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Diffusion MRI of the human brain: signal modeling and quantitative analysis
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Diffusion MRI of the human brain: signal modeling and quantitative analysis
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DIFFUSION MRI OF THE HUMAN BRAIN: SIGNAL MODELING AND QUANTITATIVE ANALYSIS by Bryce Wilkins A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (BIOMEDICAL ENGINEERING) December 2014 Dissertation Committee: Research Assistant Professor Natasha Lepor e, Ph.D. (Advisor) Professor Vasilis Z. Marmarelis, Ph.D. (Chair) Professor Richard M. Leahy, Ph.D. (Outside member) Copyright 2014 Bryce Wilkins There are three things extremely hard, steel, a diamond and to know one's self. { Benjamin Franklin It always seems impossible until it's done. { Nelson Mandela Once we accept our limits, we go beyond them. { Albert Einstein Abstract Diusion-weighted magnetic resonance imaging (DW-MRI) and specic applications such as DTI are uniquely capable of revealing the human brain's complex white-matter structure. Since its introduction two decades ago DTI has been applied to a broad range of neurological inquiry spanning neuroscience to clinical research. As the eld has matured advances in signal modeling have led to a vast number of alternative diusion sampling strategies (commonly HARDI techniques) and analysis methodologies. New analysis methods are commonly introduced alongside simulation studies for com- parison against alternatives, however dierences in signal models, simulation parameters and/or evaluation metrics often prevents a broad comparison of similar work. Conse- quently it often remains unclear whether or not new techniques are improvements over existing approaches, and if so, under what conditions. Development of real phantoms and synthetic data sets are indispensable for evalu- ating the accuracy, precision, reproducibility and noise sensitivity of DW-MRI analysis methods in a quantitative manner. While there has been a concerted eort towards this goal, there remains a need for publicly accessible DW-MRI data sets comprising realis- tic congurations of white matter pathways, with corresponding ground-truth of ber directions and software tools to permit consistent and comparable detailed evaluation of analyzed data. To this end we develop a comprehensive framework for synthesizing DW-MRI data resembling the human brain, with congurable SNR and diusion sampling patterns, and apply the resulting data to evaluation of several multi-ber DW-MRI analysis methods. The data sets and quantitative tools developed were made publicly available. It is our hope that the availability of these tools will enable a greater understanding of dierences between analysis methods and development of better techniques. iii With respect to quantitative diusion imaging metrics|such as FA and MD which re ect changes in tissue microstructure and have become indispensable to non-invasive in-vivo assessment of neuropathology|it is important to assess the extent to which MR artifacts in uence such metrics. While many geometric image distortion artifacts have been investigated and correction methods proposed, less work has examined the eect scanner drift and diusion-weighting miscalibration have on diusion metrics. We investigate these sources of erroneous signal change by developing a physical diusion phantom. Our ndings show dierences in MD caused by diusion-weighting miscalibra- tion can be comparable to dierences (in the same metric) between subjects or groups reported in clinical studies. This suggests dierences in diusion metrics cited in litera- ture may not solely re ect changes in tissue pathology. iv To my wife, and the child of our dreams. Acknowledgements No work this large could have been completed if not for the support of a large number of people; I am indebted to you all. A few of those who've crossed my path get a mention here. I am sincerely grateful to my rst and primary advisor, the late Professor Manbir Singh, for the tremendous opportunity and privilege, to do Ph.D. research. Prof. Singh rst introduced me to magnetic resonance imaging, which has become a fascinating `magnetic wilderness' more than I could imagine. I am lucky to have had an advisor who cared much for the well-being of his students, was readily accessible, and willingly supervised evening experiments at the scanner prior to permitting unsupervised access, which was a most benevolent act and signicantly aided my research. Manbir's sudden passing was a tragic loss, and left us all a heavy heart. I am indebted to Assist. Prof. Natasha Lepor e for so enthusiastically lling the role as my advisor for the last 1.5 years, welcoming me into her research group at Children's Hospital Los Angeles (CHLA), being excited about my projects, and being understanding at times I was unhappy with my progress. Thanks for sticking it out|you gave me the opportunity to realize work that I feel proud of, and for that I am forever grateful. Thank you for continuous support and advice at all stages from my qualifying exam to completion of my dissertation. I owe a huge thanks to Meng Law, M.D. for discussions on clinical relevance, positive feedback from voluntarily sharing my work with his colleagues, and ensuring on-going funding for myself and scanner access for my experiments, both without which I could not have completed my work. As a fellow Aussie from Melbourne, \Thanks, mate!" I signicantly appreciate all the excellent faculty who participated on my commit- tees: Prof. Richard Leahy, Assist. Prof. Justin Haldar, Prof. Vasilis Marmarelis and Assoc. Prof. Bartlett Mel. I enjoyed many engaging discussions with you all throughout vi my Ph.D. candidature, and am thankful for all the questions and advice which improved the quality of my work. I also thank Prof. Krishna Nayak for advice on research, coax cables from the scanner gradient hardware, and tips on project management, Dr. H. Harry Hu for being incred- ibly kind in assisting my learning of the GE HDxt 3T MRI scanner on many occasions, and Dr. Vidya Rajagopalan for helpful comments writing papers and in preparation of my qualifying exam and defense. Additionally, Don Wiggins and Kan Lee of the College of Letters, Arts & Sciences Machine Shop, for their expert services as machinists (and tolerating the mundane work I occasionally brought their way (\Thread this please...")). Colleagues, past and present, of the many labs I have had the fortune of being a member of: Prof. Singh's Biomedical Imaging Lab, Prof. Leahy's NeuroImaging Research Group, and Assist. Prof. Lepor e's CIBORG. To all members of these labs, I express a tremendous gratitude for relaxed, fun and intellectually stimulating academic environ- ments that have been a pleasure to be part of. In particular I would like to mention those I've worked alongside the most: Namgyun Lee, my strongest collaborator, whom I'm especially appreciative of for keeping up the correspondence while in South Korea (these past 2 years!) in addition to assistance with diusion phantom development, Niharika Gajawelli for help with diusion phantom `wrapping', many discussions on ber estimation error metrics, and being a willing subject for long duration DW-MRI scans on more than one occasion, Dr. Sinchai Tsao for assistance with many research assistant projects and scans in the early hours at the hospital, and Dr. Darryl Hwang for software tools I learned a great deal from and many helpful comments regarding my presentations. Thank you guys, for all the memorable times we shared. I am grateful of the Department of Biomedical Engineering, University of Southern California (USC) for many teaching assistantship's, which not only provided a stipend but oered me the chance to teach exciting topics to undergraduates, which I really enjoyed. Also, the United States Government, National Institutes of Health for research grant NIH R21-EB013456, which funded the last two years of my work. On a personal note, I would like to acknowledge The friendly and helpful sta of the Department of Biomedical Engineering, in par- ticular Mischal Diasanta and Karen Johnson, and Diana Shyco in the Department of vii Radiology at Keck Medical Center of USC, and Julia Castro at CHLA. You all helped make the journey that much smoother. Friends from Australia and the United States, whom I miss dearly, I appreciate all your support over the many years. As for my time at USC, I couldn't have hoped to meet better people. In particular Dr. C. Channing Chow II, Ranjit Raveendran, Dr. Leah Thompkins (n ee Riley), and Dr. Aamir Abid. I am fortunate to have met you all|thanks for the balance of life, the unforgettable times, and the wisdom shared. Julio Valella, my longtime mentor and friend (since 1996) who has always had my best interests at heart, who encouraged me to `dig deep' and aim for the best and most I could ever achieve, and oered honest advice (even when it hurt) over many years and on innumerable topics. The experiences have `built my backbone' and prepared me for the challenges ahead. Alisara Tareekes, my wonderful wife, for her unwavering support and patience to see this through, keeping me sane, and reminding me (often) to see the bigger picture. You have endured an arduous journey beside me, to say the least, and I am extremely fortunate for your understanding and tolerance of the past few years. Also, thank you for so graciously lending your talents to several artwork that grace the pages of this dissertation|your expertise yielded scientic illustrations more pleasing to look at than I could have achieved on my own. Most endearing of all, I thank my parents Roy and Brenda, and my brother, Ryan, for all their love, support, patience, my upbringing, and bearing with my absence so far from home, so I could grow. viii Contents Abstract iii Acknowledgements vi List of Abbreviations xii List of Tables xiv List of Figures xv 1 Introduction 1 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Previous Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.4 Dissertation Focus and Main Contributions . . . . . . . . . . . . . . . . . 4 1.5 Organization of the Dissertation . . . . . . . . . . . . . . . . . . . . . . . 5 2 Neuroanatomy and Diusion 7 2.1 Neuroanatomy and the Nervous System . . . . . . . . . . . . . . . . . . . 7 2.2 White Matter Pathways . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Molecular Diusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3.1 Free Diusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3.2 Diusion in Biological Tissues . . . . . . . . . . . . . . . . . . . . . 13 3 Diusion-Weighted MRI and Tractography 16 3.1 Diusion-Weighted MRI . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.1.1 Pulsed Gradient Spin Echo Sequence . . . . . . . . . . . . . . . . . 16 3.1.2 Twice-Refocused PGSE Sequence . . . . . . . . . . . . . . . . . . . 20 3.2 Diusion-Weighted Signal Modeling . . . . . . . . . . . . . . . . . . . . . 22 3.2.1 Diusion Tensor Imaging . . . . . . . . . . . . . . . . . . . . . . . 23 3.2.2 Beyond DTI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4 Truncated Sampling in DSI 30 4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.2 Materials and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ix 4.2.1 Single Fiber Estimation in DSI . . . . . . . . . . . . . . . . . . . . 31 4.2.2 Simulated Fiber Cup Phantom . . . . . . . . . . . . . . . . . . . . 33 4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.3.1 Truncated Sampling and Single Fiber Estimation in DSI . . . . . . 36 4.3.2 Simulated Fiber Cup Phantom . . . . . . . . . . . . . . . . . . . . 37 4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 5 Simulated DW-MRI Brain Data 43 5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 5.2 Materials and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 5.2.1 Establishing the Fiber Ground-truth . . . . . . . . . . . . . . . . . 44 5.2.2 Data Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 5.2.3 Quantitative Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5.3 Application: Comparison of six DW-MRI Analysis Methods . . . . . . . . 48 5.3.1 Data Synthesis Parameters . . . . . . . . . . . . . . . . . . . . . . 48 5.3.2 Data Analysis Parameters . . . . . . . . . . . . . . . . . . . . . . . 50 5.3.3 Comparison of In-vivo and Synthetic Data . . . . . . . . . . . . . . 53 5.3.4 Results: Fiber Orientation Estimation . . . . . . . . . . . . . . . . 53 5.3.5 Results: Tractography . . . . . . . . . . . . . . . . . . . . . . . . . 60 5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 6 Scanner Drift and Miscalibrated Diusion-Weighting in DW-MRI 75 6.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 6.2 Material and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 6.2.1 Diusion Phantom with Temperature Probe . . . . . . . . . . . . . 76 6.2.2 Data Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 6.2.3 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 6.3 Application: Signicance of Scanner Drift and Miscalibrated Diusion- Weighting in Diusion MRI . . . . . . . . . . . . . . . . . . . . . . . . . . 83 6.3.1 Results: Diusion Phantom . . . . . . . . . . . . . . . . . . . . . . 83 6.3.2 Results: Human Data Sets . . . . . . . . . . . . . . . . . . . . . . 93 6.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 7 Conclusions and Future Work 97 7.1 Summary of Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 7.1.1 Diusion Phantom Development . . . . . . . . . . . . . . . . . . . 97 7.1.2 Software Development . . . . . . . . . . . . . . . . . . . . . . . . . 98 7.1.3 Online Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 7.1.4 New Analytical Expressions . . . . . . . . . . . . . . . . . . . . . . 99 7.1.5 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 7.2 Recommendations for Further Study . . . . . . . . . . . . . . . . . . . . . 101 7.3 Final Words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 References 103 x A Fiber Estimation from Synthetic DW-MRI Data Sets 115 B Diusion Phantom Design 124 xi List of Abbreviations 1D one-dimensional 2D two-dimensional 3D three-dimensional aQBI analytical QBI BSM ball-and-stick model CAD computer aided design CSA constant solid angle CSD constrained spherical deconvolution CSF cerebrospinal uid DIFT discrete inverse Fourier transform DSI diusion spectrum imaging DTI diusion tensor imaging DW diusion-weighted DW-MRI diusion-weighted magnetic resonance imaging EAP ensemble average propagator EPI echo-planar imaging FA fractional anisotropy FOD ber orientation distribution FRACT Funk-Radon and cosine transform FRT Funk-Radon transform GFA generalized fractional anisotropy xii GM gray matter GQI generalized q-sampling imaging HARDI high angular resolution diusion imaging MD mean diusivity MR magnetic resonance MRI magnetic resonance imaging NEX number of excitations NITRC Neuroimaging Informatics Tools and Resources Clearinghouse NMR nuclear magnetic resonance ODF orientation distribution function PDF probability density function PGSE pulsed gradient spin echo QBI q-ball imaging RA relative anisotropy RF radio frequency ROI region of interest RTD resistance temperature detector SDF spin distribution function SNR signal-to-noise ratio SPGR spoiled gradient recalled echo WM white matter xiii List of Tables 5.1 Summary of parameters used for data analysis. . . . . . . . . . . . . . . . 52 5.2 Threshold values used in detecting local maxima. . . . . . . . . . . . . . . 53 5.3 Fractions of the left cingulum recovered. . . . . . . . . . . . . . . . . . . . 62 5.4 Fractions of the left inferior longitudinal fasciculus recovered. . . . . . . . 64 5.5 Fractions of the right inferior fronto-occipital fasciculus recovered. . . . . 64 5.6 Fractions of the right corticospinal track recovered. . . . . . . . . . . . . . 65 5.7 Comparison of tractography recovery of the mediolateral transcallosal bers. 71 5.8 Comparison of tractography recovery of the corticospinal track. . . . . . . 71 5.9 Comparison of tractography recovery of the anterior-posterior association bers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 6.1 Dierence in FA due to miscalibrated diusion-weighting. . . . . . . . . . 93 6.2 Dierence in GFA due to miscalibrated diusion-weighting. . . . . . . . . 94 6.3 Dierence in MD due to miscalibrated diusion-weighting. . . . . . . . . . 95 xiv List of Figures 2.1 The human cerebrum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Axial view of the authors head. . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 White matter: projection tracts . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4 White matter: commissural tracts . . . . . . . . . . . . . . . . . . . . . . 10 2.5 White matter: association tracts . . . . . . . . . . . . . . . . . . . . . . . 10 2.6 Architecture of white matter at several scales. . . . . . . . . . . . . . . . . 11 2.7 Illustration of Brownian motion in two-dimensions. . . . . . . . . . . . . . 12 2.8 Mean diusion distance in free and restricted environments. . . . . . . . . 14 2.9 Anisotropic diusion as a result of coherent structure of white matter. . . 15 3.1 Stejskal-Tanner PGSE sequence. . . . . . . . . . . . . . . . . . . . . . . . 17 3.2 Twice-refocused PGSE sequence. . . . . . . . . . . . . . . . . . . . . . . . 20 3.3 Actual DW pulse sequence from General Electric HDxt 3.0T MRI. . . . . 21 3.4 Example diusion-weighted MRI images. . . . . . . . . . . . . . . . . . . . 22 3.5 Diusion MRI sampling patterns in q space. . . . . . . . . . . . . . . . . . 23 3.6 Geometrical interpretation of the diusion tensor. . . . . . . . . . . . . . . 27 3.7 Illustration of DTI metrics. . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.1 Set of simulated single ber orientations. . . . . . . . . . . . . . . . . . . . 31 4.2 Truncated and zero-padded q-space sampling patterns. . . . . . . . . . . . 33 4.3 Simulated Fiber Cup phantom geometry. . . . . . . . . . . . . . . . . . . . 34 4.4 Clustering of ber orientations corresponding to the single voxel shown in Figure 4.3(c). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.5 Simulated Fiber Cup phantom ground-truth ber orientations. . . . . . . 35 4.6 Simulated Fiber Cup ROIs and ground-truth tracks. . . . . . . . . . . . . 36 4.7 Fiber orientation dependent error associated with truncated-sphere q- space sampling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.8 Simulated Fiber Cup ber estimation and orientation error. . . . . . . . . 38 4.9 Histograms of simulated Fiber Cup ber orientation errors. . . . . . . . . 39 4.10 Simulated Fiber Cup tractography. . . . . . . . . . . . . . . . . . . . . . . 40 5.1 Diusion gradient directions as the vertices of a tessellated sphere. . . . . 49 5.2 Qualitative comparison of data analysis obtained from in-vivo and syn- thetic noisy data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 5.3 Single-ber per voxel ber estimation results. . . . . . . . . . . . . . . . . 55 xv 5.4 Two-ber per voxel ber estimation results (grouped by N). . . . . . . . . 56 5.5 Two-ber per voxel ber estimation results (grouped by method). . . . . . 57 5.6 Three-ber per voxel ber estimation results. . . . . . . . . . . . . . . . . 59 5.7 White-matter pathways recovered by tractography: left cingulum, left inferior longitudinal fasciculus, right inferior fronto-occipital fasciculus, and right corticospinal track. . . . . . . . . . . . . . . . . . . . . . . . . . 61 5.8 Discrete voxelized equivalent of Figure 5.7. . . . . . . . . . . . . . . . . . 62 5.9 Comparison of white-matter tractography obtained from ber orientations estimated by dierent analysis methods. . . . . . . . . . . . . . . . . . . . 63 5.10 Summary of correctly estimated tractography recovered by each of the analysis methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 5.11 Ground-truth tractography through a complex three-ber crossing region. 68 5.12 Comparison of tractography through a complex three-ber crossing region, as determined by ber orientations estimated by dierent analysis methods. 70 6.1 Diusion phantom with temperature probe. . . . . . . . . . . . . . . . . . 77 6.2 The temperature probe we constructed from an RTD element. . . . . . . . 78 6.3 Platinum RTD element. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 6.4 Four-wire (Kelvin) resistance measurement circuit. . . . . . . . . . . . . . 79 6.5 T 2 -weighted image of diusion phantom. . . . . . . . . . . . . . . . . . . . 84 6.6 Scanner drift and diusion-weighting miscalibration in DW-MRI acquisition. 86 6.7 Comparison of scanner drift between non-DW and DW data. . . . . . . . 87 6.8 Correction of diusion-weighting miscalibration. . . . . . . . . . . . . . . . 89 6.9 Eect of single b = 0 acquisition on scanner drift in DW-MRI. . . . . . . . 91 6.10 Change in diusion coecient of the phantom. . . . . . . . . . . . . . . . 92 7.1 Simulated DW-MRI Brain Data Sets project on NITRC website. . . . . . 99 A.1 Details of single-ber estimation. . . . . . . . . . . . . . . . . . . . . . . . 116 A.2 Details of two-ber estimation for BSM. . . . . . . . . . . . . . . . . . . . 117 A.3 Details of two-ber estimation for CSD. . . . . . . . . . . . . . . . . . . . 118 A.4 Details of two-ber estimation for aQBI. . . . . . . . . . . . . . . . . . . . 119 A.5 Details of two-ber estimation for CSA. . . . . . . . . . . . . . . . . . . . 120 A.6 Details of two-ber estimation for FRACT. . . . . . . . . . . . . . . . . . 121 A.7 Details of two-ber estimation for GQI. . . . . . . . . . . . . . . . . . . . . 122 A.8 Details of three-ber estimation. . . . . . . . . . . . . . . . . . . . . . . . . 123 B.1 Phantom housing: Acrylic tube. . . . . . . . . . . . . . . . . . . . . . . . . 125 B.2 Phantom housing: Top cap. . . . . . . . . . . . . . . . . . . . . . . . . . . 126 B.3 Phantom housing: Bottom cap. . . . . . . . . . . . . . . . . . . . . . . . . 127 B.4 Phantom insert: Planar 90 (top) and 60 (bottom) crossing. . . . . . . . 128 B.5 Phantom insert: Spherical 60 crossing. . . . . . . . . . . . . . . . . . . . 129 B.6 Phantom assembly: CAD drawing. . . . . . . . . . . . . . . . . . . . . . . 130 B.7 Phantom photo: Complete assembled phantom. . . . . . . . . . . . . . . . 131 B.8 Phantom photo: Close-up of diusion sphere. . . . . . . . . . . . . . . . . 132 xvi B.9 Phantom photo: Temperature probe. . . . . . . . . . . . . . . . . . . . . . 132 B.10 Phantom photo: Close-up of temperature probe. . . . . . . . . . . . . . . 133 xvii Chapter 1 Introduction Diusion-weighted magnetic resonance imaging (DW-MRI) [14, 79] is a unique applica- tion of MRI in which the diusion of water molecules is used as a non-invasive probe of tissue microstructure. Since the introduction of DTI [9, 10] two decades ago, which was the rst approach to modeling three-dimensional (3D) diusion in the brain, the method has been enthusiastically applied to a broad range of neurological inquiry span- ning neuroscience to clinical research. A brief summary of applications that motivate the continued development of this method is presented next. 1.1 Motivation Applications of diusion imaging to date include, among others, graph theoreti- cal analysis of structural brain networks [22, 115], mapping the human connectome [59, 58, 124, 24], neuroanatomical studies and atlases [29], neurobehavioral modeling leading to improved models of the limbic system [28], neurosurgical planning [54] and post-surgery evaluation [146], and identifying auditory pathways [68] (which may improve the prognosis of recovering useful hearing following cochlear implantation [136]). Separately, quantitative metrics derived from DTI|including fractional anisotropy (FA) and mean diusivity (MD) which relate directly to tissue microstructure [12]|have become indispensable to many aspects of assessment and study of neurological diseases as such measures can be obtained, for the rst time, non-invasively and in-vivo. Examples of clinical research in which DTI has provided information on neuropathology include multiple sclerosis [89], Alzheimer's disease [96] and schizophrenia [137]. See [66, 84, 45] for reviews of clinical applications of DTI to white matter diseases and neuropsychiatry. 1 Alongside extensive application of DTI the foundations of diusion imaging have matured. Advances in signal modeling have led to a vast number of alternative diusion sampling strategies and analysis methodologies predominantly concerned with overcom- ing the limitation of DTI, which is unable to model complex diusion patterns that occur in white matter regions of multiple unique ber orientations. While brain-storming of new approaches is essential, without systematic evaluation it is dicult for the best ideas and knowledge learned to propagate. Furthermore, it becomes a challenge for researchers to select the most appropriate analysis method for their data. 1.2 Problem Statement While new DW-MRI analysis methodologies are frequently introduced alongside sim- ulation studies, dierences in signal models, simulation parameters and/or evaluation metrics usually prevents a broad comparison of similar work. Combined with variations in data sets (e.g. number and magnitude of diusion-weighting directions, and/or SNR of the data) the relationship between existing techniques often remains unclear, as does quantitative improvements of new methods. Overall, lack of appropriate data sets for validation and standards for quantitative evaluation has impeded advancement of the technique. We follow with a brief review of research related to addressing this problem, before describing the contributions of our work. 1.3 Previous Research The need for validation in diusion MRI was recognized early and many research groups developed diusion phantoms for such purposes. Examples have included biological phantoms constructed from rat spinal cord [25], spherical [97] or straight [111] crossings of polyester bers, and planar phantoms of various materials (e.g. water lled plastic capillaries [131], permeable Rayon bers [106], solid acrylic bers [108], or polyethylene bers [48]). These phantoms have generally consisted of simple geometry (e.g. a single crossing-ber region) for basic validation, and as such have not been used for detailed characterization and comparison of methods. 2 The `Fiber Cup' phantom [49, 107] and contest (MICCAI 2009 1 ) was purposely devel- oped to address the lack of publicly available diusion-weighted data sets. The Fiber Cup is a planar phantom with ber congurations modeled from a coronal cross-section of the brain. Overall it consists of seven ber branches having three ber crossings, one merging/diverging region, and one ber splitting region. As Fiber Cup data includ- ing tractography ground-truth and evaluation tools is publicly available, it has been frequently used for both qualitative and quantitative evaluation of DW-MRI analysis methods and tractography. Fiber Cup, however, is geared towards tractography evaluation and as such is not well suited to detailed characterization of ber estimation performance|the phantom has few crossing angles, and without a ground-truth of orientations it is impossible to quantify the accuracy of estimated bers. More recently, HARDI reconstruction challenges have been held (ISBI 2012 2 , ISBI 2013 3 ) that improve upon Fiber Cup by utilizing simulated DW phantoms, which oer considerable versatility over their physical counterparts. [40] For example, dierent signal models, ber congurations, diusion sampling schemes and noise levels can be adjusted, thereby enabling the testing of many algorithms in dierent scenarios and providing a `proving-ground' for tuning model parameters. Even though simulations oversimplify white matter architecture and the diusion-weighted signal, the approach is frequently chosen for validation of the majority of algorithms proposed in the literature, for example [133, 2, 103, 144, 128, 41, 112, 71, 92, 43, 27, 60]. The 2013 HARDI reconstruction challenge improved on the diusion model of pre- vious years' (intra-axonal and extra-axonal diusion were incorporated) and introduced metrics to evaluate the precision of estimated single ber orientations as well as the probability of false bers (false-positives and false-negatives combined), though the sim- ulated ber pathways themselves (i.e. the ground-truth) remain based on oversimplied and gross approximations to actual white matter anatomy. In related work, though not introducing specic data sets or quantitative metrics, a versatile software program `Fiberfox' was recently announced facilitating the creation of 1 http://www.lnao.fr/spip.php?rubrique79 2 http://hardi.epfl.ch/static/events/2012_ISBI/ 3 http://hardi.epfl.ch/static/events/2013_ISBI/ 3 realistic white matter synthetic (software) phantoms. [99, 101, 100] While the potential exists, Fiberfox has not (yet) been used for creation of any human brain like diusion- weighted data sets, nor quantitative metrics and tools, for comparison of DW-MRI anal- ysis methods. This brings us to the contributions of our work. 1.4 Dissertation Focus and Main Contributions This dissertation is concerned with imaging of the human brain, emphasizing non- invasive and in-vivo MR techniques for virtual reconstruction of the complex network of connections that is the brain's white-matter. Development of real phantoms and syn- thetic data sets are indispensable for evaluating the accuracy, precision, reproducibility and noise sensitivity of DW-MRI analysis methods in a quantitative manner. As dis- cussed in the previous section there has been a conscious eort towards this goal, yet still there remains a need for publicly accessible DW-MRI data sets comprising realistic congurations of white matter pathways, in addition to a ground-truth of ber directions and software tools to permit consistent and comparable detailed evaluation of analyzed data. Achieving that is the primary contribution of this dissertation. While our work is similar in vein to the HARDI reconstruction challenges, there are important dierences in the simulation models, quantitative metrics, and basis for ground-truth of bers used. Notable aspects of our methodology include: We include a free diusion compartment in the signal model to accommodate sources of isotropic diusion, and T 2 -weighting MR signal decay, both of which aect ber estimation accuracy, even in the simplest case of a single ber compart- ment. We develop the ground-truth of ber orientations from in-vivo data rather than from coarse approximations to neuroanatomy, which has the advantage of preserv- ing realistic white-matter congurations, and the ability to observe implications of ber estimation accuracy on recovering those white-matter pathways via tractog- raphy. The latter is of crucial importance to brain network and human connectome studies wherein the presence or absence of connections can fundamentally aect the interpretation of results. [13, 32, 53] 4 We utilize metrics that distinguish between false-positive (spurious) and false- negative (missing) bers. While both represent false ber detection, they have signicantly dierent impact on tractography (and therefore applications of trac- tography) and so we dierentiate between the two. We evaluate DW-MRI methods using identical data sets, whereas in some HARDI reconstruction challenges (e.g. [40]) each method is evaluated with its own cus- tomized (optimal) data set. The latter enables one to determine best-case per- formance of each method independently, whereas our approach is to identify the optimal analysis method(s) for a given set of data. A second contribution of this dissertation is investigation of the eect `scanner drift' and miscalibrated diusion-weighting have on DW-MRI metrics including FA and MD. Scanner drift refers to undesirable temporal change in the MR signal intensity caused by o-resonance 4 and is a confounding factor in long duration T 2 -weighted echo-planar imaging (EPI) acquisitions such as used for functional MRI studies (and for which data processing methods have been developed to address). [85, 119, 90] Miscalibrated diusion-weighting arises from dierences in hardware (e.g. coil resis- tance, electrical current amplier characteristics) of the independent (x-, y-, z-direction) magnetic eld generating gradient coils of the scanner. Without periodic assessment and calibration of the diusion-weighting generated by each coil, diusion metrics could unknowingly vary over time and between scanners. We explore the signicance of the above confounding factors on diusion metrics using a physical diusion phantom and human data sets. 1.5 Organization of the Dissertation This dissertation is organized as follows: Chapter 2 presents a high-level overview of the human brain, white-matter architec- ture, the phenomenon of diusion, and the biological basis for diusion anisotropy. 4 As the resonant frequency of protons depends on magnetic eld strength, an o-resonance (i.e. mismatch) between the MRI scanner RF transmit/receive center frequency and the proton resonant frequency develops as a result of the wax and wane of the scanner's `static' magnetic eld, leading to a change in the received signal amplitude. Other factors, such as temperature, also aect the proton resonant frequency and therefore contribute to o-resonance. [64] 5 This material is a reference for understanding what is being measured and how it relates to tissue microstructure. Chapter 3 presents an overview of DW-MRI including pulse sequences for acquiring diusion-weighted images and alternative approaches to modeling the data, such as DTI, HARDI techniques, and DSI. This material is essential background for under- standing how diusion in biological tissues is measured using MRI and alternative models for the data. Chapter 4 investigates the impact of truncated-sphere diusion sampling on estimated ber directions in DSI. We illustrate the impact of ber estimation dierences by developing a comprehensive simulation framework for synthesizing DW data and apply it to a simulated version of the Fiber Cup phantom. Chapter 5 details development of synthetic DW-MRI brain images based on an in-vivo acquisition, and applies the result to evaluate six dierent DW-MRI multi-ber analysis methods. The results presented show vast dierences in estimated ber pathways that are obtained with each analysis method. Chapter 6 investigates the impact of scanner drift and uneven diusion-weighting on diusion metrics. A diusion phantom with temperature probe is developed to identify drift and uneven diusion-weighting, and after correcting for these con- founds the signicance of them on diusion metrics derived from human data sets is assessed. Chapter 7 provides a summary of contributions and suggestions for future work. 6 Chapter 2 Neuroanatomy and Diusion This chapter illustrates the biological entity under study in this work, and acts as ref- erence for physical interpretation of the results of later chapters. We present a brief overview of neuroanatomy and the human central nervous system, provide examples of white-matter ber pathways, and describe molecular diusion in biological tissues. 2.1 Neuroanatomy and the Nervous System The human cerebrum (Latin for brain) is undoubtedly one of the most complex biologi- cal systems|it integrates and processes our senses, helps maintain homeostasis, provides computation for independent thought, stores our memories, and enables voluntary con- trol of our muscles, among other things. Figure 2.1 illustrates the cerebrum viewed laterally, with principal ssures dividing the cerebral cortex into four main lobes. The cerebral cortex is the outermost layer of the cerebrum and is often referred to as gray matter (GM) as it consists of the neurons (cell bodies) and contrasts the underlying white matter (WM), that consists of myelinated neuronal axons which carry signals to other neurons, muscles and glands. Figure 2.2 is an axial view of the authors head; the contrast between GM and WM is clear. While white matter appears homogenous, it is comprised of highly structured bundles of axons forming pathways, or tracts, interconnecting regions of the cerebrum and also transferring signals throughout the central nervous system. Examples of these pathways are presented next. 7 Figure 2.1: The human cerebrum, shown divided into four main lobes (frontal, temporal, parietal and occipital) by the principal ssures. From Gray's Anatomy (public domain) [55]. 2.2 White Matter Pathways There are three dierent kinds of tracts within white matter, as exemplied in Figures 2.3 to 2.5. They are: Projection tracts such as the corona radiata, extend vertically between higher and lower brain centers, and carry information between the cerebrum and the rest of the body. See Figure 2.3. Commissural tracts such as the corpus callosum, cross from one cerebral hemisphere to the other, allowing the left and right sides of the cerebrum to communicate with each other. See Figure 2.4. Association tracts such as the cingulum and fornix, connect dierent regions within the same hemisphere of the brain. They may be long bers that connect dierent lobes, or short bers that connect dierent gyri within a single lobe. See Figure 2.5. 8 Figure 2.2: Axial view of the authors head (T 1 -weighted image). The dierence in contrast of gray and white matter is easily appreciable. Note that the white matter appears homogeneous. Figure 2.3: White matter projection tracts: 1) Corona radiata 2) Internal capsule 3) Corti- cospinal tract. Adapted from [141]. 9 Figure 2.4: White matter commissural tracts: 1) Splenium of corpus callosum 2) Radiation of corpus callosum 3) Anterior commissure. Adapted from [141]. Figure 2.5: White matter association tracts: 1) Cingulum 2) Short arcuate bers 3) Column of fornix. Adapted from [141]. 10 The structure of white matter tracts is shown at various scales in Figure 2.6. The nely organized and coherent arrangement illustrated gives rise to restricted movements of randomly moving water molecules which occupy the space within (intracellular) and between (extracellular) axons. Random movement of water molecules, a phenomenon referred to as `Brownian motion', is introduced in the next section. Figure 2.6: Illustration of white matter architecture at several scales. (a) White matter ber tracts of the anterior thalamic radiation. Adapted from [141]. (b) Cross-section of ber bundle showing fasciculi surrounded by epineurium. Each fasciculus contains many myelinated axons. Adapted from Gray's Anatomy (public domain) [55]. (c) Fasciculus. (d) Myelinated axon. 2.3 Molecular Diusion Although water appears to be static, individual molecules are in constant motion, moving randomly and colliding with each other and their surrounds. Such motion was rst described by Robert Brown in 1828. [21] Brown observed the movements of grains of 11 pollen suspended in water, and while secondary to the (unobserved) movement of water molecules, the random motion he described is similar to that of molecules, and thus the molecular phenomenon is often referred to as Brownian motion. Figure 2.7 illustrates the `random walk' of three molecules undergoing Brownian motion. −40 −30 −20 −10 0 10 20 30 40 −40 −30 −20 −10 0 10 20 30 40 x displacement [µm] y displacement [µm] Figure 2.7: Illustration of Brownian motion of three molecules moving randomly in an isotropic medium in two-dimensions. Each of the paths starts at the origin, from which arrows are drawn to the end-points. Diusion can be thought of as the observable eect of many molecules undergoing Brownian motion. Before discussing diusion in biological tissues, we present the case of free (unrestricted) diusion. 2.3.1 Free Diusion In free diusion molecular displacements are isotropic, that is, molecules have equal probability of moving in any direction at any instant in time. The distribution of dis- placements, however, is non-uniform and follows a Gaussian distribution. Considering 12 displacements in one-dimension Eq. 2.1 is the displacement distribution (also diusion probability density function (PDF)) which describes the probability that a molecule initially at r 0 moves a distance r in time . P (rjr 0 ;) = 1 q (4D) 3 exp (rr 0 ) 2 4D ! (2.1) where D is a scalar constant representing the mobility of molecules, and is referred to as the diusion coecient. At body temperature (37 C), the diusivity of water is approximately 3:04 10 3 mm 2 =s. [93] Eq. 2.1 is maximal for r = r 0 meaning the most probable displacement is zero. However, if a single molecule were observed overtime, then the longer we observed, and as a result of the non-uniform displacement distribution, the further the molecule would be from its starting point. This observation was formalized by Albert Einstein in the early 1900's. Einstein investigated Brownian motion and determined the displacement that a molecule experiences on average, or equivalently, the mean displacement of an ensemble of molecules after a period of time, , to be [46, 47] = p hkrr 0 k 2 i = p 2nD (2.2) wherehi indicates an average over the ensemble and n is the number of dimensions. It is important to note that in Eq. 2.2, D is the scalar measure of diusivity for a single dimension, and is applicable to alln dimensions for free diusion in an isotropic medium such as water. 2.3.2 Diusion in Biological Tissues In white matter molecular diusivity is much lower than for free water, and considering typical diusion times of = 50 100 ms, water molecules would expect to diuse a mean distance of = 1015 m. [80] Given the scale of axons, neurolaments and other structures shown in Figure 2.6, this displacement is suciently large for water molecules to have bounced into, crossed, or otherwise interacted with, tissue components of their local environment. As a result of the interactions, the movement of water molecules is hindered and restricted|overall, the diusion process can be described as anisotropic. 13 [14] White matter, which has a high coherence of architecture, correspondingly has a high degree of diusion anisotropy. Initially it was thought anisotropy was due to myelin alone. This is now known to not be the case and anisotropic diusion has been found in non-myelinated nerve bers. [15, 142, 57] Overall, the presence of myelin has been found to modulate the degree of anisotropy. [57, 121, 122, 123, 135]. It should be noted that biological membranes are permeable, allowing for exchange of molecules between intracellular and extracellular compartments. [20, 50, 127] The concepts of free and restricted diusion introduced in the previous sections are summarized in Figure 2.8, which illustrates the relationship between mean diusion distance and the square root of diusion time (refer to Eq. 2.2). Mean diusion distance Unrestricted Permeable barriers Diusion time Restricted Figure 2.8: Mean diusion distance in free and restricted environments. In an unrestricted environment, the mean diusion distance increases in proportion to the square root of the dif- fusion time. In contrast, non-porous barriers can be seen to limit the diusion distance to a maximum. In between these, permeable barriers|which most closely characterize biological membranes|hinder but do not completely restrict, the movement of water molecules. Figure 2.9 illustrates how the coherent architecture and membranes of white matter leads to highly anisotropic diusion of water molecules. 14 (a) (b) Figure 2.9: Anisotropic diusion in white matter. (a) Myelinated axon. (b) Example `random walk' of a single molecule undergoing Brownian motion inside an axon. The resulting diusion is highly anisotropic. 15 Chapter 3 Diusion-Weighted MRI and Tractography In this chapter the fundamentals of DW-MRI are introduced, beginning with a descrip- tion of how the MR signal is made sensitive to diusion of water molecules, then following with signal modeling. For brevity, a complete review of MR physics and signal generation from the roots of nuclear magnetic resonance (NMR) phenomenon is not provided in this dissertation. For such deep perspective the reader is referred to texts such as [1, 82, 19, 83], and for thorough treatment on the eect of diusion on the MR signal refer to [23, 81]. 3.1 Diusion-Weighted MRI 3.1.1 Pulsed Gradient Spin Echo Sequence In 1965 Stejskal and Tanner introduced the pulsed gradient spin echo (PGSE) sequence, in which two identical gradient pulses are placed either side of the 180 refocusing RF pulse of a conventional spin-echo sequence, as shown in Figure 3.1. [125, 126] For clarity we do not show details of signal readout, though it is typically accomplished using single- shot EPI, which allows for very fast image acquisitions. [88] As represented in Figure 3.1, we assume (without loss of generality) that the diusion- weighting gradients are rectangular, even though in practice this cannot be realized due to gradient hardware slew-rate limitations; we also take the origin of time (t = 0) as the beginning of the rst gradient pulse. The diusion gradient pulses have amplitudeg (t) = G, duration, and separation (which corresponds to the diusion time,). Application 16 DIFFUSION GRADIENTS MAG. FIELD STRENGTH, B SPIN PHASE NET M RF Excitation In-phase De-phasing Re-phasing 90° Δ 180° Refocus Signal without diffusion G with diffusion Figure 3.1: Fundamental Stejskal-Tanner PGSE sequence. Colored sections are: spin excitation (green), diusion-weighting (orange) and signal acquisition (blue). Gradient pulses are shown for the idealized case of rectangular pulses (shaded) of amplitude G, duration , and separation (corresponding to the diusion time ). The gradient pulses create a linear variation in the magnetic eld strength, B. The rst gradient pulse de-phases spins so they have a position dependent phase; the second gradient pulse re-phases the spins. If spins do not move during time , the phase of spins will be coherent after re-phasing, and the net transverse magnetization M will be maximal. of the gradients produces a spatially varying magnetic eld, B(x;y;z), causing spins to precess at dierent frequencies, !, according to Larmor precession (! = B). While diusion-weighting gradients can be applied along any direction, here we con- sider the x-axis, with a spin at x = x 1 during the rst gradient pulse and at x = x 2 during the second gradient pulse. We further assume the spins to not move during appli- cation of the gradient pulses. Given the linear relationship between spin precession rate and the applied magnetic eld the rst gradient pulse induces a phase shift of 1 of the spin transverse magnetization according to 1 = 1 (x) = Z 0 g (t)x (t)dt = Gx 1 (3.1) 17 where is the gyromagnetic ratio for hydrogen nuclei (42 MHz/Tesla). The second gradient pulse induces a phase change 2 = 2 (x) of 2 = 2 (x) = Z + g (t)x (t)dt = Gx 2 (3.2) The sign of 1 is inverted by the 180 pulse, and so the net phase change is = 2 1 = G (x 2 x 1 ) (3.3) which reveals a simple linear relationship between displacement x 2 x 1 and net phase change. It is easy to see that for static spins i.e. x 2 =x 1 , then = 0. Alternately, for spins that are diusing, x 2 6=x 1 , which yields = G (x 2 x 1 )6= 0. It is important to note that the diusion displacement x 2 x 1 is random and therefore is a random variable. This means the net result when considering an ensemble of spins is phase incoherence leading to signal loss, and not a net phase change. [8, 86, 30, 110] In summary, application of the rst `de-phasing' gradient pulse can be seen as `tag- ging' spins with a phase that depends on their position. If spins do not move during the diusion time = , then application of the second `re-phasing' gradient pulse will lead to phase coherence (i.e. all spins will be in-phase) and no signal loss. On the other hand, if spins are diusing the end result after the `re-phasing' pulse will be some degree of phase incoherence|the net spin transverse magnetization M will be reduced, and a signal loss will result. This is illustrated schematically in the bottom half of Figure 3.1. For a population of spins, the diusion-weighted spin-echo signal S (v), which describes the signal loss due to diusion at location v (a voxel in 3D space), is given by [62] S (v) =S 0 (v)he i iS 0 (v) (3.4) where S 0 (v) is the signal intensity in the absence of a diusion-weighting gradient, i.e. G = 0, andhi represents the ensemble average over the spin population. If we denote the conditional probability density for a spin, initially at r 0 diusing a distance r in time asP (rjr 0 ;), and also denote the phase change in a general gradient 18 direction G as (r 0 r) = G > [r (t = 0)r (t = )] then Eq. 3.4 can be written as [86] S (v) =S 0 (v) Z P (rjr 0 ;)e i(r 0 r) dr (3.5) Furthermore, considering an isotropic medium which has conditional probability den- sity of diusion displacement given by the Gaussian expression of Eq. 2.1, then the Stejskal-Tanner relationship is obtained [126] S (v) =S 0 (v)e bD(v) (3.6) whereD(v) is the scalar diusivity term introduced in Eq 2.1 (and is valid for all direc- tions for an isotropic medium) at the position v and b is the diusion-weighting factor (also referred to as b-value) introduced by Le Bihan et al. [79], and dened as b = ( kGk) 2 3 = ( kGk) 2 (3.7) where = (=3) is now the eective diusion time. The term=3 accounts for spins diusing while the gradients are turned on, that is, the assumption that the spins are static during application of the gradient pulses is now relaxed. The preceding material has introduced the classic Stejskal-Tanner PGSE sequence. As is most common EPI is used for acquisition of the k-space image data, as it allows images to be acquired very quickly. However, EPI is prone to image artifacts including ghosting, signal dropout, and image distortion (translation and shearing). This last artifact is partially caused by eddy-currents induced in the scanner by the strong and rapidly alternating diusion gradients. [19] A number of techniques have been introduced to address eddy-current-induced dis- tortion, including a twice-refocused RF spin echo sequence with two bipolar diusion gra- dient pairs. [114] This method is frequently implemented on MRI scanners for DW-MRI, and so is discussed in the following section. 19 3.1.2 Twice-Refocused PGSE Sequence The twice-refocused PGSE sequence is a modication of the classic Stejskal-Tanner PGSE sequence (Figure 3.1) in which the pair of diusion gradient pulses are sepa- rated into two pairs, each with a 180 refocusing pulse. This decomposition of the diusion gradients minimizes eddy currents prior to EPI readout, leading to reduced image distortions. The twice-refocused sequence is shown schematically in Figure 3.2, in which we illus- trate how the diusion-weighting can be generated along any direction G in 3D using a combination of orthogonal gradient pulses: G = G x ^ i +G y ^ j +G z ^ k, where ^ i, ^ j, ^ k are unit vectors in directions of the frequency encoding, phase encoding, and slice selection directions, respectively. Details of signal readout using EPI [88] are also shown. FREQUENCY PHASE SLICE RF Excitation 90° 180° δ 1 δ 2 δ 3 δ 4 180° Fat saturation Refocus Signal Frequency encoding Phase encoding Refocus De-phasing Re-phasing Slice selection G x G y G z Figure 3.2: Twice-refocused, fat saturated, EPI-based PGSE sequence. Colored sections are: fat suppression (yellow), spin excitation (green), diusion-weighting (orange) and signal acqui- sition (blue). A set of four diusion-sensitizing gradients (shaded, with durations 1 , 2 , 3 , 4 ) are used to compensate for their self-induced eddy-currents, leading to reduced image distortion. Also included in Figure 3.2 is a fat saturation pulse, which nulls the signal from fat, which has a slightly dierent resonant frequency compared to water. Without fat suppression a fat-derived `ghost' image should appear shifted relative to the water-derived image. Suppression of signal from fat is accomplished by selectively exciting fatty tissues using an RF pulse on-resonance with the Larmor frequency of protons in fat, followed 20 by `spoiler' gradients to dephase the transverse magnetization. Slice selective excitation of water molecules follows immediately after. General Electric HDxt 3.0T DW Pulse Sequence For completeness, Figure 3.3 shows an actual twice-refocused PGSE DW pulse sequence from a General Electric HDxt 3.0T scanner. Figure 3.3: The author captured this image of an actual twice-refocused PGSE DW sequence, corresponding to one 64 64 image acquired on a General Electric HDxt 3.0T MRI. An Agilent MSO-X 3054A oscilloscope was connected to the RF and gradient ampliers of the scanner via coax cables; the horizontal scale is 10 ms/div. The individual traces (top-to-bottom) are: RF (yellow), frequency encoding (green), phase encoding (blue), and slice selection (magenta). The diusion-weighting applied in this case consists of components in frequency, phase and slice directions. Phase-encoding gradient `blips' during image readout are just visible. In Figure 3.3 it is possible to identify all the features of diusion-weighted image acquisition previously introduced: two 90 pulses (fat saturation and slice selective exci- tation) are visible, in addition to the two 180 refocusing pulses. The EPI readout (frequency and phase encoding) is easy to spot. Also shown in this image but not pre- viously discussed are `crusher' gradients placed just before and after each refocus pulse. Crusher gradients are used to eliminate 2 -pulse components of the refocusing pulses. 21 Example DW-MRI Images Before ending this section we illustrate in Figure 3.4 how anisotropic diusion in white matter is revealed by changes in the DW signal S (v) obtained by applying diusion sensing gradients G in dierent directions. For a given direction G, the signal S (v) is attenuated most (least) in regions having tissue microstructure parallel (perpendicular) to the diusion sensing direction. Figure 3.4: Diusion-weighted images of the same axial slice of the same subject, illustrating the dependence of S (v) on the direction G (approximately indicated by the red arrows). Locations in white matter where tissue microstructure is parallel (perpendicular) to G appear dark (bright). This completes the review of diusion-weighted acquisition by MRI. In the following section we discuss modeling of DW-MRI data, inference of white matter orientation, and diusion metrics. 3.2 Diusion-Weighted Signal Modeling The most general framework from which to illustrate the dierent diusion sampling schemes and analysis methods of DW-MRI is q-space. [23] In Figure 3.5 we illustrate three dierent diusion sampling schemes, for which dierent analysis methods apply. In the following section we discuss DTI in detail, including metrics characterizing diusion anisotropy and estimating white matter ber orientation. 22 Figure 3.5: Examples of diusion MRI sampling schemes presented inq-space. (a) Low angular resolution sampling of minimum 6 DW measurements, as could be used in DTI. (b) Single-shell HARDI sampling as in aQBI and FRACT. (c) Truncated-sphere Cartesian grid sampling as in DSI. The red dots represent diusion-weighted data points of unique gradient direction and b-value; scan time increases with the number of data points acquired. 3.2.1 Diusion Tensor Imaging As mentioned at the beginning of this dissertation diusion tensor imaging (DTI), intro- duced by Basser et al. in 1994 [10, 9, 12], was the rst approach to modeling 3D dif- fusion of water molecules in the brain. The fundamental assumption of DTI is that water molecules diuse freely in white matter, permitting molecular displacement to be modeled by a third-order Gaussian|the one-dimensional (1D) diusion displacement distribution function Eq. 2.1 is generalized to 3D P (rjr 0 ;) = 1 q (4) 3 jDj exp (r r 0 ) > D 1 (r r 0 ) 4 (3.8) where D is a 3 3 covariance matrix physically representing diusivity in 3D andjj represents the determinant. It can be shown that for free diusion D is symmetric and positive-denite. [39, 102] From Eq. 2.2 and using notation x;y;z to represent orthogonal directions in n = 3 dimensions and the symmetry property of diusion, D can be written: D = 1 6 hkr r 0 k > kr r 0 ki 2 4 D xx D xy D xz D xy D yy D yz D xz D yz D zz 3 5 (3.9) 23 In DTI, D is a rank-2 tensor, referred to as the diusion tensor. To compute D, which has six degrees of freedom, at least six DW measurements, S, are required in addition to a reference non-DW measurement S 0 . [12, 72] Measurements, S, are acquired along non-collinear directions G j (j = 1;:::;N) by application of dierent amplitudes (G x , G y and G z ) of the gradient pulses (refer to Figure 3.2) for each direction j. The diusion tensor D can then be calculated for every position v (a voxel in 3D space) by solving the system of equations S (v;b;j) =S 0 (v)e b^ g > j D(v)^ g j (3.10) where ^ g j = G j kG j k and j = 1;:::;N (3.11) andb is the diusion-weightingb-value, dependent on the pulse sequence and not neces- sarily given by Eq. 3.7 (which applied to the classic Stejskal-Tanner PGSE sequence). Before proceeding we simplify notation by dropping the v with the implicit under- standing that the expression applies to measurements of a single voxel, and can be applied to all such voxels independently. In addition, we prefer to write S j rather than S (j). Lastly it should be noted that althoughb depends onkGk (refer Eq. 3.7) therefore requiring us to write b (j) = f (G j ), in the case of DTIkG j k is xed for all j and so b is constant. We now discuss computing the diusion tensor, D, from measurementsS 0 andS j (for j = 1;:::;N) (where S j is implicitly understood to be the diusion attenuated signal obtained from diusion-weighting b applied in direction j). 24 Computing the Diusion Tensor The most straight-forward approach to solving Eq. 3.10 is to linearize the system and express the equations in matrix form. By taking the logarithm we can write in matrix notation 2 6 6 6 6 6 6 4 1 b ln S 0 S 1 1 b ln S 0 S 2 . . . 1 b ln S 0 S N 3 7 7 7 7 7 7 5 | {z } S = 2 6 6 6 6 6 6 4 (G1x) 2 2G1xG1y 2G1xG1z (G1y) 2 G1yG1z (G1z) 2 (G2x) 2 2G2xG2y 2G2xG2z (G2y) 2 G2yG2z (G2z) 2 . . . . . . . . . . . . . . . . . . (GNx) 2 2GNxGNy 2GNxGNz (GNy) 2 GNyGNz (GNz) 2 3 7 7 7 7 7 7 5 | {z } G 2 6 6 6 6 6 6 6 4 Dxx Dxy Dxz Dyy Dyz Dzz 3 7 7 7 7 7 7 7 5 | {z } D (3.12) where S is a vector of measured data, G is a matrix formed from the known diusion sensing directions, and D is a vector of the unknown diusion tensor elements (see Eq. 3.9). ForN = 6, Eq. 3.12 has a unique solution and estimates of D can easily be obtained by inverting G ~ D = G 1 S (3.13) However, considering that S is corrupted by noise, greater than the minimum number of measurements should be acquired. The literature suggestsN 30 is sucient, beyond which there is negligible further reduction in the variance of computed tensor metrics (discussed later). [73] For N > 6, Eq. 3.12 is over-determined and so estimating D becomes a minimization problem. The simplest solution is a non-weighted linear least squares approach which estimates D by minimizing the sum-of-squared errors between the measurements S and what would be obtained given the linear relationship. It can be computed as ~ D = (G > G) 1 G > S = G + S (3.14) where G + is the Moore-Penrose pseudoinverse of G. The above approaches, while elegant and computationally fast, neglect important constraints based on physical characteristics of the diusion tensor. For example, the eigenvalues of D, which physically represent diusivity of water molecules in orthogonal directions, should be positive. Furthermore, linearization of Eq. 3.10 as in Eq. 3.12 assumes homoskedasticity in the elements of S; however this is no longer the case|the 25 logarithmic transform has introduced heteroskedasticity such that elements of S have higher variance for lower signal and vice versa. The solution to these problems adopted in our work is to t the non-linear model Eq. 3.10 to the data directly, which obviates the need to transform the data and preserves the homoskedasticity, and has been shown to improve parameter estimation when the SNR is very low. [74] Moreover, we use a Cholesky representation of the tensor, D = U > U, to ensure positive semi-deniteness of D and thus physically realistic eigenvalues. We provide an overview of the approach here. The matrix U is upper triangular with non-zero diagonal elements U = 2 4 u 1 u 4 u 6 0 u 2 u 5 0 0 u 3 3 5 (3.15) which we represent in compact vector form u = [u 1 ;u 2 ;u 3 ;u 4 ;u 5 ;u 6 ] > . Each component of D = [D xx ;D xy ;D xz ;D yy ;D yz ;D zz ] > of Eq. 3.12 can be expressed as a function of u as follows D (u) 2 6 6 6 6 6 6 4 D xx (u) D xy (u) D xz (u) D yy (u) D yz (u) D zz (u) 3 7 7 7 7 7 7 5 2 6 6 6 6 6 6 4 u 2 1 u 1 u 4 u 1 u 6 u 2 2 +u 2 4 u 2 u 5 +u 4 u 6 u 2 3 +u 2 5 +u 2 6 3 7 7 7 7 7 7 5 (3.16) In the constrained estimation of the diusion tensor, the minimization of a least squares objective function is carried out by searching for an optimal point in the space of u, rather than D. The objective function is f (u) = 1 2 N X i=1 0 @ S i S 0 exp 2 4 6 X j=1 b G ij D (u) j 3 5 1 A 2 (3.17) Any unconstrained optimization routine can be used to solve Eq. 3.17. Refer to [75, 76] for further details of the Cholesky decomposition and related approaches to diusion tensor estimation. 26 Geometrical Interpretation Having estimated elements of the diusion tensor D, which is in fact a covariance matrix describing the displacement of diusing water molecules in 3D and is symmetric and positive semi-denite, we gain further insight into the physical and geometric meaning of D by eigenvalue decomposition. The tensor D can be factorized as D = E E 1 (3.18) with E = [e 1 e 2 e 3 ] and = 2 4 1 0 0 0 2 0 0 0 3 3 5 (3.19) dening the matrix of orthonormal eigenvectors e i and a diagonal matrix of eigenvalues i (fori = 1; 2; 3). A 3D ellipsoid can be constructed from the eigenvectors and eigenvalues as in Figure 3.6, overall representing the probabilistic iso-surface of diusion displacement characterized by D. λ 1 , e 1 λ 2 , e 2 λ 3 , e 3 Figure 3.6: Geometrical interpretation of the diusion tensor. The principal axes of the ellipsoid are dened by the eigenvectors e i and eigenvalues i of D. Compare the ellipsoid shape and orientation to the image of restricted diusion in white matter, Figure 2.9(b). In general, the eigenvalues are sorted according to the convention 1 > 2 > 3 , so that the rst eigenvalue e 1 (corresponding to 1 ) denes the direction of greatest diu- sivity and is assumed to be tangential to the underlying white matter ber orientation. From such estimates of ber orientation, ber-tracking algorithms [95] can be used to generate non-invasive virtual reconstruction of neuronal connectivity [36, 94, 11], referred to as tractography. Example of human brain tractography is presented in Chapter 5. 27 Scalar DTI Metrics In addition to the directional information above, rotationally invariant scalar metrics can be derived from the eigenvalue decomposition of D. Diusion Size The overall mean-squared displacement of water molecules is character- ized by the mean diusivity (MD), and can be calculated as MD = trace (D) 3 = D xx +D yy +D zz 3 = 1 + 2 + 3 3 = (3.20) Note that MD does not describe the shape of the diusion prole, but rather its size. Diusion Anisotropy The degree of diusion anisotropy is most commonly dened according to measures based on the normalized variance of the eigenvalues. Two common metrics are fractional anisotropy (FA) and relative anisotropy (RA), and can be calculated as [12] FA = r 3 2 q 1 2 + 2 2 + 3 2 p 2 1 + 2 2 + 2 3 (3.21) and RA = r 1 3 q 1 2 + 2 2 + 3 2 (3.22) The FA metric is appropriately normalized so that it take values in the range zero (when diusion is isotropic: 1 = 2 = 3 ) to one (when diusion is constrained to one axis only: 1 6= 0 and 2 = 3 = 0). Note that the RA metric needs an additional factor of p 1=2 to ensure it scales from zero to one. While both these metrics indicate a degree of coherence of tissue microstructure, FA is most commonly used in the literature. Figure 3.7 illustrates how these metrics reveal changes in tissue microstructure. 3.2.2 Beyond DTI Diusion tensor imaging is limited to modeling a single-ber orientation per voxel and is therefore incapable of resolving complex intra-voxel geometry such as crossing-bers 28 Figure 3.7: Axial, coronal and sagittal slices of the human brain illustrating directional and scalar metrics of DTI. Left-to-right: Directionally-encoded fractional anisotropy, scalar fractional anisotropy, and mean diusivity. [4, 51, 133], which are thought to occur in at least one-third of voxels in white matter [16]. To overcome this problem, many alternative multi-ber analysis methods have been proposed. The alternatives include high angular resolution diusion imaging (HARDI) methods, such as a family of q-ball imaging ([27, 43, 63, 91, 132]) and many other variants of the methods listed here, and spherical deconvolution approaches ([42, 130, 128]), which sample single or multiple shells in q-space, and methods based on Cartesian sampling schemes of q-space, such as DSI ([134, 138, 139]), DSI with partial sampling schemes ([78, 143]) and related variants [26]. This list is not exhaustive and a great many other model and non-model based methods exist; see [7, 60] for a more comprehensive list and theoretical dierences. 29 Chapter 4 Truncated Sampling in DSI Diusion spectrum imaging [138] samples the three-dimensional q-space to yield an ensemble average propagator (EAP) describing the local diusion of water molecules. To reduce scan time, sampling is frequently limited to a truncated-sphere within a Carte- sian grid ofq-space. In two-dimensional imaging it has been shown [18] that the corners of k-space contribute to the resolution of an image, provided an interpolation method such as zero-padding is used. It is not well known how signicant the corner samples of q-space are to the problem of ber orientation estimation, including whether or not the error in estimating a single ber is dependent on its orientation. The objective of this work was to investigate the consequence of truncated-sphere q-space sampling on estimated ber directions in DSI. 4.1 Overview The eect of truncated-sphere acquisition schemes (see Figure 3.5) frequently used in DSI is simulated using fully-sampled (777) q-space data (343 samples), truncated-sphere sampling (203 samples), and zero-padding (to 212121 samples), in both noise-free and noisy (SNR = 30) scenarios. We determine the error in resolving a single ber direction via simulation by sys- tematically varying the orientation of a single ber in a model of the DW MR signal, and after synthesizing the signal use DSI to obtain the EAP from which we compute the orientation distribution function (ODF) and subsequently estimate the ber orienta- tions. The orientation error of a single ber is the absolute angular dierence in degrees 30 between the known (simulated) and estimated ber orientations. The systematic varia- tion of ber orientation allows us to examine the in uence of corner q-space samples on estimated ber orientations. To assess the signicance of orientation estimation errors on tractography a sim- ulation of the `Fiber Cup' phantom [49, 107] was created. The Fiber Cup phantom was originally developed for quantitative comparison of multi-ber diusion models and tractography algorithms. While several Fiber Cup data sets are available they are inap- propriate for DSI, and hence could not be used in this work. 4.2 Materials and Methods In this section we present the approach to determining the eect of truncated q-space sampling on single ber estimation, and then present development of the simulated Fiber Cup. 4.2.1 Single Fiber Estimation in DSI To determine the eect of truncated sampling on single ber orientation we simulated the DW signal for a systematically oriented ber, and then estimated its orientation from the reconstructed ODF. The directions of the single ber are shown in Figure 4.1. Figure 4.1: Set of single ber orientations (indicated by dots), one of which is indicated by an arrow. 31 In Figure 4.1 the angles ' and dene the orientation of a single ber. Due to spherical symmetry' and were limited over the ranges 0'=2 and 0=2. The angular step between ber directions (d' and d) was xed at 1:5 , resulting in 3721 orientations. A single ber tensor model mixed with isotropic (free) diusion was used to generate noiseless q-space data according to [144] Eq. 4.1. S (q) =S 0 h f 0 e q > D 0 q + (1f 0 )e q > Dq i (4.1) where the isotropic compartment size and diusivity was xed at f 0 = 0:25 and D 0 = 2:20 10 3 mm 2 =s, respectively. The eective diusion time = =3 was determined from parameters in [144]: = = 80 ms=35 ms. Each single ber tensor D was obtained according to the method presented later in Section 5.2.2, using eigenvalues (see Figure 3.6(c)) set tof 1 ; 2 ; 3 g =f1:70; 0:30; 0:30g10 3 mm 2 =s based on nominal values observed in diusion measurements of white matter axons and used by Tuch in [132]. The non-diusion weighted signal was assumed constant at S 0 = 1:0. A 7 7 7 Cartesian grid of q-space was selected to result in an optimal truncated sampling with maximum diusion-weighting b max = 4000 s/mm 2 as proposed by Kuo in [77]. Fully sampled q-space data (323 q-space samples) had a maximum diusion- weighting ofb max = 6800 s/mm 2 along diagonal directions. A mask limited the samples to within a truncated-sphere was applied to result in DSI-203 data (203q-space samples). Both the full and truncated data sets were zero-padded to 21 21 21, preserving the conjugate symmetry of the data. The combinations of truncated and zero-paddedq-space give rise to four dierent sampling patterns, shown in Figure 4.2. From theq-space data the EAPs were obtained by 3D discrete inverse Fourier trans- form (DIFT). For a truncated data set the missing points were zero-lled before applying the 3D DIFT. The ODFs were obtained by integration of the EAPs along 642 radial projections. Lastly, the orientation of the single ber was identied by the direction of global maxima on the ODF. For all sampling pattens in Figure 4.2 the eect of noise was also investigated; a Rician distribution of noisy diusion data [56] was obtained with SNR = 30 based on the non-diusion weighted signal, S 0 . Thus in total eight dierent combinations of 32 Fully Sampled Truncated Sphere No zero-padding Zero-padding No zero-padding Zero-padding 343 q-space samples 203 q-space samples q x q y q z q x q y q z q x q y q z q x q y q z Figure 4.2: Four dierentq-space data sets used in evaluating the eect of truncated sampling on estimated ber directions in DSI; actualq-space samples (red) and zero-padded points (blue). parameters were simulated, and for the noisy cases, the simulation was repeated 100 times with independent noise realization. 4.2.2 Simulated Fiber Cup Phantom A simulation of the Fiber Cup phantom was created [140] to investigate the impact of estimated ber orientation errors resulting from truncated sampling, on tractgora- phy. Starting with the geometry of the Fiber Cup [49], Adobe Illustrator 1 was used to create an accurate vector drawing of the phantoms seven ber pathways; each path- way was represented by nine parallel ber bundles, illustrated as orange lines in Fig- ure 4.3(a). A 64 64 pixel grid (Figure 4.3, blue lines) simulates the imaging resolution and denes boundaries at which the continuous bers were cut into voxel-sized segments. Figure 4.3(c), highlights a single voxel containing 8 ber segments, grouped (by color) into three signicantly dierent orientations. An Adobe Illustrator ExtendScript program was written to output the coordinates of the start and end of all individual ber segments, from which the orientation and length of each segment could be determined. While each voxel may contain any number of ber segments, as in Figure 4.3(c), the number of signicantly dierent ber orientations is at most three, and so a clustering algorithm was used to group similar orientations, see Figure 4.4. 1 http://www.adobe.com/products/illustrator.html 33 (a) (b) (c) Figure 4.3: Simulated Fiber Cup phantom geometry, at three levels of detail. (a) Each ber pathway represented by 9 parallel ber bundles (orange lines). (b) Complexity of ber crossings, and pixel grid (blue lines). (c) Single voxel containing 8 ber segments, grouped (by color) into three signicantly dierent orientations. Figure 4.4: Clustering of ber orientations corresponding to the single voxel shown in Fig- ure 4.3(c). The clustered orientations dened the ground-truth of ber orientations for each voxel, while the lengths of clustered segments established the fraction of each ber orien- tation within a voxel. Figure 4.5 shows the complete ground-truth of ber orientations for the simulated Fiber Cup phantom. Given the ber orientations of each voxel the diusion-weighted signal was synthe- sized according to a multi-tensor version of Eq 4.1. For a single voxel,k, accommodating n k unique ber orientations, the diusion-weighted signal is: S k (q) =S k 0 2 4 n k X i=1 f k i e q > D k i q 3 5 (4.2) 34 Figure 4.5: Ground-truth of ber orientations for the simulated Fiber Cup phantom, overlaid on partial volume image. where S k 0 represents the maximal non diusion-weighted signal of voxel k, and is equiv- alent to the fraction of that voxel occupied by ber pathways. The individual ber weightings (f k i such that P i f k i = 1:0) were determined from the relative lengths of all ber segments within a voxel. Each single ber tensor D k i was computed using the single ber eigenvalues listed in Section 4.2.1, with the rotation matrix being determined from the orientation of ber i belonging to voxel k. Fully sampled q-space data was simulated as described in Section 4.2.1 and subse- quently masked and/or zero-padded to give the four dierent q-space sampling patterns of Figure 4.2. In each case, noise (SNR = 30) was added to allow comparison of noise-free and noisy results. The eight simulated data sets were processed according to DSI, and at most three ber orientations were estimated from local maxima of the ODF for each voxel. Fiber orientation estimation errors were calculated as the average absolute error between known ber directions and (the closest) estimated ber orientations, for each voxel. As the ground-truth provides the number of bers in each voxel the number of false-positives (spurious bers) and false-negatives (missing bers) were also calculated. From the estimated ber orientations, deterministic streamline tractography was generated from all voxels (3 seeds/voxel) using a vector-interpolation method. The 35 resulting track paths were ltered by 16-single voxel ROIs (Figure 4.6(a)) located in regions of unique (single) ber orientation, thus yielding 16 sets of track paths. From each set of tracks, the track with minimum l 2 -norm (spatial separation) with respect to the corresponding ground-truth ber (Figure 4.6(b)) was selected as the \best" estimate of the ber through that ROI. (a) (b) Figure 4.6: (a) Simulated Fiber Cup ber bundles (orange lines) with 64 64 voxel grid (blue lines) and 16 single-voxel ROIs (white squares) used to lter tractography. (b) Ground-truth bers corresponding to the ROIs in (a), used in selection of \best" ber tracks resulting from each processed q-space data set. 4.3 Results 4.3.1 Truncated Sampling and Single Fiber Estimation in DSI Figure 4.7 illustrates the spatial variation in error when estimating the orientation of a single ber, in both the noise-free and noisy (SNR=30) cases, with and without zero- padding, and with fully-sampled (343q-space samples) and truncated-sphere (203q-space samples) sampling schemes. Figure 4.7 clearly illustrates a ber-orientation dependent error, which is reduced for full-sampling compared to truncated-sphere sampling, as would be expected. Also, in all cases, zero-padding is shown to uniformly distribute error that accumulates along orthogonal directions throughout all directions, and reduces the average ber orientation error compared to non-zero-padding. Lastly, the results show that the corners ofq-space, 36 Noise-free Data Noisy Data (SNR = 30) No zero-padding (7x7x7) Zero-padding (21x21x21) No zero-padding (7x7x7) Zero-padding (21x21x21) Fully Sampled (343 q-space samples) Error ± σ 4.9° ± 2.6 3.2° ± 1.5 5.2° ± 2.3 3.5° ± 1.3 Truncated Sphere (203 q-space samples) Error ± σ 5.3° ± 2.9 3.6° ± 1.8 5.7° ± 2.6 3.9° ± 1.5 0 0.5 1 0 0.5 1 0 0.2 0.4 0.6 0.8 1 0 0.5 1 0 0.5 1 0 0.2 0.4 0.6 0.8 1 0 0.5 1 0 0.5 1 0 0.2 0.4 0.6 0.8 1 0 0.5 1 0 0.5 1 0 0.2 0.4 0.6 0.8 1 0 0.5 1 0 0.5 1 0 0.2 0.4 0.6 0.8 1 0 0.5 1 0 0.5 1 0 0.2 0.4 0.6 0.8 1 0 0.5 1 0 0.5 1 0 0.2 0.4 0.6 0.8 1 0 0.5 1 0 0.5 1 0 0.2 0.4 0.6 0.8 1 0 5 10 15 Error [degrees] Figure 4.7: Fiber orientation dependent error when estimating the direction of a single ber. The error and standard deviation () listed below each plot is calculated over all ber orientations (Figure 4.1). which are removed in truncated-sphere sampling and have the worst SNR (due to reduced signal amplitude) compared to the center samples ofq-space, still contribute to reducing ber orientation error, even in the case of noisy data. This is most evident by the increased error in estimating ber orientation for bers oriented towards the corners (' =4 and =4) of the spatial (and thus frequency) domain. Even when zero- padding is used with truncated q-space, errors still accumulate along the directions of \missing" q-space data. Our results are consistent extensions to the two-dimensional (2D) k-space ndings reported by [18]. 4.3.2 Simulated Fiber Cup Phantom Results of the simulated Fiber Cup phantom include the average estimated ber orien- tation error per voxel, the number of false-positives and false-negatives (which together give the number of incorrectly estimated bers), and the \best" track paths recovered by processing each q-space data set. 37 Estimated ber orientation error Figure 4.8 illustrates the estimated ber orientations for the complete simulated Fiber Cup. The number of false-positives and false-negatives were computed by comparing, in each case, the number of estimated and known ber orientations (Figure 4.5). Noise-free Data Noisy Data (SNR = 30) No zero-padding (7x7x7) Zero-padding (21x21x21) No zero-padding (7x7x7) Zero-padding (21x21x21) Fully Sampled (343 q-space samples) False Pos. 2 81 5 422 False Neg. 114 47 115 23 Truncated Sphere (203 q-space samples) False Pos. 59 36 114 44 False Neg. 108 119 97 109 0 10 20 30 40 50 60 Error [degrees] Figure 4.8: Simulated Fiber Cup ber estimation (colored ber direction vectors) and average orientation error (grayscale background) indicated per voxel, along with the total number of false-positives and false-negatives. Figure 4.8 shows that the greatest ber orientation estimation errors occur in crossing ber regions. Specically the region of 3 crossing bers has the highest error as often only a single ber direction is estimated in that region|when two or more bers are almost parallel, individual local maxima of the ODFs cannot be resolved. This leads to fewer ber directions being estimated than are present, which is recorded as a false-negative. When zero-padding is applied to fully sampled q-space (which contains a greater number of low SNR data points than truncated q-space), the number of false-positives increases substantially due to more local maxima in the ODFs (which result from noisy EAP). Simultaneously, and vice versa, the number of false-negatives decreases sub- stantially as more ber directions are estimated. Thus, the option of zero-padding can 38 be seen to reduce false-negatives at the expense of increased false-positives. Compar- ing the grayscale backgrounds of the fully sampled results (upper row in Figure 4.8) reveals a small reduction in average ber estimation error when zero-padding is used. False-positives, however, will lead to spurious track paths and therefore may reduce the quality and condence of tractography. When zero-padding is applied to truncatedq-space data, the number of false-positives decreases and the number of false-negatives slightly increases. These trends, which are opposite those mentioned above, are a result of the spherical truncation ofq-space remov- ing high-frequency low SNR data, which leads to smoother ODFs with fewer local max- ima. The greater number of false-negatives, especially when compared to the fully sam- pled zero-padded result, will lead to a reduced number of track paths, and in some cases ber pathways in the ground-truth may not be recovered. Figure 4.9 presents histograms of the average ber orientation error (i.e. the non- zero grayscale backgrounds of Figure 4.8); each histogram consists of 981 data points. The gure illustrates the slight reduction in ber orientation error achieved when zero- padding is applied|in each case the use of zero-padding causes a population of voxels with orientation error 10 , to move to a lower orientation error. Noise-free Data Noisy Data (SNR = 30) No zero-padding (7x7x7) Zero-padding (21x21x21) No zero-padding (7x7x7) Zero-padding (21x21x21) Fully Sampled (343 q-space samples) Truncated Sphere (203 q-space samples) Average Orientation Error [degrees] Average Orientation Error [degrees] Average Orientation Error [degrees] Average Orientation Error [degrees] Average Orientation Error [degrees] Average Orientation Error [degrees] Average Orientation Error [degrees] Average Orientation Error [degrees] Figure 4.9: Histograms of ber orientation errors shown pictorially in Figure 4.8. 39 Tractography Figure 4.10 presents the 16 \best" tracks that passed through the single-voxel ROIs of Figure 4.6(a). Noise-free Data Noisy Data (SNR = 30) No zero-padding (7x7x7) Zero-padding (21x21x21) No zero-padding (7x7x7) Zero-padding (21x21x21) Fully Sampled (343 q-space samples) l 2 -norm 14.17 8.99 15.14 6.45 Truncated Sphere (203 q-space samples) l 2 -norm 14.65 5.48 19.14 4.60 Figure 4.10: Simulated Fiber Cup tractography with l 2 -norm averaged over the length of the tracks shown indicated below each plot. The tractography results show the surprising result that tracks obtained from noisy zero-padded data were better (spatially closer to the ground truth) than what could be recovered from noise-free data. This unexpected outcome can partially be explained by the limited angular resolution of the ODF. Each ODF is constructed from 642 radial projections of approximately uniform sepa- ration, yielding an angular resolution of 8 . As a result of this limited sampling, ber directions not aligned parallel with one of the 642 radial projections will always expe- rience a nite orientation error, even in noise free conditions. Straight ber pathways, including those in the simulated Fiber Cup phantom, have consistent ber orientation, and so bers along such paths will be assigned the exact same ber orientation estima- tion, and estimation error, in noise free conditions. As tractography progresses along such a pathway, the track will experience consistent deviation away from the actual ber path, and eventually the ber will exit the path and be terminated. This is particularly 40 evident from the noise-free fully sampled zero-padded result in Figure 4.10, in which orange and cyan colored tracks stop abruptly for no apparent reason. In the case of noisy data, noise adds randomness to the estimated ber orientations, which are on average are better aligned to the (straight) ber pathway. As a result, track paths are able to propagate along the ber pathway more successfully than in the noise-free case. Ultimately this is responsible for the apparent best tractography result coming from a noisy truncated zero-padded data set. The problem of insucient angular resolution of the ODF can be overcome at the cost of increased computation by simply increasing the number of radial projections, or randomly rotating a given set of projections, and computing the average ber orientation resulting from the set of ODFs [33]. 4.4 Conclusion By simulation of a single ber orientation, we have shown that the corners of q-space contribute to reduced ber orientation estimation error, and combined with zero-padding, produced the most spatially uniform pattern of ber orientation error. This nding should be of particular interest to ber tractography applications using ber orientations estimated by DSI. However, the average increase in orientation error from fully sampled to truncated sphere cases is small (0:4 ) and may be justied given that truncated sampling requires 40% less data acquisition time than fully sampled data. The simulated Fiber Cup phantom proved valuable in assessing dierences in esti- mated ber orientations, including numbers of false-positives and false-negatives, under noise-free and noisy scenarios. Most notably it was found that zero-padding can reduce the number of false-negatives in fully sampled data, or reduce the number of false- positives in truncated data. Regardless of the presence of noise, the application of zero-padding reduced the average ber orientation estimation error, as was the result in the single ber simulation, and tractography results improved. An unexpected observation was the eect of limited angular resolution of the ODF on tractography|tractography appeared worse with noise-free data due to consistent ber orientation errors causing tracks to propagate out of ber pathways. Noisy data 41 leads to randomness of ber orientation errors, which in this case had the side-eect of enabling tracks to successfully continue along ber pathways. 42 Chapter 5 Simulated DW-MRI Brain Data In completing the work of Chapter 4 we developed a versatile software for synthesizing DW-MRI data according to a multi-tensor model [3, 133], and applied it to a simulated version of the Fiber Cup phantom [107, 49]. In this chapter we extend the simulation to accommodate T 2 -weighted image contrast and a free (isotropic) diusion compart- ment. We also develop a ground-truth of ber orientations from a whole-brain HARDI acquisition, which in conjunction with the simulation, allows us to to generate sets of whole-brain synthetic diusion data, consisting of realistic congurations of ber cross- ings and SNR. We illustrate application of the ground-truth and synthetic data to ber estimation and tractography of several multi-ber diusion analysis methods. 5.1 Overview An important step towards further clinical application of tractography is a broader under- standing of how the data acquisition, analysis method and ber-tracking algorithm, each aect track reconstruction. Also, it is advantageous to know which approaches yield the most complete and accurate tractography for DW data acquired with a particular set of parameters (i.e.b-value, number of diusion sampling directions, etc.), and to be knowl- edgeable of the dierences and limitations of available methods. While tractography are strongly dependent on the ber-tracking algorithm itself, results are fundamentally determined by the analysis methods ability to resolve crossing bers and provide accurate estimates of their individual orientations. Therefore, standardized quantitative metrics 43 that adequately describe ber estimation errors, in addition to erroneous ber detec- tion (either false-positives (spurious bers) or false-negatives (missing true bers)), are needed. This chapter address these needs by developing sets of simulated DW brain images and quantitative metrics, which we then use to compare the performance of six DW-MRI analysis methods. The simulated data is derived from a ground-truth of ber orientations that we estimate from an in-vivo data acquisition. This is a signicant improvement to the planar ber arrangement of Chapter 4 as realistic congurations of ber crossings and entire white-matter pathways are preserved. The six evaluated methods are: 1. a two-compartment ball-and-stick model (BSM) [17], 2. non-negativity constrained spherical deconvolution (CSD) [128], 3. analytical q-ball imaging (QBI) [43], 4. analytical q-Ball Imaging with Funk-Radon and cosine transform (FRACT) [60], 5. analytical q-Ball Imaging within constant solid angle (CSA) [2], and 6. generalized q-sampling imaging (GQI) [144]. We investigate these approaches using sets of 20, 30, 40, 60, 90 and 120 evenly distributedq-space samples of a single shell, and focus on an SNR and diusion-weighting common to clinical studies. 5.2 Materials and Methods In this section we establish the ground-truth, dene the data synthesis model, and present the quantitative metrics for comparison of results. 5.2.1 Establishing the Fiber Ground-truth A 28-year-old right-handed male volunteer without any history of neurological disease was scanned on a GE 3T HDxt scanner (General Electric, Milwaukee, WI, USA), equipped with an 8-channel head coil. The subject signed an informed consent form 44 for which the imaging protocol was approved by the Institutional Review Board of the University of Southern California. A DW data set was acquired by a twice-refocused PGSE sequence with TE/TR = 83.4 ms/16100 ms, acquisition matrix = 128128, ASSET acceleration factor of 2, voxel size = 2.42.42.4 mm, 60 contiguous slices, 150 diusion gradient directions with diusion-weighting b = 1000 s/mm 2 , and 10 non-diusion weighted volumes. The acquisition took approximately 43 minutes. Without eddy-current or motion correction 1 the diusion data set was processed by the probabilistic multi-ber BSM implemented in the program `bedpostx', a part of the diusion toolbox in the FMRIB Software Library 2 [17, 120]. Up to three bers were estimated per voxel. To reduce the possibility of false minor bers resulting from data over-tting, a threshold of 0.1 was applied to second and third ber volume fractions. Images of number of bers/voxel were inspected to ensure known crossing regions (as explored later in Section 5.3.5) retained 2 or 3 bers after thresholding. Our synthetic DW data are derived from the ber volume fractions (f 1 , f 2 , f 3 ) and orientations (v 1 , v 2 , v 3 ) estimated for each voxel and output by `bedpostx'. Because of dierences between the BSM and our data synthesis equation, Eq. 5.1, the isotropic compartment fraction (f 0 ) was not used. Instead, the ber fractions were normalized ( P 3 k=1 f k = 1) and f 0 was iteratively determined per voxel: beginning with f 0 = 0, f 0 was gradually increased until the GFA [132] of the synthetic data reduced to within 0.00005 of the GFA of the corresponding in-vivo data. Anatomical T 1 -weighted SPGR images (TE/TR = 2.856 ms/7 ms) were acquired with a voxel size of 1.01.01.0 mm. The anatomical volume was registered to the mean non-diusion weighted volume and subsequently segmented into WM, GM and cerebrospinal uid (CSF) using default options in SPM 3 [52]. The high-resolution tissue 1 The diusion-weighted data was inspected for eddy-current and motion related artifacts, and only minor artifacts were found. Even so, we evaluated eddy-current correction but the post-processing caused smoothing of the data which we considered detrimental to accurately resolving crossing-bers, especially in the case of small crossing angles. 2 FSL v5.0.2.2, http://www.fmrib.ox.ac.uk/fsl 3 SPM v8, http://www.fil.ion.ucl.ac.uk/spm 45 probability maps were then down-sampled to the resolution of the DW data, and each voxel was classied as WM, GM, or CSF according to its most probable tissue type. 5.2.2 Data Synthesis Diusion-weighted data were synthesized according to a multi-tensor model [3, 133] accommodating three crossing bers per voxel in addition to a free-diusion compart- ment. For any given voxel the signal model is: S (b; g j ) =S 0 " f 0 e bD 0 + (1f 0 ) 3 X k=1 f k e bg > j D k g j # (5.1) where S 0 simulates T 2 -weighting, f 0 and D 0 are the volume fraction and diusivity, respectively, of the isotropic free-diusion compartment, f k and D k are the volume fraction and diusion tensor, respectively, of the k th ber in the voxel,b is the diusion- weighting, and g j is a unit vector representing thej th gradient direction. Altogether the volume fractions satisfy f 0 + (1f 0 ) P 3 k=1 f k = 1. Each ber's diusion tensor, D k , is computed by rotating a default single tensor, D x . That is, D k = R x (v) D x R x (v) > , where v is a vector dening the desired ber orien- tation, R x (v) is the rotation matrix that aligns the vector x = [1 0 0] > oriented along the x-axis to v, and D x is the single-ber tensor model with diusivities in orthogonal directions given byf 1 ; 2 ; 3 g. The single ber tensor and rotation matrix are given in Eqs. 5.2 and 5.3, respectively. D x = 2 4 1 0 0 0 2 0 0 0 3 3 5 (5.2) R x (v) = (x + v) (x + v) > (x > v + 1) 2 4 1 0 0 0 1 0 0 0 1 3 5 (5.3) Complex Gaussian noise was added to the synthesized DW signal (Eq. 5.1) to achieve a Rician distribution of noisy magnitude diusion data [56], according to Eq. 5.4. E b; g j = s S b; g j + n 1 p 2 2 + n 2 p 2 2 (5.4) 46 wheren 1 andn 2 are independent and identically distributed Gaussian random variables with zero mean and standard deviation n = S 0 =SNR, in which S 0 is the mean signal from a homogeneous white-matter region of the S 0 image, andSNR is the desired SNR of the magnitude data, E. 5.2.3 Quantitative Metrics Analyzed synthetic DW data sets were evaluated against the ground-truth in terms of ber orientation error, rate of spurious bers (false-positives), rate of missing true bers (false-negatives), and fraction of similar (overlapping) and dissimilar (non-overlapping) voxelized tractography. The ber orientation and false rate metrics are consistent with those used in similar work. [40] Individual Fiber Orientation Error The ber orientation error is the angular separation in degrees between pairs of esti- mated and actual ber orientations, and lies in the range 0 90 . We report individual ber orientation errors for the unique pairing of estimated and actual ber orientations that yields the minimum total orientation error (for each voxel). Each estimated ber orientation is paired with only one actual ber orientation from the ground-truth. The orientation error, , is calculated as Eq. 5.5. = 180 cos 1 (v estimated v ground-truth ) (5.5) where the unitary vectors v estimated and v ground-truth are the orientations of the estimated ber and closest ground-truth ber, respectively, and denotes the dot product. False-positive and False-negative Rate The number of incorrect bers (either false-positives or false-negatives) is computed as the signed dierence between the number of estimated and actual bers (given by the ground-truth) on a voxel-by-voxel basis. In this way, +1 indicates a single spurious ber, whereas -2 indicates two missing bers. 47 The total number of false-positives and false-negatives is tallied separately 4 and expressed as a percentage of the actual number of bers present. False rates >0% indi- cate false-positives (spurious bers) whereas false rates <0% represent false-negatives (missing bers). As an example, for any number of voxels containing 3 bers (as given by the ground-truth), a false rate of -66% indicates (on average) 2 of 3 bers are not detected. Fraction of Similar and Dissimilar Tractography We examine the eect of ber orientation errors on tractography and the estimation of white-matter pathways by computing the fraction of overlapping and non-overlapping track paths with respect to a ground-truth tractography derived from the in-vivo data. The overlapping and non-overlapping fractions were calculated from the union and set- dierence, respectively, of discrete voxelized equivalents of the tractography. Isolated spurious tracks unlikely to represent white-matter pathways were eliminated by applying a threshold to the number of track points per voxel. 5.3 Application: Comparison of six DW-MRI Analysis Methods Here we dene parameters used for the data synthesis and analysis, provide a brief qualitative comparison of results obtained from processing the in-vivo and corresponding synthetic data, and nally present a discussion of the quantitative ndings comparing the analysis methods. 5.3.1 Data Synthesis Parameters Synthetic diusion-weighted human brain data sets were synthesized according to Eq. 5.4, in which the ber orientations and associated volume fractions were obtained as described in Section 5.2.1. Approximately 250 voxels classied as WM and possessing FA values within [0.85, 0.95] were selected to determine the diusivitiesf 1 ; 2 ; 3 g = f1:70; 0:17; 0:17g 10 3 mm 2 =s for the default single-ber tensor D x . The free diusion 4 Occurrence of spurious bers in one voxel do not \make up" for missing bers of a dierent voxel. 48 Figure 5.1: Diusion gradient directions as the vertices of a tessellated sphere used in the data synthesis and analysis steps. In each case (a)-(f) the number of vertices on the sphere is twice the number of diusion-weighting directions, N due to the symmetry of diusion (a measurement in direction g = [g x ;g y ;g z ] > is equivalent to a measurement in directiong). parameterD 0 was computed for each tissue classication (WM, GM and CSF) separately by averaging diusivities over similarly classied voxels, givingfD WM 0 ;D GM 0 ;D CSF 0 g = f0:68; 0:96; 2:25g 10 3 mm 2 =s. To investigate the eect of angular sampling on ber estimation and tractography, we evaluated six sets of sampling patterns consisting of dierent number of gradient directions, namely N = 20, 30, 40, 60, 90 and 120 directions, as shown in Figure 5.1. The spatial distribution of gradient directions were based on minimization of electrostatic energy of antipodal pairs of charged particles on the sphere, as computed by [38]. These gradient directions are publicly available in Camino 5 , an open-source diusion MRI toolkit. [37] 5 http://cmic.cs.ucl.ac.uk/camino 49 For each sampling pattern, four data sets were generated with b = 1000 s/mm 2 and independent noise realization according to 5.4, to yield a SNR of 18 in the magnitude data. One noisy non-DW image was simulated for every 10 DW images. 5.3.2 Data Analysis Parameters To ensure the 6 multi-ber analysis methods were evaluated fairly the parameters of each method were individually optimized for each of the diusion sampling patterns, with the following objective in mind: maximize the detection of true bers (to permit accurate assessment of the error associated with estimating multiple bers per voxel), with limited number of false-positives (which otherwise increase spurious and false tractography). We are able to adjust parameters towards achieving this objective because the ber ground- truth is known. More formally, the following constraints were imposed: 1. The false-positive rate for single-ber voxels should not exceed 10%, and be approx- imately equal among methods and constant over all sets of diusion-weighting directions, and 2. The false-positive rate for two-ber voxels should not exceed 5%, averaged over all two-ber crossing angles. Following is a brief description of the implementations and software used for the data analysis; the nal set of parameters used is listed in Table 5.1. Ball-and-stick model (BSM) analysis was conducted as per analysis of the in-vivo data (Section 5.2.1), with default parameters for estimating up to 3 bers per voxel. Constrained spherical deconvolution (CSD) performed using the software pack- age MRtrix 6 [129] which implements non-negativity constrained normal and super- resolved spherical deconvolution. The order of the spherical harmonics for estima- tion of the single-ber response function (`estimate_response' command) and spherical deconvolution (`csdeconv' command) steps was adjusted for best results. In our analysis, super-resolved CSD was used for all but one of the diusion gra- dient sampling patterns (N = 120), for which better results were obtained for our synthetic data using normal CSD. 6 MRtrix v0.2.11, http://www.brain.org.au/software 50 Analytical QBI (aQBI) was implemented in MATLAB according to the symmetric and real-valued spherical harmonics framework proposed by [43], which makes use of the Funk-Hecke theorem to analytically evaluate the Funk-Radon transform (FRT) of QBI, in conjunction with Laplace-Beltrami regularization to improve robustness to noise. Funk-Radon and cosine transform (FRACT) QBI uses an alternative to the FRT used in aQBI and thus was readily implemented by modifying our aQBI implementation: the eigenvalues used in computing the FRT were exchanged with values derived from the kernel function the FRACT. See [60] for details of the FRACT kernel and its associated spherical harmonics eigenvalues. Constant solid angle (CSA) QBI as a MATLAB implementation was provided by Imam Aganj and is fully described in [2]. We found it necessary to use a spherical harmonics order of 4 for all numbers of gradient directions to avoid noisy orientation distribution functions resulting in a high rate of false-positive bers. Generalized q-sampling imaging (GQI) was implemented in MATLAB according to [144]. The optimal mean diusion length ratio, , was experimentally deter- mined; we found that values greater than those ( = 1:0 - 1.3) recommended for b-values of 3000{4000 s/mm 2 achieved best results for our low b-value synthetic data (b = 1000 s/mm 2 ). Estimated ber orientations were obtained dierently depending on the analy- sis method. For BSM, ber orientations were output directly by the FSL program `bedpostx'. For CSD, the MRtrix command `find_SH_peaks' was used to determine the orientations of up to three largest peaks of the ber orientation distribution (FOD) function, which were in turn taken as ber orientations. For the remaining methods a similar peak nding task was accomplished with an in-house MATLAB code by recon- structing the ODF (for aQBI, FRACT or CSA) or spin distribution function (SDF) (for GQI) on a 4 th -order regular icosahedron tessellation of the sphere having 2562 radial projections. In these cases estimated ber orientations include an additional error of up to 2:7 , as bers are constrained to the orientations of these projections. 51 Analysis Number of diusion-weighting directions method 20 30 40 60 90 120 BSM [17] N = 3 N = 3 N = 3 N = 3 N = 3 N = 3 CSD [128] l max = 6* l max = 8* l max = 8* l max = 10* l max = 12* l max = 12 aQBI [43] l = 4 l = 6 l = 6 l = 8 l = 10 l = 12 = 0:006 = 0:006 = 0:006 = 0:006 = 0:006 = 0:006 FRACT [60] L = 4 L = 6 L = 6 L = 8 L = 10 L = 12 = 0:006 = 0:006 = 0:006 = 0:006 = 0:006 = 0:006 = 0:10 = 0:40 = 0:40 = 0:45 = 0:45 = 0:45 CSA [2] l = 4 l = 4 l = 4 l = 4 l = 4 l = 4 1 = 2 = 0:01 1 = 2 = 0:01 1 = 2 = 0:01 1 = 2 = 0:01 1 = 2 = 0:01 1 = 2 = 0:01 GQI [144] = 1:66 = 1:85 = 1:90 = 1:95 = 2:05 = 2:10 Table 5.1: Summary of parameters used for data analysis. Notation for parameters is summarized here: N = maximum number of bers; l max , l, L = maximum degree of the spherical harmonic series; = weighting of Laplace-Beltrami regularization term; = parameter of the FRACT method; 1 , 2 = thresholding parameters of the CSA method; = diusion sampling length ratio for GQI method. The `*' indicates super-resolved non-negativity CSD. 52 Analysis Number of diusion-weighting directions method 20 30 40 60 90 120 CSD 0.30 0.35 0.30 0.35 0.45 0.40 aQBI 0.00 0.00 0.00 0.00 0.00 0.00 FRACT 0.30 0.70 0.70 0.70 0.70 0.70 CSA 2.10 1.90 1.75 1.55 1.45 1.33 GQI 0.92 0.92 0.92 0.92 0.92 0.92 Table 5.2: Threshold values used in detecting local maxima, in terms of mean value of the applicable spherical function (SDF or ODF). All peak detection approaches required the use of a threshold (minimum value of local maxima) to eliminate minor peaks resulting from noise, which would otherwise lead to false-positive bers; Table 5.2 summarizes the threshold values used. 5.3.3 Comparison of In-vivo and Synthetic Data A qualitative comparison of GFA, color FA, and tractography obtained from analysis of the in-vivo data and an example set of synthetic noisy data (SNR = 18) is shown in Figure 5.2, which also illustrates the `whole-brain' extent of our synthetic data. In Figure 5.2 the GFA images are almost identical since GFA of the in-vivo data was approximated in computingf 0 for the synthetic data model as described in Section 5.2.1. The color FA image from the synthetic data is marginally brighter (higher FA) compared to that of the in-vivo data, particularly in regions bordering ventricles (see circles). Minor dierences can be seen in tractography of the left cingulum as a result of small dierences in estimated ber orientations. Overall, the close correspondence of synthetic data results to those of the in-vivo data allows us to gain condence in the simulation framework. 5.3.4 Results: Fiber Orientation Estimation In this section the ber estimation ability of each method is presented, for voxels con- taining one, two or three bers each. In Figs. 5.3 to 5.6 each data point is the mean value of the quantity indicated, wherein the average is taken over all (four) independent noise realizations and all similar voxels as categorized by the ground-truth. Only voxels satisfying the following criteria in the ground-truth were included in results: 53 Figure 5.2: Qualitative comparison of results obtained from processing the in-vivo data (row 1) and synthetic noisy data (SNR = 18) (row 2); both data sets consisted of 150 DW volumes and 10 non-DW volumes. The three columns are: GFA (left), color FA (middle) and tractography of the left cingulum (right). Corresponding images have identical intensity scales. Color in the images corresponds to orientation of the principal eigenvector in the color FA image, and local orientation of the ber in the tractography image. 1. voxel must be classied as WM, 2. individual bers must have a volume fraction of at least 15%, and 3. the free-diusion compartment size is no more than 30%. The purpose of the criteria is to lter-out voxels in which estimated ber orientations are governed by noise, leading to random magnitude of ber orientation error. False- positive/-negative rates are given in terms of the percentage of individual bers in the ground-truth satisfying the criteria. For clarity, error bars are not shown in gures here, but can be seen in an expanded version of the results in Appendix A. One ber/voxel Single ber/voxel orientation error and false-positive rate is presented in Figure 5.3. 54 Figure 5.3: Single-ber estimation versus number of diusion-weighting gradient directions, N. Multi-ber analysis method indicated by color according to the legend. Filled markers (circle and triangle) are mean values of the quantity indicated. (a) Individual ber orientation error. (b) Corresponding false-positive rate. Figure 5.3(a) shows that even in the simplest case of single ber estimation, an increase in the number of DW directions helps reduce ber orientation error, although beyond N = 60 there is minimal improvement. The BSM method had the least ber orientation error, as would be expected because its model most closely matches that used for data synthesis, Eq. 5.1. The CSD method also performed very well, and this can be attributed to accurate estimation of the single-ber response function over a large number of voxels having synthetic data modeled by identical bers with diusion tensor given by Eq. 5.2. Figure 5.3(b) shows approximately constant false-positive rate for all methods over the range of DW directions, and values are <10% (in accordance with our constraints for parameter adjustment). There are no false-negatives because at least one ber orientation must be present. Two bers/voxel For voxels with two-bers, ber estimation results are grouped according to the crossing angle (0 90 ) of the two bers in the ground truth; 9 bins of width 10 are used. Fiber orientation error and false-negative rate is shown in Figure 5.4 (grouped by number of DW directions) and Figure 5.5 (grouped by analysis method). In Figures 5.4 and 5.5 the false rate approaches approximately -50% for crossing angles approaching zero, indicating that only half the bers present were detected. This 55 Figure 5.4: Two ber estimation versus ber crossing angle; multi-ber analysis method indi- cated by color according to the legend. Filled markers (circle and triangle) are mean values of the quantity indicated. (a)-(f) Individual ber orientation error, with increasing N. (g)-(l) Corresponding false-negative rate, with increasing N. 56 Figure 5.5: Two ber estimation versus ber crossing angle; number of diusion-weighting gradient directions indicated by color according to the legend. Filled markers (circle and triangle) are mean values of the quantity indicated. (a)-(f) Individual ber orientation error, for each analysis method. (g)-(l) Corresponding false-negative rate, for each analysis method. 57 is as expected, as two increasingly parallel bers eventually become indistinguishable from a single ber. Figure 5.4 directly compares the diusion analysis methods. As N increases there is greater dierentiation between the methods in terms of both orientation error and false rate. As for voxels with single bers, BSM and CSD perform similarly to each other, and almost always yield the least orientation error and false rate closest to zero. Once again, this is expected as both methods are positively biased due to the simulation model. For N = 20 and 30, CSD outperforms BSM in terms of false rate for crossing angles >45 . This was found (results not shown) to be attributed to the higher-order spher- ical harmonics of super-resolved CSD, that leads to improved angular resolution and consequently ber orientation estimation, especially for low N, as was also found by [128]. We found aQBI typically had the largest orientation errors and greatest false-negative rate for allN. From Figure 5.4(g)-(l), we see that the false rate of aQBI is -50% (for all N) until the 70 80 bin, indicating that two bers cannot be resolved below crossing angle of approximately 75 . This is consistent with the results of [43], in which a crossing angle of 74 or 75 (depending on order of the spherical harmonics, L) was identied as the critical angle below which only a single maxima on the ODF starts to be detected instead of two. Several ODF sharpening techniques have been developed [44, 145] to enhance maxima of the ber orientations on the ODF, although they are not explored in this work. Our results show that FRACT substantially improves upon the FRT used in aQBI, which in part is due to FRACTs suppression of the isotropic component of the ODF, allowing anisotropic components to be more easily detected. Compared to the FRT of aQBI, FRACT generally decreased the orientation error and lowered the critical angle for detection of two bers to 55 . Furthermore, FRACT signicantly increased the detection rate of two bers; for example the false-negative rate dropped from -45% for aQBI in the 70 80 crossing angle bin (N = 60) to -20% for FRACT. Overall, we found the most dening characteristic between the methods to be the false-negative rate. Even when there is little dierence in orientation error separating the methods, there can be a substantial dierence in false-negative rate. For example, at N = 60 and considering the 70 80 bin, the dierence in orientation error between 58 Figure 5.6: Three ber estimation versus number of diusion-weighting gradient directions, N. Multi-ber analysis method indicated by color according to the legend. Filled markers (circle and triangle) are mean values of the quantity indicated. (a) Individual ber orientation error. (b) Corresponding false-negative rate. the best and worst method is approximately 3 , whereas the corresponding dierence in false-negative rate is 45%. Figure 5.5 groups the results by analysis method and in each case a clear trend of reduced orientation error and improved ber detection is seen with increasing N. Also revealed is the unique manner in which orientation error and false rate changes with N; for some methods a limited range of ber crossing angles benets from increased N, whereas other methods show improvement over a wide range of crossing angles. It is clear that some methods benet from increased N more than others. Three bers/voxel In the case of three bers per voxel, wherein each of the bers are oriented independently of each other and eectively randomly, a single crossing angle is insucient to describe the relative ber orientations. However, as the true ber orientations are known from the ground truth, we are still able to calculate the orientation error of individual estimated bers and detect false-negatives, with results shown in Fig 5.6. Figure 5.6(a) shows thatN is a signicant factor in reducing orientation error in the case of three bers per voxel. Unlike the result of Figure 5.3(a), there is more notable reduction in orientation error for N > 60 in this case. For N 60, there is little dierence ( 1 ) in orientation error between the methods. Figure 5.6(b) reveals substantial dierences in detection rate of bers between the methods, and we expect this to have signicant impact on tractography through regions 59 of crossing bers. With the exception of GQI, all methods indicate improved detection rate of bers with increasing N, although some methods show more substantial gains than others. There are no false-positives in three ber voxels because at most 3 ber orientations were estimated by each analysis method. 5.3.5 Results: Tractography Whole-brain streamline tractography was performed using an in-house `C' program. The ber tracking algorithm used is similar to the Euler integration scheme by [11], but modied to accommodate multiple ber orientations per voxel. Track propagation used a xed step size (0.2 mm) and at each step the propagation direction was calculated by tri-linear interpolation of ber orientations from the eight voxels surrounding the current position. For each surrounding voxel, only the ber orientation with smallest de ection angle (with respect to the current propagation direction) was used for interpolation. Tractography seed points (5 seeds/voxel) were distributed in all voxels classied as WM in the ground-truth, with identical seeds used in ber tracking of all data sets. From the seed points, track propagation continued throughout a track mask consisting of all voxels classied as WM, and any other voxel having partial volume WM> 2% and CSF< 40% (based on the ground-truth tissue segmentation; see Section 5.2.1). Additional track termination criteria included: maximum change in track propagation direction between steps of 45 , and maximum track curvature of 90 over 10 mm. Prominent white-matter pathways Tractography of the in-vivo data was used to identify four prominent white-matter path- ways for comparison: the left cingulum (C), left inferior longitudinal fasciculus (LF), right inferior fronto-occipital fasciculus (FOF), and right corticospinal track (CST), as shown in Figure 5.7. The pathways were identied following procedures described by [29]. Before quantitative comparison we converted the continuous ber paths (Figure 5.7) to discrete voxelized regions (Figure 5.8) by transforming fractional track coordinates to integer voxel space. 60 Figure 5.7: Four white-matter pathways identied from the in-vivo data and assumed as the ground-truth: left cingulum (red), left inferior longitudinal fasciculus (purple), right inferior fronto-occipital fasciculus (blue), and right corticospinal track (green). A registered T 1 -weighted anatomical image is inserted for reference. Orientations are as indicated. Tractography from each processed simulated data set was ltered with the same regions-of-interest used in obtaining the ground-truth pathways, and then voxelized. Voxelized results from each of the independent noise trials were then averaged. For each of the four pathways we calculated the percent voxelized region found overlapping and non-overlapping its corresponding ground-truth. Example tractography and correspond- ing voxelized regions for each multi-ber analysis method are shown in Figure 5.9. Figure 5.9 shows that all methods are capable of recovering such prominent WM path- ways with only minor dierences between results. This is unsurprising as the chosen track paths have relatively high fractional anisotropy throughout (mean FA of the in-vivo data for the regions in Figure 5.8 isfFA C ;FA LF ;FA FOF ;FA CST g =f0:42; 0:46; 0:51; 0:55g) and as such a single ber is likely to be the most appropriate model to the data. For single-ber voxels, results from Figure 5.3 indicate only a small dierence in orientation error ( 1:7 at N = 60 between the best and worst methods), and no missing bers (false-negatives) for any method, and so comparable tractography is expected. The percent overlap and non-overlap of voxelized regions relative to the ground-truth, for the 4 dierent pathways investigated and averaged over all (four) independent noise realizations, is given in Tables 5.3 to 5.6; a graphical representation of the overlapping fraction is shown in Figure 5.10. 61 Figure 5.8: Voxelized regions of the track paths of Figure 5.7. The regions are: left cingulum (red), left inferior longitudinal fasciculus (purple), right inferior fronto-occipital fasciculus (blue), and right corticospinal track (green). A registered T 1 -weighted anatomical image is inserted for reference. Orientations are as indicated. Analysis Number of diusion-weighting directions method 20 30 40 60 90 120 BSM 63.5/14.2 72.5/13.7 69.0/11.5 73.0/16.8 75.7/13.9 75.9/14.5 CSD 60.8/11.8 64.7/14.5 67.9/14.3 74.4/22.8 74.6/19.5 71.1/13.2 aQBI 58.1/10.0 54.6/8.9 61.6/11.5 63.5/12.5 64.4/10.4 65.0/7.4 FRACT 58.2/13.6 59.8/12.3 59.9/10.0 70.6/17.4 68.7/13.6 67.6/11.4 CSA 51.8/11.0 56.6/9.6 64.4/12.1 69.2/11.4 68.6/11.1 72.8/12.2 GQI 36.2/22.4 56.3/10.0 63.2/15.1 62.8/18.9 66.8/10.9 66.0/8.1 Table 5.3: Average percentage of overlapping (v 1 ) and non-overlapping (v 2 ) voxelized tractog- raphy (writtenv 1 =v 2 ) for the left cingulum relative to the ground-truth; greatest overlap for each set of diusion-weighting directions indicated in bold. 62 Figure 5.9: Tractography (columns 1 and 2) and corresponding discrete voxelized regions (columns 3 and 4) obtained from processing oneN = 60 DW data set by each of the six analysis methods (each row). White-matter pathways are: left cingulum (red), left inferior longitudinal fasciculus (purple), right inferior fronto-occipital fasciculus (blue), and right corticospinal track (green). A registered T 1 -weighted anatomical image is inserted for reference. Orientation as indicated. 63 Analysis Number of diusion-weighting directions method 20 30 40 60 90 120 BSM 68.6/65.2 76.9/54.7 75.6/40.2 78.8/47.3 82.0/33.9 84.1/36.4 CSD 73.0/70.8 77.7/74.0 76.8/56.5 85.3/85.5 82.6/60.3 87.3/82.9 aQBI 60.5/57.4 63.5/62.9 65.0/59.7 64.8/56.0 64.5/63.3 63.9/26.1 FRACT 65.9/70.5 65.6/58.5 75.4/66.3 71.0/59.6 71.4/68.4 73.0/70.7 CSA 64.5/54.7 65.9/49.1 70.8/41.8 72.8/59.5 71.6/49.2 74.6/56.4 GQI 58.0/48.9 57.9/55.9 70.2/67.4 66.3/69.9 67.6/59.8 67.5/69.0 Table 5.4: Average percentage of overlapping (v 1 ) and non-overlapping (v 2 ) voxelized tractog- raphy (written v 1 =v 2 ) for the left inferior longitudinal fasciculus relative to the ground-truth; greatest overlap for each set of diusion-weighting directions indicated in bold. Analysis Number of diusion-weighting directions method 20 30 40 60 90 120 BSM 75.0/90.1 78.9/74.6 78.9/56.9 76.6/50.4 81.7/39.1 79.9/38.5 CSD 76.8/59.3 76.8/63.2 78.8/63.4 81.1/66.0 84.8/83.7 82.1/62.9 aQBI 69.3/86.6 71.1/99.0 73.5/92.5 73.8/87.8 72.9/87.7 73.8/86.8 FRACT 68.1/69.8 70.0/70.9 75.1/67.8 75.1/71.4 72.1/61.4 75.7/76.8 CSA 68.9/51.5 65.0/52.1 74.3/51.0 70.2/53.4 70.8/47.8 75.8/59.7 GQI 25.4/27.9 50.4/32.0 79.3/74.5 67.7/58.7 76.6/75.0 71.3/55.9 Table 5.5: Average percentage of overlapping (v 1 ) and non-overlapping (v 2 ) voxelized tractogra- phy (written v 1 =v 2 ) for the right inferior fronto-occipital fasciculus relative to the ground-truth; greatest overlap for each set of diusion-weighting directions indicated in bold. 64 Analysis Number of diusion-weighting directions method 20 30 40 60 90 120 BSM 62.3/46.2 64.4/44.9 53.8/29.6 62.4/30.2 60.9/24.6 47.6/18.1 CSD 67.8/49.6 71.1/55.3 72.6/58.0 73.8/53.2 79.3/60.4 79.2/59.3 aQBI 59.0/42.7 59.9/41.4 57.3/39.9 59.1/38.2 60.0/42.9 59.3/43.4 FRACT 66.3/53.2 64.8/51.0 66.7/57.0 68.5/49.1 73.7/54.0 71.2/49.4 CSA 68.5/56.0 67.2/44.7 65.0/42.8 66.5/41.6 71.1/48.3 70.8/46.9 GQI 49.4/51.5 56.8/40.8 64.2/46.7 60.7/41.0 66.8/54.1 62.1/42.5 Table 5.6: Average percentage of overlapping (v 1 ) and non-overlapping (v 2 ) voxelized trac- tography (written v 1 =v 2 ) for the right corticospinal track relative to the ground-truth; greatest overlap for each set of diusion-weighting directions indicated in bold. 65 Figure 5.10: Summary of correctly estimated tractography recovered by each of the analysis methods. 66 Tables 5.3 to 5.6 show a general increase in the fraction of recovered white-matter pathways with higher number of diusion-weighting directions, for all analysis methods. This trend is generally not monotonic withN, and is in part due to the uncertain way in which changes in ber orientation aect propagation of track paths over long distances. Overall we nd that reduced ber orientation error leads to a quantitative improvement in estimation of white-matter pathways, even in the case of simple prominent white- matter pathways as shown here. Consistent with the ndings of ber orientation error, generally ber orientations estimated by BSM or CSD yielded the largest overlapping fraction of each ber pathway with respect to the ground-truth. A trend in the percent non-overlapping white-matter is less clear. Increasing N would assume to reduce the volume of non-overlapping regions, and this can be observed in some cases ( 25%). However, in a similar number of instances the percent non- overlapping pathways increased, and the remaining cases ( 50%) had no clear increase or decrease in non-overlapping region. This variability may be attributed to our ground- truth pathways which were established from the in-vivo data. While we necessarily interpret our ground-truth as denitive for the purpose of comparison, it is itself an approximation of reality with unknown accuracy. If our ground-truth regions were sub- sets of their respective actual pathways, the non-overlapping volumes could represent additional valid contributions to the estimated white-matter pathways. Alternatively, the non-overlapping volume may encompass a neighboring anatomically dierent white- matter pathway which is undesirable. Because of the ambiguity it is not possible to conclusively interpret the non-overlapping volume. Also, as indicated in Tables 5.3 to 5.6 the percent non-overlapping region can be comparable to, even greater than, the percent overlapping region. This nding high- lights a caveat for tractography-based region analyses: if the non-overlapping fraction is signicant, it could lead to diminished or in ated measures of signicant dierence (e.g. t-scores) between groups. Three-ber white-matter crossing region We examined tractography through a complex crossing-ber region consisting of inter- secting mediolaterally directed transcallosal bers, vertically oriented corticospinal bers, 67 and anterior-posterior association bers comprising part of the superior longitudinal fas- ciculus, as revealed by [139] using DSI. The tractography ground-truth of these pathways was recovered from the in-vivo data, and is shown in Figure 5.11. Figure 5.11: Tractography from the in-vivo data through a complex ber crossing region; intersection of callosal bers (red), corticospinal bers (blue) and association bers (green). Orientation is oblique posterior view. Inset: magnied crossing region with reduced diameter bers for clarity. A registered T 1 -weighted anatomical image is inserted for reference. Despite our in-vivo data being of relatively low diusion-weighting, which inherently limits the accuracy and ability to resolve individual bers in complex crossing regions [34, 130, 131, 132], Figure 5.11 shows three unique ber branches intersecting and suc- cessfully traversing the crossing, though not as comprehensively as in [139]. Several individual corticospinal bers show incorrect termination in the white-matter region, which was found to be due to abrupt changes in estimated ber orientation leading to discontinuity in tracking. These shortcomings notwithstanding, the result includes sig- nicant continuous tracks in each of the ber branches to remain a suitable ground-truth of a complex white-matter crossing region. Figure 5.12 illustrates the varying success of each method to traverse the crossing region and recover each of the ber pathways. For BSM and CSD there are a large 68 number of tracks in each pathway and therefore evidently more voxels in the crossing region contain three bers allowing the individual tracks to continue uninterrupted. The FRACT, CSA and GQI methods recover tracks in all three pathways, though crossing areas consist mostly of pairs of tracks and the ber bundles themselves appear mostly adjacent to each other with few intersections. The aQBI result illustrates the outcome for which a large percentage of false-negatives (missing true ber orientations) lead to parallel ber bundles unable to cross each other. Overall, the number of bers propa- gating through the three-ber crossing region correlates strongly with the false-negative rates reported in Figure 5.6(b). 69 Figure 5.12: Tractography obtained from processing one simulated N = 60 DW direction data set by each of the six DW-MRI analysis methods; intersection of callosal bers (red), corticospinal bers (blue) and association bers (green). Orientation is oblique posterior view. Inset: magnied crossing region with reduced diameter bers for clarity. A registered T 1 -weighted anatomical image is inserted for reference. 70 Analysis Number of diusion-weighting directions method 20 30 40 60 90 120 BSM 53.5/63.3 53.9/66.1 59.4/70.4 67.3/78.3 68.3/82.0 66.4/60.0 CSD 60.7/72.7 68.0/82.7 64.0/76.3 71.6/87.4 67.3/82.6 75.3/80.0 aQBI 47.2/57.7 48.8/60.2 48.3/65.3 52.9/63.0 55.0/72.2 54.5/71.3 FRACT 54.7/71.1 60.3/76.0 53.7/77.3 62.4/79.6 63.2/79.9 64.9/85.3 CSA 51.8/56.1 54.4/73.3 58.5/78.1 65.3/84.6 66.9/85.9 65.5/86.6 GQI 32.9/29.7 41.2/49.6 47.3/63.4 62.3/79.2 61.0/79.4 62.0/78.1 Table 5.7: Average percentage of overlapping (v 1 ) and non-overlapping (v 2 ) voxelized tractogra- phy (writtenv 1 =v 2 ) for the mediolateral transcallosal bers relative to the ground-truth; greatest overlap for each set of diusion-weighting directions indicated in bold. Analysis Number of diusion-weighting directions method 20 30 40 60 90 120 BSM 80.6/128.3 80.2/113.8 68.0/68.7 63.9/44.6 64.0/41.7 56.6/27.8 CSD 82.0/122.1 85.0/118.6 83.0/122.0 82.5/107.8 86.3/117.3 87.6/105.0 aQBI 78.0/128.3 77.5/121.3 78.5/125.4 79.2/121.8 80.8/131.3 82.2/135.8 FRACT 83.4/139.8 84.0/133.2 85.4/146.0 84.9/129.3 88.0/139.2 86.6/127.8 CSA 81.3/134.2 85.7/109.9 82.0/107.5 83.5/113.4 86.7/137.7 86.0/129.5 GQI 56.6/119.1 75.1/124.1 80.8/125.6 80.3/114.2 85.7/143.5 83.0/125.3 Table 5.8: Average percentage of overlapping (v 1 ) and non-overlapping (v 2 ) voxelized tractog- raphy (written v 1 =v 2 ) for the vertically oriented corticospinal track relative to the ground-truth; greatest overlap for each set of diusion-weighting directions indicated in bold. Quantitative results of the percent overlap and non-overlap of voxelized equivalents of the callosal, corticospinal and association ber bundles relative to the ground-truth is presented in Table 5.7 to 5.9, respectively. Tables 5.7 to 5.9 yield similar conclusions as to the eect of the number of diusion- weighting directions on the overlapping and non-overlapping fraction of recovered path- ways as was observed and reported earlier for the prominent ber pathways, in Tables 5.3 to 5.6. It is unexpected that results in Tables 5.6 and 5.8, both regarding the corticospinal track, show BSM to have a decreasing trend in overlapping region asN increases. These pair of results is opposite of other methods' and we are unable to rationalize this result. 71 Analysis Number of diusion-weighting directions method 20 30 40 60 90 120 BSM 3.8/1.3 5.9/1.3 13.1/2.8 14.9/1.8 32.4/11.2 41.9/8.0 CSD 17.2/5.3 35.6/15.3 33.3/10.9 54.8/20.2 62.3/31.3 65.9/27.5 aQBI 0.0/0.0 0.0/0.0 0.0/0.0 0.0/0.0 0.0/0.0 0.0/0.0 FRACT 4.1/0.6 9.0/5.9 15.6/5.8 27.3/12.6 25.1/9.1 19.8/8.7 CSA 0.0/0.0 5.8/4.4 2.5/1.3 19.2/6.0 13.4/1.4 26.8/5.5 GQI 0.0/0.0 3.0/1.5 4.5/1.8 7.6/3.1 3.0/0.0 24.8/9.0 Table 5.9: Average percentage of overlapping (v 1 ) and non-overlapping (v 2 ) voxelized tractog- raphy (written v 1 =v 2 ) for the anterior-posterior association bers relative to the ground-truth; greatest overlap for each set of diusion-weighting directions indicated in bold. Overall the three-ber crossing emphasizes the importance of accurate estimation of all present ber orientations present in order to recover track paths through complex ber crossing regions. 72 5.4 Conclusion In this work we conducted a simulation study to evaluate six multiple-ber diusion MRI analysis methods at a diusion-weighting and SNR common to clinical settings. We focused on issues relevant to white-matter tractography and quantitatively compared the methods in terms of estimated ber orientation error, false-positive and false-negative bers, and percent recovery of select white-matter pathways. A range of diusion- weighting directions, N, were investigated to analyze the eect of angular sampling. To ensure ndings were as relevant as possible to practical application, we developed the simulation from an in-vivo data set. Of the methods studied, and within the scope of clinically acquired diusion data (b 1000 s/mm 2 and SNR 18), we found the two-compartment ball-and-stick model (BSM) and non-negativity constrained super-resolved spherical deconvolution CSD methods yielded the most accurate ber orientation estimation, and greatest detec- tion rate of bers, for all N. Additionally we observed that even small improvements in ber estimation, in terms of both reduced orientation error and greater detection rate, engendered more complete recovery of white-matter pathways via tractography. While all methods were able to recover prominent white-matter pathways consisting of generally high FA and therefore minimal crossing of bers, a complex three-ber cross- ing region revealed signicant dierences. For the particular region studied we found ber orientations estimated by CSD led to a substantially greater number of tracks able to traverse the crossing region than the alternative approaches. This outcome can be attributed to the much higher detection rate of bers achieved by CSD, particularly for voxels containing three bers. We acknowledge the BSM and CSD methods are favorably biased in comparison to the remaining approaches due to the data synthesis model, which is based on rotating a single ber represented by a tensor. Also, the BSM and CSD methods used their own algorithms to estimate ber orientations, which are dierent to the discrete peak-nding technique we implemented for aQBI, FRACT, CSA and GQI. The extent of bias in the BSM and CSD results could be evaluated by using more generalized methods for data synthesis; for example, a taxonomy of multi-compartment models of white-matter 73 comprising intra- and extra-cellular water ([105]), or Monte-Carlo random-walk simula- tions using three-dimensional mesh substrates to model complex diusion environments ([61, 104]). Our results provide important information on the performance of ber estimation and subsequent tractography for a set of well-known diusion analysis methods and sampling patterns. We believe the results to be of particular interest to researchers undertaking tractography-based analyses and brain network/connectivity studies. 74 Chapter 6 Scanner Drift and Miscalibrated Diusion-Weighting in DW-MRI Diusion-weighted MRI most commonly utilizes single-shot EPI [88] for read-out of the MR signal, as was discussed in Section 3.1. It is well known that EPI entails long echo times (> 50 ms) and long data acquisition windows, and that these properties contribute to magnetic susceptibility-induced signal loss and geometric image distortions. These types of MR artifacts have been studied and reported on extensively in the literature. [70, 67, 5, 147, 69, 116, 113, 31, 87, 6] In this chapter we are concerned with other sources of undesirable signal variabil- ity, specically scanner drift and miscalibrated diusion-weighting, and their eect on quantitative diusion metrics. 6.1 Overview We investigate the eect of scanner drift and miscalibrated diusion-weighting on quan- titative diusion metrics. Scanner drift refers to undesirable temporal change in the MR signal intensity due to o-resonance, whereas miscalibrated diusion-weighting arises from dierences in scanner hardware (e.g. coil resistance, electrical current amplier characteristics, or calibration thereof) of the independent (x-, y-, z-direction) magnetic eld generating coils. We make use of a diusion phantom to observe scanner drift and DW miscalibration, which allows us to determine the severity of the artifacts in the absence of additional variability introduced by human subjects. We then develop a technique for correcting 75 miscalibrated DW in post processing, and apply the method to human data sets to assess the signicance of erroneous DW on FA, GFA and MD. 6.2 Material and Methods In this section we present the experimental apparatus, imaging protocols, data analysis methodology, and technique for correcting miscalibrated DW. 6.2.1 Diusion Phantom with Temperature Probe To observe MR signal drift due to o-resonance as exclusively as possible, i.e. without additional sources of signal variability due to human factors (e.g. motion artifacts, phys- iological artifacts of breathing, heartbeat, blood ow, etc.), and identify dierences in DW between gradient coils, a diusion phantom having isotropic diusivity is required. We constructed the phantom shown in Figure 6.1, for which the design is provided in Appendix B. To correct for erroneous diusion-weighting that may be present, an independent method of determining the `true' diusion coecient of the phantom 1 is required. In this work we acquire temperature measurements of the phantom and use an empirical relationship of the temperature-dependent self-diusion coecient of water ([65, 93, 118]) to provide independent estimates of the phantom's `true' diusion coecient. The temperature probe shown in Figure 6.2 uses a precision resistance temperature detector (RTD) element (Figure 6.3) as the temperature sensor. RTDs are well known for highly accurate, repeatable and stable temperature measurements. 2 The RTD consists of a ceramic housing with an embedded platinum 3 wire, the elec- trical resistance of which varies with temperature. Highly accurate measurements of the RTDs resistance can be achieved with a four-wire (Kelvin) connection of the RTD to a digital multimeter as shown in Figure 6.4. This setup uses two pairs of identical leads 1 Throughout this chapter `temperature of' and `diusion coecient of' refer to distilled water inside the phantom, and not any part of the spherical insert or its ber wrappings. For brevity we often do not make this distinction and it should be taken as implied. 2 In fact, the platinum RTD is used as an interpolation standard of the international temperature scale from the triple point of hydrogen (259:3467 C) to the freezing point of silver (961:78 C). [109] 3 Platinum is an MRI compatible material. [117] 76 Figure 6.1: Diusion phantom with spherical insert and temperature probe. The phantom is lled with distilled water. (one for providing a test current and the other for simultaneous voltage measurement) eectively compensating for lead wire resistance, and changes in lead wire resistance with temperature, that would otherwise signicantly reduce the accuracy of measurements. An Agilent 34461A 6 1 2 -digit integrating multimeter with valid calibration was used for the four-wire measurement setup of Figure 6.4. The Agilent 34461A uses a high-order polynomial for accurate interpolation of temperature from RTD resistance, which itself is derived from electrical current and voltage measurements. 6.2.2 Data Acquisition All MR data sets were acquired on a GE 3T HDxt scanner (General Electric, Milwaukee, WI, USA) equipped with an 8-channel head coil. 77 Figure 6.2: The temperature probe we constructed from an RTD element. Figure 6.3: Platinum RTD element having a nominal resistance of 100 (at 0 C) and tem- perature coecient of 0:00385 1 C 1 (at 0 C). The phantom was placed in the same room as the scanner for four days prior to data acquisition for acclimatization, and then imaged with the following two scans: 1. A long duration ( 80 minute) continuous acquisition for the purpose of investi- gating scanner drift, consisting of diusion-weighting b = 1000 s/mm 2 generated by each gradient coil individually (i.e. cycling through the orthogonal directions: G =f(+1; 0; 0); (0; +1; 0); (0; 0; +1)g), with two non-DW images (b = 0 s/mm 2 ) for every 12 DW images acquired, distributed evenly throughout scan. Imaging parameters were: TE/TR = 83.4 ms/1500 ms, acquisition matrix = 6464, ASSET acceleration factor of 2, voxel size = 555 mm, 6 contiguous slices, and number of excitations (NEX) = 3. 2. A DW acquisition for the purpose of detecting variance in diusion- weighting between gradient coils consisting of low (b = 100 s/mm 2 ) and high (b = 1000 s/mm 2 ) diusion-weighting generated by each gradient coil in positive and negative directions individually (i.e. six directions: G = f(1; 0; 0); (0;1; 0); (0; 0;1)g) with identical imaging parameters as above 78 Test Current (I) R LEAD I V M Sense HI Sense LO V M R LEAD V R RTD (R) Lead Resistances Precision Digital Multimeter Constant Current Source V M = Voltage measured by meter V R = Voltage across RTD resistance R R LEAD R LEAD Source HI Source LO Sense Current (pA) Figure 6.4: Electrical circuit of four-wire (Kelvin) setup for precision resistance measurements. The RTDs resistance, R, is calculated by Ohm's Law: R =V M =I. except TE/TR = 78.4 ms/1500 ms and NEX = 5. We refer to this acquisition as the DW `calibration' scan. Temperature measurements of the phantom were acquired immediately before and after the DW calibration scan, allowing the `true' diusivity of the phantom during calibration to be estimated. For the human studies a HARDI data set was acquired followed by an anatomical scan. Imaging parameters were as follows: 1. HARDI acquisition consisting of 130 unique gradient directions with TE/TR = 83.4 ms/11000 ms, acquisition matrix = 9696, ASSET acceleration factor of 2, voxel size = 2.52.52.5 mm, 45 contiguous slices, diusion-weightingb = 1000 s/mm 2 , with 1 non-DW volume (b = 0 s/mm 2 ) acquired for every 10 DW volumes, NEX = 1. The acquisition required approximately 27 minutes. 2. T 1 -weighted anatomical spoiled gradient recalled echo (SPGR) with TE/TR = 2.856 ms/7 ms, having a voxel size of 1.01.01.0 mm. 79 6.2.3 Data Analysis Diusion Coecient from Temperature Measurements In this work, diusivity values obtained from temperature measurements are used as an independent reference of the phantom's `true' diusivity. The diusivity of the phantom D was calculated from RTD temperature measurements T using the following empirical relationship from [65] D =D 0 [(T=T S ) 1] (6.1) where D 0 = 1:635 10 8 2:242 10 11 m 2 s 1 , T S = (215:05 1:20) K and = 2:063 0:051. Diusion-Weighting Correction Factors Overall, diusion-weighting correction enables compensation of miscalibrated diusion- weighting at the data analysis stage, such that `true' diusivities can be calculated from DW-MRI data in which the diusion-weighting is miscalibrated. To calculate the correction factors, we follow a procedure similar to [98], with mod- ications so the correction applies to the diusion-weighting b-value when processing acquired DW data (allowing us to compare results with-and-without correction), rather than rescaling each component (G x , G y , G z ) of the diusion-weighting gradient vectors (G) on the MRI scanner. 4 We start with the scalar introduced in [98] that relates the requested gradient amplitude G R (of an individual (x-, y-, or z-direction) gradient coil) to the actual (unknown) gradient amplitude (of the same gradient coil) played out by the scanner G P G R j = j G P j (6.2) where we have included the subscript j to denote the gradient coil for which the applies. As mentioned in Section 6.2.2, the DW calibration scan consists of DW 4 In this chapterG are unit vectors, and soGx,Gy andGz are the relative diusion-weighting provided by each gradient coil. 80 in positive and negative directions of each independent gradient coil, and thus j 2 f+x;x; +y;y; +z;zg. It is shown in [98] that the factors j can be computed as j = s D T D P j (6.3) whereD T is the `true' diusion coecient (calculated from Eq. 6.1) and has no directional dependence because diusivity of water is isotropic, and D P j is the `apparent' diusion coecient calculated from DW-MRI data in which the uncalibrated gradient amplitude G P j was applied by the scanner. Following [98], the valueD P is computed from two data sets having dierent diusion- weighting (e.g. b 1 = 100 s/mm 2 and b 2 = 1000 s/mm 2 ), and computed as D P = ln (S 1 =S 2 ) (b 2 b 1 ) (6.4) where S 1 and S 2 are the acquired DW data obtained with diusion-weightings b 1 and b 2 , respectively. Next, since b/G 2 for both the traditional Stejskal and Tanner (Section 3.1.1) and twice-refocused (Section 3.1.2) PGSE sequences, it follows from Eq. 6.2 that b R j = 2 j b P j , orb P j =b R j = 2 j . It is important to note that these expressions apply to each component of diusion-weighting (i.e. relative amplitudes G x ,G y ,G z produced by the gradient coils). 81 We now derive the b-value correction factor that encompasses scaling each com- ponent of a gradient direction vector G = G x ^ i +G y ^ j +G z ^ k (kGk = 1) and requested diusion-weighting magnitudekb R k =B R as follows kb R k kb P k (6.5) = B R q (b P x ) 2 + b P y 2 + (b P z ) 2 (6.6) = B R r b R x 2 x 2 + b R y 2 y 2 + b R z 2 z 2 (6.7) = B R r GxB R 2 x 2 + GyB R 2 y 2 + GzB R 2 z 2 (6.8) = B R B R r Gx 2 x 2 + Gy 2 y 2 + Gz 2 z 2 (6.9) = 1 r Gx 2 x 2 + Gy 2 y 2 + Gz 2 z 2 (6.10) where x , y and z are selected from the setf +x ; x ; +y ; y ; +z ; z g according to the positive or negative sense of each component G x , G y and G z . That is, x = ( x if G x < 0; +x if G x 0: (6.11) and similarly for y and z . The correction factor scales the requested diusion-weighting b-value so that dif- fusivity calculated from uncalibrated DW data yields `true' values (as determined by temperature measurements). As shown in Eq. 6.12, is equivalently a scaling of the erroneous diusivity D P . D T = ln (S 0 =S) b= = ln (S 0 =S) b =D P (6.12) 82 We emphasize that depends on the direction of the diusion-weighting and must therefore be calculated for each such direction. The correction factors can also be used to synthesize DW data with the gradient miscalibration `removed' as follows S =S 0 e ( ln(S=S 0 )) (6.13) where S corresponds to data acquired with erroneous DW, and S corresponds to data that theoretically would have been acquired had the original DW been correct. DW Signal Percent Change We illustrate signal drift and miscalibration of diusion-weighting in terms of percent signal change, as it allows the relative magnitude of signal variability to be immediately apparent. Displaying results as percent signal change is accomplished by dividing each data point by its mean value (computed over time). 6.3 Application: Signicance of Scanner Drift and Miscal- ibrated Diusion-Weighting in Diusion MRI In this section we present results of our phantom and human studies assessing the sig- nicance of scanner drift and miscalibrated diusion-weighting in diusion MRI. 6.3.1 Results: Diusion Phantom A T 2 -weighted image of the phantom is shown in Figure 6.5, with several ROIs indi- cating voxels from which data was pooled (from either non-DW or DW images) for averaging. Image contrast in homogeneous water regions is due to o-resonance caused by non-uniform magnetic eld strength, itself a consequence of dierences in the mag- netic susceptibility of water and nylon (used for the phantom's spherical insert). This contrast, while undesirable, does not interfere with the artifacts being studied here. Results follow in Figures 6.6 to 6.10. Figure 6.6(a) illustrates scanner drift, i.e. temporal changes in signal amplitude due to o-resonance, of the non-DW (S b=0 ) data. The signal change is 2% over the almost 80 minute scan duration, with the majority of change occurring in the rst 40 minutes. 83 Figure 6.5: T 2 -weighted image of diusion phantom showing voxels selected (red) for averaging data from both non-DW or DW images. Variation in image contrast in homogeneous regions is due to o-resonance caused by non-uniform magnetic eld strength. Figure 6.6(b) exemplies miscalibration of DW (S b=1000 ) between gradient coils, which manifests as vertical osets between plots. From the gure the maximum (vertical) dierence between plots is 10%. The scanner drift artifact can also be seen. Figure 6.6(c) illustrates the normalized (S b=1000 =S b=0 ) signal. As both non-DW and DW images are in uenced similarly by o-resonance, the drift artifact is eliminated. There is, however, a decreasing linear trend of 1% signal change over the last 60 minutes of acquisition, which is shown later to result from an increase in the diusivity of the phantom. Figure 6.7(a)-(c) compares scanner drift between the non-DW (S b=0 ) and DW (S b=1000 ) acquisitions, for three gradient directions G = f(+1; 0; 0); (0; +1; 0); (0; 0; +1)g. In each case the rst 15 minutes of acquisi- tion show identical percent signal change between S b=0 and S b=1000 . This is expected as o-resonance in uences all acquired data similarly. As noted earlier, during the last 60 minutes of acquisition the DW signal trends away from the non-DW signal. Based on temperature readings obtained from the RTD probe before and after the scan, we determined the temperature of the phantom increased by 0:2 C, leading to an 84 increase in diusivity and corresponding decrease in magnitude of the DW signal. The change in diusivity is examined later in this section. 85 Figure 6.6: Illustration of scanner drift and miscalibrated DW. (a) Drift in the non-DW data. (b) Drift in the DW data from gradient directions f(+1; 0; 0) = red curve; (0; +1; 0) = green curve; (0; 0; +1) = blue curveg. (c) Normalized DW signal; predominantly at with a minor decreasing trend over the last 60 minutes of acquisition due to increase in temperature of the phantom. In (b) and (c) the vertical osets between plots are a result of miscalibration in DW between the independent gradient coils. 86 Figure 6.7: Comparison of scanner drift between the non-DW (S b=0 ) and DW (S b=1000 ) acqui- sitions. (a) Drift of S b=0 and S b=1000 for G = (+1; 0; 0). (b) Drift of S b=0 and S b=1000 for G = (0; +1; 0). (c) Drift of S b=0 and S b=1000 for G = (0; 0; +1). In each case the rst 15 minutes of acquisition show identical percent signal change between DW and non-DW data. During the last 60 minutes the DW signal gradually trends away from the non-DW signal due to temperature and therefore diusivity change in the phantom. 87 From the DW calibration scan, theb-value correction factors were computed accord- ing to Eq. 6.10 and applied in Eq. 6.13 to synthesize a new DW data set with the diusion-weighting miscalibration removed. The resulting corrected data was processed identically as the acquired data, with results corresponding to those of Figure 6.6 shown in Figure 6.8. The vertical oset between plots in Figures 6.6(b)-(c) are not present in Figures 6.8(b)-(c), illustrating successful correction of diusion-weighting miscalibration. The vertical scales of the gures remain the same to aid comparison. 88 Figure 6.8: Illustration of correction of diusion-weighting miscalibration; plots (a)-(c) are comparable to those of Figure 6.6. (a) Drift in the non-DW data. (b) Drift in the DW data from gradient directionsf(+1; 0; 0) = red curve; (0; +1; 0) = green curve; (0; 0; +1) = blue curveg. (c) Normalized DW signal. The vertical osets present in Figure 6.6(b) and (c) have been eliminated. 89 Up to this point we have presented a scenario of data acquisition in which DW images are interspersed with non-DW images. As scanner drift aects both sets of images similarly, the confounding factor may be `divided out' before processing the data according to DTI, HARDI methods, or other techniques. This acquisition scenario, however, is not typical of DW-MRI scans in clinical settings in which often only a single non-DW image is acquired at the beginning of the scan, followed by a series of DW images. Such an acquisition is illustrated in Figure 6.9, which compared to Figure 6.8 exemplies scanner drift artifacts carrying over to the normalized DW signal. 90 Figure 6.9: Typical DW-MRI scans in clinical settings have only a single non-DW (S b=0 ) volume acquired at the beginning of the scan. In such cases, scanner drift cannot be 'divided out' and it carries over to the normalized DW signal where it has the potential to impact diusion metrics. (a) Single non-DW acquisition at the beginning of the scan. (b) DW acquisitions corrected for gradient miscalibration; identical to Figure 6.8(b) except axis limits. (c) Since S b=0 is constant (from (a)), scanner drift artifacts are not eliminated from the normalized DW signal (which is unlike Figure 6.8(c)). 91 To illustrate the eect scanner drift can have on calculated parameters, we examine the phantom's predicted diusivity change over the course of the almost 80 minute DW scan. Using the data of Figures 6.8(c) and 6.9(c), which are both corrected for diusion- weighting miscalibration, the diusivity of the phantom was calculated (see Eq. 3.6), with results shown in Figure 6.10. The blue data points represent the estimated diusivity when non-DW images are acquired throughout the scan, eectively tracking scanner drift, whereas the black data points illustrate diusivity whenS b=0 is assumed to remain constant. In the former case, diusivity is seen to increase from 1:9165 10 3 mm 2 =s to 1:9258 10 3 mm 2 =s, i.e. a change of +0:009 10 3 mm 2 =s, which suggests that the phantom temperature increased. 5 Figure 6.10: Change in diusion coecient of the phantom during the experiment. Blue curve: diusivity calculated with a continuous series of non-DW images to `divide out' scanner drift. Black curve: Erroneous result of diusivity change when only a single non-DW image is used; scanner drift confounds interpretation of the result. We conrmed the temperature change from RTD probe measurements before and after the scan. Temperature was found to have increased from 18:0 C to 18:2 C, which according to Eq. 6.1 predicts a diusivity increase from 1:9177 10 3 mm 2 =s to 1:9281 10 3 mm 2 =s, i.e. a change of +0:010 10 3 mm 2 =s, which is in strong agreement with the value predicted from the DW-MRI data. The assumption of constant S 0 , which is typical of clinical DW-MRI acquisitions, leads to the prediction of decreasing-then-increasing diusivity (see the black data points 5 Increase in temperature is plausible given that the phantom is subject to constant RF energy during data acquisition. 92 ROI Subject Number 1 2 3 4 5 6 7 1 0.004 0.004 0.004 0.005 0.004 0.004 0.007 L{R (0.002) (0.002) (0.003) (0.003) (0.004) (0.003) (0.004) 2 0.010 0.012 0.010 0.011 0.010 0.010 0.010 P{A (0.004) (0.004) (0.004) (0.004) (0.004) (0.004) (0.004) 3 0.015 0.017 0.014 0.015 0.016 0.016 0.016 I{S (0.005) (0.003) (0.004) (0.004) (0.004) (0.004) (0.004) 4 0.008 0.009 0.010 0.007 0.007 0.010 0.008 C (0.005) (0.006) (0.006) (0.005) (0.005) (0.006) (0.005) Table 6.1: Average absolute dierence in FA resulting from miscalibrated diusion-weighting. Results are shown for four ROIs and seven subjects. Standard deviation in parentheses. Instances where dierences are 0:01 are highlighted in bold. in Figure 6.10). Considering the last 60 minutes of acquisition, there is little overall change in diusivity. This contradicts what we expect given the measured temperature increase ( 0:2 C) over the same interval, and so we conclude it is an erroneous nding. Altogether the analysis serves as an example of potential confounds in results if scanner drift is not taken into account. 6.3.2 Results: Human Data Sets The phantom study revealed that DW miscalibration between gradient directions led to larger unwanted percent signal change than scanner drift, and so here we examine the signicance of miscalibration on quantitative diusion metrics. We calculate FA, GFA and MD from in-vivo diusion data of seven human sub- jects with and without the diusion-weighting correction factor (Eq. 6.10) applied, and report absolute dierences in these metrics averaged over four ROIs. The ROIs were selected in WM as follows: three in regions dominated (individually) by Left-Right (L{R), Posterior-Anterior (P{A), and Inferior-Superior (I{S) bers (determined by direc- tional FA images), and one in a region of crossing (C) ber bundles (determined by FA image). Results are grouped by metric and presented in Tables 6.1 to 6.3. 93 ROI Subject Number 1 2 3 4 5 6 7 1 0.010 0.009 0.009 0.009 0.009 0.009 0.008 L{R (0.001) (0.001) (0.001) (0.001) (0.002) (0.001) (0.002) 2 0.009 0.009 0.008 0.008 0.009 0.009 0.009 P{A (0.001) (0.002) (0.002) (0.002) (0.001) (0.001) (0.002) 3 0.002 0.002 0.002 0.002 0.002 0.002 0.002 I{S (0.001) (0.001) (0.001) (0.001) (0.001) (0.001) (0.001) 4 0.004 0.005 0.005 0.005 0.005 0.006 0.006 C (0.002) (0.002) (0.002) (0.002) (0.002) (0.002) (0.002) Table 6.2: Average absolute dierence in GFA resulting from miscalibrated diusion-weighting. Results are shown for four ROIs and seven subjects. Standard deviation in parentheses. Instances where dierences are 0:01 are highlighted in bold. 94 ROI Subject Number 1 2 3 4 5 6 7 1 0.040 0.045 0.045 0.045 0.044 0.044 0.047 L{R (0.007) (0.008) (0.009) (0.008) (0.009) (0.008) (0.009) 2 0.042 0.048 0.043 0.045 0.042 0.043 0.045 P{A (0.004) (0.009) (0.004) (0.003) (0.004) (0.003) (0.004) 3 0.034 0.036 0.035 0.037 0.035 0.035 0.037 I{S (0.003) (0.003) (0.002) (0.002) (0.002) (0.002) (0.003) 4 0.039 0.046 0.043 0.043 0.040 0.043 0.045 C (0.002) (0.003) (0.004) (0.003) (0.003) (0.003) (0.004) Table 6.3: Average absolute dierence in MD resulting from miscalibrated diusion-weighting. Results are shown for four ROIs and seven subjects. Standard deviation in parentheses. Instances where dierences are 0:01 are highlighted in bold. The quantitative dierences suggest DTI metrics FA and particularly MD are notably aected by miscalibration of diusion-weighting. In more than 50% of instances, FA values diered by 0:010, and in 100% of instances MD diered by 0:034. The consequence of these ndings are important because FA and MD are very frequently considered candidate `biomarkers' (i.e. indicators) of tissue microstructure, in which dierences in values are assumed to re ect real pathological changes (between regions of white matter, or between groups of subjects, or over time in longitudinal studies). Furthermore, the magnitude of the dierences we report are occasionally on-par with those cited in clinical studies (e.g. a DTI study on temporal lobe epilepsy [35]) and as a result, care should be taken when interpreting such ndings|not all dierences cited may be attributable to pathological change. 6.4 Conclusion This work investigated scanner drift and miscalibrated diusion-weighting in diusion MRI. Using a diusion phantom with temperature probe we determined that signal drift due to o-resonance contributed to 2% change in the DW signal over the course of an almost 80 minute experiment, with the majority of change occurring within the rst 30 minutes. We showed that not compensating for drift artifacts can lead to incorrect conclusions regarding changes in diusivity, particularly over long experiments. 95 Separately, we found that miscalibrated diusion-weighting contributed to 10% dierence in the diusion-attenuated signal when the DW was generated by the inde- pendent gradient coils. By use of a temperature probe to obtain an independent measure of the phantom's `true' diusivity, we were able to correct for the miscalibrated diusion- weighting in post-processing and assess the signicance miscalibration can have on diu- sion metrics. We found that miscalibration had a notable eect on FA, and particularly MD. Our results suggest that calibration of diusion-weighting is an important factor to consider when evaluating dierences in diusion metrics computed from data obtained over extended periods of time (e.g. longitudinal studies), and when pooling data from many dierent MRI scanners. In each of these cases notable dierences in diusion metrics may arise from scanner artifacts, and thus not attributed to pathological tissue changes. 96 Chapter 7 Conclusions and Future Work This dissertation has discussed imaging of the human brain, emphasizing diusion- weighted MRI techniques for non-invasive in-vivo reconstruction of the complex network of connections that is the brain's white-matter. We have primarily presented devel- opment of synthetic human brain like data sets comprising realistic congurations of white matter pathways, and shown how such data sets can be used for comparison of DW-MRI techniques. Such data sets are indispensable for evaluating the accuracy, preci- sion, reproducibility and noise sensitivity of DW-MRI analysis methods in a quantitative manner. A number of contributions have been made while completing this dissertation and are mentioned next, followed by recommendations for further study. 7.1 Summary of Contributions 7.1.1 Diusion Phantom Development Designed and constructed a diusion MRI phantom capable of housing planar (2D) and spherical (3D) diusion phantoms, including RTD-based temperature probe for high accuracy temperature monitoring and diusion-weighting calibration purposes. See Appendix B for engineering drawings and photos. 97 7.1.2 Software Development DW-MRI Data Analysis Implemented estimation of diusion tensor by LLS, NLLS and CNLS methods, according to [75]; all run using parallel processing in MATLAB. Implemented HARDI methods: DSI, QBI, aQBI, FRACT, GQI. Quantitative evaluation metrics: ber orientation estimation, false-positive/- negative rates. DW-MRI Data Synthesis Jointly developed (with Namgyun Lee) a comprehensive framework for multi-tensor diusion-weighted MR simulation, capable of arbitrary number of ber directions per voxel, any specied SNR, diusion-weighting (b-value(s)), and diusion sam- pling patterns; runs using parallel processing. Developed simulated version of the Fiber Cup phantom, to allow more extensive evaluation of DW-MRI analysis than what is possible with the data provided by the original developers. Developed sets of human brain like synthetic DW-MRI data based on an in-vivo acquisition, for application in quantitative study of diusion data analysis methods. Tractography Developed multi-ber streamline tractography program to accommodate arbitrary numbers of ber orientations per voxel, arbitrary voxel dimensions, arbitrary num- ber of seeds per voxel, masks for ber tracking region. 7.1.3 Online Resources Simulated DW-MRI Brain Data Sets Published rst online project consisting of whole-brain simulated DW-MRI brain data sets and MATLAB scripts for computing quantitative metrics. The data and evaluation 98 tools are freely available from the NITRC website as project Simulated DW-MRI Brain Data Sets for Quantitative Evaluation of Estimated Fiber Orientations 1 , as shown in Figure 7.1. Figure 7.1: My project `Simulated DW-MRI Brain Data Sets' on the NITRC website. As of June 1 st 2014 there had been 402 downloads (cumulative count|includes data, ground-truth les, quantitative metrics, etc.). 7.1.4 New Analytical Expressions A factor, , was introduced (see Section 6.2.3 for details) to correct for erroneous (i.e. miscalibrated) diusion-weighting during data post-processing = 1 r Gx 2 x 2 + Gy 2 y 2 + Gz 2 z 2 (7.1) 7.1.5 Publications Original research presented in this dissertation contributed to some of the following peer-reviewed publications: 1 http://www.nitrc.org/projects/sim_dwi_brain 99 Conference Abstracts B. Wilkins, and M. Singh. Diusion Histogram as a Marker of Fiber Crossing within a Voxel. 18 th Annual Meeting of the International Society for Magnetic Resonance in Medicine, p.1699, 2010. B. Wilkins, N. Lee, and M. Singh. Eect of Truncated Sampling on Estimated Fiber Directions in q-space Imaging. 19 th Annual Meeting of the International Society for Magnetic Resonance in Medicine, p.3927, 2011. B. Wilkins, N. Lee, and M. Singh. Development and Evaluation of a Simulated FiberCup Phantom. 20 th Annual Meeting of the International Society for Magnetic Resonance in Medicine, p.1938, 2012. N. Lee, B. Wilkins, and M. Singh. Compressed Sensing based Diusion Spectrum Imaging (CS-DSI) Tractgoraphy. 20 th Annual Meeting of the International Society for Magnetic Resonance in Medicine, p.1923, 2012. B. Wilkins, N. Lee, K. Nam, D. Hwang and M. Singh. Comparison of Novel ICA-based Approach to Existing Diusion MRI multi-ber Reconstruction Meth- ods. 20 th Annual Meeting of the International Society for Magnetic Resonance in Medicine, p.1934, 2012. B. Wilkins, N. Lee, M. Law, and N. Lepor e. Simulated DW-MRI Brain Data Sets for Quantitative Evaluation of Estimated Fiber Orientations. 22 nd Annual Meeting of the International Society for Magnetic Resonance in Medicine, p.2591, 2014. Conference Papers B. Wilkins, N. Lee, and M. Singh. Fiber Estimation Errors Incurred from Trun- cated Sampling in q-space Diusion Magnetic Resonance Imaging. Proceedings of the SPIE, p.83144A, 2012. N. Lee, B. Wilkins, and M. Singh. Accelerated Diusion Spectrum Imaging via Compressed Sensing for the Human Connectome Project. Proceedings of the SPIE, p.83144G, 2012. B. Wilkins, N. Lee, V. Rajagopalan, M. Law, and N. Lepor e. Eect of Data Acquisition and Analysis Method on Fiber Orientation and Tractography in Dif- fusion MRI. MICCAI, 16 th International Conference, 2013. 100 Journal Papers B. Wilkins, N. Lee, N. Gajawelli, M. Law, N. Lepor e. Fiber Estimation and Tractography in Diusion MRI: Development of Simulated Brain Images and Com- parison of Multi-ber Analysis Methods at Clinical b-values. NeuroImage, 2014. Book Chapters B. Wilkins, N. Lee, V. Rajagopalan, M. Law, and N. Lepor e. Eect of Data Acquisition and Analysis Method on Fiber Orientation Estimation in Diusion MRI. In Computational Diusion MRI and Brain Connectivity, Mathematics and Visualization, pages 13{24. Springer International Publishing, 2014. 7.2 Recommendations for Further Study A signicant advantage of DTI is that its metrics (e.g. FA, MD) are quantitative and can be fully described in terms of mean and variance, allowing repeatability and noise sensi- tivity to be characterized. This allows the measures to be meaningfully used as biomark- ers with dierences in metrics re ecting changes in tissue pathology over time and/or between subjects/groups in comparative studies, irrespective of MRI scanner hardware used to acquire the data. In future years, following improvements in MRI scanner hard- ware and image reconstruction techniques, HARDI acquisitions will undoubtedly replace DTI to circumvent limitations of the tensor model. When this happens new metrics quantifying clearly dened microstructural changes will be required. Progress towards such a goal will necessitate novel work in many respects including: 1. Furthering our understanding of the relationship between tissue microstructure and the diusion-weighted signal. 2. Development of more comprehensive simulation models and (software) tools for generating synthetic data sets. 3. Development of realistic physical phantoms (based on biological and/or synthetic materials). This may require methods for construction that have not been pre- viously considered, for example 3D-printing of synthetic and biological materials could lead to new level of sophistication and realism of diusion phantoms. 101 4. Development of tools for monitoring and calibration/validation of diusion- weighting of MR scanners. 7.3 Final Words Since the introduction of diusion tensor imaging two decades ago, DW-MRI has pro- foundly in uenced our study of the human brain. 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Magnetic Resonance in Medicine, 48(1):137{146, 2002. 75 114 Appendix A Fiber Estimation from Synthetic DW-MRI Data Sets This appendix expands on results of ber estimation of the synthetic brain DW-MRI data sets presented in Section 5.3.4, and uses the same denitions, acronyms, and color-coding of plots that were introduced therein. Figure A.1 contains results for single ber voxels, Figures A.2 to A.7 contain results for voxels having two bers, and Figure A.8 contains results for voxels with three bers. 115 Figure A.1: Single-ber estimation versus number of diusion-weighting directions, N. Error bars extend to the mean of samples above and below the lled marker (mean of all data points). (a)-(f) Individual ber orientation error for each method. (g)-(l) Corresponding false-positive (N) rate. 116 Figure A.2: Two-ber estimation versus ber crossing angle, for ball-and-stick model (BSM). Error bars extend to the mean of samples above and below the lled marker (mean of all data points). (a)-(f) Individual ber orientation error, with increasing N. (g)-(l) Corresponding false-positive (N) and false-negative (H) rates. 117 Figure A.3: Two-ber estimation versus ber crossing angle, for constrained spherical decon- volution (CSD). Error bars extend to the mean of samples above and below the lled marker (mean of all data points). (a)-(f) Individual ber orientation error, with increasing N. (g)-(l) Corresponding false-positive (N) and false-negative (H) rates. 118 Figure A.4: Two-ber estimation versus ber crossing angle, for analytical QBI (aQBI). Error bars extend to the mean of samples above and below the lled marker (mean of all data points). (a)-(f) Individual ber orientation error, with increasing N. (g)-(l) Corresponding false-positive (N) and false-negative (H) rates. 119 Figure A.5: Two-ber estimation versus ber crossing angle, for constant solid angle (CSA) QBI. Error bars extend to the mean of samples above and below the lled marker (mean of all data points). (a)-(f) Individual ber orientation error, with increasing N. (g)-(l) Corresponding false-positive (N) and false-negative (H) rates. 120 Figure A.6: Two-ber estimation versus ber crossing angle, for Funk-Radon and cosine trans- form (FRACT) QBI. Error bars extend to the mean of samples above and below the lled marker (mean of all data points). (a)-(f) Individual ber orientation error, with increasing N. (g)-(l) Corresponding false-positive (N) and false-negative (H) rates. 121 Figure A.7: Two-ber estimation versus ber crossing angle, for generalizedq-sampling imaging (GQI). Error bars extend to the mean of samples above and below the lled marker (mean of all data points). (a)-(f) Individual ber orientation error, with increasing N. (g)-(l) Corresponding false-positive (N) and false-negative (H) rates. 122 Figure A.8: Three-ber estimation versus number of diusion-weighting directions, N. Error bars extend to the mean of samples above and below the lled marker (mean of all data points). (a)-(f) Individual ber orientation error for each method. (g)-(l) Corresponding false-negative (H) rate. 123 Appendix B Diusion Phantom Design The diusion phantom housing is made from acrylic and consists of one tubular and two circular sections. When used for MRI experiments the assembled phantom is lled with distilled water and is leak proof. One end of the housing is permanently sealed, while the other end has a removable cap. A formed-in-place polyurethane gasket provides a seal between the acrylic tube and removable cap to prevent leaks. The removable cap has several threaded holes for installing nylon standos on to which dierent size and shape phantoms can be mounted. The conguration of standos can accommodate both planar (2D) and spherical (3D) phantoms. 124 Cast acrylic tube 6” outer diameter, wall thickness 1/8” R 3" 8 1/2" Section C-C’ Top C' C Figure B.1: Phantom housing: Acrylic tube. 125 B' B Section B-B’ Top View 1/4" Clearance t for 3/8”-16 threaded nylon rod Six places equally spaced Use size W gauge drill Ø 3/8" R 3 3/4" R 3 11/32" 7 1/2" 60° Figure B.2: Phantom housing: Top cap. 126 Clearance t for 3/8”-16 threaded nylon rod Six places equally spaced Use size W gauge drill For tapping 3/8”-16 Four places equally spaced Section A-A’ Top View 1/8” deep recess for gasket, 1/4” wide R 3 11/32" R 2 15/16" 3/8" 7 1/2" 1/8" 1/4" R 3 3/4" Ø 3/8" 60° A' A R 2 1/16" Ø 5/16" 90° For tapping 3/8”-16 Two places Ø 5/16" For tapping 1/2”-13 Ø 1/2" R 1 1/4” 45˚ Figure B.3: Phantom housing: Bottom cap. 127 60° 0.6" 90° R 2 1/16" R 1 5/8" R 2 1/2" R 1 5/8" R 2 1/2" R 2 1/16" 0.6" Four places equally spaced Ø 5/16" Four places equally spaced Ø 5/16" Figure B.4: Phantom insert: Planar 90 (top) and 60 (bottom) crossing. 128 R1.75 60° 2.50 60° 60° X Clearance fit for Ø3/8" nylon rod. Two places equally spaced. X Dimension X is nominally 19/32", but can be reduced to 1/2" if this simplifies machining and reduces cost. Figure B.5: Phantom insert: Spherical 60 crossing. 129 Figure B.6: Illustration of assembled phantom housing, with planar phantom shown on 4 standos; early design. 130 Figure B.7: Assembled diusion phantom with spherical insert and temperature probe, lled with water. 131 Figure B.8: Close-up of diusion sphere; synthetic bers wrapped around the sphere are visible. Figure B.9: Temperature probe passing through the removable bottom-cap into the vessel. 132 Figure B.10: Close-up of temperature probe inserted into diusion phantom, with temperature sensing RTD element clearly visible. 133
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Asset Metadata
Creator
Wilkins, Bryce
(author)
Core Title
Diffusion MRI of the human brain: signal modeling and quantitative analysis
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Biomedical Engineering
Publication Date
11/06/2014
Defense Date
07/02/2014
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
diffusion,diffusion analysis methods,human brain imaging,magnetic resonance imaging,OAI-PMH Harvest,white matter structure
Format
application/pdf
(imt)
Language
English
Contributor
Electronically uploaded by the author
(provenance)
Advisor
Marmarelis, Vasilis Z. (
committee chair
), Leahy, Richard M. (
committee member
), Lepore, Natasha (
committee member
)
Creator Email
bryce.wilkins@gmail.com,brycewil@usc.edu
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-c3-514770
Unique identifier
UC11298562
Identifier
etd-WilkinsBry-3067.pdf (filename),usctheses-c3-514770 (legacy record id)
Legacy Identifier
etd-WilkinsBry-3067.pdf
Dmrecord
514770
Document Type
Dissertation
Format
application/pdf (imt)
Rights
Wilkins, Bryce
Type
texts
Source
University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
(collection)
Access Conditions
The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...
Repository Name
University of Southern California Digital Library
Repository Location
USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA
Tags
diffusion
diffusion analysis methods
human brain imaging
magnetic resonance imaging
white matter structure