-likelihood reconstruction [35, 36]. A n y comprehensive review of this subject would require a chapter to discuss this task. Consequently, we would like to review and discuss only the two most common A C algorithms: the Chang algorithm [37] used with F B P , and ordered-subset expectation-maximization reconstruction algorithm ( O S E M , ordered-subset version of the M L E M algorithm) [35]. Besides these two common algorithms, there are other A C algorithms referred to [38, 39]. 2.2.1 The Chang A lgor i thm The Chang algorithm is the best-known and most commonly employed method to perform A C with F B P [37]. The Chang algorithm is a post-reconstruction correction method, which performs attenuation compensation by multiplying each point in the reconstructed emission image by the correction factor for the point. The correction factor is the reciprocal of the transmitted fraction, which is calculated by averaging transmitted fraction (TF) over acquisition angles. A s discussed previously, the T F for each point at each acquisition angle can be calculated by using Equation 1.3. That is, the correction factor for each point ( x 1 , y') in the slice (C (x1, y')) is calculated as follows: Chapter 2 Attenuation Correction Methods 28 (2.3) where P is the number of projection angles 6i, TF is calculated by using Equation 1.3 and Equation 1.1, which is used to convert x' and y' into t' and s' respectively for each 6i.[l] Chang algorithm includes the multiplicative or zeroth-order Chang method. A s mentioned above, with the zeroth-order chang method, the zeroth-order corrected image is obtained by multiplying each point in the reconstructed emission image and the correction factor for the point. The multiplicative Chang method is performed in the following way: First, the zeroth-order corrected image is projected as the process of imaging. Second, projection differences are obtained by subtracting the estimated projections from the actual emission projections. Third, error image is reconstructed from the differences by using F B P . A t last, after attenuation correction for the error image is performed in the same manner as the zeroth-order correction, the corrected error image is added to the zeroth-order estimate of the image to obtain the first-order estimate [1]. Typically only the first-order correction is performed, but higher-order estimates can be obtained by repeating the process any number of times. However, in this process the method does not converge (i.e., reach a definite solution and then not change with further iteration). Also , the higher-order estimates are degraded by elevated noise [41]. Chapter 2 Attenuation Correction Methods 29 2.2.2 Ordered-Subset Version of the M L E M Algorithm (OSEM) M L E M is the second algorithm we wi l l illustrate. In maximum-likelihood expectation-maximization ( M L E M ) , the noise is modeled as Poisson distribution, and the source distribution that most likely gives the emission data is determined by using the expectation-maximization algorithm. B y incorporating attenuation in the system matrix used to model imaging, A C is performed as part of the reconstruction process [1]. Generally the reconstruction process by using M L E M costs more time than that by using F B P . In order to speed up reconstruction process, ordered-subset algorithm is employed with M L E M . The ordered-subset version of the M L E M algorithm ( O S E M ) [35] can accelerate reconstruction to the extent that reconstruction time is now clinically acceptable. In O S E M , the projections are divided into several separate subsets, and the estimate of the source distribution is updated by using the projections within the given subset. O S E M is so successful that it is now being used routinely for selected applications instead of F B P . 2.3. Review of Artifacts and Corrections Although the fast development of transmission source/detector configurations provides us with many different choices, we still have many problems that need to be overcome in the transmission scan measurements. These problems include: Chapter 2 Attenuation Correction Methods 30 (1) Missing counts in a transmission scan (2) Truncation effect in a transmission scan (3) Cross talk effect of transmission data measured simultaneously with emission data. 2.3.1 Review of Artifacts 2.3.1.1 Missing Counts in a Transmission scan In clinical practice, the total radioactivity of transmission source is around several thousand M B q or even higher, and the radioactivity decays by time. A s discussed previously, due to attenuation effect, most of photons from transmission source are attenuated and only a small portion of them reaches on the detector. Especially when a highly attenuating patient is being scanned, only a very small number of transmission photons can go through the center of field of view (FOV) where the patient has greater attenuation, and consequently few counts are recorded in some center camera bins. To clearly present the above idea, a simple computer simulation was performed as follows: a theoretical uniform transmission source was modeled to irradiate the attenuation phantom shown in Figure 2.2 and the strength of the irradiation was 200 photons/bin, the counts recorded in the center bins of the detector were even less than 3 counts as shown in Figure 2.3. In reality, we have to consider noise effect and detector Chapter 2 Attenuation Correction Methods 31 efficiency, so it is not difficult to understand why some center camera bins record zero counts. As illustrated in Equation 1.5, the existence of zero counts means that attenuation coefficients in some pixels should be infinite values, which tend to appear during the reconstruction of the attenuation map. Consequently the missing counts effect results in the loss of the quality and quantitative accuracy of the reconstructed attenuation map. Transmission source > Transmission collimator Collimated transmission photon beams — Detector Detector collimator Figure 2.2: The geometry of a transmission scan. Chapter 2 Attenuation Correction Methods 32 Figure 2.3: Theoretical transmission profile data with Poisson noise effect 2.3.1.2 Truncation Effect in a Transmission Scan Theoretically the whole body of a patient should be placed in the F O V during the transmission and emission acquisition so that no information about the patient w i l l be lost. However, since the size of a detector and the size of transmission source are both limited, they result into the limited F O V . Consequently, even i f a patient is placed at the center of the F O V , it is not guaranteed that the whole body of the patient is always in the F O V . That's why truncation effect appears in a transmission scan at some angle positions. Figure2.4 shows the truncation effect caused by the limited F O V . Chapter 2 Attenuation Correction Methods 33 Transmission source Transmission collimator Detector lllllllllll n Detector collimator Truncated area Truncated area Figure 2.4: Truncation due to limited size of FOV Besides the limitations of the F O V , the position and alignment of detector and transmission source should also be considered. In clinical practice, many truncation events occur due to improper patient position or misalignment of transmission source and detector. Figure 2.5 shows the truncation effect due to improper position of a patient, and Figure 2.6 shows the truncation effect due to misalignment of detector and transmission source. Chapter 2 Attenuation Correction Methods Transmission source" v h o O D 0 D D I 3 D B D o l ~* Transmission collimator Detector Detector collimator Truncated area Figure 2.5: Truncation due to improper position of a patient Figure 2.6: Truncation due to misalignment of detector and transmission source Chapter 2 Attenuation Correction Methods 35 Heart is outside the range of transmission scan Figure 2.7: Truncation in the axial direction (the thorax phantom is from [41]) Besides that truncation effect in the transverse slice, truncation also may occur in the axial direction. As shown in Figure 2.7, although there is no transmission truncation in the transverse slice, the area of interest (i.e., heart) is not completely covered by the transmission scan. It means that truncation effect also occurs in the axial direction. Truncation of transmission data results in "cupping artifact" in the reconstructed attenuation maps. This artifact is the result of the pile up of information from the truncated region near its edge when the reconstruction is restricted to reconstruct only the region within the F O V at every angle [1]. Chapter 2 Attenuation Correction Methods 3 6 2.3.1.3 Cross-talk effect of transmission data In clinical practice, attenuation maps can be reconstructed from transmission data measured simultaneously with a SPECT emission scan. G d 1 5 3 and T c 9 9 m are usually used as transmission source and emission source respectively. Since the energy of transmission photons from G d 1 5 3 is 100 keV and the energy of the emission photons from Tc 9 9 m i s l40 keV, Theoretically the two different kinds of photons could be easily separated by using multi-energy window technique. However, due to the scatter effect and the limited energy resolution of a detector, the scattered emission photons (140 keV emission photons) are recorded in the lower energy window (lOOkeV transmission window) as shown in Figure2.8. Scattered emission photons Energy Transmission photopeak Emission photopeak Figure 2.8: Schematic presentation of the energy spectrum of the simultaneous transmission/emission acquisition. Spectrum of photons scattered in detector (Nal) is not shown since the number of these photons is small compared to the number of photons scattered in the object. Chapter 2 Attenuation Correction Methods 3 7 Consequently the total counts measured in the transmission energy window must be more than the true transmission counts recorded in the transmission energy window. If the data measured in the transmission energy window is used as the true transmission data in the reconstruction of an attenuation map, the values in the reconstructed attenuation map must appear to be lower than the true attenuation coefficients. The effect that the photons with higher energy are recorded in the lower energy window is referred to as the "cross-talk effect". 2.3.2 Review of Artifact Corrections 2.3.2.1 Missing Counts Effect Correction In order to correct missing counts effect, two methods have been investigated. One method is to increase the strength of transmission source and the scanning time of a transmission scan. As discussed previously, the missing counts effect is caused by the lack of photons recorded by the detector. If a hot transmission source is employed to irradiate a patient for a long transmission scan, the detector can record enough counts so that the missing counts effect can be easily eliminated. However, due to the cost of transmission source, the strength of transmission source cannot be increased up to the strength we expect. Similarly, due to the limitations of clinical operations and dose to patients, the scanning time of a transmission scan cannot be increased up to the length we expect, either. Chapter 2 Attenuation Correction Methods 38 Alternative method is to employ the filter that is able to remove those zero counts. For example, we can use a gaussian filter or average the counts in neighbor pixels to eliminate zero counts. Unfortunately, the filters result in the loss of spatial resolution. This kind of tradeoff should be considered before we employ any filter to correct the missing counts effect. 2.3.2.2 Truncation Effect Correction In order to correct truncation effect, we should consider at least three factors: the size of FOV, the position of a patient, and the alignment of transmission source and a detector. If the patient can be placed at the center of the FOV, which should be large enough to accommodate the whole body during the scan, and make proper alignment between transmission source and the detector, the truncation effect could be avoided. While it is theoretically possible to completely eliminate truncation effect, it is factually difficult to create such perfect operating situation in the environment of limited clinical operations. In order to overcome this difficulty, recently there has been strong interest in the method that can reduce truncation effect by constraining the reconstruction to the actual area of the body [42]. Since the goal of transmission measurement is to estimate attenuation maps for AC of the emission images. An important question is not whether the attenuation map is distorted by truncation, but rather whether AC is degraded by the Chapter 2 Attenuation Correction Methods 39 truncation effect. It has been reported that even though the attenuation map is significantly distorted outside the full sampled region (FSR, the region within the F O V at every angle), the total attenuation effect for locations in the F S R is estimated fairly accurately with iterative reconstruction [16, 17]. Thus, a reasonably accurate A C can be obtained for organs (such as a heart) within the FSR. However, in reality this method has not solved truncation problem completely. For example, it has been reported that truncation may have an impact on the detection of cardiac lesions [43]. 2.3.2.3 Cross-talk Effect Correction In order to correct cross-talk effect, methods have been developed to estimate the cross-talk effect of transmission data caused by emission radiation when the emission and transmission data are measured simultaneously [6,8,32]. In the following section, a simple cross-talk correction method is presented. The method is called a conventional cross-talk subtraction method, which is used with Siemens Profile System. Chapter 2 Attenuation Correction Methods 40 Figure 2.9: Schematic of the photon spectrum in the four energy windows. As shown in Figure 2.9, transmission and emission data can be acquired simultaneously by employing four different energy windows. In the conventional cross-talk subtraction method, only the data recorded in the following three different energy windows are needed. 1) 100 keV transmission window 20% 2) 86 keV scatter window 8% 3) 116 keV scatter window 12% Chapter 2 Attenuation Correction Methods 41 Then the estimate of transmission data is calculated as Equation2.4. transmission estimate = experimental transmission - coefficient x (scatter 1 + scatter2) (2.4) where scatter! corresponds to the scatter data recorded in 86 kev scatter window, scatter2 corresponds to the scatter data recorded in 116 kev scatter window, and the coefficient is equal to 1.1, which is determined by using clinical data. Although the conventional subtraction method can correct the cross talk effect, the method results in the increase of the noise effect in the data. The increased noise effect not only further decreases the attenuation map quality and quantitative accuracy, but also increases the artifacts related to zero or few pixel counts. In order to overcome this problem, this work presents a new cross-talk correction method, ordered-subset version of the M L T G S algorithm ("OSTGS") . The new method not only explicitly accounts for the estimate of cross talk, but also can accelerate to such the extent that reconstruction time is clinically acceptable. Chapter 3 Simulation Experiments 42 3 Simulation Experiments Different methods can be employed to obtain attenuation map as discussed previously. A n attenuation map obtained by O S T G S seems to be better than the maps obtained by other methods. In practice, it is hard to tell which method is the best, because the true attenuation map of the patient is unknown. In order to investigate and improve the A C correction, a simulation method modeling emission and transmission measurements accurately is performed and presented in this work. The simulation method involves emission simulation and transmission simulation. In emission simulation, a mathematical cardiac torso ( M C A T ) phantom and S I M S E T algorithm are employed to model the S P E C T emission imaging. The M C A T phantom is a 3 D digital phantom. Users can design the activity and the attenuation distribution of the phantom. The SEVISET package employs Monte Carlo techniques to model the physical processes and instrumentation used in emission imaging. S I M S E T package involves Photon History Generator ("PHG") module, Collimator Module, Detector Module, and Binning Module. In S I M S E T , the core module is the P H G , which models photon creation and transport through patients. The Collimator Module receives the photons from the P H G and tracks them through the modeled collimator, which can be modeled as a parallel-hole collimator, fan beam collimator or cone beam collimator. The Detector Module receives photons either directly from the P H G module or from the Chapter 3 Simulation Experiments 43 Collimator module, and tracks the photons through the specified detector. The Binning Module is used to process photon and detection records. Each module can create a Photon Flistory File to record information on the photons tracked. [44] In transmission simulation, a transmission scan can be modeled by Monte Carlo simulation or analytical simulation. In Monte Carlo transmission simulation, SBVISET can be used to model transmission source, transmission physical processes and instrumentation used in transmission scan. Although transmission can be modeled by using this method, its application is limited by the two problems. One is that long time is required to run the Monte Carlo transmission simulation. The other is that SEVISET is not designed to model a transmission source collimator. Analytical simulation is an alternative transmission simulation method instead of the Monte Carlo simulation. The Analytical transmission simulation runs much faster than the Monte Carlo transmission simulation, and creates data similar to those obtained by the Monte Carlo transmission simulation. The major difference between the Monte Carlo simulation and the analytical simulation is that the scatter effect in a transmission scan is not modeled by the analytical transmission simulation. This is because the scatter effect in a transmission scan is negligible, as validated by the analysis of the experiment "transmission scatter effect in a transmission scan" in Section 3.2.2.1. Chapter 3 Simulation Experiments 44 3.1. E m i s s i o n Simulat ion 3.1.1. M C A T Phantom Studies us ing T c 9 9 m labeled hexakis-2-methoxyisobutyl isoni t r i le ( M I B I ) are rout inely performed i n order to assess myocard ia l b l o o d perfusion. A M C A T phantom was used to mode l M I B I injection o f about 2 5 m C i . In general, 5% o f the total act ivi ty enters into the heart and the act ivi ty density i n the heart is about 5 times as much as that i n the body. F igure 3.1 shows the act ivi ty distr ibution ( A ) and the attenuation dis tr ibut ion (B) i n the same transverse slice of the M C A T phantom. Table 3.1 presents the act ivi ty dis tr ibut ion o f the who le 3 D digi ta l phantom used in our s imulat ion experiments. A B Figure 3.1: A, Activity distribution in the transverse slice of M C A T phantom. B , Distribution of attenuating medium in the transverse slice of M C A T phantom. Chapter 3 Simulation Experiments 45 M C A T organ Volume (ml) Activity concentration (uCi/ml) Total activity in organ (mCi) Liver 1774 5.0 8.87 Stomach 397 1.0 0.40 Kidney 286 0.0 0.0 Spleen 176 4.0 0.70 Right lung 1932 0.3 5.80 Left lung 1882 0.3 5.65 Heart myocardium 250 5.0 1.25 Chambers 388 1.0 0.39 Total activity 23.06mCi Table 3.1: The activity distribution used in simulation experiments. 3.1.2. Camera System In our simulation experiments, the camera system is modeled as a 50cm by 50cm detector with a parallel hole, low-energy high-resolution (HIRES) collimator. The detector rotates around the phantom clockwise at the radius of 32 cm. Stop and shoot model is used to record emission projections. Similar to a standard S P E C T detector, the energy resolution of the modeled detector is assumed to be 10% and its intrinsic spatial resolution is 3 mm. For the parallel-hole H I R E S collimator, the acceptance angle is 2.7°. The acceptance angle of a collimator is defined as the maximum angle between the direction of an outgoing photon and normal to the collimator surface. As illustrated in Figure 3.2, the hole diameter and septa length of the parallel hole H I R E S collimator are 1.13 and 24.1 mm respectively. Chapter 3 Simulation Experiments 46 hole diameter septa length Figure 3.2: The cross section of a parallel collimator 3.1.3. Data Acquisition Emission data acquisition in our simulation experiments is modeled on acquiring data by employing multi-energy window technique. As shown in Table 3.2, emission data were recorded in four different energy windows. Energy window Energy window percentage Energy widow range Emission window 20% 140kev(126-154) Transmission window 20% 100kev(90-110) Chapter 3 Simulation Experiments 47 Upper transmission scatter window 12% 116kev(109-123) Lower transmission scatter window 8% 86kev(83-90) Table 3.2:The four different energy windows used in emission simulation. In each energy window, each projection of emission imaging was saved in a 128 pixels byl28 pixels matrix where the pixel size was 0.3906 cm/pixel. The total emission projections over 360° for each energy window were 128 projections (64 projection/head for dual-head system). The acquisition time of each projection was 20 seconds. This means that each projection modeled by S I M S E T has the same number of photons emitted from body as real situation in 20 seconds. In order to save time and computer memory, only 24 slices of the phantom (containing heart and liver) were simulated. Since the emission data simulated by employing S I M S E T was the probability distribution, the simulated emission data was adjusted accordingly to the scale of the counts in clinical M I B I emission data. In clinical MIJ3I emission data, the count density in the heart region of each projection recorded in 140 keV emission windows is around 80 counts/pixel. The accordingly adjusted emission simulation data was presented in Figure 3.3. Chapter 3 Simulation Experiments 48 20 60 80 100 120 - 180 - 1620 140 40 120 10<$0 80 80 60 20 120 0 5 10 15 20 5 10 15 20 - 50 - 30 - 16 - 45 20 20 - 14 - 40 25 - 12 35 4 0 40 20 10 •-. 30 60 60 25 8 15 20 so 80 6 15 10 100 100 4 10 5 2 5 120 120 o 0 0 5 10 15 20 5 10 15 20 A. 140keV window B.lOOkeV window C.116keV window D. 86keV window Figure 3.3: Emission data of the projection (at the projection angle of 34° with respect to the x-axis in Figure 1.1) acquired in four different energy windows 3.2. Transmission Simulations In order to make transmission simulation consistent with the emission simulation, the same configuration of camera and phantom in the emission simulation is also used in the Monte Carlo transmission simulation and the analytical transmission simulation: (1) The M C A T phantom is also used as an attenuation phantom in transmission simulation. G d 1 5 3 is used as transmission source; and T c y v m i s used as emission source 99  • Chapter 3 Simulation Experiments 49 (2) The camera system is a 50cm by 50cm detector equipped with a parallel-hole H I R E S collimator. The energy resolution of the detector is assumed to be 10% and the intrinsic camera resolution is about 3 mm. (3) The detector rotates around the phantom clockwise at the radius of 32 cm. The distance between the detector and the transmission source is 115 cm. Stop and shoot model is used to record transmission projections. Each projection is saved in a 128 pixels by l28 pixels matrix where the pixel size is 0.3906 cm /pixel. (4) Transmission photons are recorded by employing the same multi-energy window technique as the acquisition of emission data. The four different energy windows include: emission energy window (20% energy window centered at 140 keV), transmission energy window (20% energy window at lOOkeV), 116keV scatter window (12 % energy window at 116 keV) and 86keV scatter window (8 % energy window at 86 keV). (5) The total transmission projections over 360° for each energy window are 128 projections (64 projection/head for dual head system). The acquisition time of each projection is 20 seconds. (6) In order to save time and computer memory, only 24 slices of the phantom (containing heart and liver inside) were simulated. Chapter 3 Simulation Experiments 50 3.2.1. Monte Carlo Transmission Simulation In the Monte Carlo transmission simulation, S I M S E T can be used to model transmission physical processes. S I M S E T package currently is designed for emission simulation, not for transmission simulation, but it is still possible to simulate transmission scan by modifying parameter files. The Monte Carlo transmission simulation was performed in the following way: First, an attenuation index map shown in Figure 3.4 was created for S I M S E T instead of real attenuation coefficient distribution. Because S I M S E T can't recognize the real attenuation coefficient and attenuation coefficient varies with energy, attenuation index numbers were input in the attenuation map of the M C A T phantom instead of real attenuation coefficients. A s shown in Table 3.3, the attenuation index numbers were used to identify different materials such as water. 20 40 60 80 100 120 Figure 3.4: Attenuation distributions in the M C A T phantom Chapter 3 Simulation Experiments 51 M a t e r i a l I n d e x n u m b e r A i r • W a t e r 1 B l o o d 2 B o n e 3 B r a i n 4 H e a r t 5 L u n g (5 M i u s c l e 7 Table 3.3: Attenuation index number table Second, transmission source was modeled for S I M S E T . Although S I M S E T package has no transmission source simulation option, transmission source can be modeled by simulating special activity source. For example, radioactive source in S I M S E T can be positioned in the phantom or outside the phantom. Transmission source can be modeled as shown in Figure 3.5 A i f we regard the radioactive source outside the phantom as the transmission source. Moreover, different kinds of transmission source such as sheet transmission source or multi-linear transmission source can be modeled. 20 40 60 80 100 120 Figure 3.5: A, A transverse slice of sheet transmission source (rotating around MCAT phantom). B, Transmission sinogram of one slice over 360°' Chapter 3 Simulation Experiments 52 Finally, with the attenuation index map and the modeled transmission source, the Monte Carlo transmission simulation was performed in the same way as simulating emission imaging. A s discussed previously, the Monte Carlo simulation can accurately simulate transmission scans, but transmission sinogram was obtained (as shown in Figure 3.5 B) at the expense of long running time, which may be more than a week. Moreover, a transmission collimator was not simulated in this Monte Carlo transmission simulation y. For the above reasons, our computer simulations adopted the analytical transmission simulation as an alternative method instead of the S I M S E T Monte Carlo transmission simulation. 3.2.2. Analytical Transmission Simulation This section consists two parts. The first part presents three transmission measurements, which are used to determine the parameters of transmission scan. The second part describes how the analytical transmission simulation is performed. 3.2.2.1. Transmission Experiments Analytical transmission simulation should be performed under a complete and correct understanding of transmission physics processes. For this reason, we performed three experiments to determine parameters of transmission scan. Chapter 3 Simulation Experiments 53 The first experiment is referred to as the "transmission scans of two collimated linear transmission sources". We expect to validate the idea that multiplication of two different gaussian functions can be used to represent the total effect of transmission source collimator and detector collimator. The second experiment is referred to as the "transmission scans when an object is placed at different distances from the detector". We expect to validate the idea that the blurring effect is determined not only by the transmission source collimator and detector collimator, but also by the distance between object and the surface of detector. The third experiment is referred to as the "transmission scatter effect in transmission scan". We expect to validate the idea that most of scattered transmission photons are not recorded since the transmission source collimator and detector collimator block scattered transmission photons before they reach the detector. Consequently, transmission scatter effect in transmission scan can be negligible. 1) Transmission scans of two collimated linear transmission sources Figure 3.6 shows a parallel-hole HIRES detector collimator and a standard parallel hole, high-sensitivity (HS) transmission source collimator, which are employed in a transmission scan. In theory, a parallel hole only allows the photons parallel to the hole septa through the collimator. In reality, due to the limited collimation of the collimator, the photons in the acceptance angle of the collimator can reach the detector Chapter 3 Simulation Experiments 54 and therefore be recorded as counts. This effect is referred to as the "blurring effect", which results in the loss of image resolution. In general, a 2D gaussian function can be used to represent the blurring effect of a collimator. Such the 2D gaussian function can be characterized by the full width of half maximum ( F W H M ) . The F W H M can be calculated by using Equation 3.1. where a is the acceptance angle, which is the ratio of hole diameter to septa length i f the angle is very small. The distance is the distance between the object in F O V and the detector. FWHM = the distance x tga (3.1) Transmission source ^ [j U U U' U"U¥U;lTlTirTrTj-* Parallel hole HS transmission collimator Transmission photon beams Detector The distance between transmission source and surface of detector Parallel hole H R E S detector collimator Figure 3.6: The geometry of transmission scan Chapter 3 Simulation Experiments 55 For the purpose of simulating the blurring effect of a transmission scan, the combined effect of the detector collimator and the transmission source collimator should be considered. In the analytical transmission simulation, each gaussian function of these two different collimators was simulated, and then the multiplication of these two gaussian functions was used to represent the combined effect of these two collimators. In order to validate this idea, the "two collimated linear transmission source scan" experiment was designed as shown in Figure 3.7 Transmission source Collimated transmission • f i photon beams * / i Detector -Parallel hole HS transmission collimator The distance between transmission source and surface of detector = 115 cm Parallel hole FIRES detector collimator Figure 3.7: The transmission scan geometry of two collimated linear transmission sources. In this experiment, two collimated linear transmission sources were placed parallel to each other. The collimation of the transmission source beam on the source side was achieved by using the parallel-hole HS collimator with 2.45 mm diameter holes Chapter 3 Simulation Experiments 56 and with 22.8mm long septa. The experimentally determined acceptance angle of the HS collimator was 5.3°. The sources were opposite to the detector of the Siemens dual-head camera equipped with a parallel-hole H R E S collimator, which had 24.1 mm long septa and 1.13 mm diameter holes. The experimentally determined acceptance angle of the H R E S collimator was 2.7°. In this configuration, the distance between the detector and source was 115 cm. A single projection of two collimated linear transmission sources was recorded in a 128 pixels by 128 pixels matrix where the pixel size was 0.4795 cm. One data zone was selected to study as Figure3.8 Figure 3.8: A single projection image of two collimated linear transmission sources As discussed above, the blurring effect can be described by a gaussian function. There are two methods to calculate the F W H M of the gaussian function: one is to obtain the F W H M by calculating profile-by-profile as shown in Figure 3.9 and Table 3.4; the Chapter 3 Simulation Experiments 57 other method is to obtain the F W H M by summing the selected data into one profile as shown in Figure 3.10 and Table 3.5. G O Figure 3.9: A. A profile in the selected data zone. B. Gaussian function fit to the projection of the first collimated linear source. C. Gaussian function fit to the projection of the second collimated linear source Chapter 3 Simulation Experiments 58 Index of slice in the selected area F W H M of the first gaussian function (pixel) F W H M of the second gaussian function (pixel) No 1. slice 11.7367 10.4146 No2.slice 10.6718 10.9348 No3. slice 10.8665 9.6485 No4. slice 10.2759 8.9074 No5. slice 10.5754 10.0595 No6. slice 9.8112 10.5176 No7.slice 9.2643 9.5469 No8. slice 10.1601 11.0007 No9.slice 10.2856 10.6005 NolO.slice 10.1546 10.1425 N o 11. slice 10.8181 9.3099 Nol2.s l ice 11.0802 11.1229 No 13. slice 9.9180 9.9570 Nol4.s l ice 9.9245 10.1566 No 15.slice 9.8963 10.5171 Nol6.s l ice 9.5133 11.2190 Nol7.s l ice 9.8701 9.9275 No 18. slice 10.5679 10.2930 Nol9.s l ice 10.2905 11.0495 Chapter 3 Simulation Experiments 59 No20.slice 11.3567 11.3678 No21. slice 9.5969 10.1960 Average F W H M = 10.3220 pixels =  4.949 cm Pixel size = 0.4795cm/pixel Table 3.4: FWHMs of gaussian functions describing different profiles in the selected data zone Figure 3.10: A. The profile obtained by summing selected data into one profile. B. Gaussian function fit to the projection of the first collimated linear source. C. Gaussian function fit to the projection of the second collimated linear source Chapter 3 Simulation Experiments 60 Index of slice in the selected area by summing data into one slice F W H M of the first gaussian function (pixel) F W H M of the second gaussian function (pixel) N o 1. slice 10.3762 10.3876 Average F W H M = 10.3819 pixels = 4.978cm Pixel size = 0.4795cm/pixel Table 3.5: FWHM of gaussian function obtained by summing selected data into one profile In order to compare the experimentally acquired data with the simulated data, a collimated line transmission source scan was simulated as shown in Figure 3.11. The 50cm by 50 cm detector equipped with a parallel-hole H R E S collimator was modeled in this simulation. Transmission source Collimated Transmission photon beams Detector Parallel hole H S transmission collimator The distance between transmission source and surface of detector =115 cm Parallel hole H R E S detector collimator Figure 3.11: One collimated line transmission source scan geometry in simulation Chapter 3 Simulation Experiments 61 Figure 3.12 showed a theoretical projection of the linear transmission source. The theoretical projection was recorded in a 128 pixels by 128 pixels matrix where the pixel size was 0.3906 cm /pixel. Theoretical projection of one collimated linear transmission source in ideal situation Figure 3.12: A theoretical projection of one collimated linear transmission source For the purpose of simulating the blurring effects of the two collimators used in the transmission scan, a HS gaussian function and a H R E S gaussian function were used to represent the effects of the HS collimator and H R E S collimator respectively as shown in Figure3.12. The F W H M s of the two gaussian functions were calculated by using Equation 3.1, where the distance is the distance between the source and the detector (115cm). Chapter 3 Simulation Experiments 62 Figure 3.13: A, 2D HS gaussian function; B. 2D HS gaussian function in surf image format. C, 2D HRES gaussian function. D, 2D HRES gaussian function in surf image format The combined effect of HS and H R E S collimators was represented by the multiplication of the HS gaussian function and the H R E S gaussian function, which was referred to as the " H S - H R E S " gaussian function shown in Figure 3.14. Figure 3.14: Multiplication of HS gaussian function and HRES gaussian function (the "HS-HRES gaussian"). A. 2D HS-HRES gaussian; B. 2D HS-HRES gaussian in surf image format Chapter 3 Simulation Experiments 63 The theoretical projection of the linear transmission source with blurring effect (shown in Figure 3.15 C) was simulated by convolving the theoretical projection (shown in Figure 3.15 A ) with the 2D H S - H R E S gaussian function (shown in Figure3.15 B) . 20 40 s o ao -i oo 120 o ^ c o n v o l u t i o n c Figure 3.15: The convolution of theoretical projection and HS-HRES gaussian function. A, 2D theoretical projection. B, 2D HS-HRES gaussian in surf image format. C, The projection with blurring effect As shown in Figure 3.16, a line was drawn through the blurring projection and a profile study was performed. The F W H M of the H S - H R E S gaussian function was calculated by using this profile study. Chapter 3 Simulation Experiments 64 20 40 60 SO 100 120 A B Figure 3.16: A, One line through the blurring projection. B, FWHM of simulated HS-HRES collimator obtained from profile study B y comparing the F W H M s from the above simulation and from the experimentally acquired data as shown in Table 3.6, we conclude that the H S - H R E S gaussian function can be used to simulate the combined blurring effect of the HS transmission source collimator and the H R E S detector collimator. Method to achieve F W H M F W H M (cm) Average F W H M obtained by calculating slice-by-slice of experimentally acquired transmission data 4.949 F W H M obtained by adding transmission data into one profile 4.978 F W H M obtained by by simulating H S - H R E S gaussian function 4.95 Table 3.6: Comparing the FWHMs from transmission simulation and from experimentally acquired transmission data Chapter 3 Simulation Experiments 65 2) Transmission scans when an object is placed at different distances from the detector As discussed above, the type of the collimator impacts the blurring effect in a transmission scan. In reality, the blurring effect is impacted not only by the type of the collimator, but also by the position of the object in F O V . transmission source with HS collimator > blocked transmission photon • \ beams F * l \ solid cylinder transmission source with HS collimator > blocked transmission photon beams solid cylinder unblocked transmission j photon beams unblocked transmission t photon beams ¥• Detector with HRES collimator Detector with HRES collimator Figure 3.17: A, Transmission scan geometry when the cylinder is placed at the center. B, Transmission scan geometry when the cylinder is placed closer to the source. As illustrated in Figure3.17, a much bigger portion of transmission photon beam is blocked i f the solid cylinder is placed closer to source. The extent to which the transmission photon beams are blocked by the cylinder depends on not only the size of the cylinder, but also the distance between the cylinder and detector. This leads to the inference that the spatial resolution of transmission image decreases as the distance between cylinder and the detector increases. In order to validate the idea, transmission Chapter 3 Simulation Experiments 66 scans were experimentally acquired with one solid rod and two solid rods, which were placed at the different distances from the detector. The rods made of A l with 2.5 cm diameter were used in these experiments. In the transmission scans with one rod, the rod was placed at two positions: 20 cm to the detector and 58 cm to the detector. Figure 3.18 and Figure 3.19 show transmission geometry (A), projections (B), and profiles(C) of these two one-rod transmission scans. Chapter 3 Simulation Experiments 67 Figure 3.18: A, One-rod transmission scan geometry where the rod was placed at 20 cm to the detector. B , Transmission projection of one-rod transmission scan. C, A profile of the one-rod transmission projection. Chapter 3 Simulation Experiments 68 Transmission source one solid rod Detector t * a iv j ,* i y H S collimator Distance from transmission source to surface of detector = 115 cm distance from rod to the surface of detector = 58 cm H R E S collimator 100 120 UO Figure 3.19: A, One-rod transmission scan geometry where the rod was placed at 58 cm to the detector. B , Transmission projection of one-rod transmission scan. C, A profile of the one-rod transmission projection B y comparing the profiles of these two one-rod transmission projections as shown in Figure 3.20, we conclude that the distance between the object and the detector ("Distance") impact the resolution and the blurring effect of transmission image. Chapter 3 Simulation Experiments 69 A . line profilej when the rod of 58 cm to B . l ineWofile of transmission when theVod was 20 cm to the detector of transmission projection Jkvas placed at the position detector; the placed projection at the position of 120 140 Figure 3.20: Comparing the two profiles of one-rod transmission projections. This conclusion was confirmed by the second series of experiments where the resolution was measured by the space between two rods. In the transmission scans with two rods, the two rods was parallel placed at two positions: 16 cm to the detector and 50 cm to the detector. The space between the two rods was 1.3 cm. Figure 3.21 and Figure 3.22 show transmission geometry (A), projections (B), and profiles(C) of these two-rod transmission scans Chapter 3 Simulation Experiments 70 Transmission soufee> Space between two solid y?* rods =1.3 cm ; \ Two solid rods' Detector MUIIWIIJ HS collimator Distance from transmission source to surface of detector = 115 cm Distance from rods to the surface of detector = 16 cm H R E S collimator Figure 3.21: A, Two-rod transmission scan geometry where the rods were placed at the position of 16 cm to the detector. B, Transmission projection of two-rod transmission can. C, The profile of two-rod transmission projection. Chapter 3 Simulation Experiments 71 Transmission souree-* Space between two solid! ;/ A ; f-J rods =1.3 cm Two solid rods Detector -4- mm H y. HS collimator Distance from transmission source to surface of detector = 115 cm Distance from rods to the surface of detector = 50 cm - H R E S collimator 40 60 80 100 120 II , , ! \ - f j J r / \ . 20 40 60 100 120 140 B Figure 3.22: A, Two-rod transmission scan geometry where the rods were placed at the position of 50 cm to the detector. B, Transmission projection of two-rod transmission can. C, The profile of two-rod transmission projection B y comparing the two rods profiles of these two-rod transmission projections as shown in Figure 3.23, we reconfirmed the conclusion that the Distance impacts the resolution and the blurring effect of transmission image. Chapter 3 Simulation Experiments 72 profile • A . line when the position of of transmission projection t^o rods was placed at the 16 cm to the detector; B . line projection ^ placed at \ profile of transmission when the two rods was [the position of 50 cm to the 120 140 Figure 3.23: Comparing the two profile studies of two rods transmission projections. Given the above analysis, we conclude that not only the type of collimator but also the position of the object in F O V impact the blurring effect and the resolution of transmission images. The extent of blurring depends on the distance between the object and the detector because the distance determines the F W H M of the H S - H R E S gaussian function. 3) Transmission scatter effect in transmission scan As discussed in section 2.3.1.3, the cross-talk effect results in the loss of quality of attenuation maps since some scattered emission photons are recorded in the transmission energy window. In order to correct the cross-talk effect, the estimate of the Chapter 3 Simulation Experiments 73 cross-talk was obtained by employing multi-energy window technique. For example, as indicated in Equation 2.4, the 1.1 times of the total scattered emission photons in 116 keV scatter window and 86 keV scatter window gave the estimate of total scattered emission photons recorded in transmission energy window. The question is whether all the photons recorded in scatter windows are scattered emission photons when emission and transmission data are measured simultaneously. In order to answer this question, a transmission experiment was designed. In this experiment, transmission data were measured in two different situations by employing multi-energy window technique. First, we measured transmission data without any object in F O V as shown in Figure 3.24. Transmission source ^ hnnnnnn niTuTTOT H S collimator photon beams —•Hf- JV Collimated transmission .•• \ ; Distance from transmission source to surface of detector = 115 cm Detector H R E S collimator Figure 3.24: Transmission scan without any object in the FOV Chapter 3 Simulation Experiments 74 The transmission data were recorded in three different energy windows: 20% transmission window centered at 100 K e V , 8% scatter window at 86 keV, and 12% scatter window at 116 keV. With the transmission data, the mean ratio of the transmission photons recorded in transmission energy window to the transmission photons recorded in other two scatter windows ("Mean Ratio") was calculated by using Equation 3.2: _ . total transmission photons recorded in transmission energy window m o x Mean Ratio = — (3.2) total transmission photons recorded in!16keV and 86 keV scatter windows 300 200 100 20 40 60 80 100 120 A. one projection in transmission energy window(100 keV window) 20 40 60 80 100 120 20 40 60 80 100 120 B. Addition of one projection in 86 keV window and one projection in 116 keV window 20 40 60 80 100 120 C. one projection in 86 keV scatter window D. one projection in 116 keV scatter window Figure 3.25Transmission projections without thorax phantom recorded in different energy windows Chapter 3 Simulation Experiments 75 The Mean Scatter ratio was equal to 0.13078 in the instance where transmission data was measured without any object in the F O V . Second, we measured the transmission scan with a thorax phantom (in which there was no activity) at the center of the F O V as shown in Figure 3.26. The transmission data was recorded in the same three different energy windows as the first measurement. The Mean Ratio was equal to 0.13228 in the instance where transmission data was measured with a non-activity thorax phantom at the center of the F O V . Transmission source *" Collimated transmission \, ', M \ • photon beams "f- \ , / ; J ; j \} i *-%t •.f\S >-Thorax phantom without any activity — • Detector HS collimator Distance from transmission source to surface of detector = 115 cm H R E S collimator Figure 3.26: Transmission scan with non-activity thorax phantom at the center of the FOV. Chapter 3 Simulation Experiments 76 250 20 200 40 150 60 100 SO 50 100 120 20 40 60 80 100 120 A. one projection in transmission energy window(100 keV window) 20 40 60 80 100 120 B. Addition of one projection in 86 keV window and one projection in 116 keV window 40 20 40 30 60 20 80 • 10 100 120 20 40 60 80 100 120 C. one projection in 86 keV scatter window 20 40 60 80 100 120 D. one projection in 116 keV scatter window Figure 3.27: Transmission projections with a non-activity thorax phantom recorded in the different energy windows. B y comparing the Mean Ratios in Table 3.7, we realized that the Mean Ratio almost did not change in the two different situations. This led to the inference that the Mean Ratio did not depend on the object in the F O V . We also realized that 13 % transmission photons were recorded in 116 keV and 86 keV scater windows. Obviously these photons were not scattered transmission photons. They were recorded in the two scatter windows due to the limited energy resolution of the detector. Given the above analysis, the experimental results implied that most of scattered transmission photons were blocked by collimators used in transmission scan and thus the Chapter 3 Simulation Experiments 77 transmission scatter effect is negligible. Consequently, transmission scatter effect did not have to be simulated in the analytical transmission simulation. Experimental situation The mean scatter ratio transmission scans without thorax phantom 0.13078 transmission scans with non-activity thorax phantom 0.13228 Table 3.7: The Mean Ratios of two different measurements 3.2.2.2. Analytical Transmission Simulation Method Pursuant to the experimental validation above, a transmission scan can be simulated by using analytical calculation. The analytical transmission simulation is performed in the following way: 1) A theoretical transmission blank scan is created. 2) The transmission data are calculated by using a I D simple system matrix, the theoretical blank data and the true attenuation map of the M C A T phantom. 3) Physics effects are added into the theoretical transmission data and blank data. These physics effects include blurring effect, Poisson noise and integer count effect. Chapter 3 Simulation Experiments 78 1) The creation of a theoretical transmission blank scan In this work, two kinds of transmission blank scan data were simulated: a uniform-type transmission blank scan and a profile-type transmission blank scan. Figure 3.28 shows a uniform-type transmission blank scan with the strength of 500 counts/pixel ("Uniform Blank"). The strength of blank scan is the same in all irradiated area. The total counts recorded in one projection are 1,356,000 counts. The simulated blank scan corresponds to the 20-second-per-projection blank scan with 24 M B q transmission source i f the efficiency of photon reception is assumed to be 100%. Figure 3.28: A, 2D projection of the Uniform Blank. B, The profile of the 2D projection. A s shown in Figure 3.29, a profile-type transmission blank scan with the maximum strength of 500 counts/pixel ("Profile Blank") is presented. The strength of transmission activity is tailored to fit attenuation distribution of a body. The total counts recorded in one projection are 895,930 counts, which are 66 % of the Uniform Blank. Chapter 3 Simulation Experiments 79 Compared with the Uniform Blank, the Profile Blank not only allows the activity distribution to be optimized to the attenuation distribution in the body, but also save the cost of transmission source. A B Figure 3.29: A,2D projection of the Profile Blank. B, The profile of the 2D projection 2) The creation of the theoretical transmission data If we have three parameters: system matrix, blank data and a true attenuation map, theoretically the theoretical transmission data can be calculated by using Equation 3.3. Ti = Bi exp (- £ jUj * lij) (3.3) where T, is transmission scan, 5, is blank scan, / / , is attenuation map, l,j is the element of system matrix representing contribution that pixel j makes to camera bin i. Chapter 3 Simulation Experiments 80 In the analytical simulation, a parallel system matrix was used as the system matrix. The parallel system matrix is a l D simple system matrix, which assumes perfect parallel collimation. The parallel strips of the center bin at the different projections are presented in Figure 3.30. Figure 3.30: The parallel strips of the center bin at the different projections. A, The parallel strip of the center bin at the first projection. B, The parallel strip of the center bin at the 15th projection. C, The parallel strip of the center bin at the 25th projection. In the analytical simulation, the Uniform Blank (shown in Figure 3.28) or the Profile Blank (show in Figure 3.29) was used as the blank scan. The M C A T attenuation phantom related to 100 keV(shown in Figure 3.31) was used as the true attenuation map. Figure 3.31: MCAT attenuation phantom related to 100 keV Chapter 3 Simulation Experiments 81 Figure 3.32 and Figure 3.33 show the theoretical transmission data for the Uniform Blank and for the Profile Blank respectively. 1 PR 1 • 100 120 140 A B Figure 3.32: The theoretical transmission data obtained by using the Uniform Blank. A. 2D transmission projection. B. The profile of 2D transmission projection. / X / / / \ / / ! / { A / \ 0 20 40 60 100 120 140 B Figure 3.33: The theoretical transmission data obtained by using the Profile Blank. A. 2D transmission projection. B. The profile of 2D transmission projection. Chapter 3 Simulation Experiments 82 3) Physics effects are added into the theoretical transmission and blank data. In reality, transmission physics processes are influenced by several physics factors such as blurring effect, Poisson noise and integer count effect. In order to accurately simulate a transmission scan and a blank scan, we must consider and add these physics factors into the analytical transmission simulation. (1) Blurring effect In the transmission scan, blurring effect, which is determined by the total spatial resolution of the imaging system, degrades the quality of transmission image. The total resolution was calculated by using the collimator spatial resolution and the camera intrinsic spatial resolution as illustrated in Equation 3.4 total spatial resolution - ^(collimator resolution) + (intrinsic resolution) (3.3) The collimator spatial resolution was calculated by using Equation 3.1, where the distance was the distance between the phantom and the detector. A s discussed previously, the blurring effect in transmission scans was impacted not only the type of collimator but also the distance between the object in F O V and detector. In our simulation experiments, the distance was 32 cm because the detector was simulated to Chapter 3 Simulation Experiments 83 rotating around the phantom clockwise at the radius of 32 cm. With the distance (32cm) and the acceptance angle of each collimator, the H S collimator resolution (the F W H M of the HS gaussian function) was 3.44 cm, and the H R E S collimator resolution (the F W H N of the H R E S gaussian function) was 1.5 cm in our simulation experiments. The total collimator resolution (the F W H M of the H S - H R E S gaussian function) was 1.37 cm by measuring the F W H M of the multiplication of the HS and the H R E S gaussian functions (as validated by the "transmission scans of two collimated linear transmission sources" experiment in Section 3.2.2.1). The camera intrinsic spatial resolution depends on the characteristics of the camera used. A s mentioned previously, the S P E C T camera intrinsic resolution in our simulation experiments was about 3 mm. Therefore, the total spatial resolution was 1.3703 cm by using Equation3.3. With the total spatial resolution as F W H M , a new 2D gaussian function was generated. A s shown in Figure 3.15, the blurring effect was simulated by convolving the theoretical transmission and blank projection with the new 2D gaussian function, where the F W H M was 1.3703 cm. Figure 3.34 and Figure 3.35 show the blurring blank projections (A), the blurring transmission projections (B), and their profiles(C) in these two different situations: employing the Uniform Blank and employing the Profile Blank. Chapter 3 Simulation Experiments 84 A . 2D transmission projection with B . Line profile of 2D blurring blurring effect transmission projection 600 500 400 300 200 100 0 II -\ I 1 L _ J i _ )l—•- 1 1 — I — - • , I 0 20 40 60 80 100 120 140 C. Line profile of 2D blurring transmission blank projection Figure 3.34: A, The 2D Uniform Blank projection with blurring effect. B, 2D transmission projection with blurring effect. C, Their profiles of 2D blurring transmission and blank projections. Chapter 3 Simulation Experiments 85 A . 2D transmission projection with B . Line profile of 2D blurring blurring effect transmission projection C. Line profile of 2D blurring transmission blank scan projection Figure 3.35: A, The Profile Blank projection with blurring effect. B, 2D transmission projection with blurring effect. C, Their profiles of 2D blurring transmission and blank projections. (2) Poisson noise During the S P E C T measurement, the statistic noise, which follows the Poisson distribution, occurs when the photons are recorded as counts by the detector. Generally the Poisson noise effect in transmission data is stronger than the one in blank data because the blank data has higher counts. The difference in the number of counts Chapter 3 Simulation Experiments 86 between the blank data and the transmission data is caused by attenuation effect. This is because the transmission data is attenuated by a patient, but the blank data not. Additionally, the difference of counts is related to the scanning time. More scanning time, more counts. For example, 2 minutes per projection during a blank scan and 20 seconds per projection during a transmission scan. In our simulations, Poisson noise effect was accordingly added into the transmission and the blank data respectively. Figure 3.36 and Figure 3.37 show the blank projections with blurring effect and Poisson noise (A), the related transmission projections (B), and their profiles(C) in these two different situations: employing the Uniform Blank and employing the Profile Blank. Chapter 3 Simulation Experiments 87 h H 400 20 40 eo 1 s o 100 120 A . The 2D Uniform Blank projection with R r e l a t e d ? ? t ™ 1 8 ™ ^ ™ projection blurring effect and Poisson noise W l t h b l u m n § e f f e c t a n d P o i s s o n n o i s e C. The line profiles of 2D transmission and blank projection with blurring effect and Poisson noise Fi gure 3.36: A , The 2D Uniform Blank projection with blurring effect and Poisson noise. B, The related 2D transmission projection. C, Their profiles of 2D blurring transmission and blank projections. Chapter 3 Simulation Experiments 88 20 40 BO I 80 100 -I20 A . The 2D Uniform Blank projection with R r e l a t e d ? ? t r a n s m i s s i o n projection noise C . The line profiles of 2D transmission and blank projection with blurring effect and Poisson noise Figure 3.37: A, The 2D Profile Blank projection with blurring effect and Poisson noise. B, The related 2D transmission projection. C, Their profiles of 2D blurring transmission and blank projections (3) Integer count effect In the analytical calculation, fractions are introduced into the simulation and used as count numbers. However, counts are recorded as integers in real measurements. In order to simulate this integer count effect, rounding method was employed to our analytical transmission simulations. Chapter 3 Simulation Experiments 89 In summary, the analytical transmission simulation not only simulate a transmission scan and a blank scan, but also model various physics effects such as blurring effect in the scans by adding these physics factors into the theoretical transmission data and blank data. Additionally, the transmission scatter effect in a transmission scan is not modeled in the analytical simulation. 3.3. Creation of "experimental" data Clinical ly transmission scan can be measured simultaneously with S P E C T emission scan. G d 1 5 3 and T c 9 9 m are usually used as transmission source and emission source respectively. A s discussed in section 2.3.1.3, due to the cross-talk effect, not only transmission photons but also a portion of scattered emission photons wi l l be recorded in the transmission energy window (100 keV window) shown in Figure 2.9,. In order to simulate such the "experimental" data, the scattered emission photons recorded in the transmission energy window (cross-talk) are added to the transmission photons recorded in the transmission energy window. 3.4. Data Analysis Method In the case of computer simulations, the true object attenuation distribution is known. This enables one to evaluate the quantitative accuracy of the image Chapter 3 Simulation Experiments 90 reconstruction. The accuracy of the reconstruction was measured by using the relative standard deviation. The simulated amount of attenuation within each region (for example, left lung region) is distributed uniformly and thus all pixels in the given region should ideally reconstruct to the same value. However, due to factors such as statistical noise, each reconstructed pixel value would be slightly different from those in the neighboring pixels. The relative standard deviation of the whole phantom was obtained by measuring the standard deviation of reconstructed pixel values to the true pixel attenuation values for each region. The equations used are given below: where £ris the relative standard deviation of the whole phantom, eRis the relative standard deviation each region, and NRegion is the total number of regions in the phantom. 8R is calculated by using Equation 3.5. Ok £R = XR (3.5) Chapter 3 Simulation Experiments 91 where XR is the true attenuation value in the region R and OR is the standard deviation within the region R. CTR is calculated as, where xx is the reconstructed pixel value within the zth pixel, xR is the true simulated attenuation value in the same pixel and NR is the total number of pixels in the region R. With this accuracy evaluation, it should be obvious that values closer to 0 imply that reconstructed maps have less deviation from the true attenuation distribution, while larger values indicate a larger degree of deviation. 3.5. Results and Discussion - Simulation Experiments 3.5.1. Reconstruction of Attenuation Maps without Emission Cross-talk (3. Effect With the Uniform Blank with physics effects and its transmission data, simulated uniform-type sinogram was obtained as shown in Figure 3.40 A . In the same way, simulated profile-type sinogram was obtained as shown in Figure 3.40 B . Chapter 3 Simulation Experiments 92 Due to the Poisson noise effect, some "non-reasonable" values (such as negative values and infinite values) would appear in the obtained sinograms. As indicated in Equation 1.5, the negative values at some pixels occurred i f the number of counts recorded at the pixels of transmission projections was more than that at the same pixels of blank projections. Theoretically the negative values should not occur in the sinograms. In order to eliminate the negative values, zero values were used to replace them. Similarly, as indicated in Equation 1.5, the infinite values at some pixels occurred i f the number of counts recorded at the pixels of transmission projections were zero, while the counts recorded at the same pixels of blank projections were not zero. Obviously dividing by zero is impossible in computing. In order to avoid this computing difficulty, very small values such as 1 0 3 were used to replace the zero counts in transmission projections. Although the infinite values were avoided, some values out of reasonable range still remained. In this work, zero values were used to replace the values out of reasonable range. A B Figure 3.38: The sinogram simulated with physics effects. A. The simulated uniform-type sinogram; B. The simulated profile-type sinogram. Chapter 3 Simulation Experiments 93 With the sinograms, an attenuation map can be generally reconstructed by using F B P (with 0.5 order Butterworth filter and 0.7 cutoff frequency) or O S E M . The quality of the reconstructed attenuation map is impacted by not only the shape of blank but also the strength of blank. A question is what is the extent of this impact. In the following sections, in order to analyze how the shape of blank and the strength of blank impact on the quality of reconstructed attenuation map, the blank and transmission data were simulated in several different situations. Then attenuation maps were reconstructed by using F B P and O S E M method respectively: 1) A s shown in Figure 3.41, reconstructed F B P and O S E M attenuation maps were obtained by employing the uniform-type blank. The strength of the uniform blank is 500 counts/pixel. 2) As shown in Figure 3.42, reconstructed F B P and O S E M attenuation maps were obtained by employing the profile-type blank. The maximum strength of the profile blank is 500 counts/pixel. 3) A s shown in Figure 3.43, reconstructed F B P and O S E M attenuation maps were obtained by employing the uniform-type blank. The strength of the uniform blank is 2000 counts/pixel. 4) A s shown in Figure 3.44, reconstructed F B P and O S E M attenuation maps were obtained by employing the profile-type blank. The maximum strength of the profile blank is 2000 counts/pixel. Chapter 3 Simulation Experiments 94 5) As shown in Figure 3.45, reconstructed F B P and O S E M attenuation maps were obtained by employing the uniform-type blank. The strength of the uniform blank is 10000 counts/pixel. 6) As shown in Figure 3.46, reconstructed F B P and O S E M attenuation maps were obtained by employing the profile-type blank. The maximum strength of the profile blank is 10000 counts/pixel. Figure 3.40: A, The reconstructed FBP (A) and OSEM (B) attenuation map was obtained by the profile blank. The maximum strength of the profile blank is 500 counts/pixel. Chapter 3 Simulation Experiments 95 Fi gure 3.42: A, The reconstructed FBP (A) and OSEM (B) attenuation map was obtained by employing the profile blank. The maximum strength of the profile blank is 2000 counts/pixel. Chapter 3 Simulation Experiments 96 A B Figure 3.43: A, The reconstructed FBP (A) and OSEM (B) attenuation map was obtained by employing the uniform blank. The strength of the uniform blank is 10000 counts/pixel. A B Figure 3.44: A, The reconstructed FBP (A) and OSEM (B) attenuation map was obtained by employing the profile blank. The strength of the uniform blank is 10000 counts/pixel. A further question is how well these reconstructed attenuation maps compare with the true attenuation map shown in Figure3.30. Since the true attenuation coefficient at each phantom pixel is known, it is possible to quantitatively assess the accuracy of attenuation map reconstruction. In order to evaluate the quantitative accuracy of the reconstructed attenuation maps shown above, the relative standard deviation was employed. Chapter 3 Simulation Experiments 97 Simulated situation F B P O S E M Uniform blank with the strength of 500 counts/pixel 0.43395 0.44392 Profile blank with the max strength of 500 counts/pixel 0.45115 0.40052 Uniform blank with the strength of 2000 counts/pixel 0.30066 0.28851 Profile blank with the max strength of 2000 counts/pixel 0.29402 0.28014 Uniform blank with the strength of 10000 counts/pixel 0.29103 0.24944 Profile blank with the max strength of 10000 counts/pixel 0.28511 0.24763 Table 3.8: The relative standard deviations of the reconstructed attenuation maps in several different simulated situations B y analyzing the results presented in the Table 3.8, i f the uniform-type blank and the profile-type blank are of the same strength, the attenuation maps reconstructed from the two kinds of blanks almost have identical relative deviation. This leads to the inference that the shape of blank does not impact the quality of reconstructed attenuation maps for these two kinds of blanks. We also realized that the relative standard deviation significantly decreases as the strength of the blank increases. The reason is that the influence of Poisson noise decreases as the strength of blank increases. Therefore, we Chapter 3 Simulation Experiments 98 conclude that quality of a reconstructed attenuation map increases as the strength of blank increases. 3.5.2. Reconstruction of attenuation maps with emission cross-talk effect As we discussed above, the factor impacting the quality of reconstructed attenuation map is the strength of blank, not the shape of blank. In order to compare the performances of cross-talk correction methods for the instances where different strengths of blanks were employed, the "experimental" data was simulated in the following two different situations: 1) A s shown in Figure 3.47, the first "experimental" transmission data was simulated by adding the emission cross-talk to the transmission data which was obtained by employing the profile blank with the max strength of 500 counts/pixel. Chapter 3 Simulation Experiments 99 Figure 3.45: The first "experimental" transmission data was simulated by adding emission cross-talk and transmission data which was obtained by employing profile blank with the max strength of 500 counts/pixel. 2) As shown in Figure 3.48, the second "experimental" transmission data was simulated by adding the emission cross-talk to the transmission data which was obtained by employing the profile blank with the max strength of 2000 counts/pixel. Chapter 3 Simulation Experiments 100 Transmission with cross-talk Figure 3.46: The second "experimental" transmission data was simulated by adding emission cross-talk and transmission data which was obtained by employing profile blank with the max strength of 2000 counts/pixel. With the "experimental" data above, the attenuation maps were reconstructed by using F B P without cross-talk correction (as shown in Figure 3.49), F B P with subtraction cross-talk correction method ( "FBPS" as shown in Figure 3.50), O S E M with subtraction cross-talk correction method ( " O S E M S " as shown in Figure 3.51) and O S T G S method (as shown in Figure 3.52). Chapter 3 Simulation Experiments 101 A B Figure 3.47: The reconstructed attenuation maps by using F B P without cross-talk correction. A, The attenuation map obtained by using the first "experimental" data. B , The attenuation map obtained by using the second "experimental" data. Obviously, due to the cross-talk effect, the reconstructed attenuation coefficient distribution in the heart and liver regions was much lower than the true attenuation coefficient distribution. In clinics, studies using M I B I are routinely performed to assess myocardial blood perfusion. This leads to the inference that the heart region is the ROI in the M I B I studies. Consequently, cross-talk correction is required to perform. Fi gure 3.48: The reconstructed attenuation maps by using F B P with subtraction cross-talk correction. A, The attenuation map obtained by using the first "experimental" data. B , The attenuation map obtained by using the second "experimental" data. Chapter 3 Simulation Experiments 102 20 40 e o s o 100 120 20 40 eo eo 100 120 A B Figure 3.49: The reconstructed attenuation maps by using O S E M with subtraction cross-talk correction. A , The attenuation map obtained by using the first "experimental" data. B , The attenuation map obtained by using the second "experimental" data. 20 40 eo s o 100 120 20 AO s o s o 100 120 B Figure 3.50: The reconstructed attenuation maps by using OSTGS. A , The attenuation map obtained by using the first "experimental" data. B , The attenuation map obtained by using the second "experimental" data. In order to evaluate the performance of the cross-talk corrections above, the relative standard deviation was employed as shown in Table 3.9. Chapter 3 Simulation Experiments 103 Reconstruction method The first "experimental" data The second "experimental" data F B P S 0.57943 0.32178 O S E M S 0.42251 0.31496 O S T G S 0.33018 0.29783 Table 3.9: The relative standard deviations of the reconstructed attenuation maps by using different reconstruction methods in two different "experimental" situations. B y analyzing the results shown in Table 3.9, we realized that the relative standard deviations of O S T G S attenuation map were lower than those of other reconstructed attenuation maps in the two different situations, especially at lower strength of blank such as the strength of 500 counts/pixel. It meant that O S T G S attenuation maps were closer to the true attenuation maps. Consequently, we concluded that O S T G S was the better cross-talk correction method than the conventional subtraction method, especially at lower strength of transmission. We also realized that the relative standard deviation decreases as the strength of blank data increases. This leads to the inference that the quality of reconstructed attenuation map increases as the strength of blank data increases. 3.5.3. Reconstruction of Emission Images with A C In order to analyze the effects of different attenuation corrections on the emission image reconstruction, A C was performed by employing the attenuation maps presented above. In our simulation experiments, emission images were reconstructed by employing Chapter 3 Simulation Experiments 104 O S E M with different attenuation corrections. These O S E M emission images were listed as follows: 1) The O S E M emission image without A C ("OSEM_without_AC") . 2) The O S E M emission image with attenuation correction by using the F B P attenuation map ( " O S E M _ w i t h _ A C _ F B P " ) . 3) The O S E M emission image with attenuation correction by using the F B P S attenuation map ( " O S E M _ w i t h _ A C _ F B P S " ) . 4) The O S E M emission image with attenuation correction by using the O S E M S attenuation map ( " O S E M _ w i t h _ A C _ O S E M S " ) . 5) O S E M emission image with attenuation correction by using the O S T G S attenuation map ( " O S E M _ w i t h _ A C _ O S T G S " ) . 6) The O S E M emission images with attenuation correction by using true attenuation map ( " O S E M _ w i t h _ A C _ T R U E " ) . Figure 3.53 and Figure 3.54 show the OSEM_wi thou t_AC (A), the O S E M _ w i t h _ A C _ F B P (B), the O S E M _ w i t h _ A C _ F B P S (C), the O S E M _ w i t h _ A C _ O S E M S (D), the O S E M _ w i t h _ A C _ O S T G S (E) and the O S E M _ w i t h _ A C _ T R U E (F) in the two different situations: employing profile blank with the max strength of 500 counts/pixel and employing profile blank with the max strength of 2000 counts/pixel. Chapter 3 Simulation Experiments 20 40 60 80 100 120 A . O S E M w i t h o u t A C 20 40 60 80 100 120 C . O S E M w i t h A C F B P S 20 40 60 80 100 120 E . O S E M _ w i t h A C O S T G S 20 40 60 80 100 120 B . O S E M w i t h A C F B P 20 40 60 80 100 120 D . O S E M w i t h A C O S E M S 20 40 60 80 100 120 F . O S E M _ w i t h _ A C T R U E Figure 3.51 When profile blank with the max strength of 500 counts/pixel was employed, OSEM emission images were obtained with different attenuation corrections Chapter 3 Simulation Experiments 106 20 40 60 80 100 120 A . O S E M without A C 20 40 60 80 100 120 C. O S E M with A C F B P S 20 40 60 80 100 120 E . O S E M with A C O S T G S 20 40 60 80 100 120 B . O S E M with A C F B P 20 40 60 80 100 120 D . O S E M with A C O S E M S 20 40 60 80 100 120 F. O S E M with A C T R U E Figure 3.52 When profile blank with the max strength of 2000 counts/pixel was employed, OSEM emission images were obtained with different attenuation corrections. Chapter 3 Simulation Experiments 107 B y comparing the images above, we realized that the shape of heart in the O S E M emission image with A C by using O S T G S attenuation map seems to be closer to that of the O S E M emission image with A C by using true attenuation map. In order to quantitatively analyze the results, the accuracies of the above images compared with the O S E M emission image with A C by using true attenuation map were evaluated by employing the relative standard deviation as shown in Table 3.9. Emission images with A C The first "experimental" data The second "experimental" data O S E M _ w i t h _ A C _ F B P 0.64 0.35 O S E M _ w i t h _ A C _ F B P S 0.85 0.28 O S E M _ w i t h _ A C _ O S E M S 0.46 0.24 O S E M _ w i t h _ A C _ O S T G S 0.33 0.24 Table 3.10: The relative standard deviations of the reconstructed OSEM emission images by using different attenuation correction methods. Table 3.10 indicated that the performance of the O S E M _ w i t h _ A C _ O S T G S was better that those of the other images, especially for the instance where the profile blank with the max strength of 500 counts/pixel was employed. Also , Table 3.10 showed that the O S E M _ w i t h _ A C _ O S E M S could obtain the similar result to the O S E M _ w i t h _ A C _ O S T G S at very high strength of blank such as 2000 counts/pixel. Chapter 4 Phantom Experiments 108 4 Phantom Experiments In conjunction with the previous described computer simulations, a series of experimental tests of cross-talk correction methods were performed. In this chapter, a brief description of how the phantom experiments were performed wi l l be given. Also , the results w i l l be summarized and analyzed. 4.1. The Brief Description of the Phantom Experiments In the phantom experiments, the thorax phantom was filled with water and two big water bags were attached modeling as breasts. Moreover, two small cylinder tumors were modeled in the phantom. One cylinder tumor was 8 ml in volume; the other tumor was 12 ml in volume. The smaller tumor contained 2.15 M B q in activity and was attached to the heart of the phantom. The bigger tumor contained 3.3 M B q in activity and was attached to the spine of the phantom. Besides the activity in the tumors, the heart part of the phantom contained 16 M B q in activity, while the rest part of the phantom didn't contain any activity. The camera system was a 64 cm by 64 cm dual-head detector with a parallel-hole H I R E S collimator. Each camera head rotated 90°around the water filled thorax phantom clockwise at the radius of 32 cm. One head started from 45°, and the second head start from - 4 5 ° . It meant that the total extent of detector rotation was 180°. The transmission data and emission data were measured Chapter 4 Phantom Experiments 109 simultaneously. The measured data were recorded in four different energy windows (20% energy window centered at 140 keV, 20% energy window at 100 keV, 8% energy window at 86 keV and 12% energy window at 116 keV). For each energy window, projection data were saved in a 128 pixels byl28 pixels matrix where the pixel size was 0.4795cm. The total projections over 180° for each energy window were 64 projections (32 projections/head for the dual-head system). Acquisition time of each projection was 40 seconds. 4.2. Results and Discussion - Phantom Experiments 4.2.1. Reconstruction of Attenuation Maps With the transmission and blank data experimentally acquired, attenuation maps were reconstructed by employing different reconstruction methods, and then smoothed by employing the gaussian filter with 1.5cm F W F I M . Figure 4.1 shows the reconstructed attenuation maps of the No.8 and No. 12 transverse slices by using F B P (A), the reconstructed attenuation map by using O S E M (B), the reconstructed attenuation map by using F B P S (C), the reconstructed attenuation map by using O S E M S (D) and the reconstructed attenuation map by using O S T G S (E). Chapter 4 Phantom Experiments 110 20 4 0 GO BO IOO 120 20 40 eo ao 100 -120 A . F B P attenuation map of No.8 slice 20 4 0 eo s o 100 120 B . O S E M attenuation map of No.8 slice 20 40 60 SO "I OO 1 20 C. F B P S attenuation map of N0.8 slice 20 40 s o 0 0 100 120 F B P attenuation map of No. 12 slice O S E M attenuation map of No. 12 slice 20 40 60 SO 10O -1 20 F B P S attenuation map of No. 12 slice Chapter 4 Phantom Experiments 2 0 4 0 GO SO IOO 1 2 0 20 40 SO SO 1 OO 1 20 40 60 SO 1 OO 1 D . O S E M S attenuation map of No.8 slice O S E M S attenuation map of No. 12 slice E . O S T G S attenuation map of No.8 slice O S T G S attenuation map of No. 12 slice Figure 4.1: The reconstructed attenuation maps of the No.8 and No.12 transverse slices by using FBP (A), O S E M (B), FBPS (C), OSEMS (D) and OSTGS (E). Since the thorax phantom is filled by water, the attenuation coefficient distribution we expect is uniform. A s shown in Figure 4.1 A and B , the attenuation coefficient distributions in the F B P and O S E M reconstructed attenuation maps without cross-talk correction are not uniform (especially, the reconstructed attenuation coefficients in the breasts are higher than those in heart). Among the above attenuation maps, the reconstructed attenuation maps by using O S T G S show uniform attenuation Chapter 4 Phantom Experiments 112 distributions and clearly visible spinal cord. Also, the reconstructed attenuation coefficients of the phantom in O S T G S attenuation maps are about 17 m" 1, which is the attenuation coefficient of water at 100 keV. Given the analysis above, we conclude that O S T G S attenuation map is the closest to the map we expect. 4.2.2. Reconstruction of Emission Images With the reconstructed attenuation maps above, emission images with A C were reconstructed by employing O S E M method. Since the heart is not suspended vertical to the transverse slice, cardiac activity images can't be displayed completely and correctly with the coordinate system used previously. Consequently, the coordinate system should be adjusted accordingly. For example, a new coordinate system can be obtained by rotation. In the new coordinate system, the cardiac short axis, horizontal long axis and vertical long axis are regarded x, y and z axes respectively. The rotated emission images were obtained by rotating the reconstructed O S E M emission images. These rotated emission images were listed as follows: 1) The rotated O S E M emission image without A C ("R_OSEM_without_AC") as shown in Figure 4.2. 2) The rotated O S E M emission image with attenuation correction by using the F B P attenuation map ( " R _ O S E M _ w i t h _ A C _ F B P " ) as shown in Figure 4.3. Chapter 4 Phantom Experiments 113 3) The rotated O S E M emission image with attenuation correction by using the O S E M attenuation map ( " R _ O S E M _ w i t h _ A C _ O S E M " ) as shown in Figure 4.4. 4) . The rotated O S E M emission image with attenuation correction by using the F B P S attenuation map ( " R _ O S E M _ w i t h _ A C _ F B P S " ) as shown in Figure 4.5. 5) The rotated O S E M emission image with attenuation correction by using the O S E M S attenuation map ( " R _ O S E M _ w i t h _ A C _ O S E M S " ) as shown in Figure 4.6. 6) The rotated O S E M emission image with attenuation correction by using the O S T G S attenuation map ( " R _ O S E M _ w i t h _ A C _ O S T G S " ) as shown in Figure 4.7. The following Figures show the No.63 cross sections of the cardiac vertical long axis (A), the No. 12 cross sections of the cardiac horizontal long axis (B), the No.74 cross sections of the cardiac short axis (C) and profiles of the No.74 cross section of the cardiac short axis (D) of the above rotated images. In order to easily compare the following images, the same scales for the same cross sections were used. Chapter 4 Phantom Experiments 1 Figure 4.2: The No.63 cross sections of the cardiac vertical long axis (A) , the N o . 12 cross sections of the cardiac horizontal long axis (B), the No.74 cross sections of the cardiac short axis (C) and profiles of the No.74 cross section of the cardiac short axis (D) o f the R _ O S E M _ w i t h o u t _ A C . Figure 4.3: The No.63 cross sections of the cardiac vertical long axis (A), the No. 12 cross sections of the cardiac horizontal long axis (B), the No.74 cross sections of the cardiac short axis (C) and profiles of the No.74 cross section of the cardiac short axis (D) of the R_OSEM_with_AC_FBP. Chapter 4 Phantom Experiments 1 Figure 4.4: The No.63 cross sections of the cardiac vertical long axis (A), the No. 12 cross sections of the cardiac horizontal long axis (B), the No.74 cross sections of the cardiac short axis (C) and profiles of the No.74 cross section of the cardiac short axis (D) of the R_OSEM_with_AC_OSEM. Chapter 4 Phantom Experiments 1 1 7 Figure 4.5: The No.63 cross sections of the cardiac vertical long axis (A), the No. 12 cross sections of the cardiac horizontal long axis (B), the No.74 cross sections of the cardiac short axis (C) and profiles of the No.74 cross section of the cardiac short axis (D) of the R_OSEM_with_AC_FBPS. Chapter 4 Phantom Experiments 118 Figure 4.6: The No.63 cross sections of the cardiac vertical long axis (A), the No. 12 cross sections of the cardiac horizontal long axis (B), the No.74 cross sections of the cardiac short axis (C) and profiles of the No.74 cross section of the cardiac short axis (D) of the R_OSEM_with_AC_OSEMS. Chapter 4 Phantom Experiments 119 C D Figure 4.7: The No.63 cross sections of the cardiac vertical long axis (A), the No. 12 cross sections of the cardiac horizontal long axis (B), the No.74 cross sections of the cardiac short axis (C) and profiles of the No.74 cross section of the cardiac short axis (D) of the R_OSEM_with_AC_OSTGS. A s shown in F igure 4.2, images shows very l o w act ivi ty distr ibution since the loss o f act ivi ty due to attenuation is not recovered. A l s o , the profi le plot shows that there is Chapter 4 Phantom Experiments 120 strong activity outside of heart compared with the activity in heart. In reality, only the heart part of the phantom contained activity, while the rest part of the phantom didn't contain any activity. A s shown in Figure 4.3 and Figure 4.4, the loss of activity is recovered partially, not completely. This is because that cross-talk effect causes lower reconstructed attenuation coefficients in the reconstructed attenuation map. Additionally, the profile plots show that there is non-uniform activity distribution in the heart. As shown in Figure 4.5 and Figure 4.6, the loss of activity is over recovered. This is because subtraction cross-talk correction increases noise effect, which caused some higher reconstructed attenuation coefficients as shown in Figure 4.1 (C) and (D). As shown in Figure 4.7, the loss of activity is recovered correctly by employing O S T G S attenuation map. Also , the profile plot shows that the activity distribution in the heart is more uniform compared with other profiles of the images. Given the analysis above, we reconfirm that O S T G S can offer high quality attenuation map since O S T G S can efficiently correct cross-talk effect by incorporating the cross-talk estimate into the O S T G S formula. Chapter 5 Conclusion and Future Work 121 5 Conclusion and Future Work This chapter provides a brief review of this work and proposes future improvements to the attenuation correction. 5 . 1 . Review of the Work This work presents O S T G S , a new reconstruction method of attenuation distribution. The primary advantage of O S T G S is that the method explicitly accounts for cross talk estimate and thus yields better estimate of attenuation map by avoiding the subtraction of the cross-talk estimate. The fact that reconstruction time is clinically acceptable is another advantage of O S T G S . For example, an attenuation map can be obtained within a minute by using O S T G S in the system of 500MHz Pentium II C P U and 640 S D R A M . Computer simulations were performed to compare the performance of O S T G S with the reconstruction methods which involves the use of subtraction cross-talk correction. In the computer simulations, emission data was simulated by employing S I M S E T and transmission data was simulated by using analytical calculation. Thus, the "experimental" data, which was obtained with simultaneous transmission/emission acquisition protocol, was generated by combining the simulated emission data and transmission data. With the "experimental" data, the reconstructed attenuation maps with Chapter 5 Conclusion and Future Work 122 cross-talk correction were obtained by using the reconstruction methods involving conventional subtraction cross-talk correction and O S T G S . The reconstructed attenuation maps obtained based on the "experimental" data were compared with true attenuation map by using relative standard deviation shown in Table 3.9. The comparative analysis indicates that O S T G S , as a cross-talk correction method, is better than conventional subtraction cross-talk correction. The comparison of emission slices with A C in computer simulations and the phantom experiments also reveals that O S T G S attenuation map not only reduces the artificial extra-cardiac activity but also accurately recovers the lost cardiac activity due to attenuation. 5.2. Future Work Currently O S T G S method has shown reasonable results based upon the simulations and experiments performed. It is still possible to improve this method further. Moreover, although attenuation correction has been investigated since S P E C T appeared about two decades ago, there are many problems remaining. Some of the future works are listed as follows. Chapter 5 Conclusion and Future Work 123 5.2.1. Fully Three Dimensional Attenuation Reconstruction with Collimator Blurring Correction In the current attenuation map reconstruction technique, a I D simple system matrix is used. The simple system matrix assumes perfect parallel collimation. In reality, a true collimator has a limited acceptance angle from which incoming photons originate. Consequently the collimator is unable to achieve perfect parallel collimation. The problem is complicated by the fact that two different collimators (i.e. transmission source collimator and detector collimator) are employed in the transmission measurements. If such the model discussed above can be incorporated in to fully 3D system matrix used in the attenuation reconstruction, this improvement w i l l result in the improved spatial resolution in the reconstructed attenuation map. 5.2.2. Truncation Correction The truncation problem has been investigated in nuclear medicine imaging in order to improve image quality and quantitation. While it was reported that improvement in truncation correction can be obtained by constraining the reconstruction to the actual area of the body [33], in fact the truncation problem is yet to be solved. For example, it has been shown that truncation may impact on the detection of cardiac lesions [36]. It is Chapter 5 Conclusion and Future Work 124 necessary to investigate truncation effect further and find a new truncation correction method. 5.2.3. Cross-talk Estimation Currently the estimate of cross-talk can be obtained by employing multi-energy window technique. As indicated in Equation 2.4, the cross-talk estimate is approximately equal to 1.1 times the sum of counts recorded in the two side scatter energy windows. Obviously, the above method cannot give us the accurate cross-talk estimate, which is required fundamentally for cross-talk correction. A new method called A P D can be employed to accurately estimate cross-talk with emission peak data only. The new method is performed by using scatter calculation. With the reconstructed attenuation map without cross-talk correction, A P D method can obtain the first-order cross-talk estimate by using scatter calculation. Then, the reconstructed attenuation map can be updated by performing cross-talk correction with the first-order cross-talk estimate. Higher-order cross-talk estimate can be obtained by repeating the above process any number of times. Better performance is expected to be obtained with the improvement of estimating cross-talk. Bibliography 125 Bibliography [1] Michael A . K i n g , Stephen J. Gl ick, P. Hendrik Pretorius, R. Glenn Wells, Howard C. Gifford, and Manoj V . 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