Science, Faculty of
Physics and Astronomy, Department of
DSpace
UBCV
Bohnet, Barry
Master of Science - MSc
University of British Columbia
Several methods have been developed to decrease scanning times including gradient-and-spin echo (GraSE) imaging, and compressed sensing. One of the benefits of both techniques is that they can decrease scan time to varying degrees by changing the GraSE factor or sampling level. An electronic phantom was created to model a single slice multi-echo magnetic resonance imaging (acquisitions using different GraSE factors, sampling levels, and signal-to-noise ratios (SNRs). The image included two types of prostatic tissue, peripheral zone and transitional zone, both with the signal composed of two different T2 decay values. LWI analysis was applied to the resulting images and the resulting T2 decay values and luminal water fraction (LWF) measurements were compared with a modeled multi-echo spin echo sequence without noise. The results of comparing the error with the theoretical scan times suggest that if scan times are comparable, the errors will be comparable. As the SNR decreased preference shifted towards using lower GraSE when possible. Next a celery phantom was scanned because it shows two T2 components with values similar to those found in an in-vivo prostate. 3D GraSE Images were acquired with different GraSE factors and had simulated undersampling to various degrees. Measurements of the reconstructions were compared to the fully sampled GraSE factor 3 images and agreed with the previous result that similar scan times lead to similar errors, showing no preference for a higher or lower GraSE factor. Lastly one patient had 3D multi-echo GraSE images taken of their prostate. Images had a simulated undersampling of various levels and reconstructions were compared to the fully sampled GraSE factor 3 images. Again, similar scan times resulted in similar error measurements. For the celery and human prostate scans, selected combinations of GraSE factor and sampling levels were simulated with one hundred different sampling schemes each in order to observe the variability due to the chosen sampling scheme. Following our method of fully sampling the central 33% of k-space and randomly sampling the remainder, the results strongly suggest that different sampling schemes of the same level have little effect on the resulting LWF and T2 measurements.
COMPARISON OF COMBINATIONS OF VARIOUS LEVELS OF GRASE AND COMPRESSED SENSING ON LUMINAL WATER IMAGING MEASUREMENTS IN MRI by Barry Bohnet B.Sc., The University of Alberta, 2010 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES (Physics) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) August 2020 © Barry Bohnet, 2020 ii The following individuals certify that they have read, and recommend to the Faculty of Graduate and Postdoctoral Studies for acceptance, a thesis/dissertation entitled: COMPARISON OF COMBINATIONS OF VARIOUS LEVELS OF GRASE AND COMPRESSED SENSING ON LUMINAL WATER IMAGING MEASUREMENTS IN MRI submitted by Barry Bohnet in partial fulfillment of the requirements for the degree of Master of Science in Medical Physics Examining Committee: Piotr Kozlowski, Radiology Supervisor Stefan Reinsberg, Physics Supervisory Committee Member iii Abstract Several methods have been developed to decrease scanning times including gradient-and-spin echo (GraSE) imaging, and compressed sensing. One of the benefits of both techniques is that they can decrease scan time to varying degrees by changing the GraSE factor or sampling level. An electronic phantom was created to model a single slice multi-echo magnetic resonance imaging (acquisitions using different GraSE factors, sampling levels, and signal-to-noise ratios (SNRs). The image included two types of prostatic tissue, peripheral zone and transitional zone, both with the signal composed of two different T2 decay values. LWI analysis was applied to the resulting images and the resulting T2 decay values and luminal water fraction (LWF) measurements were compared with a modeled multi-echo spin echo sequence without noise. The results of comparing the error with the theoretical scan times suggest that if scan times are comparable, the errors will be comparable. As the SNR decreased preference shifted towards using lower GraSE when possible. Next a celery phantom was scanned because it shows two T2 components with values similar to those found in an in-vivo prostate. 3D GraSE Images were acquired with different GraSE factors and had simulated undersampling to various degrees. Measurements of the reconstructions were compared to the fully sampled GraSE factor 3 images and agreed with the previous result that similar scan times lead to similar errors, showing no preference for a higher or lower GraSE factor. iv Lastly one patient had 3D multi-echo GraSE images taken of their prostate. Images had a simulated undersampling of various levels and reconstructions were compared to the fully sampled GraSE factor 3 images. Again, similar scan times resulted in similar error measurements. For the celery and human prostate scans, selected combinations of GraSE factor and sampling levels were simulated with one hundred different sampling schemes each in order to observe the variability due to the chosen sampling scheme. Following our method of fully sampling the central 33% of k-space and randomly sampling the remainder, the results strongly suggest that different sampling schemes of the same level have little effect on the resulting LWF and T2 measurements. v Lay Summary A technique called Luminal Water Imaging (LWI) can be used in the diagnosis and grading of prostate cancer. This magnetic resonance imaging (MRI) scan is long, which can lead to patient motion and poor results. Gradient-and-spin-echo imaging and compressed sensing are two different MRI methods that can be used to reduce scan times to varying degrees. The goal of this thesis is to compare different combinations of these approaches at different levels in order to optimize the scanning protocol. This is tested on an electronic phantom, a celery phantom, and an in-vivo prostate. Compressed sensing involves under-sampling the image with a random sampling scheme and using a reconstruction algorithm in order to fill in the missing information. As a side project 100 different sampling schemes were used for the same sampling level, and the results were compared to find the importance of sampling scheme choice. vi Preface In-vivo prostate scans were acquired under protocol H02-70400. The idea of comparing combinations of various levels of GraSE MRI and compressed sensing and the effect on Luminal Water Imaging was developed by Dr. Piotr Kozlowski. It built upon the work of Dr. Shirin Sabouri who developed the Luminal Water Imaging technique with Dr. Kozlowski, and the work of Dr. Henry Chen who worked on the optimization of compressed sensing in Myelin Water Imaging. All simulations and computations were done using Matlab 2019. Scripts for generating electronic phantoms were developed by the author, with help from Dr. Kozlowski. Scripts for compressed sensing reconstruction were adapted by the author from scripts written by Dr. Henry Chen, and scripts for applying the Luminal Water Analysis were adapted by the author from scripts written by Dr. Shirin Sabouri. Analysis of the results was all done by the author. A majority of the simulations were run on Compute Canada Servers which helped to save a tremendous amount of time, and the Matlab SPGL1 package was used for compressed sensing reconstructions. vii Table of Contents Abstract ......................................................................................................................................... iii Lay Summary .................................................................................................................................v Preface ........................................................................................................................................... vi Table of Contents ........................................................................................................................ vii List of Tables ..................................................................................................................................x List of Figures ............................................................................................................................... xi List of Symbols ........................................................................................................................... xiii List of Abbreviations ................................................................................................................. xiv Acknowledgements .................................................................................................................... xvi Dedication .................................................................................................................................. xvii Chapter 1: Introduction ................................................................................................................1 1.1 Prostate Cancer ................................................................................................................. 1 1.2 Magnetic Resonance Imaging .......................................................................................... 2 1.2.1 Nuclear Magnetic Resonance ................................................................................... 3 1.2.2 T1, T2, and T2* Relaxation ......................................................................................... 4 1.2.3 Bloch Equations ........................................................................................................ 5 1.2.4 Slice Selection ........................................................................................................... 7 1.2.5 Frequency Encoding ................................................................................................. 7 1.2.6 Phase Encoding ......................................................................................................... 8 1.2.7 k-space ...................................................................................................................... 8 1.2.8 Spin Echo .................................................................................................................. 9 viii 1.2.9 Gradient Echo ......................................................................................................... 11 1.2.10 Gradient and Spin Echo .......................................................................................... 12 1.2.11 2D vs 3D Imaging ................................................................................................... 14 1.3 Compressed Sensing ...................................................................................................... 15 1.3.1 Incoherent Sampling ............................................................................................... 15 1.3.2 Wavelet Decomposition .......................................................................................... 16 1.3.3 Reconstruction ........................................................................................................ 17 1.4 Luminal Water Imaging ................................................................................................. 19 1.4.1 T2 Measurements .................................................................................................... 19 1.4.2 Luminal Water Fraction .......................................................................................... 21 1.5 Statistics ......................................................................................................................... 22 1.5.1 Signal-to-Noise Ratio.............................................................................................. 22 1.5.2 Normalized Mean Square Error .............................................................................. 23 1.5.3 Coefficient of Variation .......................................................................................... 23 Chapter 2: Electronic Phantom ..................................................................................................24 2.1 Method ........................................................................................................................... 24 2.2 Results ............................................................................................................................ 27 2.3 Conclusions .................................................................................................................... 30 Chapter 3: Celery Phantom ........................................................................................................32 3.1 Method ........................................................................................................................... 32 3.2 Results ............................................................................................................................ 34 3.2.1 Effects of Sampling Schemes ................................................................................. 34 3.2.2 Effects on Luminal Water Imaging Analysis.......................................................... 35 ix Chapter 4: Prostate ......................................................................................................................39 4.1 Method ........................................................................................................................... 39 4.2 Results ............................................................................................................................ 40 4.2.1 Effect of Sampling Scheme .................................................................................... 40 4.2.2 Effects on Luminal Water Imaging Analysis.......................................................... 41 Chapter 5: Discussion ..................................................................................................................44 References .....................................................................................................................................47 Appendices ....................................................................................................................................50 Appendix A - Additional results from electronic phantom ..................................................... 50 Appendix B - Additional Results for Celery Phantom ............................................................. 52 B.1 T2Short ROI Measurements .......................................................................................... 52 B.2 T2Long ROI Measurements ........................................................................................... 53 B.3 Luminal Water Fraction ROI Measurements ............................................................. 54 Appendix C - Additional Results for Prostate Scans ................................................................ 55 C.1 T2Short ROI Measurements .......................................................................................... 55 C.2 T2Long ROI Measurements ........................................................................................... 56 C.3 Luminal Water Fraction ROI Measurements ............................................................. 57 x List of Tables Table 2-1 Electronic phantom modelling parameters ................................................................... 25 Table 3-1 Acquisition parameters for celery phantom MRI acquisitions ..................................... 32 Table 4-1. Scanning parameters for prostate imaging scan .......................................................... 39 Table 5-1 Mean value of the short T2 component in the ROI of the celery .................................. 52 Table 5-2 Median value of the short T2 component in the ROI of the celery............................... 52 Table 5-3 Standard deviation of the short T2 component in the ROI of the celery ...................... 52 Table 5-4 Mean value of the long T2 component in the ROI of the celery................................... 53 Table 5-5 Median value of the long T2 component in the ROI of the celery ............................... 53 Table 5-6 Standard deviation of the long T2 component in the ROI of the celery ....................... 53 Table 5-7 Mean value of the LWF in the ROI of the celery ......................................................... 54 Table 5-8 Median value of the LWF in the ROI of the celery ...................................................... 54 Table 5-9 Standard deviation of the LWF in the ROI of the celery ............................................. 54 Table 5-10 Mean value of the short T2 component in the peripheral zone of the prostate ........... 55 Table 5-11 Median value of the short T2 component in the peripheral zone of the prostate ........ 55 Table 5-12 Standard deviation of the short T2 component in the peripheral zone of the prostate 55 Table 5-13 Mean value of the long T2 component in the peripheral zone of the prostate ............ 56 Table 5-14 Median value of the long T2 component in the peripheral zone of the prostate ......... 56 Table 5-15 Standard deviation of the long T2 component in the peripheral zone of the prostate 56 Table 5-16 Mean value of the LWF measurement in the peripheral zone of the prostate ............ 57 Table 5-17 Median value of the LWF measurement in the peripheral zone of the prostate ......... 57 Table 5-18 Standard deviation of the LWF measurement in the peripheral zone of the prostate 57 xi List of Figures Figure 1-1. Reconstructed 512x512 images using various levels of zero-filling in k-space. Top Left: Fully sampled k-space. Top Right: Central 256x256 sampled k-space. Bottom Left: Central 128x128 sampled k-space. Bottom Right: Central 64x64 sampled k-space. Signal intensity was normalized for the individual images. ............................................................................................ 9 Figure 1-2: Pulse sequence for a single echo spin echo MRI ....................................................... 11 Figure 1-3 Pulse sequence of a single echo gradient echo MRI ................................................... 12 Figure 1-4 Diagram of a single echo GraSE MRI with a GraSE factor of 3 ................................ 13 Figure 1-5 An example of second level wavelet decomposition. Wavelet of the image in the top left results in coefficient maps for horizontal, vertical, and diagonal wavelet transforms. In order to better visualize the maps, individual map intensities were normalized .................................... 17 Figure 1-6. Signal decay curve and T2 distribution. Left: A signal decay curve from a simulated multi-echo spin echo sequence. Right: The resulting T2 distribution following the fit of the regularized non-negative least squares fit. .................................................................................... 20 Figure 1-7. A graphical description of the luminal water fraction ............................................... 22 Figure 2-1: The Electronic Phantom: A simple 2D image was created to mimic the shape of a cross-sectional image of a human prostate ................................................................................... 24 Figure 2-2 Examples of electronic phantom reconstructions ....................................................... 28 Figure 2-3 NMSE results for peripheral zone LWF measurements for the different signal-to-noise ratios .................................................................................................................................... 29 Figure 2-4 NMSE results from electronic phantom measurements in both peripheral and transition zone for an SNR of 300 ................................................................................................ 30 xii Figure 3-1. Coefficient of variation results comparing the mean parameter in the region of interest for one-hundred different sampling schemes. .................................................................. 34 Figure 3-2. NMSE measurements for each parameter in the celery scans. .................................. 36 Figure 3-3. NMSE measurements compared to total scan time for the various parameters ......... 38 Figure 5-1 Normalized mean square error measured against the ground for luminal water fraction measurements in the electronic phantom ...................................................................................... 50 Figure 5-2 Normalized mean square error measured against the ground truth on Long T2 values of the electronic phantom.............................................................................................................. 51 xiii List of Symbols ALong Area under the peak of long T2 AShort Area under the peak of short T2 Ncomp Number of peaks in the T2 distribution T1 Spin-Lattice Relaxation T2 Spin-Spin Relaxation T2-Long Geometric mean T2 of the long T2 component T2-Short Geometric mean T2 of the short T2 component xiv List of Abbreviations 2D Two-Dimensional 3D Three-Dimensional ANOVA Analysis of Variance COV Coefficient of Variation CPMG Carr-Purcell-Meiboom-Gill CS Compressed Sensing FOV Field of View GF Grase Factor GMT2 Geometric Mean T2 GRASE Gradient and Spin Echo LWF Luminal Water Fraction LWI Luminal Water Imaging MRI Magnetic Resonance Imaging NE Number of Echoes NMR Nuclear Magnetic Resonance NMSE Normalize Mean Square Error NNLS Non-Negative Least Squares PSA Prostate Specific Antigen PZ Peripheral Zone RF Radio Frequency ROI Region of Interest SNR Signal to Noise Ratio xv SPGL1 Spectral Projected Gradient for L1 Minimization TE Echo Time TR Repetition Time TZ Transition Zone ZF Zero Filled xvi Acknowledgements "If I have seen further, it is by standing upon the shoulders of giants" – Isaac Newton A great big thank you to my supervisor Dr. Piotr Kozlowski, who encouraged me to take on this challenge and has been supportive throughout the whole endeavor. His guidance and constant positivity are inspiring, and it has been a true privilege to have learned from him. I must also thank Andrew Yung who taught me everything I know about MRI. He has always found the time and the patience to answer my questions and to check in and make sure that I was doing well along the way. A special thanks to Dr. Henry Chen and Dr. Shirin Sabouri, friends who always believed in me, and were always there to support me, especially when it came to quantum mechanics… A thanks as well to Dr. Stefan Reinsberg and Dr. Firas Moosvi. While you might not know it, your input and discussions at group meetings were motivating and helped me to understand more than I thought I could. As always, a huge thanks to my parents who have always been supportive no matter what I do. And, of course, a giant thank you to my loving partner Jenny who has always been there for me and our wonderful daughter Amber who has been an endless supply of stories and huggles. xvii Dedication To my wife Jenny and my daughter Amber, thank you for all the love and support. I couldn’t have done it without you. 1 Chapter 1: Introduction Magnetic Resonance Imaging (MRI) is a widely used, non-invasive imaging technique. While it can be effective and versatile, one of the most limiting factors in MRI is the length of time it can take to acquire images. There are several methods for decreasing the imaging time of scans including Gradient-and-Spin-Echo (GraSE) imaging, and Compressed Sensing (CS). One of the benefits of these two methods is that they can be used simultaneously, and both can be applied to decrease scan times by varying degrees. The goal of this thesis is to apply various levels of combined GraSE imaging and compressed sensing in order to speed up a multi-echo T2-weighted imaging scan, and to investigate the effects on Luminal Water Imaging (LWI) analysis. In order to explore this question, results from three different experiments are used: the first is simulation data of an MRI scan of a prostate, the second is phantom data acquired by scanning a celery bunch, and the third is from an in-vivo scan of a human prostate. 1.1 Prostate Cancer According to the Canadian Cancer Society (1), prostate cancer is projected to be the fourth most diagnosed cancer in Canada in 2020, and the most diagnosed cancer in men. It is expected to account for almost 10% of the cancer deaths in men, making it the third most deadly cancer for Canadian males. Prostate cancer tends to grow slowly, and often responds well to treatment. Approximately 93% of those diagnosed with prostate cancer survive longer than 5 years after 2 diagnoses. Survival rates are much higher, nearly 100%, in patients who are diagnosed before the cancer has metastasized, suggesting that early diagnosis is key to survival. Generally, once symptoms of prostate cancer appear, patients are given a blood test to measure levels of Prostate Specific Antigen (PSA) and a digital rectal exam is performed to manually feel for abnormalities. Neither of these tests are conclusive, but they can suggest the presence of prostate cancer. The next step in diagnosis is a biopsy where 8-12 cores can be taken in order to confirm prostate cancer. From here LWI MRI can be used in the diagnosis and grading of the prostate cancer and a treatment plan can be made. Treatment can consist of either radiation, surgery, or a wait-and-see approach (2). 1.2 Magnetic Resonance Imaging MRI is an imaging modality that has been used for nearly 50 years and is based on the principles of Nuclear Magnetic Resonance (NMR). The first use of NMR was by Isidor I. Rabi in 1938 (3,4). Then in 1946, two research groups led by Felix Bloch and Edward M. Purcell were able to independently detect NMR signal in condensed matter (5-7). The first NMR image was acquired by Paul Lauterbur (8) in 1971, and in 1977 the first MRI scan of a human was carried out by Raymond Damadian (9). 3 1.2.1 Nuclear Magnetic Resonance While quantum mechanics is commonly used to derive the basic principles of NMR, classical mechanics is often sufficient to describe MRI physics (10). The Larmor equation (Equation 1.1 below) is used to describe the precessional frequency of a non-zero spin, charged particle in a magnetic field. This, where ω is the Larmor frequency, γ is the gyromagnetic ratio of the particle of interest, and B is the strength of the magnetic field. In the case of MRI, the particle of interest is most often the hydrogen nucleus, i.e. proton, due to its presence in water and the high water content in the human body. The gyromagnetic ratio of proton is 42.58MHz/T, which means in a 3 Tesla magnetic field, the resonant frequency is 127.74MHz. 𝜔 = 𝛾 ∙ 𝐵 Equation 1.1 In Rabi’s experiments in 1939 (3,4) he placed a molecular beam of particles with non-zero spin in a strong magnetic field. By doing this, the effects of the interactions between the spins would be dominated by the strength of the magnetic dipole and the main magnetic moment would align itself with the field. By applying a radio frequency (RF) pulse near the beam at the Larmor frequency of the particles, the magnetic moment of the beam could be tilted and will begin to precess about the direction of the external magnetic field. This oscillation of the magnetization moment creates a measurable signal that can be detected by an RF coil. 4 1.2.2 T1, T2, and T2* Relaxation Once the excitation pulse is turned off, the excited spins will begin to return to their equilibrium state and the net magnetization vector will eventually align itself with the direction of the magnetic field once again. This change back to the equilibrium state is called relaxation and can be described by three different time constants, T1, T2, and T2* (11). T1 relaxation, also called the longitudinal relaxation, is a process by which the magnetization vector returns back to the equilibrium orientation along the magnetic field. This longitudinal relaxation can be modeled with an exponential recovery equation with T1 as the time constant (This can be seen in section 1.2.3). T2 relaxation, also called the transverse relaxation, is the process of the decay in the coherence of the spins. The magnetization vector is a macroscopic sum of many individual spin magnetic moments. While the excitation pulse can bring these spins into phase to create the magnetization vector, the individual spins precess at slightly different frequencies due to molecular interactions, and as time passes the distribution of these spins begins to broaden which reduces the amplitude of the magnetization vector. This process can also be described as an exponential decay function with T2 being the time constant. The decay of the transverse magnetization vector can more broadly be described by T2* decay. While T2 decay is related to the transverse relaxation processes, T2* is the rate of spin-spin dephasing caused by T2 as well as local static field inhomogeneities. It is defined by equation 1.2 where T2´ is the transverse decay due to static field inhomogeneities. Factors such as changes in 5 tissue composition, or air-tissue interfaces can cause small inhomogeneities in the magnetic field which can result in much faster decay of the transverse magnetization. The spin echo scanning sequence (described in more detail in section 1.1.9) can be used to reverse the T2´ decay, leaving a signal dependent solely on T2 decay. 1𝑇2∗ =1𝑇2′ +1𝑇2 Equation 1-2 Another explanation of T2* decay can be found in the lyrics of Greg Crowther’s song “Twinkle Twinkle T2*: (12): Twinkle twinkle T2*, how I wonder what you are! X-Y signal soon decays, why do the spins go out of phase? Twinkle, twinkle, T2*, something pulls those spins apart. Spin-spin crosstalk sets T2, but by then T2* is through. A brief duration here is sealed by an inhomogeneous field. Twinkle, twinkle, T2*, Now I know just what you are! 1.2.3 Bloch Equations The Bloch equations were developed by Felix Bloch (6) to describe nuclear magnetization as a function of time when T1 and T2 decays are present. In MRI, commonly the direction of the external magnetic field is assigned as the z-direction, simplifying the equations 1-3 through 1-5. 6 𝑑𝑀𝑥𝑑𝑡= 𝜔0𝑀𝑦 − 𝑀𝑥𝑇2 Equation 1-3 𝑑𝑀𝑦𝑑𝑡= −𝜔0𝑀𝑥 − 𝑀𝑦𝑇2 Equation 1-4 𝑑𝑀𝑧𝑑𝑡= 𝑀0− 𝑀𝑧𝑇1 Equation 1-5 Here, Mx, My, and Mz refer to the magnetization vectors along the x, y, and z directions, while M0 refers to the equlibrium magnetization along the z-direction. The easiest equation to solve is the case of the longitudinal magnetization since it can be solved as a simple linear differential equation. The general solution to this can be seen in equation 1-6 below. 𝑀𝑧(𝑡) = 𝑀𝑧(𝑡0) ∙ 𝑒−(𝑡−𝑡0) 𝑇1⁄ + 𝑀0(1 − 𝑒−(𝑡−𝑡0) 𝑇1⁄ ) Equation 1-6 In the case of magnetization immediately following the 90° flip angle RF pulse, longitudinal magnetization is zero at time t0, and the equation 1.6 is simplified to equation 1.7. 𝑀𝑧(𝑡) = 𝑀0(1 − 𝑒−(𝑡−𝑡0) 𝑇1⁄ ) Equation 1-7 The solutions to equations 1-3 and 1-4 can be written as equations 1-8 and 1-9, and by combining these solutions we can solve for the total transverse magnetization, MT = Mx+iMy which can be seen in equation 1-10. 𝑀𝑥 = 𝑒−𝑡 𝑇2⁄ (𝑀𝑥(0) cos(𝜔𝑡) + 𝑀𝑦(0)sin (𝜔𝑡)) Equation 1-8 7 𝑀𝑦 = 𝑒−𝑡 𝑇2⁄ (𝑀𝑦(0) cos(𝜔𝑡) − 𝑀𝑥(0)sin (𝜔𝑡)) Equation 1-9 𝑀𝑇 = 𝑀𝑇(0)𝑒𝑖𝜔𝑡−𝑡 𝑇2⁄ Equation 1-10 By combining the equations for the longitudinal and transverse magnetizations the signal produced by an MRI sequence can be predicted. 1.2.4 Slice Selection MRI can be performed on an entire volume within the magnetic field, but in order to only observe signal from one slice of the sample a linear magnetic gradient can be applied. For the sake of simplicity, we let this be the z-direction, making the images in the x-y plane. By varying the magnetic field linearly, the Larmor frequency begins to vary linearly as well along the z-direction. Since the excitation of spins only occurs within the range of Larmor frequencies corresponding the bandwidth of the applied RF pulse, the slice of interest can now be spatially selected. 1.2.5 Frequency Encoding After a slice of interest has been excited by the RF pulse and an echo is about to be formed (spin echo and gradient echo methods are further described in sections 1.2.7 and 1.2.8 respectively), a linear magnetic gradient can now be applied in a direction orthogonal to the slice selective direction. For simplicity, let this be the x-direction. Once again, this affects excited spins and will cause the Larmor frequency to vary linearly along the x-direction. Well before the echo, a negative gradient can be applied to de-phase the spins, and during the echo a positive gradient can be applied that will allow the spins to re-phase at the center of the echo and then begin to de-8 phase again. The signal in the collected echo will then be comprised of spins oscillating at different frequencies, which will be spatially dependent. This echo can be transformed from the frequency domain into the spatial domain by using a Fourier Transform. This results in a projection of the signal along the x-axis only; in order to obtain a 2D image, phase encoding is commonly used. 1.2.6 Phase Encoding Phase encoding is a way in which a second spatial dimension can be collected. Once an excitation pulse is completed, a phase encoding gradient can be applied in the y-direction in order to change the initial phase of the spins along this axis. By collecting a series of echoes in the frequency domain with different strengths of phase encoding gradients, these varying initial phase values can be used to provide spatial information. Similar to the frequency encoding direction, the spatial information can be retrieved from the phase encoding direction using a Fourier transform. 1.2.7 k-space This collection of echoes with different phase encoding steps is referred to as k-space. k-space has an interesting feature where every point in k-space has information about the entire image space. This means that a sub-sample of k-space can still provide a reasonable image. In fact, the outer portion of k-space corresponds to high frequency components of the image, meaning that changing these values to zero (zero-filling) results in a more blurred image. 9 An example of the effect of this can be seen in figure 1-1 where an image was inversely Fourier transformed to obtain its k-space. A central portion of this k-space is assumed to have been sampled and then zero-filled to 512x512, then Fourier transformed into the image space. Figure 1-1. Reconstructed 512x512 images using various levels of zero-filling in k-space. Top Left: Fully sampled k-space. Top Right: Central 256x256 sampled k-space. Bottom Left: Central 128x128 sampled k-space. Bottom Right: Central 64x64 sampled k-space. Signal intensity was normalized for the individual images. 1.2.8 Spin Echo Based off the original Hahn Echo (13), a spin echo scan is one of the most basic types of NMR sequences. Following a 90° excitation pulse applied with a slice selective gradient, (and for 10 imaging) a phase encoding gradient is then applied. Spins will begin to de-phase and T1 relaxation will occur slowly drawing the net magnetization vector back toward the +z-direction. A small gradient is applied along the readout direction in order to prepare for the frequency encoding during the echo. At a time τ after the excitation pulse is applied, a 180° flip angle pulse is used to refocus the spins and form an echo. During the echo a gradient is applied in the readout direction in order to obtain our frequency encoding. A diagram of this sequence can be seen in figure 1-2 The signal in a single voxel for a spin echo scan can be described by equation 1-11. The signal depends on the T2 decay value and the initial transverse magnetization M0. 𝑴𝑻(𝑻𝑬) = 𝑴𝟎𝒆−𝑻𝑬/𝑻𝟐 Equation 1-11 In order to more accurately measure T2 quantitatively a multi-echo spin echo sequence can be used which will give more points along the T2 decay curve. It begins as a normal spin echo sequence, but at a time 3τ after the excitation pulse a 180° RF pulse is applied once again allowing the spins to come back into phase and create another echo at a time 4τ after the excitation pulse. This continues with more 180° RF pulses and more echoes being acquired, generally until well after the signal from transverse magnetization decays to zero. In the case of our multi-echo prostate scans, a total of 64 echoes were collected. 11 Figure 1-2: Pulse sequence for a single echo spin echo MRI 1.2.9 Gradient Echo In a gradient echo, the set-up is similar to that for the spin echo sequence. A slice selective gradient is applied simultaneously with an excitation pulse, and then a gradient is applied in the phase encoding direction. This is where the two sequences begin to differ. While the phase encoding gradient is applied, a negative gradient is applied in the readout direction in order to de-phase the spins in that direction. Immediately after, a positive gradient is applied in the readout direction for twice as long as the negative gradient. This allows the spins to come back into phase and de-phase again, creating the gradient echo. 12 The gradient echo can also easily be adapted to be used as a multi-echo sequence. After the last positive gradient is applied in the readout direction, a negative gradient can be applied which will again refocus and eventually de-phase the spins creating a second gradient echo. Negative and positive gradients can continue to be applied in an alternating pattern to give any number of gradient echoes. The amplitude of these echoes is affected by the T2* relaxation process. Figure 1-3 Pulse sequence of a single echo gradient echo MRI 1.2.10 Gradient and Spin Echo The Gradient and Spin Echo (GraSE) technique (14) is a scanning method that combines both gradient and spin echo techniques in order to acquire several lines of k-space following a single 13 excitation pulse. The number of k-space lines acquired per excitation pulse is called the GraSE factor (GF). Essentially the sequence can be thought of as a multi-echo gradient echo sequence with a spin echo in the center. Figure 1-4 Diagram of a single echo GraSE MRI with a GraSE factor of 3 The sequence starts off the same as the gradient echo, with a slice selective gradient combined with a 90° excitation pulse followed by a gradient applied in the readout direction. This time, however, the gradient can be positive because at a time τ after the excitation pulse, a 180° refocusing pulse will be applied, which reverses the sign of the accumulated phase. Following this, alternating positive and negative gradients can be applied to give us a set of gradient echoes with a spin echo in the center at time 2τ. In between the echoes a small gradient is applied in the 14 phase encoding direction in order to “jump” the acquisition from one k-space line to the next. This leads to the acquisition of k-space in a zig-zag pattern. One of the advantages of GraSE imaging is that it can be performed with either T2 or T2* weighting. Since the center of k-space plays an important role in determining the image contrast, by using the spin echoes to collect the center of k-space the image will be mostly influenced by the T2 weighting. Similarly, a stronger T2* weighting can also be achieved by using the gradient echoes to make up the center of k-space. 1.2.11 2D vs 3D Imaging The volume of interest in MRI can be acquired using two different methods, 2-dimensional multi-slice, or 3-dimensional slab acquisition. In a 2-dimensional multi-slice imaging sequence, a gradient is applied in the slice encoding direction during the excitation pulse, and only one slice is excited by the RF pulse. As long as neighbouring slices are not overlapping, they should, theoretically, be unaffected by the excitation pulse. This means that during the relaxation phase for this slice, another slice can be imaged, which can reduce the overall scan time. 3D MRI acquisitions start off similarly, but the slice selective gradient would have a lower intensity and the excitation pulse would cover the total volume being imaged. A new gradient pulse in the slice selective direction is then added to work as the phase encoding gradient. From there on, the scanning procedure is the same. The benefit of using 3D MRI is that it increases the signal-to-noise ratio because each echo contains signal from the entire volume instead of just a 15 single slice, and the repeated scans with multiple values of the second phase encoding gradient are equivalent to multiple averages. 1.3 Compressed Sensing Compressed sensing (CS) is a signal processing technique that recreates a signal beyond the Shannon-Nyquist limitation using a sparsely sampled dataset. It relies on three things; incoherence of the undersampling artifacts, sparsity in the transform domain (a mostly-zero-filled matrix), and non-linear reconstruction (14). While CS is a relatively new technique in MRI, having been proposed in 2007, it has been shown to work well in a variety of imaging methods, including myelin water imaging (16, 17). Luminal water imaging is based on the same principles as myelin water imaging, which makes compressed sensing a promising approach in LWI. 1.3.1 Incoherent Sampling The k-space resulting from an MRI acquisition is made up of 2 components; frequency encoding and phase encoding. In order for compressed sensing to work, there must be an incoherently sampled dataset which means that one or both of these encoding directions needs to be under sampled. Because the frequency encoding direction is created by the collection of the signal echo, undersampling in this dimension would not reduce the scan time, making it, essentially, free information. The number of phase encoding steps, however, is directly related to the imaging time making it a prime candidate for under sampling. 16 In section 1.2.17 it was mentioned that the center of k-space is where the majority of the contrast information in an MR image is derived from. Because of this, when developing an undersampling scheme it is important to ensure that a certain amount of the center of k-space is acquired. In the creation of sampling schemes for the experiments in this thesis, the central 33% of k-space is always fully sampled. The remaining k-space lines chosen to be sampled follow a gaussian distribution about the center of k-space. This results in a denser sampling of the inner k-space and more sparsely sampled periphery. In CS MRI, several different sampling methods have been used including Poisson-disc distributions (18,19), variable density sampling (20), and central k-space with random distributions (16, 21). The decision to mix central k-space with a gaussian distribution was made as an attempt to improve upon the central k-space with random distribution design. 1.3.2 Wavelet Decomposition The chosen sparsifying transform used in our compressed sensing method is a wavelet decomposition. A wavelet transform of an image creates a sparse image domain that highlights the edges of structures in an image resulting in an image with only a few high intensity pixels and many low intensity pixels. Homogenous regions of an image results in wavelet transform values of zero, while edges of structures result in high value coefficients. Figure 1-5 shows an example of wavelet decomposition of the second level. A wavelet decomposition level of 3 with a Daubechies 8 wavelet was used as the sparsifying transform for all of the experiments in these thesis. 17 Figure 1-5 An example of second level wavelet decomposition. Wavelet of the image in the top left results in coefficient maps for horizontal, vertical, and diagonal wavelet transforms. In order to better visualize the maps, individual map intensities were normalized 1.3.3 Reconstruction The reconstruction method used was a Spectral Projected Gradient for L1 minimization package (22). One of the benefits to using multi-echo data is that the wavelet decomposition should be relatively stable between echoes. We can take advantage of this using a method called group-sparse reconstruction (23), in particular our reconstructions were performed with the SPGL1 package in Matlab. If yi is the k-space for the ith echo, it can be written as equation 1-12. 𝑦𝑖 = 𝐹𝑥𝑖 + 𝜂 Equation 1-12 18 Where F is the Fourier transform, xi is the image, and η is the noise. If αi is the set of wavelet coefficients for the ith echo, W is the wavelet transform, and if we use orthogonal wavelets, then WT would be the inverse wavelet transform. This allows equation 1.9 to be written as equation 1.13, which can be expanded as equation 1.14 and simplified as equation 1.15 where Φ is the transformation matrix. 𝑦𝑖 = 𝐹𝑊𝑇𝛼𝑖 + 𝜂 Equation 1.13 [𝑦1⋮𝑦𝑁] = [𝐹𝑊𝑇 0 00 ⋱ 00 0 𝐹𝑊𝑇] [𝛼1⋮𝛼𝑁] + [𝜂1⋮𝜂𝑁] Equation 1.14 ?̃? = Φ?̃? + ?̃? Equation 1.15 The assumption being made is that the wavelet coefficients of the images are sparse, so the combined vector ᾶ will also be sparse. This allows us to take advantage of the l1 minimization which can be seen in equation 1.16. argmin?̃?‖?̃?‖1 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 ‖?̃? − Φ?̃?‖2 ≤ 𝜎 Equation 1.16 where ‖𝛼‖ = ∑ |𝛼𝑗|𝑗 Equation 1.17 19 This process is applied iteratively until an error threshold is met, or until a certain number of iterations have passed. In the CS reconstructions used for all of the experiments that follow, the maximum number of iterations was 100. 1.4 Luminal Water Imaging At its most basic level, Luminal Water Imaging (LWI) (24,25) is an MRI scanning method based on the principle that an image voxel is a macroscopic view of various microscopic tissues and the signal in the image is a sum of the signal from the individual components. Of particular interest in LWI is the T2 decay, which is affected by molecular interactions as well as diffusion. If two distinct T2 decay values can be measured in a voxel, then it can safely be assumed that there are two water compartments responsible for the signal. If the proportion of the signal due to each compartment can be calculated, then a new parameter can be created that would describe microscopic features of the tissue that cannot be observed at the resolution of the image. This new parameter is called the Luminal Water Fraction (LWF) and in the case of prostate imaging, describes the amount of luminal space in a voxel which is lower in cancerous tissue than in normal tissue. 1.4.1 T2 Measurements Once a multi-echo T2 weighted scan has been acquired a regularized non-negative least squares (NNLS) algorithm is applied to the signal decay curve for each voxel (26). Assuming there are N T2 components in the voxel, the decay of the signal can be modelled by equation 1-18 𝑺(𝒕) = ∑ 𝒔𝒊𝑵𝒊=𝟏 𝒆−𝒕 𝑻𝟐𝒊⁄ Equation 1-18 20 The NNLS algorithm fits the data to a basis set of possible values, or in terms of equation 1-18, N components. In the experiments in this thesis, the basis set of possible T2 values is 101 different values distributed logarithmically from 0.026---2s. An example of a decay curve and distribution of T2 values can be seen in figure 1-6 Because of the 2 distinct peaks formed in this distribution, we could assume that this decay curve is the result of only 2 components. Figure 1-6. Signal decay curve and T2 distribution. Left: A signal decay curve from a simulated multi-echo spin echo sequence. Right: The resulting T2 distribution following the fit of the regularized non-negative least squares fit. The amplitude points on this T2 distribution are proportional to the amount of signal created by a particular T2 value. Because of the regularization, these curves are smoothed rather than discrete points, which results in more robust fitting of the noisy data. By choosing a minimum and maximum T2 value for a component (T2min and T2max), a geometric mean T2 value can be calculated using equation 1-19, where A is the amplitude. In the example of figure 1-6 there would be two T2 values calculated, and they would be designated T2short for the smaller T2 value, and T2long, for the larger value. 21 𝑇2 = 𝑒𝑥𝑝 [∑ 𝐴(𝑇2)log (𝑇2)𝑇2𝑚𝑎𝑥𝑇2𝑚𝑖𝑛∑ 𝐴(𝑇2)𝑇2𝑚𝑎𝑥𝑇2𝑚𝑖𝑛] Equation 1-19 1.4.2 Luminal Water Fraction In the case of Luminal Water Imaging, an important parameter is the Luminal Water Fraction (LWF), which is the proportion of the signal generated due to the component with the longer T2 value. Equation 1-18 can then be re-written as Equation 1-20. 𝑺(𝑻𝑬) = 𝑳𝑾𝑭 ∙ 𝒆−𝑻𝑬 𝑻𝟐𝑳𝒐𝒏𝒈⁄ + (𝟏 − 𝑳𝑾𝑭) ∙ 𝒆−𝑻𝑬 𝑻𝟐𝑺𝒉𝒐𝒓𝒕⁄ Equation 1-20 Where S is the magnitude of the signal, LWF is the Luminal Water Fraction, TE is the echo time, T2Long is the value of the long T2 component, and T2Short is the value of the short T2 component. 𝐿𝑊𝐹 = ∑ 𝐴(𝑇 2)𝑇2𝑚𝑎𝑥𝑇2𝑡ℎ𝑟𝑒𝑠ℎ𝑜𝑙𝑑∑ 𝐴(𝑇2)𝑇2𝑚𝑎𝑥𝑇2𝑚𝑖𝑛 Equation 1-21 The LWF is simply found by taking the area under the curve of the long T2 component over the total area under the curve of the T2 distribution. This can be seen graphically in figure 1-7. The formula can also be seen in equation 1-21 where A is the amplitude for a specific T2 value, and T2threshold is the chosen threshold between classifying a T2 value as the short component or the long component. This threshold was set at 200ms for all of the experiments in this thesis and was chosen based on previous works. 22 Figure 1-7. A graphical description of the luminal water fraction 1.5 Statistics 1.5.1 Signal-to-Noise Ratio The signal-to-noise ratio (SNR) is a measure used to describe the total signal of an MRI scan in relation to the noise. It is the ratio of the mean signal in a region of interest (ROI) divided by the standard deviation of noise. 𝑆𝑁𝑅 = 𝑚𝑒𝑎𝑛(𝑠𝑖𝑔𝑛𝑎𝑙)𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛(𝑛𝑜𝑖𝑠𝑒) Equation1-22 There are several ways of measuring the standard deviation of the noise; the most straightforward method would be to place an ROI in the air space outside of the object being scanned where the only signal should come from noise. Another easy way is to place an ROI in a homogenous region inside the image, where the only variation in image intensity should be coming from noise. 23 In the celery and prostate scans for this thesis (sections 2 and 3), a third method is used because the image filters applied by the scanning computer reduce the amount of noise in the air space, and there are no large homogenous regions in the images. These scans are multi-echo scans and the ROI for the mean signal is also used as the ROI to measure the standard deviation of the noise. Rather than measuring the standard deviation of noise on the first echo, a new image is created made from the difference of the last two echoes. Theoretically only very long T2 components will still be giving signal at this point, so there will be little difference in signal from the last 2 echoes, which means the difference of the 2 images should give a homogenous region made up of only noise. 1.5.2 Normalized Mean Square Error The normalized mean square error is useful statistic in these experiments because it measures error based on pixel-to-pixel changes, and it is sensitive to infrequent and large differences. 𝑁𝑀𝑆𝐸 = ‖𝑥− 𝑥𝑟𝑒𝑓‖‖𝑥𝑟𝑒𝑓‖ Equation 1-23 1.5.3 Coefficient of Variation The coefficient of variation is measure of dispersion and is the ratio of the standard deviation to the mean value of a dataset. It is a way of normalizing the variation between different data sets. 𝐶𝑂𝑉 = 𝜎𝜇 Equation 1-24 24 Chapter 2: Electronic Phantom The Bloch equations are used to describe the magnetization vector in an external magnetic field. By using the solutions to the Bloch equations, along with T2, and T2* decay values, theoretically the complex image space can be predicted. For simplicity, only T2 and T2* are considered in the model. 2.1 Method As an initial test of how the different combinations of CS levels and GraSE factors have an impact on the measurements of multi-component T2 values, an electronic phantom was developed. A simple, single-slice image was created to mimic a 2-dimensional transverse cross-section of a human prostate containing two types of prostatic tissue; transitional zone and peripheral zone (Figure 2-1). Figure 2-1: The Electronic Phantom: A simple 2D image was created to mimic the shape of a cross-sectional image of a human prostate Using the solutions of the Bloch equations, the magnitude of the transverse magnetization vector of an imaging sequence can be simulated. If multiple components are assumed, then equation 1-18 can be used, where si is the fraction of the signal due to the ith component. 25 For the electronic phantom, historical values of LWF and long and short T2 values were used from Sabouri et. al (24) and can be seen in Table 2-1. T2* values were assumed to be half of the T2 value for each component. A TE of 25ms and 64 echoes were chosen because they have been used in previous LWI experiments, and a matrix size for the image of 128 x 128 was used. MR Measurement Non-Cancerous PZ Non-Cancerous TZ T2Short (msec) 90 104 T2Long (msec) 545 548 LWF 0.24 0.20 Table 2-1 Electronic phantom modelling parameters A multi-echo spin echo image was generated using equation 1-20 for each pixel in the electronic phantom image. This image was used as the reference image to compare the simulated GraSE images to and no noise was added to this image. Next, GraSE images needed to be simulated. Equation 1-20 describes the signal generated by spin echo, but it is not accurate for the gradient echoes because it does not account for T2* effects. Equation 1-20 can be generalized as equation 2-1 where SLong/Short describes the decay of the signal generated by the long and short T2 components respectively, and S(TE) describes the total signal in the voxel. 𝑺(𝑻𝑬) = 𝑳𝑾𝑭 ∙ 𝑺𝑳𝒐𝒏𝒈 + (𝟏 − 𝑳𝑾𝑭) ∙ 𝑺𝑺𝒉𝒐𝒓𝒕 Equation 2-1 26 The decay of the signal for each component, with a given echo time for the spin echo, can then be written as the product of the T2 and T2* decay as can be seen in equation 2-2. 𝑺𝑳𝒐𝒏𝒈/𝑺𝒉𝒐𝒓𝒕 = 𝒆−(𝑻𝑬+∆𝒏) 𝑻𝟐⁄ ∙ 𝒆−|∆𝒏| 𝑻𝟐∗⁄ Equation 2-2 In this equation there is an additional Δn term to describe the offset from the echo time of the spin echo for each of the gradient echoes. Since the inhomogeneous effects that account for the majority of the T2* decay are corrected for at the time of the spin echo, the T2* effects are only dependent on the time between the gradient echo and the spin echo. The Δn term is defined in equation 2-3, where GF is the GraSE factor and En is the echo number. In this case the echo number is referring to which number the echo is after the refocusing pulse (i.e. for a GraSE factor of 3, the first gradient echo would be echo number 1, the spin echo would be echo number 2, and the last gradient echo would be echo number 3). This number is also scaled by the TE of the spin echo. ∆𝒏= 𝑻𝑬(𝑬𝒏−(𝑮𝑭+𝟏𝟐))𝑮𝑭+𝟏 Equation 2-3 Using equations 2-1, 2-2, and 2-3 GraSE images were created for GraSE factors of 3, 5, 7, 9, and 11. These equations, however, describe an image created with one type of echo from a GraSE sequence, but because of the nature of k-space each point in the image affects all of k-space and our goal is to describe a k-space made up of different types of echoes. 27 An image was then created for each echo number for a given GraSE factor. As was mentioned previously, the output from the model results in noiseless data, so in order to add a layer of reality, these images were inverse Fourier transformed to get the k-space created by each type of echo. Complex gaussian noise was added to the k-space of the GraSE images. The gaussian noise was scaled to give reconstructions with SNR’s of approximately 50, 100, 200, and 300 in the first echo. A sampling scheme defining which phase encoding steps would be acquired was generated for sampling levels of 100%, 80%, 75%, 67%, 50%, and 33% for each GraSE factor. All echoes were acquired using the same sampling scheme. Once a sampling scheme was created, the acquired lines of k-space were then designated as being acquired by a specific gradient echo or a spin echo. These designations were used to create a new k-space by compiling select lines from the full k-spaces generated for each gradient/spin echo, using the sampling scheme as a guide. The under-sampled data sets then went through the compressed sensing reconstruction, and the resulting images then had the NNLS algorithm applied to the region of interest and the Luminal Water Analysis was performed for each pixel. 2.2 Results A selection of the reconstructed images can be seen in figure 2-2 below. It is worth noting that different combinations of sampling levels, GraSE factors, and SNR levels can produce images that appear similarly noisy. 28 Figure 2-2 Examples of electronic phantom reconstructions NMSE values were calculated for all combinations of GraSE factors, sampling levels, and SNR levels using the noiseless multi-echo spin echo sequence as the reference measurement. These were calculated for T2Long, T2Short, and LWF values. In the next figure the graphs are showing the NMSE versus the normalized scan time (sampling level divided by GraSE factor times 3) for the LWF Measurements in the peripheral zone for the different SNR levels. For all the SNR values, reducing the sampling level of any GraSE factor tends to result in an increase in the NMSE value. The lower the SNR, the less clear this trend becomes, which could be expected due to an increase in error due to more noise in the data. 29 Figure 2-3 NMSE results for peripheral zone LWF measurements for the different signal-to-noise ratios A similar trend can be seen for all the measured parameters in both the peripheral and transitional zones, though this case showed it particularly well. It is also worth noting that the overall NMSE values were lower in the peripheral zone measurements than in the transitional zone. In figure 2-4 NMSE measurements can be seen for both the peripheral and transitional zone for all GraSE factors and sampling levels with the SNR at 300. 30 Figure 2-4 NMSE results from electronic phantom measurements in both peripheral and transition zone for an SNR of 300 2.3 Conclusions There are several interesting observations that can be made from data shown here. The first being that the overall trend in all situations is that given comparable scan times, the lower GraSE factor scan should be used. While similar scan times do result in similar NMSE values, as seen in figure 2-4, the results in figure 2-3 show that the differences become clearer with lower SNR values. Given that in our best-case scenario, an SNR of 300, there are similar errors between GraSE factors and sampling levels with comparable scan times, the results suggest it would be advantageous to opt for the lower GraSE factor since the SNR value may be lower. 31 These electronic phantom scans would, however, likely favor compressed sensing because the two regions in the phantom are homogeneous and compressed sensing tends to highlight the edges of structures and create blurring within. The blurring in this case would only benefit a large homogeneous structure. This is also likely why the NMSE values are particularly low. It is worth noting that the NMSE in the transitional zone is consistently higher than in the peripheral zone. This is possibly because the short and long T2 values in this region are closer together causing more overlap in the bimodal distribution and making it more susceptible to small errors than the peripheral zone. This would also explain why the errors are particularly higher for the LWF measurements than for the T2 values. 32 Chapter 3: Celery Phantom The next phase of the experiments was to scan a physical phantom. Fresh celery was chosen because it has previously been found to show two distinct T2 values with decay times similar to those of the prostate. 3.1 Method A fresh bunch of celery was scanned in a 3T Philips Elition MRI at the UBC MRI Research Centre using an 8-channel array coil. Axial scans were collected using multi-echo 3D-CPMG and 3D-GraSE imaging sequences with GraSE factors of 3, 5, 7, 9, and 13. The acquired matrix size was approximately 224x64x8 (for the GraSE images the first phase encoding direction would be a multiple of the GraSE factor). The scan parameters can be seen below in Table 3-1. The data was phase corrected, zero-filled, and the channels were combined using the Philips software to obtain reconstructed magnitude images with a matrix size of 224x224x16. Acquisition Size (Read, Phase1, Phase2) 224 x [63-68] x 8 Reconstructed Size (Read, Phase1, Phase2) 224 x 224 x 16 Echoes 32 TE (ms) 13 TR (ms) 797 Channels 8 Table 3-1 Acquisition parameters for celery phantom MRI acquisitions 33 In the process of scanning it was determined that the loading of the coil from the celery alone was not sufficient for the tuning and matching of the scanner software to work, therefore a bag of saline was placed in the scanner with the celery. In order to get a more accurate representation of the effects of under-sampling, the image processing would need to be done on the acquired, rather than the zero-filled, k-space. An inverse Fourier transform was applied across the 3 spatial dimensions of the magnitude images to transform the data into k-space and the zero-filled phase encoding lines were removed in order to estimate the acquired k-space with the phase correction and channels combined. For the GraSE factors of 3, 5, 7, and 9, one-hundred different 3D-Sampling schemes were generated for sampling factors of 80%, 75%, 67%, and 50%. The sampling schemes were then applied to the acquired k-space and the data underwent compressed sensing reconstruction. The reconstructed k-space was then zero-filled to a matrix size of 224x224x16 to match the original reconstructed images from the scanner, and a Fourier transform was applied to the frequency and phase encoding directions of the data in order to return to the image space. A binary mask of the celery was created by applying a threshold filter on the first echo of the GraSE factor 3 scan magnitude images. The threshold range was chosen to exclude the air space as well as the saline. This mask was used in order to speed up further data processing allowing the NNLS fitting algorithm to only be applied to the region of interest. Previous values for the T2 components were found to be 25-45(ms) and 100-400(ms) for T2Short and T2Long respectively (25) so a threshold value of 100ms was chosen to distinguish between the two components. 34 3.2 Results While 3D-CPMG scans were acquired with the intention of being the reference image, it was found that there was some distortion in the periphery of the images between the CPMG and the GraSE scans. The statistical method chosen to compare the data sets was again the NMSE which relies on voxel-to-voxel comparisons. Because of the distortion between the two scan methods, using the CPMG as the reference to be measured against would likely result in higher errors than would realistically be expected. Rather than registering the images, it was decided to use the fully sampled GraSE factor 3 image-set as the reference. 3.2.1 Effects of Sampling Schemes The first effect to be investigated was the importance of the sampling scheme. By performing the compressed sensing reconstruction using 100 different schemes for each sampling level, the variation due to the change in sampling schemes could be calculated. The mean value of the LWF, T2Short, and T2Long in the ROI was calculated for each of the reconstructions, and the coefficient of variation (COV) was measured. Figure 3-1. Coefficient of variation results comparing the mean parameter in the region of interest for one-hundred different sampling schemes. 35 For each of the parameters, the calculated COV was quite small (< 0.002), graphs of these results can be seen in Figure 3-1. This would imply that the variation of the mean ROI parameter values due to varying the sampling scheme is relatively small which would imply that the effects of changing sampling schemes is low compared to the effects of changing either the GrasSE factor or the sampling level. To follow up, a two-way ANOVA test was performed using Microsoft Excel, comparing the mean parameter value of the ROI for each reconstruction. This was once again done for the LWF, T2Short, and T2Long measurements. Reconstructions were grouped by GraSE factor and sampling level. The results of the ANOVA tests gave strongly significant p-values (the largest value was 1.4 x 10-13) across all measures, which include differences between GraSE factors, differences in sampling levels, and the interaction between them. For all three measurements the p-value calculated for the differences between GraSE factors was given to be zero which would indicate that the mean parameter value in the ROI is most dependent on the GraSE factor. 3.2.2 Effects on Luminal Water Imaging Analysis The next step was to compare the errors in the LWI analysis across the different combinations of GraSE factors and sampling levels. The statistic of choice is once again the normalized mean squared error with results from the LWI analysis for the fully sampled GraSE factor 3 images used as the reference. For the reconstructions with a sampling level of 50%, 67%, 75%, and 80%, and GraSE factors of 3, 5, 7, and 9 the NMSE was calculated for each of the 100 reconstructions and the mean value was used. The remaining combinations of GraSE factors and 36 sampling levels only had one reconstruction performed and the NMSE from that single reconstruction was used. Figure 3-2. NMSE measurements for each parameter in the celery scans. In general, for all the measurements the NMSE values at a given sampling level tend to increase with larger GraSE factors. These trends should be expected given that an increase in the GraSE factor results in more of the acquired k-space to be the result of gradient echoes which carry a T2* weighting that our current fitting model does not account for. The next observable trend is that a decrease in the sampling level also results in an increased NMSE. This could be expected because SNR is proportional to the square root of the number of phase encoding steps. The electronic phantom experiment (section 2) has shown that decreasing SNR will negatively impact the results of the LWI analysis. Another observation that can be made from the figure is that the NMSE values for the LWF measurements are lower than those of the T2 measurements. A reason for this could be that while the geometric mean of the long and short T2 is shifting, the distributions are not necessarily shifting far outside of the boundaries used to calculate the LWF. 37 In order to test this, the mean, median, and standard deviation of the ROI were measured for each combination of GraSE factor and sampling level and for each of the parameters. When compared to the fully sampled GraSE-factor-3 scans, for both the short and long T2 values, the mean value decreases (<2%) and the median value increases (<3%) bringing them closer together. There is also a drastic increase in the standard deviation of these components, as high as 24%. This increases with a decrease in the sampling level. For the LWF parameter, the mean and median values both increase (<3%) and the standard deviation decreases by up to 13%. The trend here is for the standard deviation to decrease with a higher GraSE factor or lower sampling level. The decrease in standard deviation for all the parameters could imply a blurring of the results due to an increase in GraSE factor and decrease in the sampling level. The full results can be seen in Appendix-B. The last observation that can be noticed in the figure is that the NMSE values for the T2 measurements for the GraSE factor 13 datasets appear to be independent from the sampling level. This is most likely due to the large amount of T2* effects in the images causing the NNLS fitting to give T2 values with consistently high errors, regardless of the sampling level. Measuring the SNR across all the celery scans shows the GraSE factor 13 scan to have the highest SNR of all the scans, which suggests that SNR as the reason for this issue. 38 Figure 3-3. NMSE measurements compared to total scan time for the various parameters While it can hardly be seen as surprising that the larger GraSE factors result in a larger error in our quantitative measurements, the goal of this project is to investigate how we can reduce scanning time. Figure 3-3 compares the NMSE of our various combinations of GraSE factors and sampling levels against the total scan time of each acquisition. The results of these plots are similar to those seen in the electronic phantom (section 2), which suggested that similar scan times lead to similar NMSE values. The exception to this of course is the GraSE factor 13 scan which seems to have NMSE values higher than the trend would predict, particularly for the T2Long and T2Short measurements. This is likely due to the two modes of the T2 distribution becoming less distinct with higher GraSE factors. This can be seen in the table of results in appendix B, where it can be seen that with a higher GraSE factor, the mean T2Long becomes shorter, the mean T2Short becomes longer, and the standard deviation of both decreases. 39 Chapter 4: Prostate 4.1 Method One patient had an MRI of their prostate using GraSE factors of 3, 5, and 7. The scans were carried out in one session on the same Philips 3T Elition MRI as the celery scans. The RF coil used was the OPTIMAL coil embedded in the MRI bed along with the abdominal array. Scanning parameters can be found in table 4-1. Scans were also acquired with a SENSE factor of 2 as another method of decreasing the scan time. Acquisition Size (Read, Phase1, Phase2) 240 x ~120 x 9 Reconstructed Size (Read, Phase1, Phase2) 240 x 240 x 18 FOV (mm) 240 x 240 x 72 Echoes 64 TE (ms) 25 TR (ms) 3074 Table 4-1. Scanning parameters for prostate imaging scan In this experiment the complex images were saved, and an inverse Fourier transform was applied across all three dimensions of k-space and the zero-filling was removed. One of the acquired slices in the second phase encoding direction was also removed because the compressed sensing reconstruction requires the under-sampled dimensions to have a length of 2L where L is the level of decomposition being used, and in our case L=3. 40 Sampling levels of 50%, 67%, 75%, 80%, and 100% were used. For the sampling levels between 50% and 80% 100 different 3D sampling schemes were created in order to observe the variability caused by using a different sampling scheme of the same level. Sampling schemes were applied to k-space and then the data underwent the compressed sensing reconstruction. The reconstructed k-space was zero-filled back to 240x240x18 to match the acquired images, and was Fourier transformed to obtain the image space. A mask was hand drawn around the peripheral zone of the prostate using the 4th echo of the GraSE factor 3 magnitude images to create an ROI. Voxels in the ROI underwent LWI analysis in order to obtain LWF, T2Long, and T2Short measurements. Previous experiments have found T2 values of 52-150ms and 300-887ms for the short and long T2 respectively. A threshold of 150ms was chosen to classify the T2 components as it was previously used (25). 4.2 Results 4.2.1 Effect of Sampling Scheme The mean value of the parameters was measured in the ROI for each of the one hundred reconstructions of the 50%, 67%, 75%, and 80% sampling levels for each GraSE factor. The mean and standard deviation was calculated across the different reconstructions for each combination of GraSE factor and sampling level. The coefficient of variation was calculated in order to measure the effects of varying the sampling scheme on the results. These CoV measurements can be seen in figure 4-1. 41 The trend shown in this figure is similar to those seen in the celery phantom, where a decrease in sampling level has a strong effect on the CoV measurements, while the GraSE factor has a minimal effect. The CoV values are larger than for the celery experiment, but still relatively small at less than 0.05. This would, again, suggest that the different sampling schemes at the same sampling level have little variation meaning that the NMSE measurements should not be influenced by a particular sampling scheme. Figure 4-1 Coefficient of variation results comparing the mean parameter in the region of interest for one-hundred different sampling schemes. 4.2.2 Effects on Luminal Water Imaging Analysis The NMSE values were again measured using the fully sampled GraSE factor 3 results as the reference measurement. These results can be seen in figure 4-2. 42 Figure 4-2 Normalized mean square error versus scan time for various combinations of GraSE factor and sampling level. The trends in this figure agree with the trends seen in both the electronic phantom and the celery phantom. The NMSE range is much higher than the previous two experiments, however they are comparable to levels seen in previous compressed sensing experiments in multi-component T2 imaging analysis, which is likely due to a lower SNR than the two previous experiments. Figure 4-3 A single slice of the prostate from each of the fully sampled GraSE factor image sets. The top of the prostate in the GraSE factor 5 images shows an artifact not present in the other two image sets. An interesting observation in the graphs is that the GraSE factor 5 scans, while having nearly identical trends to the other two GraSE factors, has a much higher NSME than the trends would expect. In looking at the images there appears to be a small oscillating artifact visible in the top 43 of the prostate which likely extends into the rest of the prostate creating an issue with the signal decay curves. An example of this can be seen in figure 4-3. Another interesting observation from figure 4-2 is that, particularly for the T2Long measurements, given comparable scan times it seems optimal to opt for the higher GraSE factor which is the opposite of the results in the electronic phantom. Looking at the effects on the individual parameters, the mean and the median of T2short decrease (<2%) with the sampling level, while the standard deviation increases up to 18% compared with the fully sampled GraSE factor 3 results. For T2Long the mean and median increase as the sampling level decreases and decrease as the GraSE factor increases (<3%). The effect on the standard deviation varied, but generally led to a decrease of 8-12%. The LWF measurements showed a large (up to 13%) decrease in the mean and median for all of the under-sampled results, and had mixed effects on the standard deviation, but generally it decreased with the sampling level. The full results can be seen in Appendix C. Given that there was likely an error in the GraSE factor 5 image-set it is difficult to make any conclusions regarding trends that could be associated with the change in the GraSE factor. 44 Chapter 5: Discussion Across all three experiments, the comparisons of the NMSE against scan time suggest that when using a combination of GraSE factors and compressed sensing in order to reduce the total scan time for a luminal water imaging experiment, as long as the scan times are comparable, the results should be comparable as well. There are, however, a few caveats to this result. One is that the electronic phantom experiment suggested that in the case of a low SNR it would be better to go with a lower GraSE factor and decreasing the sampling level. This result is somewhat contradicted by the in-vivo prostate scan which showed a trend suggesting that the higher GraSE factor would be the better choice for the measured parameters. This could be explained by the fact that there are more structures in the in-vivo prostate than in the electronic phantom which means that it will be more susceptible to errors due to compressed sensing which can cause blurring of the image. Another possibility is that the in-vivo scans were acquired with SENSE MRI which could have also affected the NMSE of the lower sampling level scans disproportionately. A third possibility could have to do with the artificial under sampling. Because the in-vivo scans were fully sampled to begin with, most of the phase encoding lines that were left out for the compressed sensing would have been phase encoding lines that were acquired with gradient echoes while many of the lines that were acquired with spin echoes would now simply be assumed to have been collected with a gradient echo. This would reduce some of the T2* error effects we observed in the electronic phantom which would downplay the issues of GraSE imaging in the in-vivo experiments. 45 Another caveat to this result is that it could change with a different sampling scheme design. While our method was to fully sample the center 33% of k-space and any more acquired phase encodes were randomly selected, different amounts of central sampling could also affect the results. The sampling schemes were also kept consistent between the echoes. It has been shown in an artificial undersampling experiment (27) that varying the sampling scheme between echoes has a negative effect on myelin water fraction measurements. These errors could be due to the much shorter T2 times of the components in myelin, therefore it could be worth trying a sampling scheme that changes echo-to-echo in Luminal Water Imaging. Going forward, the next steps to take would be to acquire under-sampled data sets following specific sampling schemes. This would create larger phase blips between consecutive echoes in the GraSE scans which could potentially create problems with eddy currents and large jumps in k-space could results in phase errors. It would also likely increase the T2* weighting in the images possibly leading to a result similar to the electronic phantom suggesting that the lower GraSE factor is the optimal choice for a set of comparable scan times. Another idea that could be investigated, would be to use a fitting function that would account for T2* as well as T2 decay. This is of course troublesome because while all of the collected echoes would be T2 dependent, only the gradient echoes would be T2* dependent leading to a mixed result in the image. The electronic phantom images used in section 2 would provide an array of data that could be tested with a fitting function that would account for this variable. 46 Lastly, luminal water imaging was shown to be effective in not only identifying prostate cancer, but also in the grading of tumors. The efficacy of combinations of GraSE MRI with compressed sensing on this aspect should also be investigated. 47 References 1. Projected estimates of cancer in Canada in 2020, Brenner DR, Weir HK, Demers AA, Ellison LF, Louzado C, Shaw A, Turner D, Woods RR, Smith LM. CMAJ. 2020;192:E199-205. 2. PDQ® Adult Treatment Editorial Board. PDQ Prostate Cancer Treatment. 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Retrieved August 23, 2020, from https://open.library.ubc.ca/collections/ubctheses/24/items/1.0320960Reson. 1969 1989;84:134–152 50 Appendices Appendix A - Additional results from electronic phantom Figure 5-1 Normalized mean square error measured against the ground for luminal water fraction measurements in the electronic phantom 51 Figure 5-2 Normalized mean square error measured against the ground truth on Long T2 values of the electronic phantom 00.20.40.60 0.5 1 1.5NMSEScan Time (arbitrary units)SNR 50GF3GF5GF7GF9GF1100.10.20.30 0.5 1 1.5NMSEScan Time (arbitrary units)SNR 100GF3GF5GF7GF9GF1100.050.10.150 0.5 1 1.5NMSEScan Time (arbitrary units)SNR 200GF3GF5GF7GF9GF1100.020.040.060.080.10 0.5 1 1.5NMSEScan Time (arbitrary units)SNR 300GF3GF5GF7GF9GF1152 Appendix B - Additional Results for Celery Phantom B.1 T2Short ROI Measurements Mean GraSE Factor 3 5 7 9 13 Sampling Level 33% 0.0544 0.0546 0.0552 0.0559 0.0555 50% 0.054886 0.055042 0.055355 0.056122 0.0551 67% 0.055025 0.055227 0.055291 0.056185 0.0551 75% 0.055079 0.055298 0.055323 0.056197 0.0552 80% 0.055094 0.05533 0.055342 0.05621 0.0552 100% 0.054969 0.055266 0.055229 0.056172 0.0551 Table 5-1 Mean value of the short T2 component in the ROI of the celery Median GraSE Factor 3 5 7 9 13 Sampling Level 33% 0.0546 0.0547 0.055 0.0563 0.0561 50% 0.055053 0.0552 0.055353 0.056374 0.0558 67% 0.055195 0.055414 0.055246 0.056349 0.0555 75% 0.05525 0.055507 0.055284 0.056363 0.0556 80% 0.055254 0.055539 0.055303 0.056351 0.0555 100% 0.055207 0.055414 0.055236 0.05633 0.0556 Table 5-2 Median value of the short T2 component in the ROI of the celery Standard Deviation GraSE Factor 3 5 7 9 13 Sampling Level 33% 0.0166 0.0167 0.0156 0.0161 0.0161 50% 0.0168 0.016675 0.016035 0.016346 0.0172 67% 0.017239 0.016999 0.016387 0.01673 0.0173 75% 0.017345 0.0171 0.016475 0.016812 0.0173 80% 0.017382 0.017146 0.016516 0.016844 0.0174 100% 0.01747 0.017188 0.016454 0.016831 0.0173 Table 5-3 Standard deviation of the short T2 component in the ROI of the celery 53 B.2 T2Long ROI Measurements Mean GraSE Factor 3 5 7 9 13 Sampling Level 33% 0.1535 0.1533 0.1521 0.1531 0.1526 50% 0.152428 0.151618 0.151679 0.152614 0.1509 67% 0.152411 0.151517 0.151756 0.152536 0.1504 75% 0.152392 0.151548 0.151796 0.15256 0.1506 80% 0.152375 0.151569 0.151783 0.152574 0.1505 100% 0.152617 0.1517 0.152039 0.152922 0.1504 Table 5-4 Mean value of the long T2 component in the ROI of the celery Median GraSE Factor 3 5 7 9 13 Sampling Level 33% 0.1474 0.1479 0.1472 0.1491 0.1488 50% 0.145907 0.14613 0.146051 0.147079 0.1462 67% 0.145073 0.145408 0.145218 0.146154 0.1449 75% 0.144935 0.145312 0.145086 0.145993 0.1449 80% 0.144905 0.145292 0.145046 0.145953 0.1449 100% 0.145003 0.145418 0.145006 0.145907 0.1449 Table 5-5 Median value of the long T2 component in the ROI of the celery Standard Deviation GraSE Factor 3 5 7 9 13 Sampling Level 33% 0.0512 0.0498 0.0465 0.0438 0.0404 50% 0.046671 0.043348 0.0448 0.044574 0.0393 67% 0.048809 0.044669 0.047153 0.047166 0.04 75% 0.049228 0.045182 0.047642 0.047886 0.0416 80% 0.049244 0.045556 0.047654 0.048109 0.0404 100% 0.051712 0.047112 0.04975 0.050461 0.041 Table 5-6 Standard deviation of the long T2 component in the ROI of the celery 54 B.3 Luminal Water Fraction ROI Measurements Mean GraSE Factor 3 5 7 9 13 Sampling Level 33% 0.6448 0.6482 0.6488 0.6463 0.6516 50% 0.646032 0.652488 0.649824 0.649455 0.6561 67% 0.644881 0.651857 0.64915 0.649351 0.657 75% 0.644307 0.651513 0.648817 0.649075 0.6567 80% 0.644238 0.651358 0.648696 0.649001 0.6572 100% 0.643733 0.6509 0.648165 0.648012 0.6569 Table 5-7 Mean value of the LWF in the ROI of the celery Median GraSE Factor 3 5 7 9 13 Sampling Level 33% 0.6901 0.6972 0.6938 0.687 0.6877 50% 0.693273 0.70036 0.696035 0.696057 0.6966 67% 0.693014 0.700547 0.697235 0.697202 0.698 75% 0.692586 0.700273 0.697111 0.697267 0.698 80% 0.692617 0.700199 0.697112 0.697248 0.6978 100% 0.692693 0.701491 0.696529 0.695584 0.6983 Table 5-8 Median value of the LWF in the ROI of the celery Standard Deviation GraSE Factor 3 5 7 9 13 Sampling Level 33% 0.2388 0.236 0.2256 0.2278 0.2243 50% 0.246309 0.241316 0.240863 0.240365 0.2351 67% 0.25318 0.247507 0.24799 0.247429 0.2392 75% 0.254699 0.248947 0.249288 0.248921 0.2406 80% 0.255061 0.249414 0.24973 0.249306 0.2405 100% 0.255491 0.250069 0.249677 0.249308 0.2406 Table 5-9 Standard deviation of the LWF in the ROI of the celery 55 Appendix C - Additional Results for Prostate Scans C.1 T2Short ROI Measurements Mean GraSE Factor 3 5 7 Sampling Level 50% 0.0556 0.0550 0.0553 67% 0.0556 0.0550 0.0554 75% 0.0554 0.0547 0.0551 80% 0.0557 0.0550 0.0552 100% 0.0552 0.0546 0.0546 Table 5-10 Mean value of the short T2 component in the peripheral zone of the prostate Median GraSE Factor 3 5 7 Sampling Level 50% 0.0545 0.0545 0.0542 67% 0.0547 0.0548 0.0544 75% 0.0545 0.0545 0.0541 80% 0.0548 0.0548 0.0542 100% 0.0548 0.0546 0.0538 Table 5-11 Median value of the short T2 component in the peripheral zone of the prostate Standard Deviation GraSE Factor 3 5 7 Sampling Level 50% 0.0082 0.0080 0.0083 67% 0.0079 0.0078 0.0082 75% 0.0079 0.0076 0.0079 80% 0.0081 0.0077 0.0081 100% 0.0070 0.0067 0.0072 Table 5-12 Standard deviation of the short T2 component in the peripheral zone of the prostate 56 C.2 T2Long ROI Measurements Mean GraSE Factor 3 5 7 Sampling Level 50% 0.2406 0.2403 0.2402 67% 0.2376 0.2375 0.2374 75% 0.2362 0.2356 0.2364 80% 0.2360 0.2358 0.2365 100% 0.2352 0.2320 0.2359 Table 5-13 Mean value of the long T2 component in the peripheral zone of the prostate Median GraSE Factor 3 5 7 Sampling Level 50% 0.2324 0.2315 0.2311 67% 0.2303 0.2297 0.2284 75% 0.2296 0.2297 0.2275 80% 0.2295 0.2302 0.2269 100% 0.2272 0.2278 0.2208 Table 5-14 Median value of the long T2 component in the peripheral zone of the prostate Standard Deviation GraSE Factor 3 5 7 Sampling Level 50% 0.0874 0.0882 0.0854 67% 0.0879 0.0872 0.0842 75% 0.0861 0.0831 0.0836 80% 0.0864 0.0830 0.0847 100% 0.0950 0.0844 0.1048 Table 5-15 Standard deviation of the long T2 component in the peripheral zone of the prostate 57 C.3 Luminal Water Fraction ROI Measurements Mean GraSE Factor 3 5 7 Sampling Level 50% 0.2139 0.2225 0.2095 67% 0.2181 0.2290 0.2161 75% 0.2222 0.2327 0.2200 80% 0.2254 0.2356 0.2211 100% 0.2256 0.2339 0.2226 Table 5-16 Mean value of the LWF measurement in the peripheral zone of the prostate Median GraSE Factor 3 5 7 Sampling Level 50% 0.2039 0.2117 0.1967 67% 0.2130 0.2206 0.2036 75% 0.2188 0.2254 0.2084 80% 0.2225 0.2289 0.2096 100% 0.2245 0.2287 0.2137 Table 5-17 Median value of the LWF measurement in the peripheral zone of the prostate Standard Deviation GraSE Factor 3 5 7 Sampling Level 50% 0.1205 0.1280 0.1232 67% 0.1249 0.1321 0.1303 75% 0.1271 0.1338 0.1338 80% 0.1289 0.1349 0.1355 100% 0.1295 0.1323 0.1384 Table 5-18 Standard deviation of the LWF measurement in the peripheral zone of the prostate
Thesis/Dissertation
eng
Physics
Vancouver : University of British Columbia Library
Graduate
Comparison of combinations of various levels of GraSE and compressed sensing on luminal water imaging measurements in MRI
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