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F20101108_AABCPO jian_b_Page_043thm.jpg 75d783f90f3ad76d58548ba2a5cb65b1 88b65645ad9cbc0bb2683a1f8f045d858e7d587f 6941 F20101108_AABCOZ jian_b_Page_032thm.jpg 3bf2859fc09076f11138abf7d726f1c8 d4297af2954d1b0a8e8313c970a59b39a44ca085 MATHEMATICAL MODELING FOR MULTIFIBER RECONSTRUCTION FROM DIFFUSIONWEIGHTED MAGNETIC RESONANCE IMAGES By BING JIAN A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2009 S2009 Bing Jian To my parents and Geri ACKNOWLEDGMENTS This dissertation would not have been possible without the help and support front many people. First and foremost, I would like to express my ininense gratitude to my advisor, Dr. Baha Venturi, who introduced me to the field of medical image analysis and assisted me in each step throughout my PhD study. The research described in this dissertation was mostly horn front the lively meetings and intense discussions with Dr. Venturi. Without his patient guidance and strongest support, I could never have reached this important point in my life. I feel so fortunate to have him as a good friend and kind mentor. I would also like to thank my coninittee nienters, Dr. Sartaj Sahni, Dr. Anand Rangaredi .Il Dr. Arunava Banerjee, and Dr. Siriphong L li.111,.'.1l I'!;I1.1 for being very generous with their time and knowledge and ahrl: heing supportive during the completion of my PhD study. The five years I have spent in the University of Florida is definitely one of the most nienorable experiences in my life. I am very grateful to all the professors who have taught me many interesting and rewarding courses in computer science, niathentatics and statistics. Especially, I would like to thank Dr. Baha Venturi, Dr. Siriphong L li.111,.'.1l I'!;I1.1 Dr. .Jay Gopalakrishnan, Dr. Arunava Banerjee, Dr. Yunnmei C'I. in~ Dr. .Jeffrey Ho, Dr. Tint Davis, Dr. Xianfeng Gu, Dr. David Groisser, Dr. Sergei Shabanov, Dr. Brett Presnell, and Dr. Richard ?. i.us! Ia for all the time and energy they have invested into my education and research. Nevertheless, any errors in this dissertation are solely my own. Most of my doctoral research activities were carried out in the Center for Vision, Graphics and Medical Imaging (CVGMI) where I have learned a lot from Dr. Baha Venturi, Dr. Anand Rangaredi .Il Dr. Arunava B lia. j. Dr. .Jeffrey Ho and my fellow students during weekly seminars and daily interaction. I thank former and current CVGMI lah nienters: Zhizhou Wang, Eric Spellnman, Timothy McGraw, Hongyu Guo, .Jie Zhang, Fei Wang, Nicholas Lord, Santhosh K~odipaka, Adrian Peter, Angfelos Barnipoutis, O'Neil Smith, Ajit Rajwade, Ozlent Subakan, Ritwik K~umar, Ting C'I. >.~ Guang C'I. 19_ Meizhu Liu, and Yuchen Xie, for making our CVGMI lah an intellectually thriving and socially enjoil l,h environment. I consider it a big fortune of me to have the opportunity to know these wonderful persons. I also want to take this opportunity to thank all my friends in the CISE department, CVGMI SIAM Gators, ITSTCITF alumni, and many old classmates. The diffusion AIRI data used in this dissertation were provided by the McE~night Brain Institute at the University of Florida. Thanks go to Dr. Thomas Mareci, Dr. Paul Carney, and other nienters in their groups for the collaboration on this aspect. In particular, I would like to thank Dr. Evren Ojzerslan for helping me better understand the diffusion AIR imaging and interpret the key niathentatical model developed in this dissertation. Evren also kindly provided me some simulated data used in the experiments and the IDL visualization program to render the surfaces paranletrized by spherical harmonics. Across the campus of the University of Florida, there are many warnthearted staff nienters whose excellent work and services make the study at ITF very pleasant. They are the wonderful staff at the CISE department, the International Center, the libraries, and many unsung heroes who deserve my deep thanks. Outside the ITF, I want to thank Dr. Behzad Dariush at the Honda Research Institute ITSA for giving me the opportunity to work with him as an intern in suniner 2006 and for his continued help. I would also like to extend thanks to Dr. Arun K~rishnan, Dr. Sean Zhou, Dr. Gerardo Hermosillo, Dr. Yoshihisa Shinagawa, Dr. Jiannming Liang, and many other colleagues, for their kind help and support during my early career in the Computer Aided Diagnosis group at Siemens Medical Solutions ITSA. Finally, I would like to express my deepest gratitude to nly parents in Chan!~ I and my girlfriend Geri, for their unwavering love, tremendous support, and firm belief in me. This dissertation is dedicated to them. The research for this dissertation was financially supported by NIH grants R01 EB007082 and R01NS42075 to Dr. Baha Vemuri. I also have received travel grants from the CISE department, the Graduate Student Council at the University of Florida, and the IEEE Signal Processing Society. I gratefully acknowledge the permission granted by IEEE, Elsevier, and Springer for me to reuse materials from my prior publications in this dissertation. TABLE OF CONTENTS page ACK(NOWLED GMENTS LIST OF TABLES. LIST OF FIGURES LIST OF ABBREVIATIONS LIST OF SYMBOLS .... ABSTRACT CHAPTER 1 INTRODUCTION Motivation. Alain Contributions. Outline . 2 DIFFUSION AIR FUNDAMENTALS BACKGROUND REVIEW 2.1 The Basics of Diffusion Physics 2.2 The Basics of Nuclear Magnetic Resonance .......... 2.2.1 Dynamics of Nuclear Spins. 2.2.2 Magnetic Resonance, Relaxations, and Bloch Equations .. .. 2.3 Measuring Diffusion using NMR .. ........ .. 2.3.1 Spin Echlo anld Diffusion Effects in NMRl. ........ 2.3.2 The Fourier Transform Relationship .. ...... :3 DIFFUSION AIR MODELING AND DIFFUSION PROPAGATOR RECON STRITCTION CLASSICS AND THE STATE OF THE ART ...... :3.1 From the BlochTorrey Equation to the S' i1: I1Tanner Equation :3.2 Diffusion Tensor Imaging . :3.2.1 Tensorial Steil1: I1Tanner Equation :3.2.2 Estimation of the Diffusion Tensor. :3.2.3 Fiber Orientation and Anisotropy Measures Derived from the Diffu sion Tensor . Problems of the Diffusion Tensor Imaging. Angular Resolution Diffusion Imaging (HARDI). Modeling Diffusivity Profiles. :3.3.1.1 Spherical harmonics series. :3.3.1.2 Generalized diffusion tensor imaging :3.3.1.3 The limitation of ADC profile. :3.2.4 :3.3 High :3.3.1 :3.3.2 MultiConipartniental Models .. .. 45 :3.3.3 Deconvolution Approaches ..... .. . 46 :3.3.4 Modelindependent QSpace Imaging Approaches .. .. .. .. 46 :3.3.4.1 Diffusion spectrum imagfingf . . 47 :3.3.4.2 Qhall imaging . ...... .. .. 48 :3.3.4.3 Diffusion orientation transform ... .. . .. 50 :3.4 Conclusion ......... .. .. 51 4 METHODS .... ._ ... .. 54 4.1 Some Mathematics on ~P, ......... ... .. 56 4.1.1 Measure and Integration on ~P, ..... .. . 56 4.1.2 The Laplace Transform on ~P, .... .. .. . 57 4.1.3 Wishart and MatrixVariate Ganina Distributions .. .. .. .. 60 4.2 The Expected AIR Signal from Wishart Distributed Tensors .. .. .. 6:3 4.3 Methods for MultiFiber Reconstruction ... ... .. 68 4.3.1 The Mixture of Wisharts Model .... .. .. 68 4.3.2 A Unified Deconvolution Framework .... .. .. .. 70 4.4 Computational Issues ......... .... 74 4.4.1 Regularization and Stability . ..... .. 75 4.4.2 Nonnegativity and Sparsity Constraints .. .. .. .. 81 4.4.2.1 L1 nminintization methods .... .. .. 81 4.4.2.2 Nonnegative least squares (NNLS) ... .. . .. 82 5 EXPERIMENTAL RESITLTS ......... ... .. 86 5.1 Simulations ......... .. .. 86 5.2 Real Data Experiments ......... .. .. 92 6 DISCUSSION AND CONCLUSIONS . ..... .. .. 101 6.1 Suninary ......... .. .. 101 6.2 Open Problems ......... . .. .. 102 6.2.1 Nonparanletric Inverse Laplace Transform ... .. .. 102 6.2.2 Adaptive Sparse Dictionary Learning .... .. .. 10:3 6.2.3 Subvoxel Fiber Bundles Classification ... .. .. 104 REFERENCES .. .......... ........... 105 BIOGRAPHICAL SK(ETCH ..... ._. . .. 118 LIST OF TABLES Table page 31 A list of diffusion anisotropic measures that can be derived from the eigfenvalues of the diffusion tensor. (D) = trace(D)/3 = (At + X2 + 3) 3 is known as the mean di~f: .It;, which indicates the average diffusivity over all directions. .. 41 41 A list of previously published fiber reconstruction methods expressed in the pro posed unified deconvolution framework. See text for meaning of symbols. Re produced from with permission. @[2007] IEEE. .. .. .. 72 51 Imaging parameters used for the optic chiasm dataset. ... .. .. 94 52 Imaging parameters used for the rat brain dataset. ... .. .. .. 98 LIST OF FIGURES Figure page 21 The diffusion propagators for cases as the free diffusion are probability density functions of Gaussian distributions. The diffusion coefficient and diffusion ten sor are closely related to the random displacement of particles. .. .. .. 25 31 An illustration of subvoxel fiber configurations arising from the introvoxel ori entation heterogeneity (IVOH). ......... ... .. 42 32 From the diffusion data to orientation distribution function (ODF) via the spher ical random transform with qball imaging (QBI). ... ... .. 49 33 A schematic illustration of the diffusion orientation transform (DOT). .. .. 51 34 Various quantitative profiles derived from diffusion weighted signals simulated from 1fiber, 2fiber, and 3fiber geometries. ..... .. . 53 41 Plots of density functions of gamma distribution Y4,1 W.r.t the noninvariant measure and scaleinvariant measure. . ..... .. 63 42 The Wishart distributed tensors lead to a Rigauttype signal decay. Reproduced with permission from [75] @[2007] Elsevier. ..... .. . 65 43 Sphere tessellations using an icosahedron subdivision model with different itera tion numbers. ......... ... .. 69 44 The illconditioning problem in the spherical deconvolution approaches. Repro duced from [73] with permission. @2007 IEEE. .. .. 77 51 HARDI simulations of 1, 2, and 3fibers (b = 1500s/mm2) VISualized in Qball ODF surfaces. Reproduced with permission from [73] @2007 IEEE. .. .. .. 87 52 Results of w on 1fiber HARDI simulation data using different deconvolution methods. Reproduced from [73] with permission. @2007 IEEE. .. .. .. .. 89 53 Mean and standard deviation of (a) angular correlation coefficient and (b) er ror angles for the twofiber simulation. Reproduced from [73] with permission. @2007 IEEE. ........ . ... . 93 54 The statistics of angular correlation coefficients and error angles for the 2fiber simulation. Reproduced from [77] with permission. @2009 Springer. .. .. .. 93 55 Probability maps computed using (a) damped least squares with GCV; (b) Min Li with quadratic constraints (e = 1) initialized from (a); (c) damped least squares with fixed regularization parameter (a~ = 0.6);( d) nonnegative least squares from a rat optic chiasm data set overlaid on axially oriented GA [114] maps. Reproduced from [73] with permission. @2007 IEEE. .. .. .. 96 56 Probability surfaces computed from a rat optic chiasm image using (a) QBI ODF, (b) DOT, (c) MOW+Tikhonov regularization, and (d) MOW+NNLS. Reproduced from [77] with permission. @2009 Springer. .. .. .. 97 57 Probability maps of coronally oriented GA images of a control and an epileptic hippocampus. Reproduced with permission from [75] @2007 Elsevier. .. .. 100 LIST OF ABBREVIATIONS ADC CNS DLS DOT DSI DTI (DTMRI) DWI (DWMRI) EAP FA FOD FT (FFT) GA GCV GDTI HARDI IVOH MLE MRI MOG MOW NNLS NMR ODF apparent diffusion coefficient, or simply diffusivity, in units of m2 1 central nervous system damped least squares diffusion orientation transform diffusion spectrum imaging diffusion tensor (magnetic resonance) imaging diffusion weighted (magnetic resonance) imaging ensemble average propagator fractional anisotropy fiber orientation distribution (fast) Fourier transform generalized anisotropy generalized cross validation generalized diffusion tensor imaging high angular resolution diffusion imaging intravoxel orientational heterogeneity maximum likelihood estimate magnetic resonance imaging mixture of Gaussian distributions mixture of Wishart distributions nonnegative least squares nuclear magnetic resonance orientation distribution function PAS PGSE RBF RF SH (SHS, SHT) SNR SPD SVD TE TR QBI QSI persistent angular structure spin displacement probability density function, probability profile pulsed gradient spin echo radial basis functions radio frequency spherical harmonics (series, transform) signaltonoise ratio symmetric positive definite singular value decomposition echo time repetition time qball imaging qspace imaging LIST OF SYMBOLS bvalue, b = 22G( /) taeB Bmatrix, B m bggT duration of diffusion gradient Dirac delta function time between diffusion gradients (apparent) diffusion coefficient, or diffusivity diffusivity profile as a function of directions g (apparent) diffusion tensor a prolate tensor with eigfenvalues At > X2 = 3 and dominant eigfenvec tor v Fourier transform inverse Fourier transform gyromagnetic ratio a Wishart (matrixvariate Gamma) distribution over ?P, with shape parameter p > 0 and scale parameter C E 7P, diffusion gradient magnitude of diffusion gradient, G = G direction of diffusion gradient, g = G/G initial spin density diffusion propagator giving the probability of a particle traveling from position z to zt in the diffusion time t, cPDF ensemble average propagator, PDF probability profile, P(r) := P(r, t = r) the manifold of 3 x3 symmetric positive definite matrices b B 6() D D(g) D Dy G G P(z/x, t) P(r, t) P(r) q displacement reciprocal vector, or wave vector, q = 7~6G {qi}=1 a diffusion imaging scheme with NV different wave vectors sampled in qspace r relative displacement R3" 3dimensional Euclidean space S2 2dimensional unit sphere in R3" S(q) spin echo signal, diffusion MR signal So spin echo signal in the absence of an applied diffusion gradient S(q)/So diffusion MR signal attenuation r effective diffusion time r = a 6/3 v a unit vector. {v~K~ k Sphere tessellation containing K sample directions. YEm spherical harmonics of degree I and order m w w = {wkKIS a vector, of nonnegtive weights. i,.e., > 0. () ensemble average Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy MATHEMATICAL MODELING FOR MULTIFIBER RECONSTRUCTION FROM DIFFUSIONWEIGHTED MAGNETIC RESONANCE IMAGES By Bing Jian August 2009 C'I ny~: Baba C. Vemuri Major: Computer Engineering Diffusionweighted magnetic resonance imaging (DWMRI) is a noninvasive imaging technique that allows neural tissue architecture to be probed at a microscopic scale in vivo. By measuring quantitative data sensitive to the water molecular diffusion, DWMRI provides valuable information for neuronal connectivity inference and brain developmental studies. The broad aim of this dissertation is to develop mathematical models and computational tools for quantifying and extracting information contained in diffusion MR images. One of the fundamental problems in DWMRI analysis is the mathematical modeling of the MR signal attenuation in a voxel in the presence of multiple fiber bundles. In this dissertation, we present a novel mathematical model and ....c.. I .nvII~ing efficient algorithms for this problem. Our model uses a continuous probability distribution over the space of symmetric positive definite matrices and is general enough to model water molecular diffusion in a variety of situations involving complex tissue geometry including single and multiple fiber bundle occurrences. We show that the diffusion MR signals and the probability distributions for positive definite matrixvalued random variables are related by the Laplace transform defined on the space of symmetric positive definite (SPD) matrices. Another interesting observation is that when the mixing distribution is parameterized by Wishart distributions, the resulting close form of Laplace transform leads to a Rigauttype fractal expression. This Rigauttype function exhibits the expected .Iiup!lletic powerlaw behavior and has been phenomenologically used in the past to explain the AIR signal decay but never with a rigorous niathentatical justification until the development of the proposed model. Furthermore, both the traditional diffusion tensor model and the niultitensor model can he interpreted as special cases of this continuous mixture of tensors model. In tackling the challenging problem of niultifiber reconstruction front the diffusion AIR images, we further develop the mixture of Wisharts (\lOW) model, as a natural paranietrization of the desired tensor distribution function, to describe complex tissue structure involving multiple fiber populations. The niultifiber reconstruction using the proposed MOW model essentially leads to an inverse problem. Computational methods for solving this inverse problem are investigated under a unified deconvolution framework which also includes several existing modelbased approaches. Finally, the theoretical framework we have developed for modeling and reconstruction of diffusion weighted AIRI has been tested on simulated data and real rat brain data sets. The comparisons with several competing methods enipirically II 1 that the proposed model combined with a nonnegative least squares deconvolution method yields efficient and accurate solution for the niultifiber reconstruction problem in the presence of intra voxel orientational heterogeneity. CHAPTER 1 INTRODUCTION 1.1 Motivation The desire to discover the .Inr I~linin and functionality of the human body, especially the brain structure and central nervous system (CNS), has been one of the driving forces behind efforts to develop sophisticated medical imaging technologies. In the mid 1940s, a breakthrough achievement was made involving the discovery of nuclear magnetic resonance (NMR). This eventually led to the invention of magnetic resonance imaging (jl RI) whose imaging capability due to the spatial localization of NMR signal was first demonstrated and implemented in the 1970s. Since then, MR imaging has advanced tremendously and become an indispensable diagnostic tools in modern medicine that is used everyday for clinical and research applications. Built on technologies enabling the fast acquisition time and high image quality, MR continues to pIIli .a important role in the diagnosis and treatment of a number of diseases associated with the brain, heart, liver, and other organs in the human body. Furthermore, "the ability to use MR imaging to noninvasively probe the individual regions of the brain that control vision, sensation, motor function, memory, language, and other processes has made this an extremely valuable modality to those engaged in virtually any kind of brainrelated research.[128]" Among the many types of MRI modalities, diffusion Weighted MRI (Diffusion MRI, DWMRI or DWI) is a unique MRI technique that permits invivo measurement of the diffusion of water molecules within tissue samples being imaged [88]. By exploiting the sensitivity of the MR signal to the random motion of water molecules, diffusion MRI is able to quantify different water diffusion characteristics in tissue samples locally. Because these diffusion characteristics may be substantially altered by diseases, neurologic disor ders, and during neurodevelopment and aging, diffusion MRI is now recognized as a very 1 See [40] for a beautifully written history of NMR and MRI. important clinical tool for brainrelated diseases. For example, it has been successfully applied to the evaluation of early ischemic stages of the brain [108]. Furthermore, the directional dependence of water diffusion in fibrous tissues, like muscle [39] and white matter in the brain [107], provides an indirect but powerful means to probe the anisotropic microstructure of these tissues. Like the idea of reconstruction the map of higfhli in a geographical region from the direction and frequency information of vehicle traffics in that area, the motion of water molecules in neuro tissues can he potentially used to draw inference about neuronal connections between different regions of the central nervous system. Because of this powerful potential, diffusion AIRI has a distinct position in the field of neuroscience research. The broad aim of this dissertation is to develop mathematical models and computa tional tools for quantifying and extracting information contained in diffusion AIR images. As opposed to a straightforward qualitative study, the process of producing accurate quantitative results necessarily involves substantially more time and effort. However, "the benefits of quantification are that fundamental research into biological changes in diseases, and their response to potential treatments, can proceed in a more satisfactory way. Problems of hias, reproducibility and interpretation are substantially reduced. [141]" 1.2 Main Contributions The most significant original contributions of this dissertation are summarized below. First, we present a novel mathematical model for diffusionattenuated AIR signal which involves a continuous probability distribution over the space of symmetric positive definite matrices. This model is general enough to model water molecular diffusion in a variety of situations involving complex tissue geometry including single and multiple fiber bundle occurrences. We make the interesting observation that the diffusion AIR signals and probability distributions for positive definite matrixvalued random variables are related by Laplace transform defined on the space of symmetric positive definite (SPD) matrices. We further show that in the case of Wishart distributions or mixtures of Wishart distributions, a closed form expression for the Laplace transform exists and can he used to derive a Rigauttype .Iimptotic fractal law for the AIR signal decay behavior which has been observed in the past [84] hut never with a rigorous mathematical justification until now. This theoretical result depicts surprising consistency with the experimental observations. Moreover, Diffusion Tensor Imaging (DTI), currently the most widely used technique, can he viewed as a limiting case of the proposed model. The measurements from diffusion AIRI provide unique clues for extracting orientation information of brain white matter fibers and can he potentially used to infer the brain connectivity in vivo using tractography techniques. Nowced .va the widely used DTI tech nique fails to extract multiple fiber orientations in regions with complex microstructure. In order to overcome this limitation of DTI, a vali'. iv of reconstruction algorithms have been introduced in the recent past. One of the key ingredients in several modelbased approaches is deconvolution operation which is presented in a unified deconvolution framework in this work. Additionally, some important computational issues in solving the deconvolution problem that are not addressed adequately in previous studies are described in detail here. Further, we investigate several deconvolution schemes towards achieving stable, sparse, and accurate solution. Experimental results on both simulated and real data are presented. The empirical comparisons II 1 that nonnegative least squares method is the technique of choice for the multifiber reconstruction problem in the presence of intra voxel orientational heterogeneity. 1.3 Outline This dissertation is organized as follows: ChI Ilpter 2 provides the background knowledge for understanding models and methods used in diffusion magnetic resonance imaging. The first two sections of this chapter briefly review hasic concepts in diffusion physics and nuclear magnetic resonance (NMR), respectively. Then the key principles for measuring the diffusion phenomena using NAIR, together with the fundamental relationship between the measured NMR signal and the statistical properties of molecular diffusion, are presented. ('!s Ilter 3 reviews existing models and methods used in diffusion magnetic resonance imaging. The first section in this chapter presents the classical BlochTorrey equation and the Steil: I1Tanner equation which are the foundations of the diffusion AIR imaging modeling. The second section in this chapter introduces the now widely used diffusion tensor imaging (DTI) method. The third section in this chapter is dedicated to the more recent high angular resolution diffusion imaging (HARDI) methods with a focus on the multifiber reconstruction techniques. The first two sections in C'! Ilpter 4 present the technical details of the proposed continuous tensor distribution model with a brief introduction to Laplace transforms and Wishart distributions on the manifold of symmetric positive definite matrices. This part builds the mathematical foundation of the proposed research. The last two sections in C'!s Ilter 4 focus on the multiple fiber reconstruction from diffusion AIRI using the proposed mixture of Wisharts model. In this chapter the proposed reconstruction method and several existing approaches in literature put into a unified deconvolution framework. Additionally, some important computational issues in solving the deconvolution problem that are not addressed adequately in previous studies are described in detail here. Further, we investigate several deconvolution schemes towards achieving stable, sparse, and accurate solutions. Experimental results on both simulations and real data are presented in ('! .pter 5. The comparisons with other approaches demonstrate the merits of the proposed continuous tensor model together with the deconvolution reconstruction framework solved using nonnegative least squares method for the multifiber reconstruction problem in the presence of intra voxel orientational heterogeneity. Finally in ('! .pter 6 we summarize the main contributions of this dissertation and discuss a few open problems for further research. CHAPTER 2 DIFFUSION AIR FUNDAMENTALS BACKGROUND REVIEW This chapter provides the background knowledge for understanding models and methods used in diffusion magnetic resonance imaging. The first two sections of this chapter briefly review hasic concepts in diffusion physics and nuclear magnetic resonance (NMR), respectively. Then the key principles for measuring the diffusion phenomena using NAIR, together with the fundamental relationship between the measured NMR signal and the statistical properties of molecular diffusion, are presented. 2.1 The Basics of Diffusion Physics On a macroscopic level, diffusion results from the microscopic random thermal agitation of the particles in a medium. This phenomenon of random thermal agitation is ubiquitous and known as "Brownian motion", named after the botanist Robert Brown who first discovered and described it in 1827. In the context of this study, diffusion refers specifically to the perpetual random translational motion of water molecules in any part of a human or animal .Inr Irh ow.J Diffusion of water molecules in biological tissues, observed as a macroscopic manifestation of Brownian motion, highly depends on many factors including restrictions due to cell membranes, and a variety of microstructure properties. The ability to measure the diffusion process of water molecules in tissues and the understanding of how it is affected by these factors are extremely useful for studying the biological microstructure. It is natural to model the random motion of particles using a probabilistic framework. A starting point of studying diffusion processes is the socalled selfdiffusion propagator. Generally, the propagator P,(z/x, t) denotes the conditional probability density that a particle initially located at position coordinate z moves to a position in a volume element dz at zl after a time interval t. The classical treatment of the diffusion propagator is to describe it as the Green's function for the diffusion equation via the Fick's law vls~= V D(VP,(2/x, t)) (21) subject to the initial condition P,(z/x, 0)) = 6(xt 2) (22) and proper boundary conditions. In (21), D is the molecular selfdiffusion tensor, a physical quantity describing the diffusion processes in a more compact form. In the case of unrestricted selfdiffusion, also known as free diffusion, the boundary condition of (21) is simply P,(z/x, t) 0 as z/ c o which yields 1 (2t 2)TD1(2t 2) Psx/xt)=exp( )(23) zdet(D)(4xt) 4t combined with the initial condition (22). An interesting observation is that Ps in (23) depends only on the displacement r but not the initial position 2, which reflects the Markov property of Brownian motion statistics. Another important observation is that the selfdiffusion tensor D is related to the time dependence of covariance matrix (rrT) by D = (rr T). (24) On the basis of the central limit theorem, the above results may be derived either from a formal Markov's method as in [33] or from an elementary random walk model as presented in [55, pp. 325]. In an isotropic medium, the diffusion tensor D is proportional to an identity matrix. Accordingly, the diffusion propagator (23) simplifies to 1 xt 21 P(z/x, t) = exp( )(25) (4xDL) 4Dt where D is the selfdiffusion coefficient given by the diagonal of the diffusion tensor D. In this case, the corresponding diffusion equation via Fick's laws is =DV2ps(z1/x t) (26) and we obtain D = trace(D)/3 = ~(rTr) (27) which agrees with the well known Einstein relation [51]. Because it is unrealistic to measure the random motion of a single particle, we shall take an in,~ in!!. ave I, ii view to depict the macroscopic behavior of a large number of molecules on a statistical basis. Starting from the selfdiffusion propagator, one can define the total probability, W(zt, t) of finding a particle at position z/ at time t as W~m/ t) W~, 0)s~xx, )dz(28) where W(2,0O) is just the initial particle density p(z). Using the notation of the displace ment vector, we can modify (28) and define the socalled ensemble average l".y..;yl.>r (EAP), also known as the displacement ly .?..:?9i;, distribution function (PDF), by inte grating the diffusion propagator over all initial positions [29, 79] P~ ) P~x+rlz, t)p(z)dz (29) where p(z) is the initial particle density. As we shall see later, this ensemble averaged propagator which cumulates all the microscopic contributions distributed in a voxel at the macroscopic scale may be measured directly by NMR. This is the fundamental principle exploited in the diffusion MR and it will be the topic investigated later in this chapter. In cases such as the free diffusion where P,(z + rlz, t) is independent of the initial position 2, the ensemble average propagator is also a Gaussian 1 rTDlr P(r, t) =exp( ). (210) d et(D 4x) 41 The diffusion propagators for the free diffusion as well as the corresponding diffusion coefficient and the diffusion tensor are summarized in Fig. 21. Diffusion tensor =DP)1rD^ r SP(r, t) = exp( ) D = (rr T dtD(4) 4t isotropic isotropic Diffusion coefficient =D~V2P 1 7 T ,, P(r, t) = exp D = (r Tr) 4Dtt Figure 21. The diffusion propagators for cases as the free diffusion are probability density functions of Gaussian distributions. The diffusion coefficient and diffusion tensor are closely related to the random displacement of particles. It is worth noting that the diffusion tensor framework and the Gaussian propagators described above are derived from the free diffusion which only represents a very limited class of diffusion phenomena. Commonly observed are a vast range of phenomena, such as restriction, heterogeneity, anomalous diffusion, finite boundary permeability. Generally, the diffusion equations are imposed with nontrivial boundary conditions depending on the nature of confining geometries and other physical properties. Solutions to diffusion equations in a nr1I~1ii of simple geometries are available in [44]. It has been shown that if the diffusion boundaries are closed, unrelaxing, and completely impermeable, then in the long run (long time of diffusion), the propagator will assume the shape of the space accessible to particles [29, 150]. Even though the limiting conditions are rarely satisfied in biological applications, the diffusionstructure relations nevertheless provide useful clues for probing microstructures, which has fostered a vast promising research field. 2.2 The Basics of Nuclear Magnetic Resonance In this section, we present the basic physical principles of nuclear magnetic resonance (NMR). The main references we consult are [29, 92, 94, 133]. 2.2.1 Dynamics of Nuclear Spins The fundamental concept of NMR is the spin aq,:(;,lar momentum, an intrinsic prop erty possessed by certain atomic nuclei. In atomic physics, the spin angular momentum p of an atomic nucleus is quantized by the nuclear spin quantum number I according to the formula 2x1p ( (211) where & is the Planck's constant. The spins possessed by an atomic nucleus are combined by the spins of the protons and the neutrons inside the nucleus. Since the spins of protons and neutrons may interact in various configurations, e.g., parallel or antiparallel to each other, the value of nucleus spin quantum number I is chosen as the one in the lowest energy nuclear state, also called the iI,. ;,,../ state nuclear spin. Depending on the nuclear structure, the number I can only be an integer, halfinteger, or zero with the following rules[92]: (1) if the numbers of protons and neutrons are both even, the ground state nuclear spin I is zero; (2) if the numbers of protons and neutrons are both odd, the ground state nuclear spin I is a positive integer; (3) if the total number of protons and neutrons is odd, then I is given by a halfinteger. Note that the nucleus of 1H, which is most abundant in nature and human body, contains a single proton, hence its nucleus quantum spin number I is 1/2; while the nuclei 12C 160, and 56Fe all have a ground state spin I = 0. A complete listing of nuclear isotopes and their NMR properties may be found at http: //www web elements com. Another intrinsic property possessed by fundamental particles is the magnetic moment. The spin angular momentum p and the magnetic moment p are proportional to each other as given by I = 7p (212) where y is called the ~i;, tion 7,: It.: H: ratio, a characteristic value of a particular atomic nucleus. According to the classical mechanical model, when a nucleus with spin angular momentum is subject to an external magnetic field, the interaction between the magnetic moment p and the field Bo will generate a torque force L trying to align the two: L = dp/dt = p x Bo. The outcome of this torque force is the precessional motion of the nucleus, in which the magnetic moment vector p rotates around the direction of the external magnetic field Bo in addition to the spinning motion of the nucleus around its own axis. The equation describing the precession is dp/ldt = ydp/dt = yL = wox p (213) where wo = 7Bo is the socalled Larmor /, ~i;, ,: ;t of precession. Note that the minus sign in (213) implies that the rotation obeys the lefthand rule. According to the quantum mechanical model, the nuclear magnetic moment p can only have 2I + 1 orientations in a magnetic field Bo, corresponding to 2I + 1 energy levels: E = p Bo. (214) Here the minus sign in (214) indicates that the magnetic energy is lowest when p is parallel to the Bo field while the magnetic energy is highest when p is antiparallel (spindown) to the external field Bo. The difference between the two energy levels is proportional to the strength of the applied field Bo: AE = yBo. (215) With I = 1/2, the 1H nucleus can either align with (spinup) or against (spindown) the applied field, corresponding to low and high energy states respectively. When an ensemble of nuclei are immersed in an external magnetic field, the distribution of the allowed orientations is described by Boltzmann statistics. The resulting macroscopic magnetization vector M~ can be defined as the vector sum of all the microscopic magnetic moments M = pi (216) i= 1 where ps is the individual magnetic moment of the ith nuclear spin, and NV is the total number of spins. At equilibrium, two macroscopic effects can be observed: (1) the overall transverse component of M~, i.e., the component perpendicular to the applied magnetic field direction, is zero because of the random phases introduced by the processing magnetic moments, (2) more nuclei relax into the parallel orientation (lower energy state) than the antiparallel orientation (high energy state), resulting in a bulk magnetization vector M~. The direction of this bulk magnetization vector M~ is aligned with the applied magnetic field direction, and its strength depends on the proton 1 1 2.2.2 Magnetic Resonance, Relaxations, and Bloch Equations The principle of NMR is that the alignment of nuclear spins can be perturbed by applying a circularly polarized magnetic field B1, called a radio fr .;, e.'.;i excitation pulse, which is rotating about Bo at the Larmour frequency wo. A typical B1 field is perpendicular to Bo and its generation takes the following form: Bl(t) = Bl(t)ei("Ot+4) (217) where 4 is the initial phase angle and can be assumed to be zero as it has no significant effect on the excitation result. The excitation property of an RF pulse is specified by the shape and duration of the envelop function Bl(t).Two most widely used RF pulses are the r Litty(;J.;<, pulse and the since pulse. 1 The term RF pulse is so called because wo/2xr is normally in the frequency range of radio waves. The field usually lasts for a few microseconds or milliseconds. Also, the strength of the field B1 is much weaker than the static magnetic field Bo. On the microscopic level, the energy absorbed by nuclei during the RF excitation makes some nuclear spins jump from the lower energy state to the higher energy state and hence changes the overall magnetization vector M~. Although the microscopic behaviors of individual spin magnetic moments are quantized, the observed bulk magnetization as a macroscopic manifestation may be treated classically. The excitation induced by B1 tips the bulk magnetization vector M~ away from the direction of Bo by a spiral precession around Bo described by the following equation dM/ldt = wox M~ = ylM x Bo. (218) Note that the processing magnetization creates a periodically changing magnetic field in the transverse plane. This effect may be detected and measured by a closely placed wire coil according to Faradovl~i's law of induction. The resulting signal then can be exploited in NMR spectroscopy and magnetic resonance imaging. At the time when the RF pulse is turned off, M~ makes an angle a, called flip nl with the static field Bo. In general, the flip angle a~ depends on the strength By (t) and the duration r of the applied RF: ~o (219) = B1r (for a constant B1 amplitude). Because the amplitude of the alternating voltage induced in the receiver coil is propor tional to the transverse magnetization component, a 900 pulse is commonly used as an excitation pulse as it produces a maximum transverse magfnetization component equal to the equilibrium magfnetization. Once the RF pulse is removed, the nuclear spins will tend to restore the equilibrium energy levels distribution by releasing excess energy into the surroundings. This process is called relaxation and has twofold effects: (1) the transverse magnetization component My, decays to zero exponentially, which is referred as spinspin relaxation, (2) the longitudinal magnetization component 14 returns to the equilibrium value ii1, gradually, which is referred as spinlattice relaxation. These two types of relaxations are phenomenologically described by the following equations dt T~ 1 ~(220) dt T2 with solutions (221) It) = 1Av(0) exp(t/T2) where the parameters Ti and T2 are known as lory~ll.:;J~.:..al relaxation time and the trans verse relaxation time, respectively. Note that both Ti and T2 depend on the molecular environment and thus can be used to characterize different samples. In the rotating reference frame where the xy axis keeps rotating with the precession frequency wo, the phenomenological descriptions of spinlattice and spinspin relaxation can be combined together into a system known as the Bloch equations [27] dM~(t) = TM(t) x B(t) R(M~(t) MI,) (222) where B(t) includes both the static field and the RF pulse, i.e., B(t) = Bo + Bl(t), and R is the relaxation matrix: R = 0 /T2 0(223) and the vector M.i = (0, 0, il,)T. The Bloch equations provide a valuable reference in describing macroscopic phenomena in NMR imaging. Note that if there were no relaxation, i.e., both Ti and T2 were approaching infinity, the Bloch equations simplify to Eq. (218), as expected. With the relaxation effect after the single RF pulse B1, the subsequent NMR signal induced by the (. 1;11 Ir;1). decaying magnetization in free precession is then detected in the time domain. It is therefore known as the free induction I/ .,ru (FID). The measured signals are conveniently represented by complex numbers where the real part and the imaginary part correspond to the xdirection and the y direction in the rotating frame respectively. By Fourier transformation the signals may be represented in the frequency domain. For a detailed discussion, the reader is referred to [29]. It is important to understand that the Bloch equations (222) only describe the dynamics of the magnetization under the effects of spinlattice and spinspin relaxation in an NMR experiment. By introducing a diffusion term, Torrey modified the original Bloch equations to reflect the effects of molecular diffusion. The modified equations are known as the BlochTorrey equations [142] and plI li a fundamental role in modeling the diffusion imaging data that we will discuss later. 2.3 Measuring Diffusion using NMR 2.3.1 Spin Echo and Diffusion Effects in NMR One assumption of the Bloch equations is the homogeneity of the magnetic field Bo in the absence of B1, which is never true in practice. The variations in the local magnetic field contribute to the gradual reduction of the transverse magfnetization in addition to the spinspin relaxation. Consequently, May decays more rapidly than the exponential decay May = il.*:tl/ predicted by the Bloch equations. However, if a 1800 pulse is applied at time t following the initial 900 pulse, then the additional decay due to the field inhomogeneity can be reversed at time t after the application of that 1800 pulse. This phenomenon of the signal restoration is called spin echo. The time between the initial 900 pulse and the echo formation is called TE (echo time) which is twice the time interval between the two RF pulses. Another factor affecting the signal reduction is the random thermal motion of the spins, or the molecular diffusion at the macroscopic level. Simply speaking, the molecular diffusion plus spatial variation of the magnetic field will induce a phase distribution in the transverse plane and hence result in signal attenuation. This signal attenuation is closely related to the amplitude of the spin displacements caused by the diffusion. Intuitively, the faster the diffusion, the more random the phases distributed, and a larger signal attenuation will be observed. This is the exact idea used in the classical S' i. H:.Tanner pulsegradient spinecho (PGSE) experiment [138], the first NMR experiment specifically designed for quantitatively measuring diffusion in a sample. In the standard PGSE pulse sequence for diffusion in.! I_;oy a pair of two identical field gradients are placed before and after the 180" pulse, to perform the "diffusion encodingt In the remainder of this dissertation, these two gradients are denoted by G with strength G = G and direction g = G/G. Each gradient pulse will last a time 6, and the pair is separated by a time a between the leading edges of the two gradient pulses. Because that the 180" pulse only cancels the phase shift of spins induced hv the first field gradient, the final detected signal will take into account the phase shift of spins induced by the molecular diffusion during the time period a which separates the two gradient pulses. Note that for a complete AIR image formation the applied diffusion gradients must he combined with a sequence of .slice .selection, S,. it;,. m e; encoding, and phase encoding for spatial localization of AIR signals. However, the details of these socalled kspace sampling techniques are beyond the scope of this dissertation and we refer the reader to [94]. 2.3.2 The Fourier Transform Relationship Let 1 be the time when the first gradient is applied, the phase shift #1 of the spin transverse magnetization induced by this gradient pulse with duration 6 is given by I1 = ]* GTzdt = q*6GTz (224) where ]* is the gyromagnetic ratio and z is the spin position supposed to be constant under the narrow pulse assumption (6 gradient pulse is applied, suppose the spin moves to a different position a + r, then the net phaseshift induced by this pair of gradients will be: S= y6GTr. (225) Note that if spins were stationary, i.e. r = 0, a perfectly refocused echo will occur. The NMR signal measurement is proportional to the total transverse magfnetization My, from a very large number of spins. Consider the millimetric resolution provided by NMR devices and the micrometric scale of the molecular diffusion, it is reasonable to neglect the intervoxel diffusion effects on a voxel scale. Let E(G, a) denote the amplitude of the "echo signal", then it can be expressed as the ensemble average E(G, a) =(exp(i )) = (exp(iyGGTr)). (226) It is important to note that here E(G, a) is 1...)~ II!. .1. ii so that E(0, a) = 1, in other words, E(G, a) should be considered as the signal decay induced only by the molecular diffusion but excluding the attenuation of the echo due to the T2 T68XatiiOn. The net phase distribution that weights the ensemble average of individual phase term exp(iyGGTT) iS the probability for a spin to travel from z to z + r during time a. Note that this probability is precisely p(z)P,(z + r)x, a) where p(z) is the spin density and Ps (x +  t) is the, diffusion, propagator westudied~ in the,,,;, prviussetio.Inetngti probability into (226) and using the substitution zt = z + r, we obtain E(G, A) = psP~ ~, x~gG~tx)md = ( p(ms: D~~)i)Psl(/7 + ~r~x~sc) epi)Trtx) (227) = P(r, A\) exp)(6igbGl))dr where P(r, t) is exactly the ensemble average propagator (EAP) defined in Eq. (29). By introducing a displacement reciprocal vector, q, defined as q = 6G, (228) 2xr we can now rewrite (227) in a more so~ lHi. form E(q, a) = ~ ,A)epi7rT~ 1[P(r, A). (229) As clearly expressed in (229), there is a simple Fourier reciprocal relationship between the spin echo signal E(q, a) and the EAP P(r, a). This Fourier relationship is fundamental in the direct reconstruction of the EAP P(r, a). The space of all possible q vectors is called qspace. By varying either of diffusion gradient G or the gradient duration 6, the signal can he measured at many sampled points in qspace and the EAP P(r, a) can he obtained by taking an inverse Fourier transform of (229) P(r, a) = FT[E(q, a)] = F[S(q)/So]. (230) Employing Eq. (230) to calculate the EAP is called "qp. .. analysis[29, 42]. Since in practice, the diffusion time a is treated as an experimental constant, the notation P(r) = P(r, a), as the displacement probability distribution function (PDF) defined on RW", is the central mathematical quantity being studied in this dissertation. CHAPTER 3 DIFFUSION MR MODELING AND DIFFUSION PROPAGATOR RECONSTRUCTION CLASSICS AND THE STATE OF THE ART In this chapter, existing models and methods used in diffusion magnetic resonance imaging are reviewed. The first section presents the classical BlochTorrey equation and the Steil: I1Tanner equation which are the foundations of the diffusion MR imaging modeling. The second section introduces the now widely used diffusion tensor imaging (DTI) method. The third section is dedicated to the more recent high angular resolution diffusion imaging (HARDI) methods with a focus on the multifiber reconstruction techniques. For published review articles on these topics, we refer the reader to [2, 89, 104]. 3.1 From the BlochTorrey Equation to the StejskalTanner Equation For free diffusion in an isotropic medium, combining Eqs. (25) and (227) yields S(q)/So = exp ((y6G)2DA). (31) There are two problems associated with this simple model. First, in commercial MRI units, long gradient durations 6 are required to produce observable dephasing effects, hence diffusion occurring during the application of the gradient pulses may not be ignored. Second, it does not take into account the effects induced by other additional gradient pulses, including imaging pulses and background residual gradients [24]. To address these issues, we resort to the following BlochTorrey equation which adds a free diffusion term to the original Bloch equations (222). Detailed math ematical derivations [24, 113, 150] show that in infinite and homogeneous media the solution to (32) with a spinecho sequence is given by dL,(t) = !1,(0) exp( k(u) TDk(u)dzL) (33) where k(t) = 7 G(u)du. (34) Note that the sign of G in Eq. 34 has to be inverted for all gradient pulses following the refocusing 1800 RF pulse. Furthermore, for an isotropic medium, the signal decay at echo time TE in a spin echo experiment is ,(E)= ,() xpD k(t)Tk(t~dt). (35) 12,ti.) IIY(I)C*0/TET By substituting the socalled ,9 II! 1.0~ factor", bvalue, defined as in [88] 0/TE into (33), we obtain a simpler expression for the signal attenuation 14,(TE) = 14,(0) exp(bD) (37) which is strictly valid for diffusion in unrestricted, homogenous, and isotropic media. Note that the bvalue is a useful quantity that characterizes the sensitivity of NMR sequences to diffusion [24]. For the most widely used PGSE sequence [138], an exact solution to Eq. (35) exists and the corresponding signal attenuation is given by S(q) = So exp(72 2G2(A 6/3)D) = So exp(bD) (38) where q = 276G and b=4xr2 q2( 2/) y2G2 _9). The time constant (A 6/3) in Eq. (38) is therefore known as the effective diffusion time where the 6/3 correction accounts for the diffusion that occurs during the application of gradient pulses. It is important to note that the S' i 11 .Tanner equation (:38) is derived from the BlochTorrey equation (:32) which is a microscopic, freediffusion physical model; while the observed diffusion AIR signal measured at a millimetric voxel is an ensembled contribution from all the microscopic displacement distribution of the water molecules in this voxel, as shown in (227). To reflect and partially bridge the gap between these two scales, one can consider replacing the physical diffusion coefficient, D, with its global, statistical counterpart, Do>,, termed the apparent diffusion coefficient (ADC) [88], in the Steil1: I1Tanner equation, which now reads S(q) = So exp(bDo;;). (:310) Although the concept of ADC requires some intravoxel homogeneity assumption, ADC has been largely used in the literature since it was introduced. [87] The Steil1: I1Tanner equation (:38) also assumes free diffusion as no boundary conditions were imposed. However, the water molecular diffusion in tissues is obviously both hindered and restricted. This fact inevitably leads to a deviation of the observed signal from the monoexponential behavior implied by Eq. (:310). More sophisticated models, such as the hiexponential model[:38, 111], the Laplace transform model [164], the Rigauttype .Iimptotic fractal expression [129, 1:30], and the stretched exponential model [22], have been investigated in literature to depict the non exponential signal decay. A brief discussion on these models with comments can he found in [11:3, Ch.2]. 3.2 Diffusion Tensor Imaging The scalar Steil1: I1Tanner equation (:38) depicts the dependence of the diffusion AIR signal on the gradient strength, however it can not describe the diffusion anisotropy which is observed in fibrous biological tissues, for example, in muscles [:39], the spinal cord [108], and the brain white matter [:37]. Diffusion tensor AIRI (DTAIRI or DTI), introduced by Basser et al. [17, 18], provides a relatively simple way of quantifying diffusion anisotropy as well as extracting fiber directions locally from multidirectional diffusion MRI data. The present section provides a brief description of DTI, more complete treatments can be found in recent review articles on DTI, for example, [25, 83, 120]. 3.2.1 Tensorial StejskalTanner Equation In Section 3.1, the solution to the BlochTorrey equation with an anisotropic diffusion tensor term (32) for free diffusion in homogeneous media measured by a spinecho sequence is given by i,(t) = !I,(0) exp( k(u)TDk(u)dzL) (311) where k~t) 7 G~~da.(312) Introducing the matrix B =k(t~k(t)Tdt (313) 0/~TE which is referred to as the Bmatrix [18], we obtain the tensorial Steil: I1Tanner relation: S(q) = So exp (trace(BD)) (314) which is the foundation of the diffusion tensor imaging (DTI). Note that to be meaningful at the voxel scale, the D in (314) should be considered as an apparent diffusion tensor Day, different from the original microscopical diffusion tensor in (32) based on the same argument provided for apparent diffusion coefficient (ADC) in Section 3.1. Let g be the unit vector representing the direction of the diffusion gradient, i.e., G = Gg, the Bmatrix can be quite accurately approximated by a pure outer product of the gradient direction scaled by the bvalue, i.e., B mbggT (315) where b = y2 2G2( 6 3) aS defined in Eq. (39). Defining the apparent diffusion coefficient D(g) along a certain direction g D(g) = gTDg (316) reveals the connection between the two versions of Steil1: I1Tanner equation as follows S(G) = S(Gg) = So exp(bD(g)) = So exp(bgTDg) or (317) S(q) = So exp(74xr2 TDq) where r = a 6/3 is the effective diffusion time and q = 276G is the displacement reciprocal vector. Substituting S(q)/So from (317) into the Fourier relationship (229) leads to the Gaussian propagator 1 rTDlr P(r, a) =exp[ ]. (318) /D(4xA)3) " The comparison of (318) and the previous result in (23) demonstrates the consistency of the Steil1: I1Tanner equation and the diffusion propagator formalism [14]. 3.2.2 Estimation of the Diffusion Tensor Taking the natural logarithm at both sides of Eq. (314) yields the simple relation trace(BD) log(So) = log(S(q)). (319) The lefthand side of Eq. (319) is linear with respect to D and log(So), therefore, the tensor components are obtained by solving a linear system of equations formed by stacking (319) with a set of measurements S(q) and corresponding Bmatrices. It is worth noting that the linear system approach is equivalent to maximum likeli hood estimate (j!1.11) when assuming the Gaussian noisemodel on the logtransformed signals. However, in practice, the more realistic noise model should be Rayleigh dis tributed rather than Gaussian distributed. Consequently, the Gaussian MLE will be biased if high qvalues are used since Gaussians are no longer good approximation to Rayleigh distributions at the tails. For more detailed discussion on modeling noise in diffusion 1\RI data, see [1:36]. Since there are six independent elements in diffusion tensor, image acquisitions from at least six linearly independent directions, together with a reference image So, are required. Usually, a more reliable measurement of the diffusion tensor requires more than six directions. Even though a diffusion tensor should be positive definite theoretically, the positivity of the estimated diffusion tensor may be destroyed by the signal noise, therefore, one might consider methods that can preserve the positivity of the estimated tensors, for example, [158]. 3.2.3 Fiber Orientation and Anisotropy Measures Derived from the Diffusion Tensor Represented by a :3 x :3 symmetric matrix, the diffusion tensor can he decomposed into: D = &QAQT (:320) where Q = (ele283) is an orthogonal matrix of eigenvectors and A = diag(Ay, X~, 3S) is a diagonal matrix of real eigenvalues ordered by At > Xa2 X3 Intuitively, the fiber orientation can he estimated by taking the directions along the peaks of the probability profile given in Eq. (:318). Finding the peaks of the probability profile in Eq. (:318) is equivalent to finding the minima of the quadratic form: rTDlr for r E S2. It iS easy to show that TTDlr has a uique minimum when r takes the direction of the dominant eigfenvector, el, provided that X1 > X2 3.XS In this case the ellipsoids generated from the isosurfaces: rTDlr = C also align to this dominant direction, hence they are widely used to visualize the estimated diffusion tensors. Table 31. A list of diffusion anisotropic measures that can be derived from the eigfenvalues of the diffusion tensor. (D) = trace(D)/3 = (At + X2 + 3) 3 is known as the mean di~ffr.: .It;, which indicates the average diffusivity over all directions. Name Definition Relative Anisotropy (RA) C ~(As (D)) Frac~tional Anisotfropy~~ (F) 2 1(s D) Volume Ratio X1 2 3 (D) 3 Prolateness Metric X1X 3 (D) Oblateness Metric 2(AgX3) 3 (D) Sphereness Metric While the largest eigenvalue and its corresponding eigenvector of the diffusion tensor describe the quantity and direction of the principal diffusion, its eigenvalues are also exploited to derive some scalar measures that are meaningful for studying the nature of diffusion anisotropy exhibited in the tissue of interest. Table 31 lists several anisotropy metrics that have been studied in the literature [15, 162]. Note that all metrics defined in Table 31 are dimensionless and rotationinvariant. Among these metrics, the fractional anisotropy (FA) has been most widely used because it is relatively insensitive to noise [15] and does not require the sorting of the eigenvalues. Additionally, FA is automatically normalized to the unit interval, FA takes the value of 0 when the diffusion tensor is totally isotropic, i.e., At = X2 3 X, Whereas it takes the value of 1 when the diffusion tensor is extremely anisotropic, i.e., X2 = 3 = 0. Anisotropy metrics are quite useful in quantitatively assessing the orientational coherence of the diffusion compartments within a voxel [120]. For example, the high FA values are typically associated with strongly aligned fibers such as axons in white matter, while the FA values are expected to be relatively low in regions of fiber intersections or in dense tissues where diffusion is restricted equally in all directions. 3.2.4 Problems of the Diffusion Tensor Imaging As previously mentioned, the diffusion tensor framework can only describe a very limited class of diffusion phenomena. Although promising results have been achieved using DTI to study regions of the brain and spinal cord with significant whitematter coherence and to map anatomical connections in the central nervous system [19, 41, 105, 106, 156] Sthe major drawback of diffusion tensor MRI is that it can only reveal a single fiber orientation per voxel and fails in regions where fiber populations cross, kiss, splay, branch or twist as shown in illustrated in Fig. 31. In a recent study [21], it was estimated that this intravoxel orientational heterogeneity (IVOH) problem affects one third of white matter voxels. In those regions, tractography applications based on the diffusion tensor model may result in artefactual reconstructions of pathi .va~ [19]. In recent years, high angular resolution diffusion imaging (HARDI) techniques have been able to address the challenges inherent in diffusion tensor imaging and will be the topic of the next sections. (a) kissing fibers (b) crossing fibers (c) splaying fibers Figure 31. An illustration of subvoxel fiber configurations arising from the introvoxel orientation heterogeneity (IVOH). 3.3 High Angular Resolution Diffusion Imaging (HARDI) Despite the promising results achieved by the diffusion tensor it! s. ./). the diffusion tensor model is known to be inadequate for resolving complex neural architectures, particularly in regions with complicated intravoxel fiber patterns [4, 56, 152, 154]. This limitation of the diffusion tensor model has stimulated exploration of both more demanding image acquisition strategies and more sophisticated reconstruction methods. Tuch et al. [151] developed a clinically feasible approach in which apparent diffusion coefficients are measured along many directions distributed almost isotropically on a spherical shell in the diffusion wave vector space. Since then, many methods that augment the angular resolution of the diffusion model have been used in the literature and are commonly referred to as high r,:ll;larr resolution diffusion :l,,nryl.e..y (HARDI). 3.3.1 Modeling Diffusivity Profiles In the original HARDI method [151], with diffusion gradients applied along many directions, the diffusivity profile is calculated by using the scalar version of the Steil1: I1 Tanner equation (:38) along each direction but does not assume any particular model. In diffusion tensor in.! I_;by the diffusivity profile is assumed to take the form of Eq. (:316) which is intrinsically a quadratic model. Since it has been shown that the diffusivity function exhibits complex local geometry in voxels with orientational heterogeneity (IVOH) [152, 154] and the diffusion tensor model is inadequate in such situations, various higher order models have been proposed to approximate the underlying diffusivity profile. 3.3.1.1 Spherical harmonics series Frank [56] introduced the use of spherical harmonics (SH) series [60] to model the diffusivity profile 00 1 l=0 z= 1 In (:321) the coefficients airn can he calculated using the spherical harmonics transform (SHT) at,>= Dg)*>gd. (:322) The spherical harmonics series (SHS) in (:321) is truncated so that only the most sig nificant terms are included in the expansion. Furthermore this SHS should only include evenorder spherical harmonics due to the positivity (D(g) > 0) and antipodal symmetry (D(g) = D(g)) of diffusivity profile. Instead of directly applying SHT, Alexander et al. [4] used linear regression to estimate air, and also II_0r 1. a hypothesis testing method to determine up to which order the SHS should be truncated. If the highest order of SHS is L, then the number of aine for all even I from 0 to L is (L + 1)(L + 2)/2, which reduces to 6 when L = 2. Descoteaux et al. [47] emploi a the realvalued spherical harmonics basis and proposed to regularize spherical harmonics coefficients using the LaplaceBeltrami operator. 3.3.1.2 Generalized diffusion tensor imaging Starting with an extension to the BlochTorrey equation, Ojzarslan and Mareci [114] proposed to use Cartesian tensors of rank higher than 2 to model the measured diffusion coefficients 33 3 D)(g) = o? D9ixi..izixi 9 (323) ii=0 iz=0 i3=0 where Dixi,...i, are the components of the Cartesian, rank1 tensor. To ensure the positivity and antipodal symmetry of D(g), I must be an even number and Diiz...il have to be realized as a totally symmetric tensor which contains (l+ 1)(l+ 2)/2 independent elements. In their work [114], the correspondence between the coefficients in the SHS and Cartesian tensors of higher ranks was derived as well, which reveals the equivalence between these two approaches. To generalize the anisotropy measures for DTI, Ojzarslan et al. [117] also proposed scalar measures in terms of variance or entropy derived from the higher order tensor coefficients. Recently, Barmpoutis et al. [10] represent a 4thorder tensor as a homogeneous polynomial of degree 4 in 3 variables, the socalled ternary quartic, and then impose the positivity of the 4thorder tensors in the estimation from diffusion MRI data based on the Hilbert's theorem which states that any nonnegative ternary quartic can be expressed as a sum of squares of three quadratic forms. Barmpoutis et al. [12] further introduced a novel parametrization of the 4thorder tensors for the simultaneous estimation and regularization of the 4thorder field while preserving the positivity of the estimated tensors. It is worth noting that another formulation of generalized diffusion tensor imaging was independently proposed by [96, 97] where the diffusion process is quantitatively characterized by a series of diffusion tensors with increasing orders. Similarly, diffusion kurtosis imaging (DK(I) has also been proposed to characterize the nonGaussian property of water diffusion by a socalled diffusion kurtosis '~~~ is [71, 86, 98]. 3.3.1.3 The limitation of ADC profile Although the measured diffusivity file can be used to indicate the complexity of the fiber structure within the voxel, it is important to point out that the maxima of the diffusivity profile may not correspond to the underlying distinct fiber orientations. von dem Hagen and Henkelman [154] observed that the peak of the ADC profile measured from a voxel containing two perpendicular fibers occurs at an angle in the middle of the two fibers but not in the direction of either. Similar observation was also reported by Tuch et al. [152], Zhan and Yang [165] in vivo. Due to this fact, the diffusivity profile can not be used directly for extracting fiber orientations, and one might still need to investigate the average diffusion propagator by taking the Fourier transform of the signal attenuation implied by the diffusivity profile as done in [115, 119]. 3.3.2 M ult i C ompart me ntal M models A direct extension of the DTI model proposed by Tuch et al. [152] assumes that the diffusion propagator takes a form of Gaussian mixture densities. Under this assumption, the signal can be modeled as a finite mixture of Gaussians as well: Scq) = So my~ exp(bgTDyg) (324) where my is the apparent volume fraction of the compartment with diffusion tensor Dj. Behrens et al. [20] introduced a simple partial volume model where the diffusion sig nal is expressed as the combination of an infinitely anisotropic component and an isotropic component. A B li. 1 Ia inference is then used to estimate the model parameters. This partial volume model was further extended in [21, 67] to allow the estimation of multiple fiber orientations. However, both extensions require complicated solution techniques to address the model selection problem properly, for example, the Markov C'I I!1, Monte Carlo (\!C\!lC) eI~! 1i; used in [67] and the automatic relevance determination (ARD) used in [21]. A slightly more complex multicompartment model, called composite and hin dered restricted model of diffusion (CHAR MED) was described in [7, 8]. Similar to the multiGaussian model, CHARMED also interprets the signal as a weighted sum of the contributions from a highly restricted compartment and a hindered compartment. Note that the diffusion in the hindered compartment is approximated by a Gaussian while the diffusion in the restricted compartment is described by a Neuman's model for restricted diffusion in a cylinder 110 . 3.3.3 Deconvolution Approaches 1 To avoid determining the number of components in the modeling stage and possible instabilities associated with the fitting of these models, Tournier et al. [14:3] emploi, I1 the spherical deconvolution method, assuming a distribution, rather than a discrete number, of fiber orientations. Under this assumption, the diffusion AIR signal is the convolution of a fiber orientation distribution (FOD), which is a realvalued function on the unit sphere, with some kernel function representing the response derived from a single fiber. A number of spherical deconvolution based approaches have followed [:3, 5, 145] with different choices of FOD parameterizations, deconvolution kernels and regularization schemes. More detailed discussion on the deconvolution approach is presented in OsI Ilpter 4 of this dissertation. 3.3.4 Modelindependent QSpace Imaging Approaches In contrast to previous approaches which assume diffusion tensor model or multi compartmental models, the socalled qspace :lr.:,:ny: (QSI) technique directly employs the Fourier relation between the diffusion measurements in qspace and the probability profile in displacement space without invoking any assumption on the underlying diffusion 1 This subsection is reprinted with permission from [7:3] process. Originally, the qspace imaging principle was used to vield the onedimensional density autocorrelation function in real space from timescale echo attenuation data in structural imaging of inanimate mate ~ 1 .h [.$, 42]. After the the emerging of the high angular resolution diffusion imaging (HARDI) [151], several in vivo qspace imaging techniques have been recently proposed to approximate the ensemble averaged diffusion propagator in :3D displacement space by performing the full :3D Fourier transform on qspace measurements using different sampling schemes. 3.3.4.1 Diffusion spectrum imaging In diffusion spectrum imaging (DSI) [159, 160], the diffusion signal is obtained by sampling a dense threedimensional Cartesian lattice in qspace with a very high number of gradient directions and different hvalues. Then the full displacement probability density function (PDF) of the diffusion, termed diffusion .spectrum in [159, 160] for each voxel is reconstructed directly based on the Fourier transform relation (229) S(q)/So = P(r)ol iexp'i2rq~ i (:325) P(r) = .F[S(q)/So]. It has been shown in [150, 160] that in an isolated system with time invariant (i.e. homo geneous) diffusion properties the Fourier transform of the EAP in the stationary state is real and positive. Thus the signal magnitude information is sufficient to reconstruct the EAP due to a homogeneous diffusion process. For the purpose of determining the fiber ori entations and better visualization, the resulting PDF is usually reduced to an orientation density function (ODF) by a weighted radial projection: ODF(u) = (u~T3_26) with u = 1, ru = r, and Z is a normalization constant[160]. Note that in order to make qspace imaging feasible in vivo, DSI adapts the twice refocused balanced echo (TRBE) sequence to reduce the eddy currentinduced artifacts created hv the 180" refusing pulses of the PGSE[127]. In addition, effectively constant gradients are emploi II in the TR BE experiment [150]. This modification violates the narrow pulse condition assumed in the conventional PGSE experiment, and actually measures the probability of a spin departing from its mean position over time 0 to TE/2 to its mean position over time TE/2 to TE. According to Tuch [150], this socalled apparent centerofmass (CO1\) propagator still preserves the orientational structure of the originally desired diffusion propagator. Recently, diffusion spectrum images have been successfully acquired on both human and small animal subjects [62, 95, 150]. These experiments have demonstrated the ability of DSI to resolve complex tissue microstructure such as intravoxel fiber crossing and divergence; However, the heavy sampling burden of DSI still makes the acquisition timeintensive and limits the wide spread application of DSI. 3.3.4.2 Qball imaging The ODF reconstruction by using radial projection in DSI captures the salient angular contrast of the diffusion function but discards all of the radial information contained in the diffusion function. Inspired by this observation, Tuch [149, 150, 153] proposed a modelindependent sampling and reconstruction scheme termed qhall imaging (QBI). QBI samples the diffusion signal S(q) only on a single sphere and then directly reconstructs a function defined on the unit sphere by taking the spherical Radon transform of the diffusion signal 2 The spherical Radon transform, also known as FunkRadon transform, sends a spherical function to another spherical function hv the following integration (R [f]) (u)= f (x)6(uTx)dx (327) 2 QBI can also be extended to multiple shell HARDI data. K~hachaturian et al. [82] describes a twoshell acquisition scheme to improve the sampling efficiency and signalto noise ratio of QBI. where u = 1. Tuch [150] showed that the spherical Radon transform of the diffusion data sampled on the sphere, (R [S])(u), closely resembles the ODF(u) obtained by the radial projection of the PDF ODF(u) = ~ u .(328) Recent studies have expressed QBI's FunkRadon transform in a spherical harmonic basis [5, 48, 66] ODF,(0, ~)= ()/ ~(,)(329) even I m=l where sim = S(, V)IMm(0,1 4)in Od~idc (330) is calculated from the spherical harmonics transform of the diffusion signal. In addition difusin sgnl S0, spherical radon transform qbl D difsimn = Jn S(04) (, 4) sinl Odd >(1) Figure 32. From the diffusion data to orientation distribution function (ODF) via the spherical random transform with qball imaging (QBI). to the spherical harmonics, other mathematical models for representing spherical functions or distributions have also been used to express and compute the qball ODF, including the mixture of vonM~ises[102], the mixture of Watson densities[126], spherical wavelets[112], and spherical ridgelets[103]. The QBI reconstruction has advantages of being efficient and modelindependent, which made it a popular high angular resolution reconstruction scheme in recent works [30, 122, 123, 148], however, the end result obtained by the radial projection of the three dimensional displacement PDF via a line integral is a convolution of the real probability values with a 0order Bessel function [149] but not the probability values themselves. This convolution gives rise to an undesirable "contamination" of the probability along one direction with probabilities from other directions and induces spectral broadening of the diffusion peaks. To address this problem, recently alternative approaches attempting to yield a more accurate approximation of the ODF using the spherical radon transform have been investigated[1, 147]. 3.3.4.3 Diffusion orientation transform The diffusion orientation transform (DOT), introduced in [116, 119], is a robust and fast modelindependent method. The key mathematical tool used in DOT is the Rayleigh expansion (planewave expansion) which expands a plane wave in terms of the product of derivatives of spherical Bessel functions and spherical harmonics: exp(iigr) = xid~rM(/8)E~/r)(331) 1,m where ji() is the 1th order spherical Bessel function and YEm is the spherical harmonic function. Inserting the Rayleigh expansion in the Fourier transform relation (229), we obtain J m~*4~d S"3 where Iz~~p) 4x q2 (2qr ) exp( 4xr22tD(p)) (333) and D(p) is the diffusivity profile (angular distribution of apparent diffusivities) that can be obtained from the Steil: I1Tanner expression. Note that the integral in Eq. (333) can be evaluated analytically by using the formulas derived in Ojzarslan et al. [119]. With these powerful tools, DOT is able to transform the diffusivity profiles into probability profiles either directly or parametrically in terms of a spherical harmonic series (Figure 33). The estimated probability function from DOT is also "impure" in the sense that the end result is the true probability values convolved with a function which cannot be specified analytically. It is worth noting that much of the blurring in the DOT is due to Mono or Multiexponential I II .ITanner Equation Transform Figure 33. A schematic illustration of the diffusion orientation transform (DOT). the monoexponential decay assumption of the MR signal, hence using the extension of the transform to multiexponential attenuation as described in [119] can alleviate the blurring, however, it in turn would necessitate collecting data on several spherical shells in the qspace. 3.4 Conclusion To conclude this chapter, we simulate diffusion weighted MR signals using a restricted diffusion model [137] and compute different quantitative profiles described above. The signal simulation is based on the exact form of the MR signal attenuation derived from the diffusion propagator for particles diffusing inside cylindrical boundaries [137] which can be considered as a simplified model for diffusion inside real neural tissues. For diffusion within a cylinder of length I and radius r, the signal attenuation with diffusion coefficient D obtained by Soderman and Jansson [137] is as follows: E(q, 8, a) = Kmr Sn22) n=0 k=1 m=0 [(12r/T1) (2;7qlr cos 8)2] x[1 (1)" cos(2;ql cos 8)] [J (2;qr sin 8)]2 12 [am , (2xrqpsinO)2] 2~(~, m 9 x~~~k ex 2 7 D In Eq.(:334), 8 is the angle between the direction of the magnetic field gradient and the symmetry axis of the cylinder, a is the time separation, Jr~ is the mth order Bessel function, Gkm, is the Akuz solution of the equation J,,(ca) = 0 with the convention colo = 0, and K,an's are constants defined by K,an = 21,,>.21,,> where 1A is the indicator function on a set A. The gradient directions were chosen to point toward 81 vertices sampled on a unit hemisphere from the secondorder icosahedral tessellation. The orientations in our 1, 2 and :$fiber configurations are specified by the azimuthal angles of $1 = :300, 2a = {200, 100"} and #:3 = {200, 750, 1350} respectively. Polar angles for all fibers were taken to be 8 = 900, so that a view from the x axis will clearly depict the individual fiber orientations. as illustrated in Fig. 34. Three .straightforward observations can be maade from this Agure: (1) the di~fu~son tensor model is not able to cheeracterize the IV/OH; (2) the di~f;,. .it;, 14.e..01. does not ;;.:. I1 the correct Aber orientations; (3) the diffusion ye~..;yl.r'l~l derived profile~s are calpuble of resolving comp~lex: tissue microstructure~s .such the profiles presented in the right three columns correspond to the targeted results of three modelindependent techniques: DOT, QBI, and DSI, respectively SThe actual results of these three profiles were computed from the method proposed in this dissertation. See details in later chapters. Figure 34. Various quantitative profiles derived front diffusion weighted signals simulated from 1fiber, 2fiber, and 3fiber geometries. Simulated Fiblers Traditional DTI Diffusivity Profile Plr = ro) SP(r)dr SP(r)r dr )tC CHAPTER 4 METHODS The fundamental formula in the diffusion tensor imaging (DTI) is the tensorial Steil: I1Tanner relation [18]: S(q) = So exp (trace(BD)) (41) It is derived as the solution to the BlochTorrey equation with an anisotropic diffusion tensor term (32) for free diffusion in homogeneous media i,(t) = !I,(0) exp( k(u) TDk(u)dzL) (42) where k(t) = 7 G(u)dia (43) and B =k(t~k(t)Tdt. (44) As discussed before, the BlochTorrey equation only describes the microscopic phenomena and the measured diffusion MR signal at the voxel level is an macroscopic quantity. Hence Eq. (41) is only valid by assuming the homogeneity of diffusion tensors in a voxel. And the diffusion tensor estimated from the Eq. (41) is called the apparent diffusion tensor. In this work, we model the measured diffusion MR signal in a voxel using the ensem ble sum of the individual transverse magnetization vectors: where each individual 14, is modeled by Eq. (42) with its own diffusion tensor. Suppose the diffusion tensors in a voxel are distributed according to a density function f(D), then combining (41) and (45), we propose the following tensor distribution model for the diffusion weighted MR signal: S(q) =So ex~'.p (trace(BD)) f (D)dD (46) where ?P, denotes the manifold of n x n symmetric positivedefinite matrices, and by default, refers to the the manifold of 3 x 3 symmetric positivedefinite matrices throughout this work. The key postulation in the proposed model (46) is that each voxel is associated with an underlying probability distribution defined on the space of diffusion tensors. Clearly, Eq. (46) is a more general form of multicompartmental models and simplifies to the diffusion tensor model when the underlying probability measure is the Dirac measure. In this chapter, we will first review the necessary mathematics for studying the integration on ~P,. Interestingly, Eq. (46) is exactly the Laplace transform of a prob ability distribution on ?P, whose formal definition is given later in this chapter. Since diffusion tensors are used to depict the time dependence of the covariance matrices of random molecular displacements, it is natural to choose this distribution as the Wishart distribution [163] on which we also present a short discussion in this chapter. The signal decay associated with a Wishartdistributed random tensors is no longer a Gaussian, but a Rigauttype .Ii~!lp)tic fractal expression given by the closed form Laplace transform for Wishart distributions. The technical details of this closed form Laplace transform will be derived later in this chapter. This Rigauttype .Iimptotic fractal expression has been used in previously published literature [84] to explain the MR signal decay phenomenolog ically. To the best of our knowledge, it is our statistical model that first gives a rigorous mathematical justification of this Rigauttype .Iimptotic fractal expression. Furthermore, our formulation is readily extended to a mixture of Wishart distributions to tackle the multifiber reconstruction problem. In fact, DTI and the multicompartmental models are limiting cases of our method when the tensor distribution is chosen to be a Dirac distribution or a mixture of Dirac distributions. In the last section of this chapter, out continuous tensor distribution model is then put into a unified deconvolution framework [73], and several deconvolution schemes are further designed and investigated to achieve stable, sparse and accurate solutions. 4.1 Some Mathematics on ?P, 4.1.1 Measure and Integration on ~P, To define integration on ~P,, which is not a vector space, we need to introduce some fundamental facts about the geometry and the measure on ~P,. Consider the general linear group G = GL, of nonlsing~ular Ix n rLeal mlatric~es andu define lthe auction of geG c on Ye ?, by Y Y [g] = gTYg. (47) It is easy to show that G acts transitively on ?P, according to the action [g] defined in (47), whichl imlies that~ ?, is a homogUenou~s spacet of ~lthe general Ilinear group GL,. (e [131, 139] for the definition and meaning of homogenous spaces.) In the following, we will also see that ?P, has a GL,invariant measure (volume element). By GL,invariant measure (volume element), we basically means that if dp, is a G~L,invariant volume element on ~P,, then dp,(Y) = dp,(Y [g]) should hold for any g E GL,. Since for any Y, WE ~P,, there is ag E GL, such that W = Y [g], then we should have dp,(Y) = dp,(W) for any Y, We p, which is a quite useful property for doing integration on ~P,. As a convex cone embedded in the space of symmetric matrices, ?P, has a natural induced Lebesgue measure defined as the direct product of the Lebesgue measures over the independent elements of the matrix variable, that is, for Y = (yij) E pn, dY =dy (48) where dayi is the Lebesgue measure on RW. Though dY is commonly used for integrals involving functions of matrix argument, it is not a GL,invariant volume element on ?P, which can be shown by the fact that Y = aX 4 dY = an(n+1)/2dX (49) where a is a scalar quantity. To find the relation between a GL,invariant measure and the dY, we first consider the Jacobian of the mapping induced by the group action with respect to dY. Theorem 1. [101, p.82],{189, p.19] Let J(g) be the Jacobian of the I'trry.::ll Y H Y[g] = gT~g with respect to dY for g e GL,, i.e. J(g) = dW/dY  for Y e Sym, and W = Y[ g]. Then J(g) =  det(g)i" (410) Theorem 2. [189, p.18] Let dp, be the measure on ?, I.lb: nE1 dpl = dpl(Y) = (det Y) z dY. (411) The&IUn dp, is th G~L,invariant volume element on ~P,. Note that a special case of this measure when n = 1 is just the scaleinvariant dp(x) = (1/x)dx = d log x for x e (0, 00). 4.1.2 The Laplace Transform on ?P, In this section we first give the definition of the Laplace transform on pn which pIIl we a central role in our model, and then we briefly discuss two kinds of matrix argument special functions that are used later for describing our model. For definition of Laplace transforms on ~P,, we follow the notations in [139, p.41]. Definition 1. [189, p.41] The Lap~lace I,,r,,,4,rm of f : ~P, C,@ denoted by 2f, at the / iiiili. matrix: Z E Cox" is 1, Ji, .1 by: ff (Z) f i(Y) ex rnaceYZ)]dY (412) where dY = dayi 1 < i < j < n. An equivalent definition of the Laplace transform for matrixvariate functions is also given in [101, p.255]. The Laplace transform converges in the right half plane, Re(Z) > Xo, for a sufficiently nice function f, where Re(Z) denotes the real part of Z and A > B means A Be ~P,. Theorem 3. [65, pp.479480]. The inversion formula for this Lap~lace I,,r,:,,,,rm is: (2i~+)2ff(Z) exp [trace(YZ)] dZ f (YfrY r (413) ne=xo 0, otherwise. Here dZ = n dzzy and the :,.J Iyeal is over ;; ./ i~ii. matrices Z with ix~ed real part. If f is the density function of some probability measure FT on ?P, with respect to the dominating measure dY, i.e. dPT(Y) = f(Y)dY, then Eq. (412) also defines the Laplace transform of the probability measure FT on ?P, which is denoted by MFT. Note that the L/aplace tr tfr, tsI, can also be 7. fr...l ~ by replacing the dY with the invariant volume ele ment dp,(Y) discussed in the preceding section and I,;.;;.9. y1 f (Y) to f (Y) (det Y) il" ) accordingly. In the rest of this work, the Lap~lace i,r,,,, 4,>rm & of a matrixvariate function 1~ :;,.. on ~P, will be interpreted as integration with respect to the invariant measure dp, unless il.. 11/l~ stated. In order to understand the property of the model being discuss later, we digress for moment to study two important matrix argument special functions on ~P,, namely, the power function and the gamma function. The most basic special function on ?P, is a generalization of the complex power function y*~, y7~E P1 = RW+, s EC and is defined as follows: Definition 2. [189, p.89] The power function p,(Y) for Ye EP,, and a = (sl, .. s) E ps(Y = (et Y)" ,(414) j=1 where Yi is the jth leading principal minor of Y. Proposition 1. [189, p.89] If Ye E and U is an upper t,..attyul;,lrr matrix;, then ps (Y[U]) = p,(Y)p,(I[U]). The proof for this proposition is given in [139] where other interesting properties of power functions are also discussed, for example, power functions are eigfenfunctions of invariant differential operators on ~P,. The ordinary gamma function of a complex number z with positive real part is defined by: F(z) = tz etdt= to (u)dlog(t) = 2(tz)(1) (415) which can be viewed as the Laplace transform of a power function evaluated at the identity. Similarly, the multivariate Il.::::::;.; function for ~P,, denoted by F,(s), is defined by: Definition 3. [189] En~) P(Y)ex(race(Y))dyl(Y), (416) Re( si) > (j )2 ,.,n. In fact, the multivariate gamma function En(s) can be expressed as a product of ordinary gamma functions 01 = 0 as given in the following theorem: Theorem 4. [61, pp.19/, [189, pp.41] j= 1 i= j Let 2(p,)(Z) be the Laplace transform of the power function p,(Y), note that the multivariate gamma function F,(s) is defined as (p,) (I). Herz [65] proved that M(p,)(Z) is absolutely convergent for any complex symmetric matrix Z with Re(Z) > 0 and leads to the following quite useful identity: Theorem 5. [61, pp.19/, [189, pp.41], [101, pp.254] 2 (ps)(Z) =~ p. Y) e~xp(tmrrac(YZ7))d,() (Z )u() (418) for Re(Z) > 0 and Re(C" s;) > (j 1)/2, j=1,.. n. 4.1.3 Wishart and MatrixVariate Gamma Distributions In Theorem 5, let n = 1, a = k > 0, Z = 0 > 0, the fumetion f (Y)= Y"e~le (Y > 0) (419) Okf (k) integrates to 1 with respect to the invariant measure Y dY and actually gives the density ftmetion of a gamma distribution of a shape parameter k and scale parameter 8. For general n, let s = (0,. ., p) and Z = El > 0, then we have p,(Y) = p,,, !,) (Y) = (det Y)" and ii, (etC)P,()I (det Y)Y exp(trace (YE ))dp(Y) = 1 (420) where Hence, the ordinary gamma distribution can be naturally generalized to the matrix variate case as follows [91]: Definition 4. {91]l For EE E ailnd for p i~n AZ = { 1, "1 } U (" ,0 ther W~ishart distribution y,,c with scale parameter E and shape parameter p is [n 1s dy,,r (Y) = F,(p)l (det Y)p(n+1)/2 (det E) exp(trace(E Y)) dY. (422) It is easy to see from the above definition that the matrix variate gamma distribution is closely related to the Wishart distribution, a probably more well known name, in multivariate statistics [6, 61, 109]. Definition 5. [61] A random matrix: Ye E is said to have the (central) Wishart distribution W,(p, E) with scale matrix: E and p degrees of freedom, a < p, if the joint distribution of the entries of Y has the following 1 ,.'.11/ function with respect to the L/ebesgue measure dY. f (Y) = c (det Y)(pa1)/2 (det E)p/2 exp taeE ),(3 with E E ?P, and c = 2up'/20a(p/2)] Note that the correspondence between the two notations (422) and (423) is simply given by yp/2,2E = W,(p, E). In the rest of this dissertation, we will keep using the notation y,,c but will refer to Wishart distribution and matrixvariate gamma distribution interchangeably. The Wishart distribution is one of the most important probability distribution families for nonnegativedefinite matrixvalued random variables. It has been typically used for describing the covariance matrix of multivariate normal samples in multivariate statistics [109]. Its importance in the study of multivariate statistics can be seen from the following property: Theorem~ ~ ~~I 6. {61, p.8,19 p.88 L i be a random variable in R", for i = 1, 2, .. ., n, where n < p. And suppose that X1,...,X, are I,,;,linall;i independent and distributed according to NV(0, ); that is, normal with mean 0 and covariance matrix: EE ?P, Let X = (X1,...X,) E Rnxp and Y = XXT. Then with ] <..7.11,*/,*///l 086, Y iS in ~Pn Gnd iS distributed I. I;,.../ by (428). As a natural generalization of the gamma distribution, the Wishart distribution preserves the following two important properties of the gamma distribution [61, 91, 101]: Theorem 7. [61] The Lap~lace I,,r, J~rm of the (generalized) Il.::::::;.; distribution y,, is exp(trac( 8Y)) dyl:(Y) (det(I, + OE))" (424) where (8 + El) E ~P,. Theorem 8. [61] Let Y be a random variable (matrixc) with a (matrix;variate) Il.::::::;.; distribution y,,, then its expected value Eax C(Y) is pE. However, it should be pointed out that the expected value pE does not yield the max imum value of the density function in (422) or (423). For instance, the mode of gamma density function expressed in (422) occurs at (p 1)E. Interestingly, if the dominating measure is chosen to be the GL,invariant measure, dp(Y) = (det Y)(n+1)/2dY, then the corresponding density function, which is dy,rc(Y) = F,(p)l (det Y)" (det E)" exp(trace(E Y)) dp (425) does reach its maximum at the expected point pE. Proof. Let X = YE1, then dy,rc d(det X)" exp(trace(X)) dX dX =p(dlet X)"X exp(tIrace(X)) (dlet X)"exp(trac~e(X))I (426i) =(det X)" exp( trace(X) )(pX I) When X = YE1 = pl, the above derivative becomes zero which implies that the density function reaches maximum when Y = pE. O The difference between the density functions of gamma/Wishart distribution with respect to the two different carrier measures is illustrated in Figure 41. density functions of y4 density functions of Y1 Figure 41. Plots of density functions of gamma distribution Y4,1 W.r.t the noninvariant measure and scaleinvariant measure. The xaxis in the right figure is on a log scale. Clearly, the expected value 4 corresponds to the peak of the density function w.r.t. invariant measure but not for the case of the noninvariant measure . 4.2 The Expected MR Signal from Wishart Distributed Tensors Substituting the general probability distribution in Eq. (46) by the Wishart (matrix variate gamma) distribution dy,,s, we obtain S(q)/So = 1, + BE (427) by applying the closedform Laplace transform of Wishart distribution stated in Theorem 7. If the Bmatrix is approximated by B = by g T, then we further have S(q)/So = 1, +BE"= (1 + (bgT S)p (428) 1 + (bgTCg), we first prove the following To establish the second equality, 1, + BE useful results on the Schur Theorem 9. [124] Let P Then complement in linear algebra and theory of matrices. AB" ~ ~;r (1) if A / 0 then P = A D CA1IB. (2) if D / 0 then P = A BD1C D. Note that in Theorem 9, the matrix D CA1B (or A BD1C) is called the Schur complement of A (or D) in P [166]. Let PI = Then 1 + by TCg is the Schur complement of I in P, and we have det P= det Idet(1 + bgTCS STC In an analogous manner, I + b(Eg)gT is the Schur complement of 1 in P which yields det P = det(I + b(Eg)gT)det@1). Combining the above two equations and noting that B = by gT, we have 1, + BE= 1, + EB= 1, + bCggT= 1 +(bgTC) Note that the well known identity 1,2 + BE = 1,, + EB can also be derived by applying Theorem (9) on the Schur complements of e Consider the family of Wishart distributions 7;>z and let the expected value he denoted by D = pE. In this case, Eq. (428) takes the form: S(q) So (1 +(bg TDglp)" (429) This is a familiar Rigauttype2 .Iinji**lle'~ fractal expression [129]. The important point is that this expression implies a signal decay characterized by a powerlaw in the largeq, hence largfeb region exhibiting .I inia.,l'tic behavior. This is the expected .Iiin!llani s behavior for the 1\R signal attenuation in porous media [135]. Note that, although this 2 The phrase "Rigauttype" is used to distinguish this function from Rigaut's own for mula [129] function. Although slightly different, the Rigauttype function shares many of the desirable properties of Rigaut's own function such as concavity and the .Iiinia.,'tic linearity in the loglog plots. form of a signal attenuation curve had been phenomenologically fitted to the diffusion weighted MR data before [84], to the best of our knowledge, the proposed Wishart distribution model is the first rigorous derivation of the Rigauttype expression that was used to explain the MR signal behavior as a function of bvalue. Therefore, this derivation could be useful in understanding the apparent fractallike behavior of the neural tissue in diffusionweighted MR experiments [84, 118]. Here a question naturally arises. How to choose the parameter p? In our model, we choose p to be 2 based on the following theoretical consideration. The .imptotic form of the signal attenuation due to diffusion/scattering from inhomogeneous materials obeys the well known DebyePorod law [46] which states that is a powerlaw. And in threedimensional porous media the signal decays as E(q) ~ q4. Since the pvalue is defined to be the exponent of the bvalue in Eq. (429) and b is proportional to q2, the most meaningful choice should be p = 2. Rigauttype asymptotic fractal_ Gaussian expected sigrmnal; intermediate surns all 10000 tensors OO Figure 42. The Wishart distributed tensors lead to a Rigauttype signal decay. Reproduced with permission from [75] @[2007] Elsevier. 6i5 0 .1 00 0 0.0100o 0. 0010 0. 0001 0.0 0. 10 Y r(in1') The relation (429) can be empirically validated by the following simulation. From a Wishart distribution with p = 2, we first draw a random sample that contains a random rank2 tensors {D1,...,D,} and then simulate the corresponding multiexponential signal decay using a discrete mixture of tensors: E.(q) =S(q)/So = exp (bgTDig). (430) i= 1 For each sample size n, we plot the signal decay curve by fixing the direction of diffusion gradient q and increasing the strength q = q. The relation between signal decay behavior and the sample size is illustrated in Figure 42. The left extreme dotted curve depicts the signal decay from a monoexponential model, where the diffusion tensor is taken to be the expected value of the Wishart distribution. The right extreme solid curve is the Rigauttype decay derived from (429). Note that the tail of the solid curve is linear indicating the powerlaw behavior. The dotted curves between these two extremes exhibit the decay for random samples of increasing size but smaller than 10,000. The dashed curve uses a random sample of size 10,000 and is almost identical to the expected Rigauttype function. As shown in Figure 42, a single tensor gives a Gaussian decay, and the sum of a few Gaussians also produces a curve whose tail is Gaussianlike, but as the number of tensors increases, the attenuation curve converges to a Rigauttype .imptotic fractal curve with desired linear tail and the expected slope in the double logarithmic plot. If p is an integer, then a well known result is that the gamma distribution y,, also describes the distribution of the sum of p independent exponentially distributed random variables with parameter a. It follows from the central limit theorem that if p (not necessarily an integer) is large, the gamma distribution y,~, can be approximated by the normal distribution with mean pa and variance pa2. More precisely, the gamma distribution converges to a normal distribution when p goes to infinity. A similar behavior is exhibited by the Wishart distribution. Note that when p tends to infinity, we have S(q) iSo exp(bgTg ( 1 which implies that the monoexponential model can be viewed as a limiting case (p 00) of our model. Furthermore, the Taylor expansion of (429) in the long wavelength or lowq regime leads to a quadratic decay: S(q)/So = 1 4xr2 TDq(A 6/3) + O(q4) _432) which can also be derived from the diffusion tensor imaging model [14]. Therefore, Eq. (429) can be seen as a generalization of Eq. (317). By the linearity of the Laplace transform, the biexponential and multiexponential models can be derived from the Laplace transform of the discrete mixture of Wishart distributions as well. When a Wishart distribution used as the mixing distribution in Eq. (46) with a prespecified p parameter, Eq. (428) can be rewritten as S(q)So) ,,, 43 Let K be the number of diffusion measurements acquired at each voxel, the above equation can be further expressed in a matrix form: 1 \ (S2) B,  2Bz C,, 1 (434) (SK) Bz Ex  2Bzz / Ezz 1 where Bij and Egy are the six components of the matrices B and E, respectively. The lin ear system formulated in (434) clearly II_01; a linear regression method for estimating diffusion tensors from diffusionweighted images, which is very similar to the traditional diffusion tensor estimation methods [16]. Note that in this model the final estimation of diffusion tensor D is given the expected value of the Wishart distribution y,,4, i.e. D = pE. It is worth noting that ]rn Ilw alternative methods which involve nonlinear optimiza tion and enforce the positivity constraint on the diffusion tensor, as in [34, 158], can be applied to the direct nonlinear problem: mmi (S(q) So(1 + trace(BE))")2 (435) formulated from the Wishart model proposed here. Similarly, the resulting diffusion tensor field can also be analyzed by numerous existing diffusion tensor image analysis methods [161]. 4.3 Methods for MultiFiber Reconstruction 4.3.1 The Mixture of Wisharts Model The density of a simple Wishart distribution as a function of diffusion tensors reaches one single diffusion maximum at its expected value, therefore, a single Wishart model can not resolve the intravoxel orientational heterogeneity. Clearly, the Laplace transform relation between the MR signal and the probability distributions on ?P, naturally leads to an inverse problem: to recover a distribution on ?P, that best explains the observed diffusion signal. The difficulties entailed in inverting the Laplace transform are well known (see [54] for "cogent reasons for the general sense of dread most mathematicians feel about inverting the Laplace it~ lI!1its~! ), especially for noisy data in high dimensional space, which is the case in our application domain. In order to make the problem tractable, the following simplifying assumptions are made: First, just as a discrete mixture of tensors model can be adapted, so can a discrete mixture of Wisharts model where the mixing distribution in Eq. (46) is given as f (D) =~ 7, (D) (436) i= 1 where r; .: (D) is the density function of the matrixvariate gamma distribution dr; Note that in this model the set of (pi, Es) is treated as a discretization of the 7 dimensional parametric family of Wishart distributions. Hence, the number of components in the mixture, NV, only reflects the resolution of this discretization and should not be interpreted as the expected number of fiber bundles. Secondly, it is further assumed that all the pi take the same value, pi = p = 2, which is a reasonable assumption based on the analogy between Eq. (429) and the Debye Porod law of diffraction [135] in 3D space. Since the fibers have an approximate axial symmetry, it also makes sense to assume that the two smaller eigenvalues of diffusion tensors are equal. In practice, the eigfenvalues of Di = p~i are fixed to specified values (Ai, A2, 3S) = (1.5, 0.4, 0.4)p2m MSCOnSIStent With the values commonly observed in white matter tracts [152]. Due to this rotational symmetry, the discretization of ?P, forming the mixture of Wisharts is reduced to a spherical tessellation (Figure 43). Accordingly, the prominent eigenvector of each Es can be taken from the unit vectors uniformly distributed on the unit sphere. Because of the antipodal symmetry, the sampling is actually performed on the projective plane, i.e. Only half of a normal spherical tessellation is used. (a) 92 vertices (b) 162 vertices (c) 252 vertices Figure 43. Sphere tessellations using an icosahedron subdivision model with different iteration numbers. Note that all the above assumptions leave us with the weights w = (I,) as the unknowns to be estimated. Given K measurements with qj, the signal model equation: Scq) = SoC it (1 + trace(BEi))P. (437) i= 1 leads to a linear system Aw = s, where s = (S(q)/So) contains the normalized measure ments, A is the matrix with Aji = (1 + trace(Bj~i)) and w, = (I, ) is the weight vector to be estimated. The details of solving Eq. 437 will be the central topic of the upcoming sections . After obtaining the vector w, the diffusion weighted MR signal attenuation model E(q) = S(q)/So can be analytically expressed using the general form as in Eq. (443). As a result, the displacement probability P(r) can then be computed by the Fourier transform P(r) = f(S(q)/So) exp(iq r) dq where r is the displacement vector. The P(r) function defined above describes the probability for water molecules to move a fixed distance and has been emploi II in the DOT method [119]. Note that when the mixture of Wisharts model is used, the resulting P(r) can be approximated as a mixture of oriented Gaussians P(r, ) =~ exprl(qTUqr) dF(D) exp(iq r) dq exr"p(qT) qr)ep(iq r)dq dF(D) 1 (438) eC xp(Tar Dr/47) d(D i' 1~l~ where Di = p~i are the expected values of r; In this case, many of the quantities produced by other methods including the radial integral of P(r) in QBI [149] and the integral of P(r)r2 in DSI [160] are analytically computable. This fact provides us the opportunity to understand these quantities and evaluate their performances in resolving complicated local fiber geometries. 4.3.2 A Unified Deconvolution Framework Interestingly but not surprisingly, the proposed mixture of Wisharts model (Eq. 437) can be cast into a general deconvolution framework S~q)/o = ~ q, ) f x~dx(439) that unifies several modeldependent HARDI reconstruction approaches in literature. The signal in Eq. (439) is expressed as the convolution of a probability density function f(x) and a kernel function R(q, x). The integration is over a manifold MZ/, each point, x, of which contains information indicating fiber orientation and anisotropy. The convolution kernel, R(q, x) : RW3 X iZ/1 HWmodels how the signal receives response from a single fiber. In order to handle the intravoxel orientational heterogeneity, the volume fractions represented by a continuous function f(x) : MRR I models the distribution of fiber bundles. Hence, the deconvolution problem is to estimate f(x), given the specified R(q, x) and measurements S(q)/So. A common approach in practice is to represent f(x) as a linear combination of NV basis functions: f(x) = E wyfj(x). The choice of convolution kernels and basis functions often depend on the underlying manifold MZ/. A simple example is to let MZ/ be the unit sphere (or more precisely, the projective plane due to the antipodal symmetry), which leads to the spherical deconvolution problem [3, 143]. Several other approaches start from the manifold of diffusion tensors, but again reduce to the spherical deconvolution problem since only rotationally symmetric tensors are considered. [5, 125, 143]. Following [5, 125, 143], we choose the standard diffusion tensor kernel in our model. However, it is the Wishart basis function that distinguishes our method from these related methods. It is worth noting that the Wishart basis reduces to the Dirac function on ?P, when p = 00 and thus leads to the tensor basis function method [125] as well as to the FORECAST method [5], which both estimate fiber orientations using the continuous axially symmetric tensors and hence resemble our method very closely. Table 41 summarizes some of the existing multifiber reconstruction methods in the above described unified deconvolution framework. The first approach listed in Table 41 corresponds to the well known diffusion tensor imaging method Basser et al. [18] where the mixing distribution in the convolution integral is taken to be a single Dirac distribution S(q)/BSo~ = ep(qTDq)6(DD l^)dD exp(qT ^) (4 40) Table 41. A list of previously published fiber reconstruction methods expressed in the proposed unified deconvolution framework. See text for meaning of symbols. Reproduced from with permission. @[2007] IEEE. Method MZ/ x f (x) R(q, x) unknown Basser et al. [18] ?,n D 6(D; D) exp(qTDq) D Tuch et al. [152] ~P, D I,, JC(D; Dk) exp(qTDq) { Anderson [5] S2 Uv Clm~m exp(qTDyq) {czm) Alexander [3] S2 U, ,i, e? ex(( ) t Jian et al. [75] pn D where ?P, is the manifold of 3 x3 positive definite matrices, 6(x; xo) denotes the Dirac delta function centered at point xo and the problem is to estimate the unknown D that best fits the diffusion tensor model. Instead of the single Dirac delta function used in the diffusion tensor model, all other methods listed in Table 41 express the fiber orientation distribution f(x) as a linear comblination of K basis functions f(x) C~= I E to fk(x). For. example, the second approach listed in Table 41 corresponds to the multitensor model [152] K K S(q)/So = exp(qTDq) C (D; Dk)dD expl(qTDkf) 1 In the original multitensor model proposed by [152] the objective is to find a set of Kt tensors Dk and corresponding fractions no by performing conventional nonlinear optimization algorithms. Even though the eigfenvalues of Dk are Set tO Specified values to favor physiological solutions, it is usually desirable to fix the number K to be a small number, 2 or 3, to ensure the stability of the nonlinear optimization, which in turn introduces a model selection problem. Due to its close relation to the diffusion physics, the diffusion tensor response kernel R(q, D) = exp(qTDq) (442) is still widely used in a number of recently proposed reconstruction methods. However, unlike the traditional diffusion tensor model and multitensor model where K is set to a small number (1,2,or 3), these extended diffusion tensor based models tend to let K be a relatively large number in order to gain a quite good coverage of the underlying manifold MZ/. In this sense, the number K only depends on the resolution of the discretization on the manifold MZ/ and should not be interpreted as the number of expected fiber populations. In order to avoid the difficulty of directly working on the manifold of diffusion tensors ~P,, many models take the similar assumptions used by our model. Some models [75, 125], only consider a set of K rotationally symmetric tensors, DI, with specified eigenvalues , A = { At > a2 =3 }, but varying dominant eigenvectors, {vl, .. ., vK } obtained from a sphere tessellation. For this reason, the underlying domain of deconvolution is reduced to the sphere from the manifold of diffusion tensor, ~P,. It is also worth noting that the response kernel exp((v q)2) used in [3, 143] is just the special case of the exp(qTDyq) when X2 = 3 = 0. However, it is the Wishart basis function that distinguishes our method from these related methods since Yp,Dy aS a COntinuOUS distribution on ?P, implicitly takes contributions from the entire ?P, even there are only K discrete Dy. It is also interesting to note that the Wishart basiS p,~Dyx reduces to the Dirac function on 7P, when p 00o and thus the models in [5, 125] can be treated as special cases of our model. Finally, except for the single diffusion tensor model and the multitensor model [152] where both the volume fraction I, and the parameter of basis functions (Dk) are unknown, all the other methods lead to a problem that can be expressed in the general form of Aw = s + rl, (443) where, on the righthand side, s is a column vector containing NV multidirectional mea surements, {si} = {S(qi)/So}, and rl represents the noise, while on the lefthand side, A = {aik) is an 1VX K matrix given by aik, = MI RGi, x) k(x)dx and w is the un known weight vector. Note that the integral to compute the entries of A may have an analytical solution as in [75] or needs to be numerically approximated as in [3], depending on the choices of convolution kernels and basis functions. But once the response kernel R(q, x) and the basis function are specified, the matrix A can be fully computed (or approximated) and only w, a column vector containing K unknown coefficients, remains to be estimated. It is worth noting that a further spherical harmonics (SH) expansion on w in [5, 143, 146] leads to a parametric deconvolution of w in terms of SH coefficients. However, it is straightforward to change this parametric deconvolution back to the direct nonparametric deconvolution of w. In our opinion, it is more natural and easier to inter pret and handle the weight vector than the SH coefficients vector. For this reason, we will only discuss the nonparametric deconvolution of w, in this work. 4.4 Computational Issues In the terminology of signal representation theory, the NVx K matrix A in Eq. (443) as a collection of vectors ai E R"N, i = 1... K is usually referred as a "J..i.o ,:.re;, and the column vectors (ai) are called "rlia;,, ". The deconvolution problem is then equivalent to representing the signal s = Ci f as a linear combination of atoms in the dictionary A. If we require all the weights to be nonnegative, then the linear combination is restricted to a conic combination. In addition, a parsimonious representation for signal is preferred if only a few atoms are able to produce the signal s. In the normal clinical setting, the number of diffusion MR image acquisitions, K, is rarely greater than 100. On the other hand, a high resolution tessellation with NV > 100 is usually preferred for an accurate reconstruction. This situation yields a II rectangular system matrix A in (443) and in this case the dictionary A is called overcomp~lete. An underdetermined system of equations may have infinitely many solutions in the least squares sense. To make things even worse, most deconvolution models used in literature result in extremely illconditioned linear systems whose standard least squares solutions may be r I l inOy unstable. This disturbing consequence of the numerical stability issue can he a real concern as was shown in [5]. Since the weights w = {m, } K1 correspond to volume fractions, they are expected to be nonnegative. Negative weights are not physically meaningful and should be penalized by adding a regularization term or excluded by imposing an explicit nonnegativity constraint. In addition, it is reasonable to assume that most white matter voxels only contain contributions from relatively few fiber bundles. Therefore, apart from a few significant peaks, we expect w has a sparse support, i.e. most entries of w are expected to be zero (or very small). It is important to note that the sparsity property also has advantages in the optimization process required to find the maxima of the water molecule displacement probability function, which represent the fiber orientations in the multifiber reconstruction problem. 4.4.1 Regularization and Stability The first question one needs to consider before solving the deconvolution problem is how to measure the goodness of a solution, in our case, the discrepancy between Aw and s. Commonly used measures include the L2 norm, the L1 norm, or more generally, an L, norm for 1 < p < 00. If all the elements of A,w,s are assumed to be nonnegative, then, some information divergence e.g. K~ullbackLeibler (K(L) divergence can also be used to measure the distance between Aw and s after appropriate normalization. Under the assumption that the measurement errors are independent and follow an identical normal distribution, the maximum likelihood estimate of w naturally leads to the L2 Ilorm as a goodness measure and is equivalent to the corresponding least squares (LSQ) problem [26] that minimizes the residual sum of squares (PI) min Aw s2 4 ) The solution of (P1) in the least squares sense is given by w, = A+s where A+ is the pseudoinverse of the A given by A+ = (ATA)1AT ifA has full colunin rank (5 SAT(AAT 1 ifA has full row rank. 45 Direct methods are adequate here since in our application the size of the linear system in Eq. (444) is not that large to require iterative methods. 1\oreover, since the matrix A is independent of the spatial location, the pseudoinverse is only computed once, and hence the computational burden is light. Despite its simplicity and efficiency, a direct solution to the linear system is highly susceptibility to noise, especially when A is illconditioned, which is usually the case in our application. To illustrate this illconditioning phenomena, we study two different convolution models under the unified framework of Eq. (439). The first model involves the use of radial basis functions and a Gaussian kernel with a degenerated rank1 covariance matrix, as in [:3]. The second model uses a standard diffusion tensor kernel weighted by a mixture of Wishart distributions as discussed in [75]. The reason for choosing these two models for the purpose of comparison is due to the fact that the delta function basis as used in [5, 125, 14:3] is actually a special case of the Wishart basis when p c o. The further use of spherical harmonics expansion on w in [5] is equivalent to a parametric deconvolution of w which is different front the nonparanletric deconvolution of w as discussed in this work. Consider an example situation wherein we are given 81 gradient directions chosen to point toward the vertices sampled on a unit hemisphere front a secondorder icosahedron tessellation of the sphere. To fully specify the matrix A, we still need to specify its column space which is dependent on the number of discretization points on the sphere, i.e., the discretization resolution on the sphere. We will illustrate the illconditioning with two values of resolution (column size) namely, 81 and :321. Thus, the matrix A will be of size (81, 81) in one case and (81, 321) in another. As seen from Figure 44, both the models x17 2x 0 12x 0 10~ P 6t size(A): 81 x 821 E o asize(A): 81 x 81 +size(A): 81 x 321 20 40 60 80 100 ,0 100 200 300 a in the radial basis functions p in the Wishart distributions The linear system constructed in (443) is often extremely illconditioned with very high condition number. The plot on the left shows the profile of condition numbers when the A matrices are constructed from the radial basis function and the tensor kernel model as in [3]. The plot on the right shows the case with a standard diffusion tensor kernel weighted by a mixture of Wisharts. In both scenarios there are K = 81 diffusion gradient directions. Two tessellation schemes of different resolution levels (NV = 81 and NV = 321) are considered for each model. Reproduced from [73] with permission. @[2007] IEEE. Figure 44. result in extreme illconditioning of matrix A. The plot on the left shows the case where the A matrices are constructed from the radial basis function model. Note the condition numbers are of the order of le + 06 and steadily increase with increasing o, the value of radial basis function parameter. The plot on the right shows the case where the A matrices are constructed from the mixture of Wisharts model [75]. The condition numbers in this case are an order of magnitude less than those for the radial basis function model and quickly converge to an upper limit. Note that when the parameter p in Wishart model goes to infinity, the resulting system converges to the one obtained in [5, 125, 143]. II I.!y methods aiming at reliable multifiber reconstruction in the presence of noise have been emploiv I including lowpass filtering [143], minimum entropy [144], and the maximum entropy spherical deconvolution [3]. Recently, Tournier et al. [145] proposed to combine the spherical deconvolution with Tikhonov ,ell;,lar,.:~r..rHn, a very popular technique for solving illconditioned problems [140]. In the general framework of Tikhonov regularization, the original problem in Eq. (443) is reformulated as finding the estimate w that minimizes (P2) min Aw s2 + I~l2 _46) where T is a suitably chosen regularization operator that imposes a constrain function Tw2 On W), and a~ is a nonnegative scaling factor that controls the balance between the residual term and the penalization term. In the statistical literature, the method using Tikhonove regularization is also known as ridge regression. Typically the matrix operator T is chosen as a discrete approximation to some derivative operator if smoothness is the desired property of the underlying solution. In our case, we take T to be the identity operator, I, which leads to a zeroth order Tikhonov regularization, giving preference to solutions with smaller manitudes. K~awata et al. [81] described an application of the zeroth order Tikhonov regularized leastsquares method to 3D reconstruction of optical microscope tomography data. They also pointed out that the WienerHelstrom filtering [64] or the minimum meansquare error (1111510;) deconvolution [80] can also be derived from the zeroth order Tikhonov regularization by letting the signaltonoise ratio be a~ and assuming that the signal and the noise are uncorrelated stochastic processes. Minimization of the objective function in Eq. (446) with T = I yields the following normal equation: ATs = (ATA + a~TTT~ 47 with an explicit solution given by w = (ATA + a~TT 1ATs 48 The socalled damped least squares (DLS) method[157], which is equivalent to the zeroth order Tikhonov regularization technique, is emploi II in [75] to reconstruct the mixture of Wisharts representing a continuous distribution over diffusion tensors, where the solution is given by w = (ATA + alI) ATS (449) =AT(AAT + I18 In equation (449), (ATA + alI)1AT (or AT(AAT + ~)1) iS called the damped least squares inverse of A and a~ is called the damping factor. The damped least sqaures inverse can be expressed in terms of the truncated Singular Value Decomposition (SVD) [58] as: (ATA + l) ATi a cGi UiiT _50) where r is the rank of A, ai's are singular values of A, and vi(ui)'s are the associated left(right) singular vectors of A. The nonnegative damping factor a~ controls a tradeoff between accuracy and stability. The determination of a suitable a~ to prevent the solution from the under regularization or the overregularization is often am rl i difficulty in the use of Tikhonov regularization. A similar damping factor appears in the well known and widely used LevenbergMarquardt algorithm for solving nonlinear least squares problems and can be adjusted at each iteration. Recently, a Damped Least Squares method was used to regularize the fiber orientation distribution [132], where the optimal damping factor a~ is determined by minimizing the Generalized Cross Validation (GCV) criterion [155]. The underlying statistical model used in the GCV method is that the components of s are subject to random errors of zero mean and covariance matrix a2I, Where a may or may not be known. Let A,+ denote the damped least squares inverse of A with damping factor a~. The predicted values of s from the solution in (449) is then li = Aw = AA,+s = Pos (451) where the symmetric matrix P, = A(ATA + alI)1AT iS called the f'illa. 0..: matrix;. When a2 is known, it was II_tr 1.. in Craven and Wahba [45] that a~ should be chosen as the minimizer of an unbiased estimate of the expected true mean square error l1 81 2 a2 (452) where ni is the number of rows in A. If a2 is unknown, then the socalled generalized crossvalidation (GCV) function given by ms s /[trace(I P,)]2 45:3) may be used to estimate the proper value of c0. The main reason of using this GCV function is in that the minimizer of (453) is .Iiingdo' tically the same as the minimizer of (452) when nz is large [59]. The GCV criterion is a onedimensional function of the regfularization parameter (damping factor c0) and can he computed from the SVD of A and Pa~. 1\ore efficient numerical methods based on the bidiagonalization of 4 [5:3, 6:3, 68] have also been devel oped; see [26] for more discussions on the minimization of GCV function. However, GCV does not guarantee the real optimal solution due to the theoretical limits of the GCV method, as pointed out in [155], "the theory jllr;fi;in4 the use of GCV is an .Iiini!!lle'~ one. Good results cannot he expected for very small sample sizes when there is not enough information in the data to separate signal from in~ ss Additionally, it is important to note that the regularization scheme discussed here is performed voxel by voxel, which is quite different from some other regularization schemes in the context of smoothing diffusion tensor field [4:3, 121] where a spatial coherence term is usually added in the minimization problem. The presented unified deconvolution framework could be combined with such spatial regularization schemes as well (e.g, [125]). However, the discussion of spatial regularization schemes is not the main focus of this work. 4.4.2 Nonnegativity and Sparsity Constraints 4.4.2.1 L1 minimization methods The socalled Lo norm is the most straightforward measure of sparsity by counting the number of nonzero entries. However, the naive exhaustive search for the sparsest solution with smallest Lo norm is known to be NPhard. A recent strand of research has established a number of interesting results, both theoretical and experimental, on stable and efficient recovery of sparse overcomplete representations in the presence of noise (see [32, 49] and references therein). One such important result is that, under appropriate conditions on A and s, minimizing L1 norm of the solution often recovers the sparsest solution. This phenomenon is referred to as LI/Lo equivalence. It is shown in [50], that when the dictionary A has a property of mutual incoherence (defined in [50]) and that when the ideal noiseless signal s has a sufficiently sparse representation with respect to A, minimizing the L1 norm of the solution, ( w 1 = Ci i' I ) often recovers the sparsest solution and is locally stable, i.e., under the addition of small amounts of noise, the result has an error which is at worst proportional to the input noise level. Unfortunately, it turns out that our systems consistently have very high mutual coherence values, in the range of 0.95 to 1, which makes these nice theoretical hounds inapplicable to our problem. To investigate the performance of L1 minimization based methods in the context of our problem, we experimentally tested the L1MAGIC package a collection of MATLAB routines for solving the convex optimization problems central to sparse signal recovery [31]. Among the several programs implemented in L1MAGIC package, we are particularly interested in the following two programs: *MinL1 with equality constraints. The program can he mathematically stated as (P3) min wI1 subject to Aw = s. (454) 1 http://www.11lmagic.org This program is also known as the ba~si~s pursuit [:36] and the solution is the vector with smallest L1 norm (wI1 = Ci 'II ) that explains the observation s. Candes et al. [:31] have shown that if a sufficiently sparse wi exists such that Awi = .s, then this technique will find it and when s, A, w have realvalued entries. Note (IS) can he recast as an linearprogramming (LP) problem. *MinL1 with quadratic constraints. In this formulation, one finds the vector with minimum L1 norm that comes close to explaining the measurements s, i.e., (P4) min I~11 subject to Aw s2 < e (455) where e is a user specified parameter. Candes et al. [:31] have shown that if a sufficiently sparse wi exists such that s = Aw + e, for some small error term e2 e, then a solution w, to (P4) Will be clOSe to wi in the sense that  w wil  < C , where, TC is asmll ,,cnsant It has1, also~,,, been shown, that,,, (P,,, eTCRtR second order cone programming (SOCP) problem. The experimental results for these two programs are reported in Chapter 5. Other programs implemented in L1MAGIC but not investigated here include minimum L1 error approximation, minimum total variation (TV) with equality constraints and minimum TV with quadratic constraints, etc. 4.4.2.2 Nonnegative least squares (NNLS) A direct solver may produce a solution with many negativevalued components, which is not physically meaningful. For example, the zeroth order Tikhonov regularization is able to suppress the large spurious negative spikes in the weight vector w as expected, however, there are still many negative entries in the resulting w, vector. Note that the above L1 minimization methods (P3) and (P4) do not explicitly enforce the nonnegativity constraints either. It has been shown in image reconstruction literature [80, 81] that constraining the solution to the nonnegative space could drastically reduce the ambiguity of the solution and hence improve the final reconstruction results. It has been recently reported in [146] that a spherical deconvolution technique imposed with a nonnegativity constrain on the estimated FOD does not require a lowpass filtering which was previously I _r. .1 in [143]. The least squares problem subject to a nonnegativity constraint is formulated as (Ps) min Aw s2 Subject to w > 0. (456) The nonnegative least squares (NNLS) problem is an important special case of least squares problems with linear inequality constraints (LSI) [85, Ch. 23]: min Ax b2 Subject to CTx > d. (457) The LSI problem is essentially a quadratic programming problem that minimizes a concave quadratic function in a linearly bounded convex feasible hyperspace. Let ce denote the ith column vector of the matrix C. The separating hyperplane defined by the ith constraint is {x : cTx = di}. Note that the vector ce is orthogonal to this separating hyperplane and points to the feasible halfspace {x : cTx > di}. Given a feasible point x, it may exactly sit on some separating hyperplanes, the indices of these hyperplanes form the set 8 (equality) and the complementary set of 8 is called S (slack). The optimal solution Ai to the LSI problem should meet the following K~uhnTucker conditions [99]: jri > 0 if ie E i.e. cN X= di (458) jri = 0 if i E S, i.e. TcN > di where jr defined by Cjr = AT(AAi b) is the dual vector for this problem. Proof. [85, Ch. 23] Let g = AT(AAi b) denote the gradient vector of the quadratic cost function ~(x) = IAx b2 at ~i. The K~uhnTucker conditions in (458) lead to fii (4 59) iLES which implies that the negative gradient g at the solution is expressed as a nonnegative (ysi > 0) linear combination of the outward pointing normals ci to the constraint hyperplanes on which ~i lies (ie E ). Consider an arbitrary perturbation n of 1i, which yields a new value of the cost function: #(Ai + u) = #(~i) + UTg + AU2/2. Note that for &i + n to remain feasible, U~cs > 0 for i e must hold, hence a~g = Ci, eg ye c > 0 and it follows that ~(A + u) > ~(i) for any feasible perturbation n of ~i. O Based on the K~uhnTucker conditions in (458), Lawson and Hanson [85, Ch. 23] developed an active set strategy [57] to find the optimal solution to the NNLS problem. In this classic algorithm, a series of least squares problems without constraints are solved sequentially according to the tentative status of an active set p, which contains only positive components and is initialized to an empty set. A positive component is added to P in the main loop, followed by a possible deletion of some components from P in the inner loop. On termination, x will be the solution vector and y will the dual vector, both satisfying the K~uhnTucker conditions in (458) with C = I and d = 0. An interesting observation is that the active set strategy emploi II in this algorithm tends to find a sparse solution if there exists one, even though the sparsity constraint is not explicitly imposed. This behavior was also reported in many other applications [28, 69, 78, 93]. Furthermore, even though the algorithm finds the solution to (P5) it eratively, it has been proved in [85] that the iteration alrws converges and terminates in a finite number of steps. Hence, this algorithm does not require the tuning of cutoff parameters and the output is not sensitive to the initial guess. The number of iterations to reach the full convergence, as expected, depends on the amount of noise in the measure ments. However, a fairly satisfactory solution can often be achieved well before the full convergence, since the solution gets improved smoothly with iteration. Since the number of the insertion/deletion operations is proportional to the size of the solution vector, an active set algorithm can be slow for large scale problems. In the Algorithm 1: The active set algorithm for NNLS [85, Ch. 23] Input : AE RmX", bE Rm Output: Ai > 0 such that ^< = arg minIAx b2 1 begin 2 x := 0; y := AT(b Ax) 3 Z:= {1,2,..., n}; p:= 0 4 while Z / 0 and maxiez yi > 0 do .5 3 = arg maxiez yi B Move index j from Z to P 7 Solve the least squares  Cj,? zyAj b where Aj is the column j of A s zj =0Ofor j EZ 9 while min(zy) < 0 do to a = miniap Xi/(Xi ze) 11 x = x + a~(z x) 12 Find index j in P such that xj = 0 13 MOVe index j from P to Z 14 Solve the least squares  C ,~ zy~ A b 1s zj = 0 for j EZ 17 y = AT(b Ax) Is end context of our reconstruction problem, the size of the matrix is relatively small, at most hundreds by hundreds, which makes the active set method still a reasonable choice over some other algorithms for large scale problems [35]. However, in real reconstruction of volume data, we do have to solve the NNLS problem voxelwise. Unlike the unconstrained least squares and regularized least squares where the pseudoinverse or the damped inverse can be computed only once and reused for multiple righthand sides, the active set method usually has to solve different sequences of subproblems in its inner loop. To alleviate this problem, a recent variant of NNLS algorithm [23] is able to avoid the many unnecessary recomputations by rearranging the calculations in the standard active set method on the basis of combinatorial reasoning. This socalled fast combinatorial NNLS (FCNNLS) has been tested in our experiments and proved much faster than the standard NNLS algorithm in the real MR volume data. CHAPTER 5 EXPERIMENTAL RESULTS In this chapter, we present experimental results on both simulations and real data to validate the proposed model and methods we have investigated in Chapter 4. Portions in this chapter are reprinted, with permission, from [73, 75, 77]. 5.1 Simulations In order to compare the performance of the deconvolution methods described in the previous chapter, a series of experiments were first performed on simulated data. Throughout this chapter, the signals were simulated by using the exact form of the MR signal attenuation derived from the diffusion propagator for particles diffusing inside cylindrical boundaries [137] which can be considered as a simplified model for diffusion inside real neural tissues. For diffusion within a cylinder of length I and radius r, the signal attenuation with diffusion coefficient D obtained by Soderman and Jansson [137] is as follows: OC 0 002Kamr2(2?iqr)4Sin2(20)a m E(q, 8, A)=22 n=0 k=1 m=0 [(nr/1)a a(2r cos 8)"2 x[1 (1)" cos(2;ql cos 8)] [J (2;qr sin 8)]2(51 12 [am, 2xqpsin)!2] 2a: m 2 In Eq.(51), 8 is the angle between the direction of the magnetic field gradient and the symmetry axis of the cylinder, a is the time separation, Jm is the mth order Bessel function, akm, is the kth solution of the equation J (a) = 0 with the convention a~lo = 0, and Kam's are constants defined by Kam = 21">o21m>o where 1A iS the indicator function on a set A. As in [119, 154], in our simulations we used the following parameters: 1 = 5 mm, r = 5 pm, D = 2.02 x 103 mm2 S, a = 20.8 ms, 6 = 2.4 ms, b = 1500 s/mm2 and the infinite series were truncated at n = 1000 and k, m = 10. To simulate 2 and 3fiber Figure 51. HARDI simulations of 1, 2, and :$fibers (b = 1500s/nnn2) ViSualized in Qhall ODF surfaces using [5, (21)] with spherical harmonics expansion terminated up to I = 8. The orientation configurations used in the simulation: azimuthal angles 01 = :300 2 = { 200, 1000 }, 3 = { 200, 750, 1:350 }; polar angles are all 900. Reproduced with permission from [7:3] @[2007] IEEE. geometries, we simply used an additive model based on the Eq.(51) by mixing signals simulated from two or three cylinders with known orientations. The gradient directions were chosen to point toward 81 vertices sampled on a unit hemisphere from the secondorder icosahedral tessellation. The orientations in our 1, 2 and :$fiber configurations are specified by the azimuthal angles of $1 = :300, 2a = {200, 100"} and #:3 = {200, 750, 1350} respectively. Polar angles for all fibers were taken to be 8 = 900, so that a view from the x axis will clearly depict the individual fiber orientations. Figure 51 shows the corresponding orientation distribution function profiles computed from a modelindependent qhall ODF formula derived in [5, Eq. (21)] with spherical harmonics expansion up to I = 8. For the experiments using simulation data, the deconvolution methods described in the previous chapter were all implemented using MATLAB as follows: The pseudoinverse method w = pinv(A)s as in (P1) where piny is MATLAB's builtin function for computing pseudo inverse of a matrix; The damped least squares method as a solver of the Tikhonov regularization problem (P2) aS Well aS the GCV criterion were implemented using the MATLAB regularization toolbox developed by Hansen [6:3]; The two different L1 minimization methods (?3) ndil (P4) were imp~lll~lemete using the L1MAGIC MATLAB package; *The nonnegative least squares method was implemented by the the 1\ATLAB optimization toolbox function, Isquonneg. The first experiment was done on the 1fiber HARDI simulation data shown in Figfure 51 without noise. The deconvolution problem to be solved was formulated with the matrix A being constructed by using the Wishart model with p = oc and the tessellation containing :321 unit vectors sampled from a hemisphere. Since the given signal, on the righthand side, is 81 dimensional, the matrix A is of size 81 x :321 and the unknown of this underdetermined system is a :321 dimensional weight vector. Figure 52 plots the results of w obtained from these methods. The initial guess for w was set to the zero vector for all the iterative methods (P3), (P4) and (P5). A qualitative impression of these methods is clearly indicated from Figure 52. The least squares solution to (P1) contains a large number of negative weights and has a relatively large magnitude. A zerothorder Tikhonov regfularization is able to reduce the magnitude significantly but does not help achieve the sparsity and nonnegativity. By minimizing the L1 norm with equality or quadratic constraints, (P3) yields a relative sparse solution, but the magnitude and the negative values are not well controlled. The result obtained by (P4) has better sparsity and nonnegativity. Evidently, the best result is produced by solving (P5) using the nonnegative linear least squares. Among the :321 components, only two are significant spikes, both of which lie in the neighborhood of true fiber orientation (:$Oo 90o). It is important to note that the true fiber orientations do not necessarily occur at the maxima of the discrete w vector. Although all of these different results of w actually lead to very similar displacement probability functions P(r) after taking the Fourier transform, a sparse positive representation of w is much more desirable as it provides very good initial guess for estimating the extrema of P(r) which correspond to the fiber orientations. A more realistic comparison on noisy 2fiber simulation profiles is presented in Figure 51. The noisy profiles were created by adding Riciandistributed noise with increasing noise levels (o = .01,.02,...,.10). And for each noise level, we generated 100 random (P1) Pseudoinverse solution 0.2 p3) MinL1 (Pg) Tikhonov regularization 0.15 with equality constraints 001 1 0.05 4 0 100 200 300 0 100 200 300 0 100 200 300 1.5 0.15 (31.7, 90.0) (P4) with t = 0.1 (P4) with t = 0.5 0.8 1~ 0.1 0.61 (P5) 0.51 0.05~11 Nonnegative least squares 0.4 0 =  1 C / II p 'I 0.2 (23.8, 90.0) 0.5 0.05 0 0 100 200 300 0 100 200 300 0 100 200 300 Figure 52. Results of w on 1fiber HARDI simulation data using different deconvolution methods. The xaxis shows the indices of w while the yaxis presents the corresponding numerical values of w. The matrix A is of size 81 x 321 and is built from the tensor kernel model and Wishart basis with p c o, A = {0.0015, 0.0004, 0.0004}. Reproduced from [73] with permission. @ [2007] IEEE. samples. Then, we formulate the problem as in Eq. (443) using the 321 tessellation vectors and the Wishart model with p= 2. In this experiment, the following numerical methods were tested to recover the weight vector w: (a) the pseudoinverse method as in (P1); (b) the damped least squares as in (P2) (the damping factor is empirically chosen to be So which gives quite satisfactory results); (c) the damped least squares method with the damping factor determined using the GCV criterion; (d) the MinL1 norm with equality constraints as in (P3); (e) the MinL1 norm with quadratic constraints as in (P4) (e = 1.0) ; (f) the nonnegative least squares as in (Ps) 2 In all these methods, the deconvolution formulation is derived from the mixture of Wisharts model [75]. The GCV solution was used as the initial guess in both the methods (d) and (e). For the nonnegative least squares (NNLS) approach, the initial guess for w was ahrl . set to the zero vector. As discussed in OsI Ilpter 4, once the deconvolution problem formulated from the mixture of Wisharts model is solved, quantities including the probability displacement function restricted on a sphere, P(r = r), as in DOT [119], radial integral of P(r), as in QBI [149], and the integral of P(r)T2 aS in DSI [160] are analytically computable. In this experiment, the f P(r)T2dT iS chosen as the quantity to be compared and all the resulting functions are represented by spherical harmonics expansions terminated after order 1 = 8. First, to assess the noise resistance of these methods, for each noise level, we cal culated the similarity between the resulting probability profile and the corresponding noiseless profile using the angular correlation coefficient formula [5, Eq. (71)] TA"= =" (52) [E m= U1 wm2 1/2 1 1Im=1l 2 1/2 where {wm}) and {<.. } are the spherical harmonic coefficients of the two functions to be compared. The range of the angular correlation coefficient is from 0 to 1 where 1 implies identical orientational profiles. We also estimated the fiber orientations of each system by numerically finding the maxima of the spherical functions with a QuI I;N. i.ton numerical optimization algorithm and computed the deviation angles between the estimated and the true fiber orientations. As expected, the results of method (a), the pseudoinverse method (P1), degrades quickly as the noise level increases. It has also been observed that method (e), MinL1 with quadratic constraints (P4), produces much worse results than the other four methods (b,c,d,f). In general, the results of method (d), MinL1 with equality constraints (P3), arT very close to the results of method (c), the Damped Least Squares method with damping factor determined by GCV, due to the fact that method (d) starts with the solution of method (c). Figure 53 shows the mean and standard deviation of the angular correlations and the deviation angles in the twofiber experiment, respectively. In these two figures, we do not include the results of the pseudoinverse method (P1) and the results of MinL1 with quadratic constraints (P4) since both methods are extremely sensitive to the noise according to our observations. The angular correlations obtained using the four methods shown in Figure 5:3 are all very high (> 0.9). While the GCV and MinL1 with equality constraints are slightly worse with relatively larger standard deviations. It is also observed that the angular correlations produced by the NNLS method are very close to 1 and have the smallest standard deviation. The DLS method with the damping factor empirically chosen to be So also exhibits quite satisfactory stability. In terms of the accuracy of the fiber orientation estimation, it is also clear from Figure 53 that the NNLS method gives the most accurate results consistently. While there is no significant difference between the other three methods. Note that in all the cases, MinL1 methods with both equality and quadratic constraints were initialized with the solution returned by the GCV method and the NNLS method alval started with a zero vector. Finally, as a conclusion to our experiments on the simulated data, a quantitative comparison was performed among the proposed method mixture of Wisharts (\l OW) model and the two modelfree methods, namely, the Qhall ODF [149] and the DOT [119]. All the resulting P(r) surfaces were represented by spherical harmonics coefficients up to order 1 = 6 and the Qhall ODF is computed using the formula in [5, Eq. (21)]. First, to gain a global assessment of these methods in terms of stability, we calculated the similarity between each noisy P(r) and the corresponding noiseless P(r) using the angular correlation coefficient formula given in [5, Eq. (71)]. The angular correlation ranges from 0 to 1 where 1 implies identical probability profiles. Then, we estimated the fiber orientations of each system by finding the maxima of the probability surfaces with a CMI I;N Eton numerical optimization algorithm and computed the deviation angles between the estimated and the true fiber orientations. Figure 54 shows the mean and standard deviation of the angular correlation coefficients, and error angles, respectively, for the twofiber simulation. Note that among the three methods examined, only MOW results in small error angles and high correlation coefficients in presence of relatively large noise. This trend also holds for the 1fiber and the 3fiber simulations. This can be explained by noting that NNLS is able to locate the sparse spikes quite accurately even in the presence of a lot of noise. 5.2 Real Data Experiments The aforementioned reconstruction methods were tested on two real datasets provided to us by researchers with the McE~night Brain Institute at the University of Florida. The first dataset was acquired from a perfusionfixed excised rat optic chiasm at 14.1 Telsa using a Bruker Avance imaging system (Bruker NMR Instruments, Billerica, MA) with a diffusionweighted spin echo pulse sequence. Note that the rat optic chiasm is well suited for the validation of the fiber reconstruction results because of its distinct myelinated structure with both parallel and decussating (crossing) optic nerve fibers. The imaging parameters used for the optic chiasm dataset are shown in Table 51. In this optic chiasm dataset, there are 46 images acquired from 46 gradient directions at a bvalue of 1250s/mm2 and 6 additional images acquired at b a Os/mm2. Echo time and repetition time were 23ms and 0.5s respectively; a and 6 values were set to 12.4ms and 1.2ms respectively; bandwidth was set to 35kHz; signal average was 10; a volume size of 128 x 128 x 5 and a resolution of 33.6 x 33.6 x 200pm3 WaS used. The optic chiasm images were downsampled to 67.2 x 67.2 x 200pm3 TOSolution before the subsequent computation. Two sets of experiments were performed on this optic chiasm dataset. The first experiment was designed to compare the four different numerical methods discussed in Section 4.4, namely, (a) damped least squares with GCV, (b) MinL1 with quadratic constraints, (c) damped least squares with a fixed regularization parameter, and (d)  I IP2)Damped Least Squares (DLS) IP2)DLSIGCV i (P3)M nLlwth equa ty ~ IPS)Nonnegatue Least Squares O S098~ O 0.8 0.0 0 Figure 54. std. dev. of noise std. dev. of noise (a) (b) Mean and standard deviation of (a) angular correlation coefficient and (b) error angles for the twofiber simulation. At each noise level, the four methods compared here are (1) the DLS method as in (P2) (the damping factor is empirically chosen to be So); (2) the DLS method with the damping factor determined using the GCV criterion; (3) the MinL1 norm with equality constraints as in (P3) initializedd with the solution obtained by DLS+GCV; (4) the NNLS method as in (Ps). The di;11lai II values are averaged over the two fibers. Reproduced from [73] with permission. @[2007] IEEE.  DOT QBI *MOW .DOT QBI *MOW 0.05 std. dev. of noise 0.05 std. dev. of noise The plots on the left and on the right show the statistics of angular correlation coefficients and error angles for the 2fiber simulation, respectively. The di1 pl i 4I values for error angles are averaged over the two fiber orientations. Reproduced from [77] with permission. @[2009] Springer. Table 51. Imaging parameters used for the optic chiasm dataset. Imaging parameters used for the optic chiasm dataset magnetic field strength 14.1 Telsa gradient directions 46 at high bvalue, 6 at low bvalue high bvalue 1250s/mm2 low bvalue a Os/mm2 echo time (TE) 23ms repetition time (TR) 0.5s gradient pulse separation (A) 12.4ms gradient pulse duration (5) 1.2ms bandwidth 35kHZ signal averaging 10 volume size 128 x 128 x 5 voxel size 33.6 x 33.6 x 200p~m3 nonnegative least squares (NNLS). The resulting displacement probability profiles on a region of interest are shown in Figure 55. For each method, the corresponding So image is also shown in the upper left corner of each panel as a reference. For the sake of clarity, we excluded every other voxel and overlaid the probability surfaces on the generalized anisotropy (GA) maps [117]. GA is the variance of normalized diffusivity function. Higher values of GA (brighter regions) indicate higher anisotropy. As can be seen from Figure 55, the fibercrossingf in the optic chiasm region is not identifiable in either results obtained from the method using GCV (a) or the results obtained from the method of MinL1 with quadratic constraints (b), while both the DLS method (c) and the NNLS method (d) are able to demonstrate the distinct fiber orientations in the central region of the optic chiasm where ipsilateral m linated axons from the two optic nerves cross and form the contralateral optic tracts. The failure of the GCV method is due to the fact that the damping factors estimated by the GCV method may not ak li be optimal. The MinL1 method with quadratic constraints only yields satisfactory results in regions mostly populated with single fiber, which is consistent to the previous observation on the 2fiber simulation. It is also evident from Figure 55 that compared to other three numerical methods, the NNLS scheme yields significantly sharper displacement probability profiles. Similar to the experiments on the simulated data, we also compare the proposed mix ture of Wisharts (\lOW) model with two modelfree methods, namely, the QBIODF[149] and the DOT [119]. The rest of this paragraph is reprinted from [77] with permission and is copyrighted by Springer. Figure 56 shows the reconstruction results generated using four different methods, namely, (a) QBIODF, (b) DOT, (c) MOW+Tikhonov regular ization, and (d) MOW+NNLS, on the same region of interest shown in Figure 55. The fast combinatorial NNLS method [23] is used here as the NNLS solver. The computation time for this region of interest containing 1024 voxels is less than 0.5 second for all four methods on an Intel Core Duo 2.16 GHz CPU while the standard NNLS takes about 8 seconds. As seen in the Figure 56, the fibercrossingfs in the optic chiasm region cannot he identified by using the QBIODF method. Note that both the DOT method and the MOW method with two different schemes are able to demonstrate the distinct fiber ori entations in the central region of the optic chiasm. However, it is evident from the figure that compared to all other solutions, the MOW technique in conjunction with the NNLS scheme yields significantly sharper displacement probability surfaces. This is particularly horne out in the optic chiasm, in the center of each panel. The probability surfaces in the QBI and DOT models are blurred, in part, because both vield a corrupted P(r) rather than the actual displacement probability surfaces.The corrupting factor for the QBI is a zeroth order Bessel function, for the DOT method it is a function that does not have an analytic form. This corruption affects the accuracy of the reconstructed fiber orientations as evidenced in the simulated data case where the ground truth was known. Note that validating the precise angle of the fiber crossing in this real data set is nontrivial as it will need a histology stack to be created and then fiber directions estimated from this stack to be validated against those obtained from the DWAIRI data. To investigate the capability of diffusion weighted imaging in revealing the effects in local tissue caused by diseases or neurologic disorders, further experiments were carried optic nervp /)yj t;* & M// j //////###1/// optic &~s~b~9#J//QH~+$ ChiaSm ~"ff 4 optic tract ~u~~idM/98~3ft (a) Inisj J sis 3 optic nervp / / 1' / # #207// ~u~p#p/)######ly~ ChiaSm ,~d~~U~9~ aft oWLptic tract p ,,,,,,,/B ,, cmas opti nr opictrc (b) I st M I/I opicneve/ / 4 / < *4/p we .,47,ogyE. cmiasm =ws+tit ptic tract Probability maps computed using (a) damped least squares with GCV; (b) MinL1 with quadratic constraints (e = 1) initialized from (a);(c) damped least squares with fixed regularization parameter (a~ = 0.6); ( d) nonnegative least squares from a rat optic chiasm data set overlaid on axially oriented GA [114] maps. The decussations of myelinated axons from the two optic nerves at the center of the optic chiasm are readily apparent. In all the plates, the corresponding reference (So) image is shown in the upper left corner. Reproduced from [73] with permission. @[2007] IEEE. Figure 55. optic nerve ))J'di3///// L &///s,~~uodgyj optic JJ.~~~~C~~IeS ddjgJjIty cniasm (a) QBI3J optic tract opt Qicnene s //i fe /// *** &// ess tge op4 c *** ~44 &/O#$$# nam R Iaic nev optic tract d Figue 5. Prbabiity surface coptdfr optcne'e 7 944/// **>>>Ms $$l cmasm optic nerve ,)//b+/+///// Nd MSOW optictrc m~ ~ ~ a rat optic chas iae sig a QBIODF, (b) DOT, (c) MOW+Tikhonov regularization, and (d) MOW+NNLS. Note the decussation of row;linated axons from the two optic nerves at the center of the optic chiasm. Reproduced from [77] with permission. @ [2009] Springer. out on two data sets collected from a pair of epileptic/normal rat brains. The rest of this section is reprinted from [75] with permission and is copyrighted by Elsevier. Table 52. Imaging parameters used for the rat brain dataset. Imaging parameters used for the rat brain dataset magnetic field strength 17.6 Telsa gradient directions 46 at high bvalue, 6 at low bvalue high bvalue 1250s/mm2 low bvalue 100s/mm2 echo time (TE) 28ms repetition time (TR) 1.4s gradient pulse separation (A) 17.5ms gradient pulse duration (5) 1.5ms bandwidth 750M~HZ signal averaging 20 for high bvalue and 5 for low bvalue volume size 200 x 100 x 32 voxel size 150 x 150 x 300p~m3 The imaging parameters used for the rat brain dataset are shown in Table 52. The multipleslice diffusion weighted image data were measured at 750 MHz using a 17.6 Tesla, 89 mm bore magnet with Bruker Avance console (Bruker NMR Instruments, Billerica, MA). A spinecho, pulsedfieldgradient sequence was used for data acquisition with a repetition time of 1400 ms and an echo time of 28 ms. The diffusion weighted gradient pulses were 1.5 ms long and separated by 17.5 ms. A total of 32 slices, with a thickness of 0.3 mm, were measured with an orientation parallel to the longaxis of the brain (slices progressed in the dorsalventral direction). These slices have a fieldofview 30 mm x 15 mm in a matrix of 200 x 100. The diffusion weighted images were interpolated to a matrix of 400 x 200 for each slice. Each image was measured with 2 diffusion weigfhtingfs: 100 and 1250s/mm2. Diffusionweighted images with 100s/mm2 Were measured in 6 gradient directions determined by the vertices of an icosahedron in one of the hemispheres. The images with a diffusionweighting of 1250s/mm2 Were measured in 46 gradientdirections , which are determined by the tessellation of the icosahedron on the same hemisphere. The 100s/mm2 images were acquired with 20 signal averages and the 1250s/mm2 images were acquired with 5 signal averages in a total measurement time of approximately 14 hours. Figure 57 shows the displacement probabilities calculated front excised coronal rat brain 1\RI data in a, (a) control and (b) an epileptic rat. The hippocanipus and entorhinal cortex is expanded and depicts the orientations of the highly anisotropic and coherent fibers. Note voxels with crossing orientations located in the dentate gyrus (dg) and entorhinal cortex (ec). The region superior to CA1 represent the stratum lacunosuni moleculare and statunt radiatunt. Note that in the control hippocanipus, the molecular 1 .,< c and stratum radiatunt fiber orientations paralleled the apical dendrites of granule cells and pyramidal neurons respectively. In the epileptic hippocanipus, the CA1 subfield pyramidal cell 1 .,< c is notably lost relative to the control. The architecture of the dentate gyrus is also notably altered with more evidence of crossing fibers. Future investigations employing this method should improve our understanding of normal and pathologically altered neum~ .Il 1Inyl~: in regions of complex fiber architecture such as the hippocanipus and entorhinal cortex. 22 s DG S~tS$SSSA dd ~iirid Hilu Figure 57. Probability maps of coronally oriented GA images of a control and an epileptic hippocampuus. Upper left corner shows the corresponding reference (SO) images where the rectangle regions enclose the hippocampi. In the control hippocampus, the molecular 1 ;r and stratum radiatum fiber orientations paralleled the apical dendrites of granule cells and pyramidal neurons respectively, whereas in the stratum lacunosum, molecular orientations paralleled Schaffer collaterals from CA1 neurons. In the epileptic hippocampus, the overall architecture is notably altered; the CA1 subfield is lost, while an increase in crossing fibers can he seen in the hilus and dentate gyrus (dg). Increased crossing fibers can also be seen in the entorhinal cortex (ec). Fiber density within the statum lacunosum molecular and statum radiale is also notably reduced, although fiber orientation remains unaltered. Reproduced with permission from [75] @[2007] Elsevier. .: ::. CA3 SSSrI Go W~Iilus1 Oddestd//dd d o as SSlllllltfijdS IU~~1Y ~ Al1ddstaiddsra **###ekkthdd 445St (l#ba~theb66 Sanr~r __ (b) epileptic CHAPTER 6 DISCUSSION AND CONCLUSIONS 6.1 Summary Diffusionweighted magnetic resonance imaging (DWMRI) is a noninvasive imaging technique that allows neural tissue architecture to be probed at a microscopic scale in vivo. By producing quantitative measurements of on water molecular motion, DW MRI can be processed to map the fiber paths in the brain white matter. This valuable information can be further exploited for neuronal connectivity inference and brain developmental studies [87]. In this dissertation, a novel mathematical model for the diffusion weighted MR signal attenuation is presented. The key postulation of the proposed model is that at each voxel the diffusion of water molecules is characterized by a continuous mixture of diffusion tensors. An interesting observation based on this continuous tensor distribution model is that the MR signal attenuation can be expressed as the Laplace transform of the associated tensor probability distribution function. It has also been show that when the mixing distribution is parameterized by Wishart distributions, the resulting close form of Laplace transform leads to a Rigauttype fractal expression. This Rigauttype function exhibits the expected .imptotic powerlaw behavior and has been phenomenologically used in the past to explain the MR signal decay but never with a rigorous mathematical justification until the development of the proposed model [7476]. It is easy to show that both the traditional diffusion tensor model and the multitensor model are limiting cases of this continuous mixture of tensors model. Additionally, in the long wavelength or lowq regime the proposed model leads to a quadratic decay which is consistent with the traditional diffusion tensor model [14]. Another contribution of this work is on tackling the challenging problem of multifiber reconstruction from the diffusion MR images. The mixture of Wisharts model, as a natu ral parametrization of the desired tensor distribution function, is used to describe complex tissue structure involving multiple fiber populations. The multifiber reconstruction is then formulated in a unified deconvolution framework [72, 73] which also includes some other previously published approaches in this field. One central topic of this work is to investigate several different deconvolution methods in the context of multifiber recon struction in diffusion AIRI. Experiments on both simulations and real data have shown the favorable results from the proposed mixture of Wisharts (\lOW) model in conjunction with the nonnegative least squares (NNLS) method, comparing with competing models and methods from literature. 6.2 Open Problems In this dissertation, we have developed a novel mathematical model which depicts the diffusion AIR signal attenuation reasonably well, supported by both theoretical analysis and experimental results. Regrettably but fortunately, there remain a number of open problems . 6.2.1 Nonparametric Inverse Laplace Transform The key postulation of the proposed mathematical model is that at each voxel the diffusion of water molecules is characterized by a continuous mixture of diffusion tensors. Based on this assumption, we also reveal the Laplace transform relationship between the signal attenuation and the underlying tensor probability distribution. In our approach, this tensor distribution is modeled by wishart distributions or mixture of Wisharts. This choice is mainly justified by the existing closed form Laplace transform of Wishart distributions which also yields the expected Rigauttype fractal expression. Though it is tempting and maybe more convincing to directly invert the Laplace transform in a nonparametric way, one has to recognize the difficulties entailed in inverting Laplace transform in the high dimension from the noisy data (see [54] for "cogent reasons for the general sense of dread most mathematicians feel about inverting the Laplace ti Ion1. .i ts ). Recent work by Leow et al. [90] is very similar to our model in that a tensor distri bution function (TDF) is associated to each voxel and the inverse problem there is also to recover this tensor distribution function that best explains the observed diffusion AIR signal. In [90], the domain of the TDF is a fourparanleter space which is reduced front the original 6dintensional tensor space by assuming that fiber tracts are cylindrical. Then a gradient descent method is used to solve for an optimal TDF that nminintizes the difference between the observed signal and the predicted signal in the leastsquare sense. The current mixture of Wisharts (\lOW) model eniploi. .1in this work actually reduces to a spherical deconvolution model by assuming the cylinder syninetry and fixing the eigenvalues of the tensor parameter in each Wishart components. These simplifying assumptions lead to a linear system and hence can he solved efficiently. An alternative but much more compli cated deconvolution method would be a B lio Io approach where all the parameters in the MOW model including the tensor eigfenvalues and the weights can he imposed with suitable prior distributions. 6.2.2 Adaptive Sparse Dictionary Learning We have shown that several existing nmultifiber reconstruction models can he ex pressed in a unified deconvolution framework model and reformulated into the problem of seeking a sparse and positive solution to a linear system [7:3]. In most existing approaches that lie in this unified deconvolution framework, the system matrix, A, or the socalled dictionary in the context of signal representation theory, is fixed depending on the choices of the convolution kernel, the paranietrization of the volume fraction function, and the discretization scheme. The exception is a recent work proposed in [1:3] where the shape of the convolution kernel function is estimated simultaneously with the rest of the unknowns of the model. However, the basis function of this adaptive kernel is still fixed as a spline function. A significant extension to [1:3, 7:3] would be a supervised learning hased approach that infers a learned dictionary front the training data instead of a predefined dictionary as in [52, 100] and furthermore allows a sparse encoding of the diffusion signal using this learned dictionary. 6.2.3 Subvoxel Fiber Bundles Classification It has been widely recognized that the diffusion tensor model does not perform well in regions containing intravoxel orientational heterogeneity. This limitation of the diffusion tensor model has prompted the development of many multifiber reconstruction methods, including those reviewed in Section 3.3, such as higher order tensor it., 1_;ng [7 14], diffusion spectrum ill. 1_;hly [7 60], qhall ill. 1_;ity [7 49], probabilistic PASAIRI[70], spherical deconvolution[14:3], diffusion orientation transform[119], and the mixture of Wisharts model [75] proposed in this work. However, to the best of our knowledge, none of these approaches is able to differentiate between complex configurations of intravoxel fiber bundles, e~g fibers crossing, kissing, bending or twisting within a voxel. In other words, these methods still suffer from topological ambiguity, which can not he easily resolved by only estimating fiber orientations. We believe that the ability of successfully labelling voxels into distinct subvoxel fiber bundle configurations would be crucial in the next generation of fiber tractography algorithms. Recently, UCniie7lj~ v et al. [1:34] proposed a curve inference method that uses differential geometric estimates in a local neighborhood to differentiate the structures of fanning and curving fiber bundles. Barmpoutis et al. [11] presented a novel method for estimating a field of .limmetric spherical functions, dubbed tractosemas, which can he used to model .Iimmetries such as pl wing fibers. The mixture of Wisharts model proposed in this work has been shown in [9] as an excellent generative model for depicting the diffusion AIR signal attenuation. It would be an interesting future research topic to investigate if the combination of this generative model and some discriminative models can he potentially used to distinguish different subvoxel fiber bundle categories and improve the performance of the fiber pathway reconstruction. REFERENCES [1] I. Aganj, C. Lenglet, and G. Sapiro. ODF reconstruction in qball imaging with solid angle consideration. 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I After he completed his undergraduate studies in computer science at the University of Science and Technology of CluI, I in July 1999, he chose to continue his studies at the same university, which earned him a master's degree in electrical engineering in July 2002. He then joined the Ph.D. program of the Department of Computer and Information Science and Engineering at the University of Florida in August 2002. After his first year in the Ph.D. program, he started working in the field of computer vision and medical image analysis. His research interests cover a wide range of topics from computer vision and machine learning to medical imaging and computer aided diagnosis. He hopes to make contributions to our society through innovative research on where these disciplines meet. 