Advanced imaging tools for quantifying cardiac microstructure

Diffusion tensor MRI (DT-MRI or DTI) has been proven useful for characterizing biological tissue microstructure, with the majority of DTI studies having been performed previously in the brain. Other studies have shown that changes in DTI parameters are detectable in the presence of cardiac pathology, recovery, and development, and provide insight into the microstructural mechanisms of these processes. However, the technical challenges of implementing cardiac DTI in vivo, including prohibitive scan times inherent to DTI and measuring small-scale diffusion in the beating heart, have limited its widespread usage. This research aims to address these technical challenges by: (1) formulating a model-based reconstruction algorithm to accurately estimate DTI parameters directly from fewer MRI measurements and (2) designing novel diffusion encoding MRI pulse sequences that compensate for the higher-order motion of the beating heart. The model-based reconstruction method was tested on undersampled DTI data and its performance was compared against other state-of-the-art reconstruction algorithms. Model-based reconstruction was shown to produce DTI parameter maps with less blurring and noise and to estimate global DTI parameters more accurately than alternative methods. Through numerical simulations and experimental demonstrations in live rats, higher-order motion compensated diffusion-encoding was shown to successfully eliminate signal loss due to motion, which in turn produced data of sufficient quality to accurately estimate DTI parameters, such as fiber helix angle. Ultimately, the model-based reconstruction and higher-order motion compensation methods were combined to characterize changes in the cardiac microstructure in a rat model with inducible arterial hypertension in order to demonstrate the ability of cardiac DTI to detect pathological changes in living myocardium.

ADVANCED IMAGING TOOLS FOR QUANTIFYING CARDIAC MICROSTRUCTURE by Christopher Lee Welsh A dissertation submitted to the faculty of The University of Utah in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Bioengineering The University of Utah August 2015 Copyright © Christopher Lee Welsh 2015 All Rights Reserved The Univ e r s i t y of Utah Graduat e School STATEMENT OF DISSERTATION APPROVAL The dissertation of Christopher Lee Welsh has been approved by the following supervisory committee members: Edward W. Hsu Chair 5/26/2015 Date Approved Edward V.R. DiBella Member 5/26/2015 Date Approved Rob S. MacLeod Member 5/26/2015 Date Approved Sarang Joshi Member 5/26/2015 Date Approved Yufeng Huang Member 5/26/2015 Date Approved and by Patrick A. Tresco Chair/Dean of the Department/College/School o f ________________ Bioengineering and by David B. Kieda, Dean of The Graduate School. ABSTRACT Diffusion tensor MRI (DT-MRI or DTI) has been proven useful for characterizing biological tissue microstructure, with the majority of DTI studies having been performed previously in the brain. Other studies have shown that changes in DTI parameters are detectable in the presence of cardiac pathology, recovery, and development, and provide insight into the microstructural mechanisms of these processes. However, the technical challenges of implementing cardiac DTI in vivo, including prohibitive scan times inherent to DTI and measuring small-scale diffusion in the beating heart, have limited its widespread usage. This research aims to address these technical challenges by: (1) formulating a model-based reconstruction algorithm to accurately estimate DTI parameters directly from fewer MRI measurements and (2) designing novel diffusion encoding MRI pulse sequences that compensate for the higher-order motion of the beating heart. The model-based reconstruction method was tested on undersampled DTI data and its performance was compared against other state-of-the-art reconstruction algorithms. Model-based reconstruction was shown to produce DTI parameter maps with less blurring and noise and to estimate global DTI parameters more accurately than alternative methods. Through numerical simulations and experimental demonstrations in live rats, higher-order motion compensated diffusion-encoding was shown to successfully eliminate signal loss due to motion, which in turn produced data of sufficient quality to accurately estimate DTI parameters, such as fiber helix angle. Ultimately, the model- based reconstruction and higher-order motion compensation methods were combined to characterize changes in the cardiac microstructure in a rat model with inducible arterial hypertension in order to demonstrate the ability of cardiac DTI to detect pathological changes in living myocardium. iv To my rock, Amy, and our daughter, Ruby. In memory of my friend, Owen Stedham. TABLE OF CONTENTS ABSTRACT........................................................................................................................... iii LIST OF TABLES............................................................................................................... viii LIST OF FIGURES................................................................................................................ ix ACKNOWLEDGEMENTS..................................................................................................xii CHAPTERS 1. INTRODUCTION........................................... .................................................................1 2. BACKGROUND............................................. .........................................................5 2.1 Cardiac Microstructure and Function..... .........................................................5 2.2 Diffusion Tensor Imaging........................ .........................................................7 2.3 Cardiac Diffusion Tensor Imaging......... .......................................................15 2.4 Practical Considerations of Cardiac DTI. .......................................................26 2.5 Accelerating DTI Acquisition.................. .......................................................38 2.6 Conclusion................................................ .......................................................41 2.7 References................................................. .......................................................43 3. MODEL-BASED RECONSTRUCTION OF UNDERSAMPLED DIFFUSION TENSOR K-SPACE DATA.......................... .......................................................54 3.1 Abstract..................................................... .......................................................54 3.2 Introduction............................................... .......................................................55 3.3 Theory........................................................ .......................................................59 3.4 Methods.................................................... .......................................................61 3.5 Results........................................................ .......................................................68 3.6 Discussion................................................. .......................................................75 3.7 Conclusion................................................ .......................................................78 3.8 Appendix................................................... .......................................................79 3.9 Funding Sources ...................................... .......................................................82 3.10 Conflicts of Interest............................... .......................................................82 3.11 Statement of Human Studies.........................................................................83 3.12 Statement of Animal Studies.........................................................................83 3.13 References.......................................................................................................84 4. HIGHER-ORDER MOTION-COMPENSATION FOR IN VIVO CARDIAC DIFFUSION TENSOR IMAGING IN RATS......................................................88 4.1 Abstract............................................................................................................. 88 4.2 Introducti on.......................................................................................................89 4.3 Theory............................................................................................................... 92 4.4 Methods............................................................................................................ 97 4.5 Results............................................................................................................. 105 4.6 Discussion.......................................................................................................113 4.7 Funding Sources ........................................................................................... 119 4.8 Conflicts of Interest....................................................................................... 119 4.9 Statement of Human Studies.........................................................................119 4.10 Statement of Animal Studies...................................................................... 119 4.11 References.....................................................................................................120 5. EVALUATION OF MYOCARDIAL RESTRUCTURING IN RATS WITH INDUCED ARTERIAL HYPERTENSION......................................................125 5.1 Introduction.....................................................................................................125 5.2 Methods.......................................................................................................... 128 5.3 Results............................................................................................................. 135 5.4 Discussion.......................................................................................................148 5.5 Conclusion......................................................................................................153 5.6 Funding Sources............................................................................................ 153 5.7 Conflicts of Interest....................................................................................... 153 5.8 Statement of Human Studies.........................................................................154 5.9 Statement of Animal Studies.........................................................................154 5.10 References.....................................................................................................155 6. CONCLUDING REMARKS...............................................................................161 6.1 Summary......................................................................................................... 161 6.2 Future Directions........................................................................................... 163 6.3 Final Thoughts................................................................................................167 6.4 References.......................................................................................................168 vii LIST OF TABLES 3.1. Performance of DTI acceleration schemes in terms of fiber orientation, FA, and MD errors for an acceleration factor R = 2 .............................................. 71 3.2. Performance of DTI acceleration schemes in terms of fiber orientation, FA, and MD errors for an acceleration factor R = 4 .............................................. 72 4.1 Mean error in the presence of intravoxel phase dispersion..................................106 4.2 Mean error in the presence of shot-to-shot phase variation................................ 108 4.3 Mean error in the presence of shot-to-shot phase variation................................ 110 LIST OF FIGURES 2.1 A pair of gradient pulses used to sensitize MRI to diffusion....................................9 2.2 Sample diffusion-weighted images of an ex vivo heart...........................................11 2.3 Diffusion ellipsoid......................................................................................................14 2.4 Varying fractional anisotropy and mean diffusivity................................................ 16 2.5 Diffusion tensor parameter maps from an ex vivo heart..........................................17 2.6 DTI parameters maps..................................................................................................17 2.7 Diffusion-weighted GRE and SE pulse sequences.................................................. 27 2.8 Spin-echo, echo-planar imaging (EPI) pulse sequence...........................................28 2.9 Stimulated-echo acquisition mode (STEAM) pulse sequence............................... 36 2.10FA and helix angle maps from accelerated DTI data.............................................. 39 3.1 Sample diffusion-weighted images of specimen used in the study........................62 3.2 Sampling masks.......................................................................................................... 64 3.3 Flowchart of the model-based reconstruction algorithm.........................................65 3.4 FA-weighted fiber orientation maps..........................................................................69 3.5 Mean diffusivity maps................................................................................................69 3.6 Distribution of FA, MD, and primary eigenvector deviation..................................70 3.7 Results of pairwise, post hoc analysis of the five DTI acceleration schemes from Tables 3.1 and 3.2............................................................................................. 73 3.8 Scan time efficiency of model-based reconstruction............................................... 74 4.1. Spin-echo diffusion encoding schemes for higher-order motion compensation .... 95 4.2 Creation of the 3D numerical motion phantom........................................................99 4.3 Effectiveness of intravoxel phase dispersion compensation at systole and diastole.......................................................................................................................105 4.4 Effectiveness of motion compensation at systole and diastole............................. 107 4.5 Effectiveness of motion compensation and cardiac cycle consistency.................109 4.6 Diffusion-weighted images of the heart obtained on a live rat with various degrees of motion compensation.............................................................................111 4.7 DTI images obtained on a live rat using velocity-, acceleration-, and jerk-compensated diffusion encoding in the same cardiac short-axis slice..................112 4.8 Unsmoothed DTI images obtained in four live rats............................................... 114 4.9 Diagram of acceleration-compensated diffusion encoding with constant amplitude gradients...................................................................................................116 4.10 Diagram of acceleration-compensated diffusion encoding incorporated in a STEAM preparation.......................................................................................... 117 5.1. Short-axis cardiac morphology................................................................................136 5.2 Measurements of cardiac morphology.................................................................... 137 5.3 Analysis of cardiac function.................................................................................... 138 5.4 End-systole fractional anisotropy............................................................................139 5.5 End-systole mean diffusivity...................................................................................140 5.6 End-systole fiber helix angle...................................................................................141 5.7 End-systole sheet angle............................................................................................ 143 5.8 Reduced scan time fractional anisotropy................................................................144 x 5.9 Reduced scan time mean diffusivity....................................................................... 145 5.10 Reduced scan time fiber helix angle..................................................................... 146 5.11 Reduced scan time sheet angle...............................................................................147 5.12 End-systole and end-diastole sheet angle..............................................................150 xi ACKNOWLEDGEMENTS I would not be where I am today without the support of a wide range of people. I am deeply grateful to all those that have supported me throughout the years. During the past six years, my co-advisors, Dr. Edward Hsu and Dr. Edward DiBella, have provided me with invaluable knowledge, skills, and lessons that would not have been achievable elsewhere. I am very grateful for the many hours of one-on-one help I received from each of them. In addition, I would like to thank my lab mates, David Gomez, Samer Merchant, and Osama Abdullah, for their continuous support throughout my time here at the University of Utah. Rarely was there a problem that could not be solved by talking it out with one of them. I would also like to thank those from UCAIR who provided valuable support in my research, including Dr. Ganesh Adluru and Srikant Iyer. Lastly, I would like to thank my wife, Amy. Her unwavering support during the long work hours that took my away from our family was essential for me to get where I am today. I love you. CHAPTER 1 INTRODUCTION Heart diseases remain the top cause of mortality in the Western world, with approximately 600,000 deaths in the U.S. in 2014. Proper diagnosis and treatment of cardiac diseases are necessary for potential recovery and increased quality of life in the affected population. Understanding the mechanisms of cardiac dysfunction is key to providing the correct diagnosis and treatment, which are necessary for improved prognoses for patients with heart disease. Because functions of the heart are mediated by the myocardial microstructures, changes to the myofiber or sheet structures often lead to alternations in mechanical and electrical properties. Characterizing cardiac microstructure can lead to improved detection of heart disease and quantification of the extent, or stage, of disease. In addition, tracking microstructural changes over time can evaluate disease progression or the effectiveness of therapy. All of these can lead to more personal and effective health care for those with heart disease. Cardiac disease and dysfunction are traditionally evaluated using noninvasive techniques such as EKG, echocardiography, and imaging via radiological techniques. The majority of medical imaging techniques are used to evaluate cardiac morphology and global cardiac function, such as ejection fraction. These methods are beneficial for identifying failing hearts, but ultimately do not characterize cardiac microstructure and, therefore, do not identify the low-level mechanisms of dysfunction. Histological examination has been the gold standard for characterizing tissue microstructure in all types of organs, but histology is inherently destructive and invasive. Diffusion tensor imaging (DTI) has emerged as a viable alternative for characterizing biological microstructure in a nondestructive and noninvasive manner by measuring the random diffusional motion of water. DTI in cardiac applications is able to characterize the microstructural arrangement of myocyte bundles, or fibers, and laminar sheets. Cardiac DTI has the potential to correctly detect and stage disease, and provide a means to monitor progression of disease or therapy. However, applications of DTI in the beating heart still face substantial technical challenges before it is ready to be used for diagnosis and monitoring of heart diseases in a clinical setting. This work represents key improvements towards making in vivo cardiac DTI more feasible for quantifying changes in the cardiac microstructure due to disease or recovery. Novel methods for executing cardiac DTI in vivo are presented along with an image reconstruction scheme designed to accurately reconstruct diffusion tensor data from fewer MRI measurements, allowing for shorter acquisition times. These methods were verified in numerical simulations and demonstrated experimentally in live rat models. In the end, the usefulness of these methods in characterizing heart disease and dysfunction are evaluated. Chapter 2 provides a brief background in the preliminary concepts of cardiac microstructure and the use of DTI to characterize it. The chapter details the practical and technical challenges presented by cardiac DTI, particularly in in vivo applications. Recent 2 studies that employ DTI to characterize microstructural changes due to pathology are also presented. In addition, recent advances in DTI that make the technique more practical and feasible are described. Chapter 3 describes a strategy to reconstruct diffusion tensor maps directly from accelerated k-space data. This is accomplished by modifying the objective function in traditional compressed sensing to be a function of the desired diffusion tensor instead of the magnitude of the individual diffusion-weighted images. Because the objective function is a function of the diffusion model, the method is referred to as model-based reconstruction. The proposed method is compared against other more common reconstruction techniques and control cases. A quantitative comparison between the test cases was performed to determine which method produced the most accurate DTI maps from accelerated diffusion data. Chapter 4 describes a methodology for implementing higher-order motion compensation in diffusion-encoding MRI to obtain DTI measurements in the beating heart. The study compares the performance of previously established diffusion-encoding methods, those with no motion compensation and velocity-compensation, to the performance of novel diffusion encodings with acceleration- and jerk-compensation via gradient moment nulling. All methods were evaluated in a realistic numerical phantom of the beating heart and in live rats. Acceleration-compensated diffusion encoding was found to provide the best balance of motion artifact reduction and SNR preservation, which was necessary to derive accurate DTI parameter maps. A preliminary study that combines the methodologies developed in Chapters 3 and 4 is presented in Chapter 5. DTI scans were performed in transgenic rats, using 3 acceleration-compensated diffusion encoding, prior to and two weeks after induction of arterial hypertension to observe changes in the cardiac microstructure due to increased after load. In vivo DTI was essential to observe the changes in key DTI parameters that would otherwise not be detectable if a terminal study was performed. Model-based reconstruction of diffusion tensor maps was performed to show the potential of reducing acquisition time without losing the proportional amount of accuracy. The concluding chapter of this document, Chapter 6, provides a discussion regarding the advantages and disadvantages of acceleration-compensated diffusion encoding for in vivo DTI and reducing scan time with model-based reconstruction, and offers some recommendations for improvements on the methods presented herein as well as future areas of investigation. 4 CHAPTER 2 BACKGROUND 2.1 Cardiac Microstructure and Function Myofiber structure of the heart is an important determinant of its function [1]. The distribution of myofiber orientation within the heart wall is the main determinant of stress distribution and myofiber shortening throughout the wall [2], and therefore, of cardiac perfusion [3] and structural adaptation [4], [5]. Myofiber structure also plays a key role in electrical propagation inside the heart [6]. Myofiber architecture is known to be altered in some cardiac diseases, such as ischemic heart disease and ventricular hypertrophy [7]. Therefore, detailed knowledge of myocardial fiber microstructure promises to lead to better understanding of the heart function in health and disease. Anisotropy is one of the most consistent observations in studies of the heart. It is present in cardiac material and functional properties at essentially all scales. This includes, in the molecular scale, the arrangement of collagen fibers and actin-myosin contractile structures at the subcellular level, the arrangement of myocytes with respect to their neighbors at a cellular level, and the observable texture of the cardiac muscle at an organ level. For this reason, fiber orientation is an intrinsic part of cardiac structure, and affects its local material properties, mechanical and electrical behaviors, and other functions of the heart. The ability to extract fiber structure information from an organ or samples of tissue is vital to explain these effects. Over time, many have proposed mathematical and theoretical models for different aspects of the heart as technological advances make fiber structure information available. Notable examples in biomechanics include constitutive characterization of tissues and its subsequent use in functional modeling of the whole heart. Experimental observations like mechanical testing of myocardial tissue have shown that mechanical properties are dependent on the tissue microstructure such as fiber orientation, the sheet-like formation of fibers (i.e., lamination), and the associated arrangement of the extracellular matrix. The mechanical properties have been described through several mathematical formulations of constitutive behavior [8], [9]. To reach meaningful results from the application of these models, information on organ geometry and tissue anisotropy are both necessary [10], [11]. The above structure-function relationships also apply to cardiac electrophysiology [12], and should be reflected in simulation of electrical propagation and coupled electro­mechanical modeling. It is well established that electrical conductivities of cardiac tissues also exhibit anisotropy [13], [14] and that those are determined by tissue microstructure, in particular, the local orientation and lamination of cardiac fibers. In general, anisotropic description of tissue properties is a crucial component for coupled, electro-mechanical modeling of the heart [15], which requires the integrative modeling of electrical activation, force development and mechanical deformation based on anisotropic tissue properties. For example, anisotropic cardiac tissue properties have been used to produce comprehensive models seeking to provide explanations for the basic mechanisms for ventricular contraction, expansion, and torsion [16]. 6 2.2 Diffusion Tensor Imaging The power of MRI is derived from its sensitivity to the molecular dynamics of water, which in turn closely follows the microstructure of tissues. By generalizing the principles of diffusion MRI to describe anisotropic diffusion in 3D space, diffusion tensor imaging (DTI) can be used to characterize myocardial structures. In the heart, although the exact biophysical mechanism is incompletely understood, it has been suggested that water diffusion anisotropy arises from the combined effects induced by the cardiomyocytic membrane, extracellular connective tissue, and microvasculature [17]. Mathematical descriptions of the macroscopic and microscopic consequences of molecular diffusion were originally provided by Fick and Einstein, respectively [18], [19]. Torrey [20] then incorporated anisotropic translational diffusion in the MRI Bloch equations as an additional source of signal attenuation. About a decade later, Stejskal and Tanner [21] solved the Bloch-Torrey equation for the case of free anisotropic diffusion in the principal frame of reference. The pioneering work on combining MRI and diffusion anisotropy came from the rigorous formalism of the diffusion tensor by Basser et al. [22], [23]. In this section, the physical basis of DTI and its experimental design strategy will be discussed. 2.2.1 Diffusion and the MR Signal In general, there are two types of diffusion that are of interest in MRI: movement of molecules from regions of higher to lower concentrations, and the random or Brownian motion of molecules due to thermal energy. For distinction, the latter is often referred to as self-diffusion. For the sake of simplicity, from this point forward, the term "diffusion" 7 will refer to self-diffusion, particularly the self-diffusion of water. In statistical mechanics, the average displacement along a given axis, say the x-axis, (x), of diffusing water molecules is related to the diffusion coefficient, D, via the Einstein's equation <x> = V2DA, (21) where A is the diffusion time (e.g., time between leading edges of diffusion encoding gradient pulses) and D is the diffusion coefficient. In biological tissue, D decreases, compared to free water, due to obstructions imposed by microstructure (e.g., cell membranes, fibers, etc.). These obstruction effects are generally anisotropic (i.e., not uniform in all directions), which gives rise to a preferred direction of water diffusion because, intuitively, water molecules will diffuse fastest in the direction parallel to tissue fibers. DTI can be utilized in cardiac tissue in order to characterize its fiber structure, such as fiber orientation or the organization of fibers. In MRI, linearly varying magnetic field gradients (or simply gradients) are used to manipulate the resonance frequencies of the individual magnetic moments, or spins, that contribute to the detected signal. Specifically, the relative frequency, with respect to a spin located at the origin, at which a spin located at r precesses can be expressed as « ( r, 0 = -Y G(t) ■ r(t), (2 2) where y is the gyromagnetic ratio (for 1H) and G is the applied 3D gradient field. Now, consider the scenario in Fig. 2.1, where a spin is first subjected to a gradient pulse of +G amplitude, followed by an equal but opposite (-G) pulse. Suppose during the first pulse, the spin is located at rx, and as such, it would acquire a phase of 44 = -Y G • 5, (2 3) 8 9 G V A 5 -G V A > Figure 2.1: A pair of gradient pulses used to sensitize MRI to diffusion. An individual spin is tagged with a given phase during the first gradient pulse, depending on its spatial location. The second gradient pulse undoes the phase tagging for stationary spins. For spins that move or diffuse between the pulses, the resulting phase after the second pulse is proportional to the distance moved between the two gradient pulses. where 5 is the duration of the diffusion gradient pulses. Furthermore, suppose the spin has moved during the time between the two gradient pulses and is located at r2 during the second pulse, the spin would acquire an additional phase The phase accumulated by the spin is, therefore, proportional to the distance the spin has moved from rx to r2. If a spin has not moved, the cumulative phase will be zero. The effect of diffusion in the presence of a sensitizing gradient on the MRI signal can be found by solving for the expected value of the phase dispersion of an individual spin, which is a random process, according to = y G ■ r2 5. (2.4) Consequently, the cumulative or net phase is 0 n e t 0 2 + 0 1 = - y G ■ r2 5 + y G ■ rx 5 = -y G ■ ( r 2 - r i ) 8 (2.5) I = Iq J e x p ( - i^ n e t )P ( r2 | r i)d $ , (2.6) where I0 is the diffusion-independent signal and P( •) is the probability density function of the diffusion, which in the case of free or unrestricted diffusion is a Gaussian distribution with a standard deviation specified by the Eq. (2.1), a = V2DA. It can be shown that when diffusion is encoded using a pair of rectangular pulses of opposite polarity and with magnitudes equal to G = |G|, like those in Fig. 2.1, the Stejskal-Tanner expression for diffusion can be derived I = I0 ex p (-•y2G252(A - 5 /3 )D) = I0 exp(-bD), (2 7) where b = y2G252(A - 5/3) is the so-called diffusion-weighting factor. Therefore, diffusion manifests itself in the acquired image as a loss or attenuation of signal. In turn, the diffusion coefficient D, better known in MRI as the apparent diffusion coefficient (ADC), can be computed from MRI signals acquired with and without the diffusion encoding gradients, I and I0, respectively, according to D = ( - 1 /b ) ln (I/I0). (28) To illustrate the effect of diffusion encoding and the underlying myocardial fiber structure, Fig. 2.2 shows a nondiffusion-weighted image, I0, along with a diffusion-weighted image, I, of a human heart sample. Because the amount of diffusion-induced MRI signal attenuation is dependent on the rate of diffusion along the direction of the encoding direction, the fact that different regions of the cardiac left ventricle have different intensities is an indication that the underlying myocardial fibers are oriented in different directions. 10 11 Figure 2.2: Sample diffusion-weighted images of an ex vivo heart. Nondiffusion-weighted cardiac sample (top-left) shown alongside diffusion-weighted images of the same sample encoded in the x-direction (top-right), y-direction (bottom-left), and z-direction (bottom-right) with a b-value of 2000 s/mm2. 2.2.2 MRI of Anisotropic Diffusion The orientation-dependence of the effect of anisotropic diffusion on the MRI signal can be more easily explained by first considering a special system in which the principal axes of diffusion coincide with the laboratory gradient axes. Specifically, suppose the diffusivities are Dx, D2, and D3 along the principal axes, which are aligned with the laboratory x-, y-, and z-axes, respectively. The combined signal attenuation is given by the superposition of Eq. (2.7) applied to each axes, or / = /0 exp(- b xDt - byD2 - bzD3) , (2 9) where b; = y2G! 52(A - S/3) is the diffusion weighting factor associated with each 12 i = x, y, z axis. Moreover, provided that the diffusion encoding gradients in different axes are identical in timing but differ only in their relative amplitudes, Eq. (2.9) reduces to Dx 0 0 ' I = I0 exp (-b uT 0 D! 0 u ), (2.10) 0 0 Dz where u is the unit vector denoting the composite gradient direction (e.g., u = [1 0 0 T, [0 1 0 T and [0 0 1 T for the x-, y- and z-directions, respectively). Implicit in Eq. (2.10) I = I0 exp I - b gTRT R g = I0exp ( - b gTD g ), (2.11) is that G = ^ G ! + G^ + G| should be used in Eq. (2.7) for computing the diffusion weighting factor. The obvious limitation of Eq. (2.10) is that, more often than not, the principal diffusion axes do not coincide with the laboratory axes. In the general case when the coordinate systems are not aligned, Eq. (2.10) can be modified by mapping the laboratory axes onto the diffusion coordinate system via the transformation u = Rg, resulting in D! 0 0 - 0 D2 0 .0 0 d3. where g is the directional unit vector (in laboratory coordinates) of the diffusion encoding gradient, and "D-i 0 0 1 Dxx Dxy Dxz (2.12) is the rank-2 tensor that characterizes the diffusion in 3D space, otherwise known as the diffusion tensor. Since diffusion cannot physically be negative, the principal diffusivities (i.e., the diagonal terms of the diffusion tensor) must be non-negative, which results in the diffusion tensor being positive semi-definite. Many diffusion tensor-fitting algorithms incorporate the positive definiteness constraint in their fitting [24]. It can be seen from D- yxD zxD D = Rt 0 D2 0 R = Dxy Dyy D N . 0 0 D3. Dxz D N NN Q 13 Eqs. (2.11) and (2.12) that the major task in DTI is to use the choices of encoding gradient directions g to selectively probe the elements of the diffusion tensor. Following similar derivations as described above, the signal attenuation due to anisotropic diffusion in the presence of time-varying gradient waveforms can be alternatively expressed as [25] where b^ (i, j belongs to x, y, z) corresponds to individual entries of the "b-matrix", b. 2.2.3 Experimental Strategy Regardless of whether the approach described by Eqs. (2.11) or (2.13) is used, the typical DTI experiment consists of acquiring a series of diffusion-weighted MRI scans encoded using one or more b-values along at least six noncollinear gradient directions, (since the diffusion tensor is a rank-2, symmetric tensor as seen in Eq. (2.12)) and estimation of the diffusion tensor on pixel-by-pixel basis via appropriate curve fitting of the observed signals to the signal attenuation equation. Given that a nondiffusion-weighted image, commonly referred to as a b0 image, is needed to estimate the diffusion-independent signal I0 the minimum scan time for a diffusion tensor experiment is, therefore, seven times longer than a conventional scan of the same anatomy. Directly, the estimated diffusion tensor bears little use for inferring the tissue microstructure, since the relevant information is embedded in the tensor elements. Mathematically, the surface of equal probability at a given time for finding water molecules, which are initially located at the origin but subject to anisotropic diffusion 14 governed by a diffusion tensor, is an ellipsoid. As an ellipsoid is described by the lengths and orientations of its major and minor axes, the diffusion tensor is related to the magnitudes and directions of the underlying principal diffusion processes by its eigenvalues and eigenvectors. Consequently, by applying the linear algebraic eigenvalue decomposition, the diffusion tensor can be converted into a product between a diagonal matrix of its eigenvalues and transformation, or rotation, matrix consisting of its eigenvectors. According to Eq. (2.12), the eigenvalues and the eigenvectors of the diffusion tensor correspond to the diffusivities as observed along the principal axes of diffusion and the orientations of the axes, respectively, as shown in Fig. 2.3. The central premise of DTI is that the direction in which water diffusion is the fastest, in other words the eigenvector of the largest diffusion tensor eigenvalue, coincides with the local tissue fiber orientation. To make the derived parameters even more intuitive in cardiac DTI, the fiber Figure 2.3: Diffusion ellipsoid. The amount of diffusion in the principal axes is proportional to the eigenvalues D1, D2, and D3. The orientations of the three principal axes are determined by the eigenvectors evl, ev2, and ev3. e\ orientations are often reported in terms of their helix angles and, to a less extent, imbrication angles. On the other hand, the diffusion tensor eigenvalues are commonly used to compute the mean diffusivity (MD) and fractional anisotropy (FA) index MD = (Di + D2 + D3)/3 , (214) 15 FA = (D! - MD2 + (d 2 - MD2 + (d 3 - MD2 (215) D! + D2 + D! To a first approximation, the MD is proportional to the size of the diffusion ellipsoid, whereas the FA is analogous to the standard deviation of its eigenvalues or the aspect ratio of the diffusion ellipsoid as seen in Fig. 2.4. The FA is a normalized quantity, with FA of zero and unity respectively denoting no and infinite anisotropy. In practice, not only are the above indices convenient quantities that capture the overall magnitude of diffusion and the degree of anisotropy, but they also have the nice feature of being rotationally invariant or, in other words, do not depend on the orientations of the diffusion principal axes. The fiber orientation helix angle and the scalar DTI MD and FA for the same specimen shown in Fig. 2.2 are illustrated in Figs. 2.5 and 2.6. 2.3 Cardiac Diffusion Tensor Imaging DTI has been used to characterize tissue structure in a number of applications, including studies of the myocardium and its functions. This section offers a brief survey of these applications, which include validations of DTI (Section 2.3.1), tissue specimen characterization (Section 2.3.2), DTI of cardiac pathophysiology (Section 2.3.3), and examples of clinical applications of DTI (Section 2.3.4) 16 Fractional Anisotropy © • i Figure 2.4: Varying fractional anisotropy and mean diffusivity. Diffusion ellipsoids with varying fractional anisotropy (FA) and mean diffusivity (MD) values. 2.3.1 Validation of Myocardial DTI As with any newly introduced imaging technique, the extent to which DTI is useful, in the current case for characterizing myocardial structures, requires it to be validated against the commonly accepted gold standard for the same measurements, which is histology. Early applications of DTI in the myocardium were soon followed by studies that directly correlated DTI-measured myocardial fiber orientations, which are technically the local direction in which water diffusion is fastest, with histology including separate studies performed on freshly excised canine right ventricular sample [26], perfused [7] and formalin-fixed whole rabbit left ventricle [27]. These studies using 17 Figure 2.5: Diffusion tensor parameter maps from an ex vivo heart. Maps representing the diffusion tensor parameters derived from the diffusion data shown in Fig. 2.2. The rows and columns correspond to the 3 x 3 diffusion tensor as seen in Eq. 2.12. The images on the diagonal are scaled from 0 to 1, while the off-diagonal images are scaled from -0.25 to 0.25. Helix Angle FA Mean Diffusivity - 90° 0° 90° 0 0.25 0.5 0 0.5 1.0 (x 10'3 mm2/s) Figure 2.6: DTI parameters maps. Maps representing helix angle (left), fractional anisotropy (FA) (middle), and mean diffusivity (right) derived from the diffusion tensor data shown in Fig. 2.5. diffusion imaging MRI have helped establish DTI as a valid alternative to histology for measuring myocardial fiber orientation. Because the studies were performed on differently prepared myocardial samples, they also suggest that DTI, at least for fiber orientation mapping, is immune to the effects of tissue preparation like fixation, for example. In the study performed on freshly excised canine ventricular sample, the fiber orientation helix angles measured by DTI and histology were found to differ on average by 2 - 5° [26]. The excellent correspondence between DTI and histology results supports the hypothesis that the first eigenvector of the MR diffusion tensor coincides with the orientation of the local myocardial fibers. Being a noninvasive technique, DTI may be uniquely suited to help address the controversy over the existence of myocardial laminar or sheet structure. The concern was that the laminar structure is not an intrinsic property of the myocardium, but an artifact introduced in the tissue preparation steps (e.g., fixation and sectioning) of histology [28], [29]. In DTI, one would intuitively expect myocardial sheets to make water molecules diffuse more freely within rather than across any laminar structure. Consequently, the existence of myocardial sheets would manifest in (a) distinct second and third eigenvalues in the myocardium, and (b) the eigenvectors associated with second and third eigenvalues to exhibit a nonrandom organization. Indeed, distinct populations of second and third eigenvalues within statistical confidence levels were observed in the canine myocardium [30]. Nonrandom second eigenvector fields were reported in fixed mouse hearts [31]. The organized appearance of the second eigenvector was also observed in ex vivo human myocardial specimens [32]. Moreover, similar organized appearance was also observed in fresh excised [26] and perfused unfixed myocardium [7], which suggests 18 neither tissue fixation nor sectioning as source of diffusion anisotropy observed in DTI. The link between the DTI second eigenvector and myocardial laminar structure is further supported by findings of a subsequent study comparing DTI and cut-face ink blots of the bovine myocardium [33], showing a parallel relationship between the eigenvectors and symmetry axes of the myocardial architecture. Specifically, the first, second, and third eigenvectors corresponded to the fiber, sheet, and sheet normal directions, respectively [33]. The use of cut-face ink blots provided a method in which fiber and sheet orientations could be measured under the same conditions when using different modalities (optical vs. MRI), by minimizing the possibility of tissue alterations between data acquisitions. 2.3.2 Applications of Cardiac DTI Since its advent, DTI has been used to characterize the normal myocardium in vitro and ex vivo across several species. In one study on healthy goat hearts [5], the helix angle was found to vary transmurally across the left ventricle (LV), with the steepest slope found in the anterior and septal sites. Similar variability of the helix angle slope was also observed in rabbit hearts [34] and mouse hearts [31]. The heterogeneity of the anisotropy index, FA, was measured in sheep [35] and was found to vary transmurally across the myocardium. The variability of cardiac microstructure was also studied across different species [36], which showed significant fiber structural differences between any of the pairs of species examined. High resolution DTI (100 p,m) was introduced [31] in mouse hearts, which allowed for more detailed characterization of myocardial microstructure. Cardiac studies using 19 high resolution DTI were consequently performed to illustrate microstructural changes in the myocardium following dyssynchronous heart failure in canines [37] and myocardial infarction in sheep [38]. Another high-resolution DTI study linking cardiac microstructure to its function was performed in rats [39] and found that wall thickening during contraction is related to changes in fiber and sheet structure configurations. DTI and 3D MRI imaging created the possibility of characterizing the organization of myocytes in 3D space, rather than in a 2D plane as is done in histology. Studies using alternative methods to DTI suggested that not all myocardial fibers are oriented circumferentially, but that there are intruding fibers that are oriented radially [40]. Other studies using histology [41] and confocal microscopy [42] found evidence that the organization of myocytes vary in 3D space. Studies using DTI on postmortem porcine hearts found that the ventricular mass is arranged as a mesh of tangential and intruding fibers and that there is no support for a unique myocardial band [17], [43], [44]. In another interesting study [45], a nonexchanging, two-component diffusion tensor model was fitted to diffusion-weighted images obtained in rat hearts ex vivo. The results suggested the existence of at least two distinct components of anisotropic diffusion, characterized by a "fast" component and a "slow" component, which exhibited highly similar orientations. It was suggested that the fast and slow components corresponded to the vasculature and cellular components, respectively, of the myocardium. 2.3.3 DTI and Cardiac Pathophysiology The presence of cardiac disease often involves multiscale myocardial structure remodeling, which is reflected by variations of some DTI parameters. Despite the lack of 20 comprehensive understanding regarding the mechanisms governing these variations, the correlation of DTI parameters with health and pathology have shown promise in potential tools for diagnosis and computational modeling of disease, and its progression. DTI studies on multiple animal models suggest sensitivity to pathology, which imply that DTI may be clinically useful in determining the extent of disease extent or the effectiveness of therapy. Previous studies have shown that fiber disarray, detectable by DTI, often accompanies cardiac disease. For instance, a reduction of tissue diffusivity was observed on isolated ischemic rabbit hearts [7]. The same observation was confirmed in another study performed on excised hearts of infarcted porcine, which also associated infarction with flatter helix angle [46]. The effects of infarction on the border and remote zones have also been studied using DTI. In a study where fiber structure of excised rat hearts was visualized in 3D [47], it was shown that infarct areas change from a normal fiber distribution pattern to orthogonally intersecting networks similar to a mesh, which extend across the infarcted area to the border zones. A similar study on porcine models of infarction [48] showed that infarct border zone, delineable by DTI contains viable myocardial strands, which may have an effect on postinfarct electrophysiology. The effects of infarction on FA, apparent diffusion coefficient (ADC), and helix angle have also been studied. When compared to its healthy state in pigs, the infarcted myocardium exhibits a decrease in FA value, increased ADC, and a flatter helix angle [49], [50]. These changes insinuate fiber disarray, which is observed accompanying fibrosis [51]. Additional observations, as well as multiple speculative explanations for their appearance have been made in separate studies, for example: FA has been suggested as an indicator of functional recovery following heart transplant in canines [52]. 21 Imbrication (or intrusion) angle increases were observed in hypertrophic mouse hearts [53]. Structural changes were observed during the progression of left ventricular myocardial infarction [54], and following surgical restoration [38]. Further, the double helix myocardial structure shifted more leftward around the infarcted myocardium, and the redistribution of fiber architecture correlated with the infarct size and left ventricular function [50], [55]. Finally, myocardial architecture is linked to initiation and maintenance of reentrant arrhythmias [6] as well as the mechanical coupling during systolic wall thickening [26], [56]. In terms of modeling of pathophysiology, DTI data have been used, within the realm of cardiac biomechanics, in a wide variety of studies. Some aim to improve our understanding in the overall structure-function of the heart [57], [58]. Others seek to measure stress, strain, and other biomechanical parameters by constructing finite element models for myocardial infarction [59]-[61], as well as computational representations of cardiomyopathy [62], and cardiac growth [63]. Additional researchers are also using DTI data to characterize the effects of fiber structure remodeling in animal disease models [54], [64], [65], and to quantify differences across species [36], or across the cardiac cycle [66]. A number of studies proposed that ventricular fiber orientation is a result of mechanical feedback [2], [4], [67], [68]. These studies applied biomechanical simulations and optimization approaches to derive fiber orientations leading to, for instance, uniform mechanical load. A study on ovine left ventricle, however, indicated difficulties to predict fiber orientations based on mechanical feedback [69]. It was suggested that detailed geometrical information is required for prediction of fiber orientation. A detailed knowledge of the ventricular fiber structure is important for understanding 22 the nature of cardiac electromechanics in healthy, disease, and intermediate conditions. During postinfarct healing, the fibers rearrange parallel to fibers outside the border zone [70]. Also, local fiber aggregation is disturbed increasing or decreasing fiber density due to edema, and may be affected by increased fibrosis. Tissue structure becomes irregular, or discontinuous, which may promote electrical function anomalies or mechanical failure [38], [70], [71]. Generally, the alteration of fiber structure in most cases is a dynamic process that accompanies healing or remodeling, and varies over time. In a study on a mouse model, the infarcted region measured lower ADC than the remote region, and the low values increased with time subsequent to infarction. Increased FA peaked after 28 days, which may be associated to the observed development of structured collagen fibers in the area [54]. At the molecular level, FA was found to be associated with decreased induction of endothelin-1 (ET-1) and caspase-3, improved adenosine triphosphate (ATP) storage in the myocardium, and functional recovery of the myocardium after ischemia [52]. Another study on infarcted sheep hearts revealed a significant reorganization of the three-dimensional aggregation of adjacent fibers in the remote zone of remodeled hearts [72]. Regardless of angle classification, a positive (rightward) shift in myocardial helix angle is observable in all layers of the remote zone, in particular the subepicardium. In conjunction with strain, DTI has been used to study hypertrophic cardiomyopathy (HCM) in humans establishing a relationship between myofiber disarray, mainly measured by FA, and hypokinesis, measured by tissue deformation [73], where HCM exhibited locally reduced diffusion FA, which indicate myofiber disarray. The same areas also showed decreased myocardial strain, especially in the direction perpendicular to fibers within the local sheet structure, which had the highest correlation between FA and 23 24 hypokinesis. 2.3.4 In Vivo Cardiac DTI The noninvasive nature of DTI presents the opportunity that it could be used to characterize myocardial structures in vivo in both animals and humans, which would be desirable to better understand both healthy heart functions as well as disease progression. Earliest works of in vivo cardiac DTI were simply to demonstrate the feasibility of the technique, which is not trivial due to complications arising from the beating motion of the heart, or to document its sensitivity to myocardial remodeling in diseases. Initial studies performed on perfused rat hearts [3] and on human hearts in vivo [74], [75] showed that not only was diffusion MRI on the beating heart technically feasible, but also fiber architecture of the myocardium imparted anisotropy on the water diffusion. Subsequent studies revealed that tissue strain in the beating myocardium affects the observed diffusion signal, which can be eliminated by either retrospective corrections [76] or averaging during acquisition [77]. (Section 2.4.3 discusses strain effects in DTI in more detail.) One natural application of in vivo cardiac DTI is to investigate the structure-function relationship of the same hearts. Studies have shown that myocardial fiber orientations obtained via DTI map well with fiber shortenings obtained by velocity MRI measurements [78], and that myocardial sheets contribute to ventricular wall thickening during cardiac contraction [79]. In [78], DTI was implemented to obtain images of fiber orientation in vivo in eight healthy subjects for comparison with strain images. The comparison showed that the fiber shortening, as measured by DTI, was more uniform over the myocardium than the measured radial, circumferential, longitudinal, or cross-fiber strain. It was also found that fiber orientation corresponded with the direction of maximum contraction in the epicardium and with the direction of minimum contraction in the endocardium and varied linearly in between [78]. In [79], DTI and phase-contrast (PC) MRI were used to acquire myocardial sheet structure and strain rate, respectively. The involvement of myocardial sheets in ventricular radial thickening during contraction was studied by registering the results of DTI and strain rate data. The sheet function in normal subjects was found to be heterogeneous throughout the ventricular myocardium, as opposed to the contribution of fiber shortening to wall thickening, which was found to be uniform and symmetric. The strain rate results showed that the sheet shear and sheet extension were most prominent in the anterior free wall and that the sheet-normal thickening was prominent near the right ventricular insertions [79]. The feasibility of in vivo imaging paved the way for DTI to be used as a tool for detecting or diagnosing cardiac pathology. To date, DTI has been utilized to evaluate the effects of several cardiac diseases, exploiting remodeling of the myocardial microstructure as a marker of these diseases. In MI, the microstructural remodeling was evident in an increase of the DTI-derived MD and decrease of FA in the infarct, and alterations of the fiber orientation helix angle in adjacent zones [46], [80]. In HCM, the myocardial fiber disarray resulted in decreased FA, which correlated with intramural myocardial strain hypokinesis [73]. In another study [47], changes in the 3D myocardial fiber architecture resulting from ischemic heart disease were visualized via tractography. Although in vivo applications of DTI are still in their early stages and the biophysics 25 linking microstructural alterations to DTI observations need to be better understood, DTI has already been shown to be a valuable tool for evaluating myocardial remodeling during cardiac pathology and recovery. 2.4 Practical Considerations of Cardiac DTI Although the general strategy for a DTI experiment is straight forward - acquire diffusion-weighted images in multiple encoding directions then fit the data to the diffusion tensor signal equation to characterize the underlying diffusion anisotropy - several factors conspire to make its implementation in practice technically challenging. Issues to consider include low signal-to-noise ratio (SNR), long scan time, hardware limitation, image distortion, etc. Many methodological developments have been undertaken and significant progress has been achieved in addressing the practical challenges of DTI, albeit most of the efforts have been targeted for DTI studies of the brain. This section describes in general terms some of these technical challenges, not intended to be an exhaustive review but as background, and discusses the special considerations needed for performing cardiac DTI. 2.4.1 DTI Pulse Sequences As explained in Section 2.2.1 and illustrated in Fig. 2.1, translational diffusion can be encoded into the MR signal by the action of a pair of equal but opposite-polarity gradient pulses. Therefore, by incorporating such a pair, a MRI pulse sequence can be turned into a so-called diffusion-weighted sequence for obtaining diffusion-weighted images (DWI). Figure 2.7 shows examples of diffusion-weighted gradient-recalled echo (GRE) and spin 26 27 RF Slice n Phase- Read • -G, TE [lJ Slice Phase- Read Figure 2.7: Diffusion-weighted gradient-recalled echo (GRE) (a) and spin echo (SE) (b) sequences. The grey boxes highlight diffusion sensitizing pulses. echo (SE) sequences with the diffusion encoding parts of the sequences highlighted. Note that due of the inversion RF pulse in the spin echo sequence, the diffusion encoding gradient pair should have the same polarity. By placing diffusion encoding gradient pulses in all imaging axes, the pulse sequence can be made sensitive to diffusion along any given direction in 3D space specified by the relative amplitudes of the encoding pulses. Regardless of the pulse sequence used to realize diffusion encoding, one immediate consequence of diffusion encoding is that the minimum TE of the sequence is lengthened (e.g., an extra 40 ms is required to generate diffusion weighting b-value of 1000 s/mm2 using a 40 mT/m gradient), which can aggravate the SNR challenge of DTI experiments. Because the GRE sequence is more prone to susceptibility or distortion artifacts, the SE sequence is preferred over GRE sequence for acquiring diffusion-weighted images. However, SE acquisitions suffer from long scan times, which are further exacerbated by the need to signal average or encode diffusion in a high number of directions to improve the accuracy of the DTI experiment. To make the DTI scan time practically acceptable, especially for in vivo applications, the diffusion-weighted spin-echo echo-planar imaging (EPI) has been used (Fig. 2.8), and to date remains to be the sequence of choice for most DTI studies, at least for brain applications. The typical scan time of an EPI acquisition is in the order of 100 ms, which is especially advantageous when hundreds or thousands of images are desired, as in high angular resolution diffusion imaging (HARDI). Although the issue with scan time is alleviated, the diffusion-weighted EPI sequence has its own set of technical challenges, including blurring arising from signal decay and susceptibility induced image distortions at tissue-air boundaries. The most notable challenge is image distortions generated by eddy currents associated with the use of the large diffusion encoding gradient pulses. The distortions vary in both appearance and magnitude as different diffusion encoding gradient directions and levels are used. If left uncorrected, the distortions cause inconsistent tissue borders across images in a DTI dataset, and are 28 90° 180° RF Slice u G„ Phase Read ... T J Figure 2.8: Spin-echo, echo-planar imaging (EPI) pulse sequence. The grey boxes highlight the diffusion-weighted gradient pulses. characterized by artificially high FA values observed at edges of the tissue. Because experimental requirements as well as pulse sequence performances vary, in addition to SE, GRE, and EPI, many different pulse sequences have been used for acquiring diffusion MRI or DTI data. For example, diffusion-encoding gradients have been used in conjunction with fast spin echo (FSE) pulse sequences. On the one hand, FSE sequences offer the advantages of speed (compared to conventional spin echo sequence) and being free of geometric distortions that are synonymous with EPI. On the other, FSE can be hampered by elevated RF power deposition associated with the use of multiple RF pulses, and ghosting and T2 blurring artifacts when the precise RF conditions (especially the inversion 180 pulses) are not met. In addition to FSE, DWI or DTI experiments have been performed using advanced MRI sequences such as spiral [81], [82], SSFP [83], [84], PROPELLER [85], [86], parallel imaging (SENSE [87] and GRAPPA [88], [89]), STEAM [90], etc. Needless to say, each pulse sequence has its own set of challenges and limitations, and the reader is referred to elsewhere [91] for a more exhaustive review of technical considerations associated with these pulse sequences. The large number of pulse sequences that have been used for DWI or DTI is a testament to the robustness of the DTI methodology and the flexibility in which it can be implemented. 2.4.2 DTI Experimental Strategy Besides the pulse sequence used, the design of the DTI experiment, which includes the size of the dataset and number of diffusion encoding directions, for example, can also have profound effects on the accuracy of the results of DTI experiments. For example, in 29 the worst-case scenario, a DTI experiment that fails to include the minimum number of noncollinear diffusion encoding gradient directions would yield indeterminable diffusion tensors. Because the typical DTI experiment consists of one or more b0 image and a series of diffusion-weighted scans encoded in different sensitizing directions, factors that naturally affect the accuracy of the obtained diffusion tensors, and the information therein, such as fiber orientations, include the individual image SNR, number of diffusion encoding directions, distribution of the directions, number and placement of the diffusion weighting b-values to be used, etc. As with the case of diffusion-weighted pulse sequence considerations discussed in the preceding section, thanks to efforts already taken, much understanding already exist on the impact of each of these parameters on the quality of DTI. The main strategic consideration in experimental DTI is to address its low SNR, which is due to the nature of both diffusion encoding via signal attenuation and T2 relaxation during the prolonged TE necessary to accommodate the diffusion encoding pulses. Moreover, the SNR issue is aggravated by the tradeoff among scan time, which is necessitated by the large dataset size, resolution and SNR. Similar to any quantitative MRI experiment, insufficient SNR can be detrimental to DTI. Low SNR can manifest in directly proportional random errors in the DTI results, as determined by, for example, the mean deviation angle from the true value in the estimated fiber orientation [92]. Noise can also result in a systematic bias of the DTI parameters, including overestimation of the FA, where sorting of noisy DTI eigenvalues gives rise to the artificial appearance of anisotropy [93]. Not surprisingly, in one way or another, all considerations in the DTI experimental strategy are related to boosting the effective SNR. 30 Perhaps the simplest way to improve the DTI accuracy is to signal average in order to improve the SNR of the individual diffusion-weighted scans. The relationships among signal averaging, scan time, and the resultant image SNR are well established, that scan time is directly proportional to signal averaging and SNR is proportional to the square root of the signal averaging. However, causes of inaccuracy in DTI include not only image noise, but also factors such as directional sampling, tensor fitting, etc. Accounting for these latter factors can improve the DTI accuracy beyond what is achievable by signal averaging individual diffusion scans alone. Indeed, in the context of the acquisition of the whole DTI dataset, increasing the number of diffusion encoding directions is in effect form of signal averaging. Employing more noncollinear gradient directions has the additional benefit of reducing the directional sampling error and is generally preferred over signal averaging in the same encoding directions. For a given scan time, determined by the combination of the number of individual diffusion-weighted scans and signal averages, the most efficient means to achieve DTI accuracy is to acquire diffusion-weighted scans in as many different encoding directions and distribute the encoding direction unit vectors as evenly spread out as possible on a unit sphere [94]. Techniques such as the tessellation of icosahedrons [95], [96] and electrostatic repulsion on a unit sphere [94] have been proposed and shown effective for optimizing the selection of encoding directions. In general, because of the finite number of variables in the diffusion tensor fitting and the square-root nature of averaging, DTI quality improvement by increasing the number of diffusion encoding diffusions is most pronounced when the number of directions is relatively low. In increasing the number of diffusion encoding directions, it should also be noted that since in tensor computation the same b0 image is 31 used in estimating the effective diffusivity in each encoding direction, the b0 image has disproportional impact on the accuracy of the whole DTI experiment. Therefore, the use of high number of diffusion encoding directions must be balanced by proportional increase in signal averaging (or NEX) of the b0 scan [94], [97]. Besides the number of diffusion encoding directions, the choice of the diffusion-weighting b-factor (or b-value) can also impact on the accuracy of a DTI experiment. Intuitively, the DTI experiment is akin to measuring the decay constant of an exponentially attenuating signal, or the slope of the signal on a semi-logarithmic plot, with the b-value as the independent variable. Intuitively, if the b-value used were too high such that there was too much attenuation, the diffusion-weighted scans would contain more image noise than tissue information. In contrast, if too low of a b-value were used, even small amount of noise in the image would have disproportionally large impact on the fitted slope or decay constant. Therefore, diffusion-weighted scans acquired with different b-values do not contribute equally to the accuracy of a DTI experiment. The implications for the DTI experimental design are two-fold. First, there exist an optimal diffusion-weighting b-value to be used in DTI scans. Empirical experience and studies [94], [97], [98] have shown that diffusion-weighted scans that achieves a factor of e_1 « 0.4 to 0.5 attenuation of the signal contribute the most to the accuracy of DTI experiment. Combined with considerations on the number and distribution of encoding directions, it is preferable to use the DTI scan time to repeat the same b-value meeting the optimal attenuation criterion at different additional encoding directions. (Note the single b-value criterion does not apply to experiments fitting alternative models than the diffusion tensor). Second, because of the unequal contribution of the images toward the 32 accuracy of DTI, a weighted curve fitting technique would yield more accurate diffusion tensor estimations than one that weighs all signal data equally [99]. Although optimal strategies for DTI acquisition are known, their practical implementation can be hampered by instrumentation limitations. More often than not, the optimal diffusion weighting b-value cannot be achieved due to the low or finite gradient strengths available, especially on clinical whole-body scanners. As a work around, one way to boost the effective b-value is to employ multiple gradients at the same time. For example, turning on two gradients of the amplitude effectively boost the b-value by a factor of V2 compared to when a single gradient is used. In this regard, while both {(1,0,0), (0,1,0), (1,0,0), (1,1,0), (1,0,1), (0,1,1)} and {(1,1,0), (1,-1,0), (0,1,1), (0,1,-1), (1,0,1), (1,0,-1)} contain six noncollinear directions and thus satisfy the criterion for minimal DTI encoding directions sets, the latter employ only two-gradient directions and is better for practical use. It is worth noting that using multiple gradients simultaneously to increase b-value must be weighted against the fact that the practice also dictates gradient directions and can interfere with the optimization of the latter. Implicit in the above discussion on encoding direction and weighting factor optimization is that the gradient waveforms of the pulse sequence are precisely known, which can be difficult in practice. For example, even with the best shimming effort, background gradient is inevitable. Unaccounted background gradient can not only set off the DTI encoding scheme off its optimal conditions, but also cause erroneous DTI estimations from errors in computing the b-matrix elements (e.g., via Eq. (2.14)) due to the associated cross-terms [24], which can be a bigger concern. Fortunately, effects of cross-terms are multiplicative in both amplitude and polarity, and a simple yet effective 33 means to eliminate them is to acquire diffusion-weighted scans with same but opposite-polarity encoding gradients, and to eliminate the cross-term contributions by taking the geometric average (i.e., square root of the produce of the two image intensities) of the scans [26]. The drawback of the strategy, obviously, is that the scan time is doubled. This is yet another example that optimization of the DTI experimental strategy often involves addressing not a single consideration but weighting and trading off among multiple counter-opposing factors. In addition to the above measures for acquiring the dataset, the accuracy of DTI can also be improved in the postprocessing. For example, by recognizing that a properly estimated diffusion tensor should bear certain characteristics of the physical entity (e.g., having only real, positive eigenvalues), appropriate numeric estimation algorithms, in this case Cholesky parameterization, can be applied to avoid bad tensor fittings produced in noisy pixels [99]. Similarly, by recognizing that noise tends to produce more variability than the underlying tissue structure in tensors estimated for neighboring pixels, denoising or other a priori information-based "regularization" techniques have been found to be useful to boost the DTI quality, often without incurring additional scans. Denoising is in effect image smoothing, and can be achieved by techniques as simple as low-pass filtering of the images. Different denoising techniques have been evaluated on both simulated and empirical diffusion and DTI images [92], including cardiac scans [100], [101]. For DTI, it has been found that vector- or tensor-based denoising is better than image-based treatment, since in the former case deviations introduced after acquisition, during the tensor fitting and diagonalization, for example, are also removed [92]. Related, the common idea behind regularization is to introduce a priori information about the 34 solution in order to smooth the diffusion-weighted images while preserving relevant details [102], [103]. Separately, sparse representation-based methods, which effectively randomize the effects of noise, have been used to denoising cardiac DTI images [101] while preserving image's useful coherent structures. Because of their estimation nature, most regularization techniques offer the benefit for being able to capture the essential DTI information from only a small subset of the original dataset, and have the potential for accelerating DTI scan times. An example of this is using compressed-sensing methodology in DTI [104], which is described in more detail in Section 2.5. 2.4.3 Special considerations for in vivo cardiac DTI Besides the same challenges facing all DTI applications, in vivo cardiac DTI requires at least three additional technical considerations, all of which stem from the physiology of the heart. First, compared to other organs, the heart undergoes large, but relatively periodic, beating motion and unattended motion can lead to pronounced ghosting and streaking artifacts along the phase encoding axis of an MR image. Because of the large diffusion encoding gradients used, motion artifacts in diffusion-weighted MRI are orders of magnitude worse than regular anatomical scans. Motion artifacts from periodically moving organs or objects can be greatly reduced by employing gated acquisition using dual cardiac and ventilation-gated MRI, for example. Indeed, cardiac gaging at the time point where bulk systolic motion is minimal has been shown effective in improving the quality of cardiac DTI [105]. Another way to address motion is to employ the so-called navigator echoes [106], which is essentially additional echoes formed by the MRI signal in the absence of phase encoding, to estimate the motion and to compensate for its effects 35 36 via postprocessing correction. A high-resolution, cardiac DTI study using a prospective navigator shows potential for in vivo DTI in humans [107]. Lastly, at least theoretically, it may be possible to reduce the sensitivity to motion in diffusion MRI by replacing the conventional unipolar diffusion encoding gradient pulses with bipolar pulses, which cancel first gradient moments, hence reducing motion sensitivity, albeit the achievable diffusion-weighting b-value by bipolar gradient pulses is also expected to be significantly reduced. One method to attain higher b-values while minimizing the effects of motion is to use stimulated-echo acquisition mode (STEAM) based acquisition in conjunction with twice cardiac-gating, where the excitation and re-excitation, or first and third, RF pulses are synchronized to the cardiac cycle [74], as seen in Fig. 2.9. Even with halved SNR A Figure 2.9: Stimulated-echo acquisition mode (STEAM) pulse sequence. The diffusion sensitizing gradients are highlighted in grey. The first and third RF pulses are synchronized to occur at the same time point in the cardiac cycle. Using STEAM allows for higher b-values given a fixed gradient strength, but at the cost of losing half of the acquired signal due to using a stimulated-echo. associated with using stimulated echoes, the approach has been found effective in mitigating motion and is increasingly used in DTI studies in humans and large animals [80], [108], [109]. Regardless of the means for compensation, the heightened sensitivity of diffusion MRI makes motion extremely challenging to correct and leaves very little room for uncorrected instrument imperfections. Second, even if the images are free of motion artifacts, motion of the heart can lead to erroneous estimates due to the strain-memory effect of the diffusion constants. Strain alters the relative distance between any two given points of a tissue. Because the diffusion-weighted MRI signal is derived in part based on the probability of spatial displacement, strain can add or subtract from the displacement, and lead to over or under­estimation of the diffusion measurements. The effects of strain on in vivo cardiac DTI measurements have long been documented [76]. Because cardiac strain can be separately quantified via, for example, tagged MRI, its effects can be subtracted to obtain pure diffusion and fiber orientation measurements from in vivo cardiac DTI data [76]. Moreover, because the effects depend on the average strain across the cardiac cycle, it has been shown possible to obtain strain-free in vivo cardiac DTI measurements by selecting the right timing delay in the cardiac-gated acquisition [77]. Lastly, recent advances in gradient hardware technology have made high-strength gradients available (up to 80 mT/m in whole-body scanners and 1500 mT/m or more in small animal systems as of the current writing), which in turn made it practically feasible to employ bipolar encoding gradient pulses capable of generating moderate but sufficient b-values, in terms of diffusion encoding, in diffusion MRI and DTI, including for cardiac imaging [110]. Bipolar diffusion pulses offer not only reduced motion sensitivity, as 37 described in the above section, they also have decreased memory effects of strain [108]. However, besides the impact on accuracy as explained in Section 2.4.2, using the relatively low b-values associated with bipolar gradient pulses can also inadvertently introduce effects of tissue perfusion and lead to additional error in the DTI experiment. Perfusion, in this case blood flow in the capillary bed, has long been known for causing additional spin phase dispersion and leading to overestimated diffusion coefficients via the so called intravoxel incoherent motion (IVIM) effect [111] in highly vascularized organs such as the liver [112]. Because the capillary flow is faster than the diffusion of water, the flow-mediated perfusion effect can be eliminated from diffusion measurements by employing sufficiently high (b > 200 s/mm2) diffusion weighting. The perfusion dependence of diffusion MRI has been theorized [111], [113] and recently empirically demonstrated [114] for the perfused heart. Together, it is clear that the specific physiology of the heart adds technical issues that need to be considered in performing in vivo cardiac DTI. Despite technological advances that have made most the known issues tractable, complete understanding and compensating the effects of motion in cardiac DTI remain works in progress. In the meantime, caution is warranted in interpreting in vivo cardiac DTI results. 2.5 Accelerating DTI Acquisition Because of its minimum dataset requirement, and the frequent use of signal averaging, including increasing the number of encoding gradient directions, to improve its accuracy, practical applications of DTI have been hampered by long scan times. Due to the repetitive nature of the DTI experiment, reduction or elimination of redundancies 38 in the data has been explored as a potential means to accelerate the data acquisition [102]. The so-called compressed sensing is an advanced technique that reconstructs MRI images from partially sampled data [115], [116]. The relative benefits of compressed sensing are demonstrated in Fig. 2.10. Conventional reconstructions, using direct inverse Fourier transform for example, of partially sampled k-space data often lead to structured artifacts such as blurring and ringing in the result images. One key feature of compressed sensing is randomized k-space sampling, which turns the effects of partial k-space sampling into incoherent, noise-like artifacts. The latter are then minimized to yield images close to what the fully sampled k-space would have achieved, but using data that were acquired in 39 Helix Angle FA - 90° 0° 90° 0 0.25 0.5 Figure 2.10. FA and helix angle maps from accelerated DTI data. Helix angle and fractional anisotropy (FA) maps from (first row) using all acquired data or the "Gold-standard" case reconstructed traditionally and (second row) using 1/16th of the acquired data, reconstructed using the model-based compressed sensing reconstruction. a fraction of the time. The ideal experiment for applying compressed sensing is one where some knowledge of the images to be reconstructed already exists. In general, compressed sensing reconstruction can be formulated as the minimization of some cost function given in the form of C(m) = ||F(m) - y||! + a ||^ (m )|li, (216) -* 2 where ||F(m) - y|| is the data fidelity term and ||^ (m )|\1 is the image sparsifying term. The fidelity term is akin to the least squares fitting between the acquired signal, y, and the estimated image, m, transformed into the measurement domain, or k-space, by the undersampled Fourier operator F(-). The image sparsifying term is used to drive the estimated image, m, to have desired features, such as spatial smoothness or sparsity, using a transform, ^ . Common examples of image sparsifying transforms are the total variation (TV) and wavelet transforms. The estimated image, m, is reconstructed by minimizing Eq. (2.16) with respect to m m = minm||F(m) - y||^ + a ||^ (m )|\t , (217) which is performed using a numerical method such as gradient descent or conjugate gradient. The contributions from the data fidelity and image sparsifying terms are controlled by the regularizing term, a. DTI lends itself well to compressed sensing because of the redundancies across the diffusion-weighted images (e.g., same organ size and shape with only diffusion contrast differences). The general cost function in Eq. (2.16) can be altered to include the series of diffusion-weighted images [117] 40 N C(mn) = ^ ( | |F (m n) - y||^ + ll^(mn)lll) . (218) n = l The reconstructed individual diffusion-weighted images, mn, can then be used to obtain the diffusion tensors similar to the standard DTI experiment. The goal of such reconstructions is to acquire DTI data in a reduced amount of time without incurring the proportional loss in accuracy associated with the scan time acceleration. 2.6 Conclusion Considering that the heart is active since before birth and for the most part throughout a person's life, myocardial microstructure changes across orders of magnitude in both time and space making structural measurements rather challenging. This short contextual survey of structural measurements via MR-DTI hopefully illustrates a versatile response to this challenge. Now, it is worth mentioning some promising examples of the future of fiber structure acquisition, and how it can have a positive effect in the way we understand cardiac structure and function. Despite its complexity, fiber structure is intimately linked to diagnosing, monitoring, and treating heart disease, which is one of the largest causes of death in the world. For this reason, many researchers and clinicians recognize the value of applying understanding of the heart on personalized medicine. This ideal includes two fundamental layers, the characterization of functional and material parameters in health and disease, as well as predicting the outcome of an intervention on a patient. The first layer involves gathering large databases of relevant information, including anatomical and diffusion tensor imaging, and establishing analysis criteria useful for classifying phenotypes and recognizing disease [118], [119]. Such an approach, would endow clinicians with information-age diagnostic tools never before seen [120]. Much of the 41 42 statistical machinery, such as the ability to merge large amounts of tensor images, is already been applied to the brain to the classification of adult and developing structures [121], [122], and their cardiac counterparts are underway [123]. Likewise, thanks to advances in computational cardiology, the ability of predicting the outcomes of treatment is no longer a dream. Computational models are becoming an important tool for planning therapeutic strategies for fibrillation [124], and for the prevention of side effects, like venous occlusion by pacemaker lead placement [125]. With the advent of more comprehensive approaches, like electro-mechanical models of the whole heart and the ability to acquire fiber structure information in vivo, predicting the outcome of surgical procedures is within sight. 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CHAPTER 3 MODEL-BASED RECONSTRUCTION OF UNDERSAMPLED DIFFUSION TENSOR K-SPACE DATA1 3.1 Abstract The practical utility of diffusion tensor imaging, especially for 3D high-resolution spin warp experiments of ex vivo specimens, has been hampered by long acquisition times. To accelerate the acquisition, a compressed sensing framework that uses a model-based formulation to reconstruct diffusion tensor fields from undersampled k-space data was presented and evaluated. Accuracies in brain specimen white matter fiber orientation, fractional anisotropy, and mean diffusivity mapping were compared with alternative methods achievable using the same scan time via reduced image resolution, fewer diffusion encoding directions, standard compressed sensing, or asymmetrical sampling reconstruction. The efficiency of the proposed approach was also compared with fully sampled cases across a range of the number of diffusion encoding directions. In general, the proposed approach was found to reduce the image blurring and noise and to provide more accurate fiber orientation, fractional anisotropy, and mean diffusivity measurements 1 © 2012 Wiley Periodicals, Inc. Reprinted, with permission, from Christopher Lee Welsh, Edward V.R. DiBella, Ganesh Adluru, and Edward W. Hsu. Model-Based Reconstruction of Undersampled Diffusion Tensor k-Space Data. Magnetic Resonance in Medicine. DOI 10.1002/mrm.24486. compared with the alternative methods. Moreover, depending on the degree of undersampling used and the diffusion tensor imaging parameter examined, the measurement accuracy of the proposed scheme was equivalent to fully sampled diffusion tensor imaging datasets that consist of 33-67% more encoding directions and require proportionally longer scan times. The findings show model-based compressed sensing to be promising for improving the resolution, accuracy, or scan time of diffusion tensor imaging. 3.2 Introduction Diffusion tensor imaging [1] (DTI) is an MRI technique that allows quantitative characterization of the geometry and organization of tissue microstructures such as fiber orientation. DTI has been applied in the brain [2], [3] to, for example, trace white matter tracts [4]-[6], map connectivity, and characterize damage caused by stroke [7]-[9]. The method has also been applied to ex vivo specimens of the heart [10]-[12] to map structural changes due to fibrosis or infarction [13]-[17]. Because the diffusion tensor is a rank 2, symmetric matrix [1], a unique solution of the diffusion tensor requires a minimum of six diffusion-weighted images sensitized in noncollinear diffusion encoding directions, plus a nondiffusion-weighted image. In this sense, the minimum scan time required for a DTI dataset, which spans both the spatial and diffusion dimensions, is seven times that of an anatomical scan acquired using the same sequence and settings. DTI suffers from low signal-to-noise ratio (SNR) because diffusion is measured as signal attenuation and from increased echo time necessary to accommodate the use of diffusion sensitizing gradients. In practice, the loss of signal is 55 often compensated for by additional signal averaging in forms of acquisition repetition or increasing the number of encoding directions, which further prolongs the scan time. Consequently, methods to accelerate the acquisition, especially those that minimize further SNR loss, are highly desirable. Perhaps the most obvious method to accelerate acquisition would be to either scan at a lower resolution or simply encode fewer diffusion directions. Scanning at a lower resolution would introduce blurring and make it more difficult to discern fine structures in an image. In contrast, although the resolution is maintained, encoding in fewer diffusion directions would sacrifice the accuracy of the diffusion tensor estimation. A possible solution to both of these problems is to accelerate acquisition by partially sampling k-space, and apply reconstruction in such a way that fine structure and diffusion tensor accuracy are preserved. When not reconstructed properly, partially sampling k-space introduces artifacts in image space, such as ghosting, field-of-view overlap, or additional noise, depending on the undersampling pattern used. Techniques have been introduced to reduce the effects of partial sampling or undersampling when DTI is acquired using multiple receive coils [18]-[23]. Rather than reconstructing each image of the multi-image acquisition separately, compressed sensing techniques [24]-[26] are capable of jointly estimating multiple acquisitions by sharing sparsely sampled data. Compressed sensing can be particularly attractive for DTI, due to the high degree of similarity or redundancy (e.g., the size and shape of the brain, including the white matter) among the acquisitions at different diffusion directions that can be leveraged to represent a transform of the data sparsely. Previously, compressed sensing based on undersampling in the diffusion encoding or q- 56 space has been applied to high-angular-resolution diffusion imaging [27], [28], which is a more general form of diffusion imaging that allows resolution of crossing fibers. A possible alternate way to undersample DTI or high-angular-resolution diffusion imaging data is in k-space [29] or spatial frequency domain. Intuitively, the nature of the desired information in the DTI dataset and the well-known relationship between image-space and k-space may offer flexibility that can be exploited for effective undersampling. For example, the fiber orientation of the brain white matter or myocardium varies relatively slowly and, therefore, the relevant information can be captured even when the outer k-space is sampled with a lower density. Regardless of the scheme of undersampling, it is important that the performance of any reconstruction method be evaluated on not only the acquisition time acceleration, but also on its ability to capture the desired information. In DTI, fiber orientation, fractional anisotropy (FA) [30], and mean diffusivity (MD) are the often sought-after parameters for assessing tissue microstructure. Therefore, ideally the performance of any compressed sensing acceleration needs to be evaluated in terms of accuracy loss in measuring these parameters with respect to a ‘‘ground truth'' or ‘‘gold standard.'' Moreover, to be considered effective, any proposed technique should retain more accuracy than alternative methods using, for example, lower resolution or fewer diffusion encoding directions to achieve the same acceleration. The goal of the current study is to investigate the validity of a compressed sensing framework for DTI that, in addition, uses the signal intensity model to directly estimate diffusion tensor fields from undersampled k-space data. The formulation bypasses the usual intermediate step of estimating diffusion-weighted images. By estimating the 57 diffusion tensor directly, the number of variables to be solved is reduced from N x d im l xdim2 xdim3 to 6 x d im l xdim2 xdim3 (excluding the nonweighted volume) where dim1, dim2, and dim3 are the spatial dimensions of a 3D acquisition, and N is the number of diffusion-weighted images acquired. In a noise-dominated system, reducing the number of unknowns can provide the more accurate estimates of the fitted parameters. As well, the model-based formulation provides a convenient platform that numerous practical considerations involved in DTI (e.g., phase errors) can be addressed in a single step. Model-based approaches have been proposed previously to compensate for eddy currents, field inhomogeneities, and motion in DTI [31], [32] and to reconstruct diffusion tensor tomography data [33] but not for accelerating acquisition. Other compressed sensing techniques using exponential models have been proposed for T1 and T2 mapping [34], [35]. The current model-based algorithm is demonstrated on a 3D DTI acquisition, which is used for high-resolution characterization of fixed specimens [36], [37]. Three­dimensional DTI, especially one acquired with a spin-echo sequence, can take many hours for ex vivo acquisitions and thus can benefit greatly from acquisition acceleration. Also, 3D acquisitions have a higher degree of data redundancy and can be undersampled in more than one dimension. For the sake of brevity, in the following sections, and unless otherwise noted, the term DTI strictly refers to 3D spin-warp spin-echo DTI. The effectiveness of the model-based compressed sensing algorithm is validated against other means to achieve comparable scan time reduction. Part o f the current work has been presented previously in a conference abstract [38]. 58 59 3.3 Theory Compressed sensing finds the target images by subjecting the estimates to a set of reconstruction constraints and minimization of the associated penalty or cost function. One form of a cost function for compressed sensing [24], [39] reconstruction of a series of N undersampled MRI k-space data, dn, is given as: where || ||2 represents the L2-norm, which produces the least squares solution [25] and can be substituted by any other type of measure of deviation between images [40], and mn is the estimated magnitude images. The terms mn and dn are the vectorized versions of mn and dn, such that for image resolution dim1 by dim2, the vectors mn and dn have dim1 x dim2 elements. The first term in Eq. (3.1) is a fidelity term that forces the solution to adhere to the acquired k-space data. The second term in Eq. (3.1) is a total variation (TV) operator [41] applied in image space to constrain the solution to generate piecewise-constant images, hence reducing erratic points due to undersampling k-space. The scalar regularization weight factor a controls the relative contributions of the fidelity and TV terms. The jth k-space element of the k-space signal model, F(mn, n ), is where Wn is the binary undersampling function for the diffusion direction n, is the image phase which is estimated and fixed from the acquired low-resolution data from each diffusion-weighted image, x is the position in image space, and kj is the position in k-space. The series of images, mn, can be obtained simultaneously by minimizing the Fj(mn, n) = Wn( k j mn(x) • • e 1 x'^i (3.2) cost function in Eq. (3.1) with respect to mn. In the current model-based DTI reconstruction, to allow direct estimation of the diffusion tensor, mn is replaced with the standard diffusion tensor intensity model [1]. This allows direct estimation of the diffusion tensor. The relationship between mn and the DTI signal is given by, mn = l0 e - b9lDgn (3.3) where I0 is the image without diffusion weighting, b is the diffusion weighting factor and gn = (gxn, gyn, gzn)T is the diffusion encoding directional vector in the 3D space spanned by the readout (x), phase (y), and slice (z) encoding directions. D is the rank 2, symmetric tensor defined as: 60 Dxx Dxy Dxz D = Dxy Dy" (3.4) Dxz Dyy Dzz_ Replacing mn yields the new k-space signal model: Fj(D,n) = Wn(kj) ^ /0(x) • e~h& D(! Bn • • e -! (3.5) X For the current formulation, the TV term can be defined as: 7V(0) = || J ( m j ! + ( m j ! || (3 6) where (mn)y and (mn) z are the partial derivatives of mn with respect to y and z. TV is not calculated in the x-direction since it is already fully-sampled. Maps corresponding to each diffusion tensor element can be estimated by minimizing Eq. (3.1) simultaneously with respect to each element of the tensor, D, via, for example, gradient descent optimization. To perform gradient descent, computational equations for the derivative of Eq. (3.1) are needed with respect to each diffusion tensor element, 61 Ds (s G {xx, yy, zz, xy, xz, y z }). 3C ( D) dD, = £ - 2 H 9 „9 ! sIoe - b^ D° " R e a l \ e ( F ^ D , n ) - djn) e ! ! ! n=l V ! h( ^ ~ ______ Cmn) y_______\ _______(fflj!_______\ 9n9n s n ygy ^ 2 + p i ) d z y j 2 + ( ^ J 2 + pi ) (3.7) where gng^ is a 3 x 3 matrix and the (0 s operator extracts the sth element corresponding to the Ds parameter map. The full derivation of this equation can be found in the attached appendix. The diffusion tensor elements are then updated iteratively using the derivative in Eq. (3.7): Dl+1 = D! - A (38) where A is the step size used in the gradient descent and r is the iteration number. 3.4 Methods 3.4.1 Dataset Fully sampled 3D, Cartesian k-space, DTI spin-echo data consisting of diffusion-weighted scans in a relatively high number of encoding directions (96 in all) and four nondiffusion-weighted "b0" images (100 x 75 x 70 matrix size, 0.5 x 0.5 x 0.5 mm3 isotropic voxel size, echo time (TE) = 39 ms, repetition time (TR) = 500 ms, and number of excitations (NEX) = 1) were acquired on a fixed, excised macaque brain hemisphere using a Bruker Biospec 7T scanner (Bruker Biospin Inc., Billerica, MA) equipped with a high-performance gradient system (max gradient amplitude capable of 600 mT/m). The acquisition time for the entire diffusion MRI dataset was ~72 h. Figure 3.1 shows the ‘‘b0'' and diffusion-weighted images of a representative 2D slice, for reference. The 62 Figure 3.1: Two-dimensional MRI coronal view of the macaque brain hemisphere used in this study. Left, nondiffusion-weighted "b0" image, b = 0 s/mm2 . Right, diffusion-weighted image, b = 5000 s/mm2 in a single diffusion direction, scaled up for better display. There is notable shading due to coil inhomogeneity. diffusion tensor solution to the entire fully sampled dataset was used as the gold standard for subsequent performance assessments. 3.4.2 Undersampling Schemes To simulate more typical DTI