BOSTON UNIVERSITY COLLEGE OF ENGINEERING. Dissertation. Adaptive Non-Fourier Spatial Encoding For Dynamic MRI PAIRASH SAIVIROONPORN

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1 BOSTON UNIVERSITY COLLEGE OF ENGINEERING Dissertation Adaptive Non-Fourier Spatial Encoding For Dynamic MRI by PAIRASH SAIVIROONPORN M.S., Boston University, 1992 B.S., KMITL, 1987 Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy 1998

2 Approved by First Reader Lucia M. Vaina, Ph.D., D.Sc. Professor of Biomedical Engineering, Boston University Associate Research Professor of Neurology, School of Medicine, Boston University Second Reader Ron Kikinis, M.D. Assistant Professor of Radiology, Brigham and Women s Hospital, Harvard Medical School Research Assistant Professor of Biomedical Engineering, Boston University Third Reader Gary Zientara, Ph.D. Assistant Professor of Radiology, Brigham and Women s Hospital, Harvard Medical School Fourth Reader S. Hamid Nawab, Ph.D. Associate Professor of Biomedical Engineering, Boston University Associate Professor of Electrical, and Computer Engineering, Boston University

3 Acknowledgments First I would like to thank my readers Lucia Vaina, Ron Kikinis, Gary Zientara, and Hamid Nawab for their time and effort. Dr. Lucia Vaina, my M.S. and Ph.D. thesis advisor, has continuously supported and helped me during my entire six years at Boston University. To her I owe so much. Dr. Ron Kikinis, my thesis advisor and most respected boss, has supported me in countless ways during the entire reseach period. I gratefully acknowledge his willingness and efforts to help, even when my work turned in a direction away from his principle area of interest. Dr. Gary Zientara has introduced me to fascinating topics in MRI, and has also provided me with much valuable advice about conducting proper research. I thank him for these things, and also for his close editing of my thesis. I am espcially indebted for all I have learned from him. I owe a large debt of gratitude to Dr. Ferenc Jolesz, the director of the MRI Division, Radiology Department, Brigham and Women s Hospital, Harvard Medical School, to allow me to conduct my research in his division, and for his support, inspiration, and wisdom. I am deeply indebted to the Royal Thai Government for providing me an opportunity to pursue my M.S. and Ph.D. studies. I thank them for their generosity. Also, thanks go to the Department of Biomedical Engineering, Boston University, for providing a supportive environment. I also thank Dr. Larry Panych, my semi-advisor, who taught me everything I know about linear system analysis in MRI, and many other topics in MRI as well. Heartfelt thanks to all of my colleagues at Brigham and Women s Hospital, especially in the Surgical Planning Laboratory and Radiology Department, for their friendship and support throughout the years. iii

4 And last, but not least, I thank my family for their continuous support during my years in Boston studying for M.S. and Ph.D. degrees. Words can not describe all they have done for me. iv

5 Adaptive Non-Fourier Spatial Encoding For Dynamic MRI (Order No. ) PAIRASH SAIVIROONPORN Boston University College of Engineering, 1998 Major Professor: Lucia M. Vaina, Professor of Biomedical Engineering Abstract In this research, an efficient adaptive encoding method for dynamic Magnetic Resonance Imaging (MRI) using near-optimal spatial encoding computed by the Singular Value Decomposition (SVD) was developed and implemented. This method was first proposed and demonstrated for this application by Zientara et al: In this research, the theory has been extended and the method refined for use in human imaging. A new gradient-recalled-echo (GRE) slice-selective pulse sequence employing SVD-based spatial encoding was developed and implemented on a commercial MRI system. A spatially selective RF excitation technique employing small flip angles was used. A multi-slice GRE SVD pulse sequence was also developed and implemented for multi-slice image acquisition within a slab. A phase-cycling technique for slab selection and phase-encoding along the echo train to encode multiple slices within the slab was employed. Encoding artifacts of the SVD-based spatial encoding method were investigated and compared to the conventional Fourier encoding method. The linear input-output (IO) system response model of MRI spatial encoding, introduced by Panych et al:, was employed to analyze encoding artifacts due to variation in the v

6 flip angles of the spatially selective RF excitation. The IO system model assumes that the system output depends only on the input. The assumption is, however, not valid in fast MR acquisition because the system output also depends on the non-equilibrium magnetization (i.e. the state) of the system. A linear input-stateoutput system response model was, therefore, introduced for investigating transient artifacts, caused by the system state in such acquisition. SVD MRI encoding efficiency, defined as the necessary numbers of SVD encodes compared to the numbers used for standard Fourier encoding, was studied. Effects on encoding efficiency due to the ideal and non-ideal encoding conditions and on changes in image contents due to translation and rotation were characterized. A dynamic MRI experiment simulating a needle biopsy, a clinical application in interventional MRI, was performed using SVD-based spatial encoding. Also multiple 2D images within a slab from a human brain were acquired by using the multi-slice GRE SVD pulse sequence demonstrating, for the first time, SVD encoding for acquiring multi-slice images for dynamic MRI. vi

7 Contents 1 Introduction 1 2 Background Conventional MRI Experiment The Slice Selective RF Excitation Pulse The Evolution Period The Data Collection Period Image Reconstruction Spatial Encoding Methods in MRI Background Non-Adaptive Technique D Separable Transform The Fourier Encoding Method The RF Impulse Encoding Method Hadamard Encoding: A Non-Fourier Method Adaptive Techniques Orthogonal Decomposition Singular Value Decomposition SVD Encoded MRI Methods and Implementations System Overview Pulse Sequence Implementation vii

8 3.2.1 Spin Echo Gradient-Recalled Echo Multi-Slice Gradient-Recalled Echo Software Processes on the Workstation Preprocessing K-Space Mapping Image Reconstruction Orthonormal Basis Set Calculation SVD Computation Using the Jacobi Algorithm Converting and Transferring RF Vectors Software Interface Non-Adaptive and Dynamic-Adaptive Algorithms Analysis of Non-Fourier Encoded MRI Using the Linear System Approach Introduction Theory The Linear Input-State-Output System Impulse Response Analysis Transient Response Analysis Experimental Results and Analysis System Response Superposition Theory System Response to RF Excitation Input The Distribution and Cumulative Distribution Functions of the SVD Flip Angle Profile Optimal Flip Angle for SVD Encoding viii

9 4.3.2 The Transient Response Zero-Input Response Analysis The Transient Artifact in Static MR Imaging The Transient Artifact in Dynamic MR Imaging Summary and Conclusions Impulse Response Analysis Transient Response Analysis Encoding Efficiency of the Non-Fourier Encoded MRI Methods Background Encoding Factor Principal Angle Factor Quantitative Analysis of Encoding Efficiency Encoding Efficiency in Cases of Ideal Encoding A Comparison of Different Encoding Methods A Study of Encoding Efficiency in the SVD Method Due to Different Image Contents Encoding Efficiency of the SVD Method at a Non-Ideal Encoding Condition Encoding Efficiency of the SVD Method in the Worst-case Scenario Effects of Translation on Encoding Effects of Rotation on Encoding Summary and Conclusions Applications of the Non-Fourier Encoded MRI in Dynamic MRI Single-Slice Spin-Echo SVD Encoded MRI Dynamically Adaptive Algorithm ix

10 6.1.2 The Keyhole Reconstruction Algorithm Multi-Slice Gradient-Recalled-Echo SVD Encoded MRI Summary Conclusions and Suggestions for Future Research Methods and Implementations Analysis of the Non-Fourier Encoding Method The Efficiency of Non-Fourier Encoded MRI Applications of the Non-Fourier Encoded MRI A Non-Fourier Encoding Method Implementation for Real Time Dynamic MRI Applications 160 B Dynamically Adaptive Algorithm For Real Time Dynamic MRI Applications 163 References ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::166 Curriculum Vitae::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::177 x

11 List of Tables 5.1 The EEF analysis of the SVD method xi

12 List of Figures 2æ1 The laboratory frame of reference æ2 Excitation of magnetization æ3 The conventional spin-echo pulse sequence diagram æ4 A spatial encoding method diagram æ5 A orthogonal decomposition method diagram æ1 The hardware system of the non-fourier encoded MRI method æ2 The software diagram of the non-fourier encoding method æ3 A SE pulse sequence diagram æ4 The two-shot slice selection technique æ5 The slice magnetization for the two-shot slice selection æ6 A GRE pulse sequence diagram æ7 A multi-slice GRE pulse sequence diagram æ8 The conjugate effect on the non-fourier encoding data due to 180 æ RF pulse æ9 A block diagram of the RF vector conversion and transfer process. 48 3æ10 The research mode user interface of the non-fourier encoding process 49 3æ11 The menu options of the multi-slice GRE pulse sequence user mode 50 3æ12 The initial process of the non-adaptive algorithm æ13 The non-adaptive algorithm for dynamic MRI application æ14 The initial process of the dynamic-adaptive algorithm æ1 A ISO system model xii

13 4æ2 Timing diagram of the ISO system model æ3 A sinc pulse shaped RF input æ4 The unit-step-input transient response model æ5 RF unit box pulses used in the superposition experiment æ6 The calculated and experimental results of the C 3 RF input pattern 66 4æ7 The calculated and experimental results of the C 4 RF input pattern 67 4æ8 RF impulse images at difference flip angles æ9 Hadamard encoding images at difference flip angles æ10 SVD encoding images at difference flip angles æ11 RF magnitude and phase inputs of the RF impulse, Hadamard, and SVD encoding æ12 Magnitude and phase profiles of the RF impulse, Hadamard, and SVD encoding æ13 The distributions of SVD flip angles at 10 æ ; 15 æ, and 20 æ æ14 The distributions of SVD flip angles at 30 æ ; 40 æ, and 50 æ æ15 The SVD images of phantoms and their flip angle distributions æ16 A SVD human brain image and its flip angle distribution æ17 The SNR of images acquired from RF impulse, Hadamard, and SVD encoding methods æ18 The ANR of images acquired from RF impulse, Hadamard, and SVD encoding methods æ19 The effect of TR to the zero-input response æ20 The effect of gradient spoiling on the transient signal æ21 The zero-input response using the volume projection pulse sequence 85 4æ22 A timing pulse diagram of a possible cause of the transient signal. 86 4æ23 The 180 æ RF slice selection pulses and their spatial profiles xiii

14 4æ24 The diagrams for acquiring the complete and zero-input responses 88 4æ25 The complete and zero-state responses of the DW and BG phantoms 89 4æ26 The SVD images reconstructed from three different responses æ27 Static MR images using Hadamard and SVD encoding methods æ28 Hadamard and SVD images using the HTL and LTH encoding orders 94 4æ29 The magnitude responses to the excitation reordering methods æ30 Dynamic MR images using difference reordering patterns æ1 A comparison of the EF measurements æ2 The distributions of the sines of the principal angle at three scenarios 109 5æ3 The comparison of eigenimages from the RF impulse, Hadamard, and SVD methods æ4 The magnitude of the projection from spatial selective encoding method at 4 and 8 encodes æ5 The magnitude of the projection from spatial selective encoding method at 16 and 32 encodes æ6 The EF analysis of the RF impulse, Hadamard, and SVD methods æ7 Four typical MR images use in the simulation æ8 The average of the EF measurement of the Fourier and SVD methods120 5æ9 The average of the EEF measurement of the SVD method æ10 The EF measurement of the SVD results at the ideal and non-ideal encoding condition æ11 The effect of the horizontal translation on the SVD image æ12 The PAF analysis on the effect of the horizontal translation æ13 The effect of the vertical translation on the SVD image æ14 The PAF analysis on the effect of the vertical translation xiv

15 5æ15 The rotation effect on the vertical encoding image using the SVD method æ16 The rotation effect on the horizontal encoding image using the SVD method æ17 The angular plot of the PAF analysis of the rotation effect æ18 The encoding and projection profiles æ1 The dynamic MRI experiment using the RF impulse method æ2 The dynamic MRI experiment using the Hadamard method æ3 The dynamic MRI experiment using the SVD method æ4 The comparison of the EF measurements from the RF impulse and SVD method in the dynamic MRI experiment æ5 The PAF analysis from the SVD method in the dynamic MRI experiment æ6 The dynamic MRI experiment using the keyhole RF impulse encoding method æ7 The dynamic MRI experiment using the keyhole Hadamard encoding method æ8 The dynamic MRI exmperiment using the keyhole SVD encoding method æ9 A multi-slice SVD image æ10 A multi-slice RF impulse image æ11 A multi-slice gradient-recalled echo image Aæ1 The hardware system of the non-fourier encoding method Bæ1 The system diagram for the non-fourier encoding method xv

16 1 Chapter 1 Introduction Magnetic Resonance Imaging (MRI) is now a conventional medical diagnostic tool used for acquiring cross-sectional images of a patient s anatomy. The advantage in MRI hardware and fast pulse sequence techniques, such as echo-planar [56], RARE [58] and BURST [57], combined with the motion artifact reduction techniques make MRI possible for imaging of moving structures, such as heart [59] and upper airways [87]. Motion artifacts in MRI can be reduced by a variety of techniques, such as physiologic gating [81], [68] and ordered phase encoding [93]. Using MRI one also has the ability to detect and quantify physical properties, such as diffusion, perfusion, flow, and temperature. Such an ability combined with fast image acquisition makes MRI applicable for the studies of dynamic processes (dynamic MRI). Dynamic MRI refers to rapid acquisition of a sequence of MR images. Some examples of the applications of dynamic MRI are the investigation of the change in contrast agent on brain tumor following the bolus injection [73], [84]; the study of the cerebral hemodynamics on the visual cortex during optic stimulation [60]; and the monitoring of the temperature distribution on target tissues under interstitial laser therapy [80], [61]. The time scale necessary for acquiring dynamic MRI varies from one image per second in laser ablation therapy to one image per minute in the bolus injection study. Dynamic MRI applications typically require not only high temporal resolution but also high spatial resolution. For example, the dynamic MRI applications of for

17 2 monitoring and controlling of minimally invasive thermal therapies conducted in our laboratory - notably laser ablation, focused ultrasound ablation [78], [65], [66] and cryo-ablation [83] therapies have these high resolution requirements. The timescale necessary for acquiring MR images in these studies is approximately two seconds. The desired image volume is 256 æ 256 or greater per slice for slices per volume. For these applications in theory, images need to be acquired at a rate of slices every two seconds or 8-32 slices per second with high tissue contrast and high sensitivity to the therapy. These particular imaging needs are not currently satisfied by any available MR techniques. Many different dynamic MRI methods have been suggested and implemented with the common goal to increase temporal resolution. The methods can be classified into two major categories: non-adaptive and adaptive imaging. Non-adaptive methods provide high temporal resolution by compromising spatial resolution or signal to noise ratio (SNR). Non-adaptive methods can be separated into two major groups as: fast pulse sequences and reduced k-space sampling techniques. The filling of k-space through data acquisition is discussed in Adaptive methods for dynamic MRI are relatively new techniques with the high potential to meet the demand for dynamic MRI applications. The current adaptive methods developed in our laboratory [7] use the spatially selective RF excitation encoding techniques. Fast pulse sequences attempt to reduce the acquisition time by reducing TR, the repetition time between spatial encoding steps, or reduce the number of spatial encoding steps. For example, fast low flip angle gradient-recalled echo (FLASH) [74] and gradient-recalled in the steady state (GRASS) [69] imaging use low flip angle RF pulses to reduce TR providing an image acquisition time range from 500 msec to several seconds. Unfortunately, the high temporal resolution of these methods is obtained by the compromise of lower image SNR. Another fast pulse

18 3 sequence approach is to acquire as much spatially encoded information as possible in a short period of time. For example, Spin echo planar and Gradient-recalled echo planar methods [55], [56] require only one or two excitation RF pulses to acquire an entire image, giving image acquisition time in tens to several hundred milliseconds for a low SNR and low spatial resolution (typically 128 æ 128) image. Echo planar methods need special gradient coils design to support a rapid switching rate and to provide very high gradient power. The temporal resolution of this method, then, is limited by magnetic coil design and power supply technology. Reduced k-space sampling methods including MR Fluoroscopy [86], [90], [76], Fourier-encoded keyhole imaging [89], [63], [64] and reduced field-of-view, (FOV) [77], attempt to increase temporal resolution by acquiring only some fraction of k- space data with the assumption of no significant change in the remainder. MR Fluoroscopy was developed for real-time visualization with specialized hardware for image reconstruction using combinations of previous and newly acquired k-space lines to quickly update image display. Fourier-encoded keyhole imaging assumes that there is no significant change in the underlying morphology responsible for high spatial frequency in an image variation, therefore, only the low frequency k-space data need to be updated while the high frequency k-space data is reused from an initial fully acquired k-space data set. This method experiences artifacts due to the change in high frequency components. The reduced FOV method performs dynamic imaging of local changes, so that a smaller FOV can be used to reduce imaging time, a spatial constraint not always satisfied in medical applications. Adaptive methods use information from the current and/or most recently acquired image(s) to minimizing redundancy in data acquisition. No methods in common use are strictly adaptive, but several methods used for studying mo-

19 4 tion of moving structures can be considered as adaptive methods. For example, to reduce moving artifact, the technique of respiratory ordered phase encoding (ROPE) [68] uses prior knowledge of the respiration pattern to reorders the phase encoding steps of a sequence. Adaptive methods can be implemented by modifying the spatial encoding of the FOV. The methods use a special orthogonal set of basis functions representing encoding profiles to acquire the minimum amount of information in which to correctly display the contents of the FOV. The orthogonal set is computed preceding each image acquisition based upon the best estimate of the current FOV. Adaptive methods, though, require an additional computational overhead. However, these methods are now practical with the availability of fast computers and high-speed communication networks. These methods can be implemented using fast pulse sequences, such as gradient-recalled echo, or spin echo pulse sequences. The transformations used to provide orthogonal sets for adaptive MRI spatial encoding techniques include Hadamard [85], [62], [70], Wavelet [75], [2] and Singular Value Decomposition (SVD) [7] transformations. SVD refers to a matrix decomposition or canonical form theorem of linear algebra [72] with many applications in signal and image processing [43], [67], [88]. Wavelet and SVD methods attempt to provide the minimum number of spatial encoding steps adaptively. The recent work by Zientara et al [7] in our lab showed that SVD encoding MRI is applicable to dynamic adaptive MRI due to a near optimal set of spatial encoding steps computed from the estimate of the recent obtained image. SVD encoded MRI is more efficient than Wavelet encoding MRI when there are changes through out the image FOV. Zientara et al [91], [92] had also developed and implemented the dynamically adaptive MRI with encoding using Multiresolution and Keyhole SVD encoding

20 5 MRI. Multiresolution SVD encoding MRI is based on the idea of using the change in image data at coarse resolution as a predictor of change at finer resolution. The method emphasizes the adaptive imaging capability of SVD encoding MRI by determining for each image in a dynamic series the near minimum necessary encoding of the FOV. Using the hypothesis that changes in FOV are significantly apparent in low resolution acquired data. Keyhole SVD encoding MRI [92], like the keyhole Fourier technique, combines information from a complete encoding of an image early in a series with incomplete current information. The work in this study is an investigation of the non-fourier encoding method based on the work by Panych et al [2] and Zientara et al [7]. The goal is first to implement the encoding method on a commercial MR system for dynamic MR applications, and then to investigate the characteristics of such encoding method including the comparison of the encoding efficiency to the conventional encoding method. Finally, the non-adaptive and dynamically adaptive MRI methods are demonstrated in a dynamic MRI experiment including the analysis of the encoding efficiency of such methods. This thesis is organized into 7 chapters as follows: Chapter 2 reviews the theoretical background of MR imaging including the Fourier and non-fourier transforms used for MRI spatial encoding. Chapter 3 describes the methods and implementations of the non-fourier encoding on a commercial MR system. Chapter 4 analyzes the characteristics of the encoding methods using the linear system approach. Chapter 5 investigates the factors which effect the efficiency of the encoding methods. Chapter 6 demonstrates the application of the methods for dynamic MRI ap-

21 6 plication in a non-real time fashion. Chapter 7 includes the conclusions and recommendations for future research.

22 7 Chapter 2 Background The basis of diagnostic MRI is proton magnetic resonance that is a phenomenon depending on the action of the hydrogen ( 1 H) nuclei which are exposed to static and radio frequency magnetic fields. The behavior of the magnetic moment, M, ~ of the nuclei in the presence of a magnetic field, B, ~ can be described macroscopically by classical mechanics or microscopically by quantum mechanics. Quantum mechanics is a necessary tool to treat some magnetic resonance phenomena, such as when the nuclear spins are mutually coupling or when there is a strong magnetic interaction with the surrounding matter. The treatment of such phenomena, which is beyond the scope of our discussion, can be found in many textbooks [10]-[13]. Fortunately, most of the basic behavior of the MRI experiment can be understood by the classical mechanics framework using the Bloch equation. The Bloch equation successfully explains the macroscopic phenomena of a MRI experiment, d ~ M dt = æ ~ M æ ~ B (2.1) In Eq. (2.1), the vector ~ M is the magnetic moment, ~ B is the magnetic field, and æ is the gyromagnetic constant. Eq. (2.1) states that the direction of d ~ M dt (i.e. the rate of change of ~ M with time) is always perpendicular to the plane containing ~ B and ~ M such that ~ M processes about ~ B. To simplify the analysis of MRI, the Bloch equation is written by neglecting the relaxation time constants, T 1 and T 2. These

23 8 time constants will be introduced later in our discussion. The following analysis of the Bloch equation is originally from [14]. In a practical MRI experiment, there are three different magnetic field components that add to the main magnetic field, ~ B 0, as shown in the following equation: 8 9 é : d M ~ = n h io = æ ~M dt ; æ ~B0 B ~ + ~ j æ è G ~ æ ~rè + B ~ 1 ~B 0 ;@ B; ~ G; ~ B ~ 1 (2.2) ~ B component represents the inhomogeneity in ~ B 0. The gradient field, ~ G:~r, is a magnetic field which changes in strength linearly along the ~r direction. ~ j is an unit vector along the ~ B 0 direction. The time varying rotating magnetic field, ~ B 1, which is due to radio frequency transmission, is used to excite the magnetic moment. To simplify, we neglect the ~ B, which only contributes an artifact to the image. Further, we introduce a new reference frame (the laboratory frame of reference), as shown in Fig. 2æ1, which rotates around the ~ B 0 direction (and call this plane of rotation the x 0, y 0 if the original is x, y) with an angular velocity,! o, which is called the Larmor frequency where,! o = æ ~ B 0 (2.3) In the rotating reference system èx 0 ;y 0 ;zèthere is no precession due to ~ B 0. Equation (2.2) then simplifies to: 8 9 é : d M ~ = dt ; = æ ~G; B ~ 1 = æ n ~M æ h ~ j æ è ~ G æ ~rè + ~ B 1 io ~ G æ ~r, ~ B1y 0, ~ G æ ~r 0 ~ B1x 0 ~B 1y 0, ~ B 1x ~M x 0 ~M y 0 ~M z (2.4) while terms on the RHS represent gradient or RF pulses applied in MR experi-

24 9 (A) B o z (B) B o z w o M M x x y y Figure 2æ1: (A) Precession of magnetic moment ( ~ M) in the presence of a magnetic field (B o ) and (B) absence of precession in the rotating frame of reference. ments. To completely describe the phenomena of magnetic resonance, the spin-lattice (longitudinal) and spin-spin (transverse) relaxation time constants, T 1 and T 2,respectively, are introduced in Eq. (2.4) : 8 é : d ~ M dt 9 = ; ~G; ~ B 1 = 2 6 4, 1 T 2 æè ~ G æ ~rè,æ ~ B 1y 0,æè ~ G æ ~rè, 1 T 2 æ ~ B 1x 0 æ ~ B 1y 0,æ ~ B 1x 0, 1 T ~M x 0 ~M y 0 ~M z Mo T (2.5) This is the general form of the Bloch equation which can be used to describe most of the MRI experiment where M 0 represents the equilibrium magnetization in the z-direction (direction of B ~ 0 ). Eq. 2.5 describes the behavior of magnetization vectors in the x 0, y 0 rotating frame of reference. If the system has been disturbed from the equilibrium using the time varying rotating magnetic field, B ~ 1, as shown in Fig. 2æ2, two processes will drive the magnetization vectors back to equilib-

25 10 rium. One of these process destroys the transverse magnetization ( ~ M x 0 and ~ M y 0 ) at the rate of 1 T 2 and the other brings the longitudinal magnetization ( ~ M z ) back to the equilibrium state at the rate of 1 T 1. The effect of the ~ B 1 excitation on the ~M is measured by the angle between the equilibrium and excited magnetization, named the flip angle (ç). B o z M 0 M z M M y 0 M x x y B 1 y Figure 2æ2: Excitation of magnetization using B ~ 0 1y displayed in the laboratory frame of reference (x 0, y 0 ). The effect of excitation is measured by the angle between the equilibrium, M0 ~, and excited magnetization M, ~ named the flip angle (ç). M z is the projection of the M ~ along the z direction, and M ~ 0 x and M ~ 0 y are the projections of M ~ along the x 0 and y 0 axes, respectively. 2.1 Conventional MRI Experiment MRI experiment consists of image acquisition and reconstruction. Image acquisition is the process of acquiring MR data using a MRI pulse sequence which directs the switching of magnetic field gradients and RF pulses for encoding data from

26 11 τ M Gy Gx N t Figure 2æ3: The conventional spin-echo pulse sequence diagram. the FOV. Image reconstruction is the process of converting or decoding the acquired MR data to form a MR image. The conventional MRI pulse sequence can be separated into three parts [1], RF excitation, evolution, and data collection or sampling. The spin-echo (SE con ) pulse sequence, as shown in Fig. 2æ3, is an example. In the slice selective part of the SE con pulse sequence, the radio frequency (RF) excitation pulse in the presence of a gradient field along the z direction disturbs the magnetic moments in the FOV along the slice selection direction (the z direction) from the equilibrium state into an excited state. The excited magnetic moments are then phase encoded via application of a field gradient in one in-plane direc-

27 12 tion (y). Finally, the MR data are collected during an applied frequency encoding gradient along the second in-plane direction (x). We can analyze these three parts of an MRI pulse sequence by using the general Bloch equation (Eq. 2.5) The Slice Selective RF Excitation Pulse In this section we analyze the effect upon the magnetic moments in the FOV due to the slice selective RF excitation. In a typical experiment, the effect of T 1 and T 2 relaxation upon longitudinal and transverse magnetizations can be neglected during this period because the RF pulse time is very short when compared to these relaxation times. Furthermore, to simplify the analysis, we use the low flip angle approximation [15]-[17] which assumes that after the excitation pulse, the ~M z ètè remains constant (ç M 0 ). This approximation is applicable when using RF excitation pulses with less than 30 æ flip angle, and gives an acceptable result even up to 90 æ [18]. Based upon these assumptions, the first two rows of equation (2.5) give: d ~ M T dt =, ~ jæè ~ G æ ~rè ~ M T + ~ jæ ~ B 1 M 0 (2.6) where ~ M T = ~ M x 0 + i ~ M y 0, ~ B1 = ~ B 1x 0 + i ~ B 1y 0, and ~j is an unit vector along the ~ B 0 direction. The solution of this non-linear differential equation [17],[14] with initial condition of Mèr;t =0è=M 0, assuming ~ B 1 ètè is non-zero from t =, T 2 ~M T è~rè =~jæm 0 Z T 2, T 2 ~B 1 ètè e ç, ~ jæ~r: R T 2 rgrèsè ds t ç to T 2,is dt (2.7) For G r èsè representing a G slice or G z constant in the appropriate in time interval equation (2.7) reduces to ~M T èzè =~jæm 0 e ë, ~ jæzgz T 2 ë Z T 2 ~B 1 ètè e ë, ~ jæzgztë dt (2.8), T 2

28 13 From equation (2.8), we see that the exponential term outside the integral depends on the z gradient. In other words, the exponential term outside the integral in equation (2.8) causes a phase dispersion (i.e. variation in phase) in ~ M T along the z direction. The phase dispersion is reversed by applying a negative G z of duration half the time length of the pulse, T, after the RF pulse period (i.e. a rephasing 2 pulse). The result after the rephasing gradient pulse is (by substituting ~ M T è T 2 è from equation (2.8) into ~ M T è0è of equation (2.10)): ~M T èzè =~j M 0 G z Z KT,KT ~B 1 èkèe ë, ~ jkzë dk (2.9) where k = æg z t and K T = æg z T 2. ~ MT èzè represents the Fourier transform of ~ B 1 ètè without phase dispersion. As an example, the ideal slice selection, ~ M T èzè =M 0 for jzj é d 2 ; with ~ M T èzè = 0elsewhere, is achieved by using B ~ 1 ètè = sincè kd è, that is the Fourier transform 2 of the step function (i.e. the idea slice selection). In general, Eq. 2.9 simply states that for the low-flip angle excitation, the excited slice profile of magnetization is the Fourier transform of the RF excitation pulse shape, and vice versa The Evolution Period There is only one gradient field, B ~ phase or G y, applied during the phase encoding period of a standard 2D SE con pulse sequence (Fig. 2æ3). Equation (2.5), then, can be solved to yield: ~M T èy; tè = ~ M T èz 0 æ d 2 èeë,~ jæææyægyætë e, t T 2 (2.10) ~M z èz; tè = 8 é é: ç ç M 0 1,e,t T 1 M 0 for jzj é d 2 ; elsewhere. (2.11)

29 14 where M ~ T èz 0 æ d è is the excited magnetic moment from the ideal RF slice selection, 2 as described in the previous section. Eq implies a phase shift in the complex magnetization, ~ MT, along the y direction. ~ Mz èz; tè is the longitudinal magnetic moment where the magnetization inside the slice returns to the equilibrium state (M 0 ) by the rate of 1 T 1 and there is no change of the magnetization outside the slice, as shown in Eq For most common MRI phase encoding methods, the MR pulse repetitions involve repeatedly changing in the G phase magnitude in such the way that a uniform sampling of k-space is accomplished [94]. The common methods use constant gradient magnetic fileds for spatial encoding along both in-plane directions. The k-space data is a 2D array of measurement data points, èk x ;k y è, where the k x parameter corresponds to the data collected during the frequency encoding period and the k y parameter is used for the different G phase energies. In the conventional method for data collection, data from frequency encoding on each G phase magnitude fill in one line of the k-space array (i.e. for a specific value of k y ). G phase magnitudes are consecutively changed on each excitation via some scheme to completely fill all the desired lines in the k-space array. The play out time of G phase (ç) is kept constant for the whole repetitions. The constant gradient magnetic filed whose strength is linearly changed with spatial location, creates linear relationship between resonance frequency and spatial location [11]. The proton density at each location is, therefore, resonance at a specific frequency. The measurement data, which represents protons density at each spatial location, thus can be linearly mapped to the resonance frequency domain. In other word, the measurement data is the frequency (k-space) domain of the FOV location. The Fourier transform of the k-space domain will therefore map the k-space data back to the FOV location.

30 The Data Collection Period After the evolution period, MR data are sampled in the presence of a frequency encoding gradient field, G freq or G x. A receiver antenna receives the radio frequency signal generated by the transverse magnetic moments. The signal acquired, Sèk x ;k y è, during the frequency encoding period can be described by an integral over the magnetization in the xy plane: Sèk x ;k y è= Z Z slice sèx; yè e ë,jèkxæx+kyæyèë e h i, TE T 2 dxdy (2.12) where k x = æg x t, k y = æg y T. sèx; yè is the distribution of signal coming from the transverse magnetization within the slice after RF excitation. TE is the time from the center of RF excitation to the center of G x. Mathematically, the signal density function, sèx; yè can be found by performing the integral 2D Fourier transform of Sèk x ;k y è: Z Z sèx; yè = 1 Sèk x ;k y èe +jèkxæx+kyæyè dk x dk y (2.13) 2ç kx ky In practice, the radio frequency signal detected from the transverse magnetization is discretely sampled at a finite number of sampling points using an A-D convertor. The sampling interval (æt) and the number of samples (N) plays a major role in MR image quality. To avoid image overlapping (aliasing) the sampling interval should be at least 1 æ:g:f OV [14] where æ is the gyromagnetic constant, G is the magnitude of the gradient magnetic field, and FOV is the size of the field of view. The number of samples, on the other hand, determines the resolution of MR image in that sampling direction. Typically, the k-space data of a MR image is not frequency band-limited and not a periodic function. A finite number of samples of Sèk x ;k y è introduces a truncation error on the MR image which appears as the

31 16 Gibbs ringing artifact [20], [21] Image Reconstruction The image reconstruction process for conventional 2D Fourier encoded MRI is simply the 2D discrete inverse Fourier transform of the sampled magnetization signal, SènæK ~ x ;mæk y èas: ~sèaæx; bæyè = 1 NM N 2X n=, N 2 M 2X m=, M 2 ~SènæK x ;mæk y èe èj2çna N è e è j2çmb M è (2.14) where a =, N 2 æææn 2, b =,M 2 æææ M 2, æx = 1, æy ægxnæt = 1, æk ægymç x = æg x æt, and æk y = æg y ç. æx and æy are spatial resolution, and æk x and æk y are sampling interval along the x and y directions, respectively. ~sèax; byè is complex-valued so it can be separated into magnitude and phase data, with standard diagnostic MR images formed from the magnitude data. Phase data is used in certain cases to analyze flow and display temperature variations in the FOV. 2.2 Spatial Encoding Methods in MRI Background Multi-dimensional spatial encoding methods can be classified, according to the implementation techniques, into two groups: spatial-selective and phase encoding, as illustrated in Fig. 2æ4. The major practical difference between the methods, besides the construction of the encodes, is in the implementation of spatial encoding along the first in-plane direction (y) (i.e. non-frequency encoding direction). For example, the standard phase encoding method uses gradient switching to encode such direction. The different spatial positions are encoded by varying the

32 17 Encoding Techniques Spatial Selective Phase (Fourier) Fourier (Non-Adaptive) Non-Fourier RF Impulse Non-Adaptive Adaptive Hadamard other OD Wavelet RROD KS SVD other Lanczos other Figure 2æ4: A diagram of spatial encoding techniques in MRI which can be separated into two groups: Spatial-Selective and Phase encoding methods. The Spatial-Selective method can be further classified into Fourier and Non-Fourier techniques. The Non-Fourier technique can be separated according to its encoding criterion as Non-Adaptive and Adaptive methods. The Non-Adaptive method uses a predefined basis set, such as Hadamard for encoding. The Adaptive method can be separated into two methods. One method computes a basis set from the contents in the FOV using orthogonal decomposition (OD) transformation whose major computational methods are Rank Revealing Orthogonal Decomposition (RROD) such as SVD; and Krylov Subspace (KS) such as Lanczos. The other method modifies the choice of encode from a predefines basis, such as with Wavelet-encoded MRI.

33 18 magnitude of the pulse gradient field on each excitation, as described in An alternative encoding method, on which the spatial-selective technique is based, is to employ spatially selective RF excitations whose magnitudes and phases are generated from either adaptively-determined or non-adaptive orthogonal basis sets. Either non-fourier or Fourier basis sets can be used in the technique to encode the first in-plane direction in slicewise acquisition. Fourier encoding (frequency encoding) is still used along the second in-plane direction. The orthogonal basis sets that can be used for encoding will be reviewed in this section. The non-fourier encoding methods can be separated into two groups, according to their encoding criteria: the non-adaptive and adaptive methods. The nonadaptive techniques utilize the predetermined orthogonal basis set for encoding, independent of the contents of the FOV. Hadamard [2] transformation is an example of such predetermined orthogonal sets used in these techniques. The Fourier encoding method is considered to be a non-adaptive technique according to its encoding criteria. It can be implemented by either the phase encoding method or spatially selective RF excitation using RF impulse (identity matrix) [8] basis sets. Unlike the non-adaptive techniques, the adaptive methods can be separated into two groups. One group employs spatial encoding determined from information obtained from the current contents of the FOV. The orthogonal sets are calculated from a recent image, or sets of images, or synthetic image data, using a matrix decomposition computation such as the SVD [7], or the Lanczos [41] algorithm. The other group modifies the choice of encode from a predefined basis, such as with Wavelet-encoded MRI [9]. Unlike non-adaptive techniques, such as the Fourier encoding method, adaptive techniques have an ability to optimize or increase the efficiency of the spatial encoding in the dynamic MRI applications by reducing the dimension of vector basis sets necessary to encode the spatial con-

34 19 tents of an image in the FOV. In the upcoming section, we will characterize the spatial encoding methods in MRI according to their encoding criteria as the non-adaptive and adaptive techniques Non-Adaptive Technique Non-adaptive MRI encoding methods can be analyzed by using the transform theory found in the image processing field [42]-[45]. The transformation of data into a different mathematical representation has been used in many image processing applications, such as image enhancement, and restoration. The Fourier transform is a widely used example of the transform method. In this section, we first introduce the transform theory based on the analysis in [20] and then characterize the Fourier, RF impulse, and Hadamard transforms which have been introduced for spatial encoding in MRI. 2D Separable Transform Let ~ f ènè be an discrete function generated from N points uniform sampling of f èxè which is a continuous function of a real variable x. The definition of ~ f ènè is ~f ènè =fèx o +næxè (2.15) where n is the discrete values 0; 1; 2;:::; N,1and æx is a sampling interval. A general one-dimensional (1D) transform of ~ f ènè can be represented as F èuè = N,1 X n=0 ~f ènègèn; uè (2.16) where F èuè is the transform of ~ fènè, u is a discrete variable with the same members as in the n variable, and gèn; uè is the forward transformation kernel. The inverse

35 20 transform can be similarly described as ~f ènè = N,1 X u=0 where hèn; uè is the inverse transform kernel. F èuèhèn; uè (2.17) For two-dimensional (2D) N æn square arrays (i.e., an MR image) the forward and inverse transforms can be presented as F èu; vè = N,1 X N X,1 n=0 m=0 ~f èn; mègèn; m; u; vè (2.18) and ~f èn; mè = N,1 X u=0 N X,1 v=0 F èu; vèhèn; m; u; vè (2.19) where gèn; m; u; vè and hèn; m; u; vè are the forward and inverse transform kernels, respectively. The kernels do not depend on the values of ~ f èn; mè or F èu; vè, but rather on their indexes (e.g. n; m; u; v). In other words, the kernels can be viewed as a series expansion of the basis functions and can be used to transform any images. The kernels, thus, are the non-adaptive functions. The forward and inverse kernels are said to be separable if gèn; m; u; vè =g 1 èn; uè g 2 èm; vè (2.20) and hèn; m; u; vè =h 1 èn; uè h 2 èm; vè (2.21) The 2D forward and inverse transforms of the separable kernels can be computed from the two separable 1D kernels as F èu; vè = N,1 X N X,1 n=0 m=0 g 1 èn; uè ~ fèn; mèg 2 èm; vè (2.22)

36 21 and ~f èn; mè = N,1 X u=0 N X,1 v=0 h 1 èn; uèf èu; vèh 2 è;m;vè (2.23) The separable 2D kernel, therefore, can be formed from any two 1D basis functions. All adaptive and non-adaptive techniques used for spatial encoding in MRI are based on the use of two 1D transform kernels. In the phase encoding MRI method (the non-adaptive technique), spatial information is Fourier encoded along the two orthogonal in-plane directions. In the spatial-selective encoding method (the non-adaptive as well as adaptive techniques), one direction is encoded using a 1D orthogonal basis set, and the other direction is Fourier encoded. The Fourier, RF impulse, and Hadamard kernels, which have been utilized for spatial encoding in MRI, are further characterized below. The Fourier Encoding Method The Fourier encoding method in MRI is based on the 2D Fourier transform kernels. The 2D Fourier forward and reverse kernels can be represented by an exponential function as and gèn; m; u; vè = 1 N eè,j2çun N hèn; m; u; vè = 1 N eè+j2çun N è e è,j2çvm N è è e è +j2çvm N è (2.24) (2.25) The conventional encoding method in MRI (the phase encoding technique) uses the 2D Fourier forward transform kernel to spatially encode an image along both in-plane directions. This encoding can be represented as ~Sèu; vè = 1 N N,1 X N X,1 n=0 m=0 e è,j2çun N è sèn; mèe è,j2çvm N è (2.26)

37 22 where sèn; mè is the proton density function inside the slice selection. The function also includes the relaxation time constant as well as the measurement instruments, such as RF coil characteristics, amplifier gains, etc. Sèu; ~ vè is measurement data whose indices are in the reciprocal domain (k-space domain) from the sèn; mè (spatial domain). To obtain the image in the spatial domain, the 2D Fourier inverse transform kernel is used for the decoding process or image reconstruction process of the acquisition data, Sèu; ~ vè, thus: ~sèn; mè = 1 N N,1 X u=0 N X,1 v=0 e è +j2çnu N è Sèu; ~ vèe è +j2çvm N è (2.27) The RF Impulse Encoding Method The RF impulse encoding method is based on the delta function often used in the analysis of the system response of linear systems. The RF impulse method is a very important tool for analyzing the response of the MR system to non-fourier encoding excitation as shown in Chapter 4. The RF vector inputs in this method are the delta function basis set. The spatial encoding profiles are, therefore, a forward Fourier transform of the delta function as described in Section In terms of transform theory, the RF impulse encoding method in MRI can be represented by the 1D forward Fourier transform of the delta function in one direction and the 1D forward Fourier transform in the other direction as ~Sèu; vè = 1 N N,1 X N X,1 n=0 m=0 rèn; uèsèn; mèe è,j2çvm N è (2.28) where rèn; uè is the 1D forward Fourier transform of the delta function æèn; wè: rèn; uè =æèn; wè e è,j2çuw N è (2.29)

38 23 where æèn; wè = 1 when n = w, and 0 otherwise. In this encoding method, the RF vectors, which are the delta kernel (æèn; wè), are used for spatially selective excitation, not rèn; uè. The reconstruction process of the RF impulse encoding method is the decoding process of the 1D reverse Fourier transform of the delta kernel in one direction and the 1D reverse Fourier transform in the other direction: ~sèn; mè = 1 N N,1 X u=0 N X,1 v=0 æèn; wèe è +j2çwu N è Sèu; ~ vèe è +j2çvm N è (2.30) The æèn; wè is an identity transform from the index w to the index n: æèn; wè e è +j2çwu N è = e è +j2çnu N è (2.31) Eq is, thus, reduced to: ~sèn; mè = 1 N N,1 X u=0 N X,1 v=0 e è +j2çnu N è Sèu; ~ vèe è +j2çvm N è (2.32) The reconstruction process in the RF impulse encoding method is, therefore, the same as in the Fourier (phase) encoding method. Hence, the measurement data, ~Sèu; vè, from the RF impulse encoding is mapped onto the same k-space domain as in the standard Fourier (phase) encoding method. Hadamard Encoding: A Non-Fourier Method Unlike the Fourier transform, which is based on a sinusoidal function, the Hadamard transform is based on a binary function whose values consist only of 1 and -1. The 1D forward and reverse Hadamard transform kernels of a N point function where N is equal to an integer power of two are gèn; uè = p 1 P n,1 è,1è i=0 b iènèbièuè N (2.33)

39 24 and hèn; uè = p 1 P n,1 è,1è i=0 b iènèbièuè N (2.34) where N =2 n, b k èzèis the k th bit of z in the binary representation. For example, if z = 5 (i.e., 101 in binary), b 0 èzè = 1, b 1 èzè = 0, and b 2 èzè = 1. The summation of b i ènèb i èuè is performed in modulo 2 arithmetic. Since the forward and reverse Hadamard transforms are identical, only one algorithm is necessary for computing transforms in both direction. The Hadamard encoding method in MRI uses the 1D Hadamard transform in one direction (the non-fourier encoding direction) and the 1D forward Fourier kernel in the other direction (the frequency encoding direction) thus: ~Sèu; vè = 1 N N,1 X N X,1 n=0 m=0 gèn; uèsèn; mèe è,j2çvm N è (2.35) where gèn; uè is a 1D Hadamard transform. The basis set for RF excitation is, however, a 1D Fourier transform of the 1D Hadamard kernel (gèn; uè) due to the Fourier transform relationship between the RF excitation and excited spatial profile of the image in the RF spatially selective encoding technique (Section 2.1.1). The RF spatially selective excitation are, therefore, the 1D forward Fourier transform of the 1D Hadamard kernel: pèn; wè =gèn; uè e è,j2çuw N è (2.36) The image data can be obtained by performing the decoding process on the 1D reverse Hadamard kernel in one direction and the 1D reverse Fourier transform in the other direction: ~sèn; mè = 1 N N,1 X u=0 N X,1 v=0 hèn; uè Sèu; ~ vèe è +j2çvm N è (2.37)

40 Adaptive Techniques Unlike non-adaptive techniques, whose basis functions do not depend on the contents in the FOV, the basis functions of adaptive techniques are computed from the contents in the FOV. An adaptive technique can increase the encoding efficiency because the technique reduces the redundancy in spatial encoding by using the basis set that attempts, in theory, to near-optimally encode according to the information in the FOV. The computation of such basis functions is based on Orthogonal Decomposition (OD). In this section, the background of OD is reviewed, and then the SVD method, which is a prominent example of OD, is characterized. Orthogonal Decomposition Orthogonal Decomposition is a mathematical method that transforms a matrix into a canonical form by determining a basis that span(s) vector subspaces, associated with the matrix, and a remaining transform matrix. Such a transform has a very prominent role in matrix computations used in linear algebra, such as computing a least squares solution of overdetermined systems of equations [46]. OD is also often used in image processing fields [42]-[45], such as in data storage and compression. In this section, the analysis of OD is in matrix form [47]-[50] instead of in the 2D function, as it was for the non-adaptive technique, because some concepts from linear algebra, as well as from image processing fields, are used for the analysis. Background There are many types of OD as described in [34]. Here, we review some of them which are potentially advantageous for the adaptive encoding technique in MRI. OD can be separated, according to the methods of decomposition, into two

41 26 major groups: the rank revealing orthogonal decomposition (RROD) methods and the Krylov subspace (KS) methods, as shown in Fig. 2æ5. Orthogonal Decomposition RROD KS RRQR RRCOD Lanczos Arnoldi SVD UTV Figure 2æ5: A diagram of orthogonal decomposition methods which can be separated into two major groups: Rank Revealing Orthogonal Decomposition (RROD) and Krylov Subspace (KS). Lanczos and Arnoldi are some examples of the KS techniques. The RROD can be further classified into the two-side, or complete (RRCOD), method and the one-side method such as RRQR. The RRCOD method, such as SVD, computes basis sets in both directions of the matrix while the oneside method only computes a basis set in one direction of the matrix. The RROD transformations reveal basis set(s) that span(s) vector subspaces of matrix. Such subspaces are also known as the range of a matrix whose dimension is defined as the rank of matrix. The rank from the RROD transformations also can be determined by the analysis of the diagonal elements of the remaining transform matrix. The other important subspace associates with matrix is the nullspace, often referred to as the noise subspace. For a S M æn matrix, where M ç N, the dimension of the range (the rank) pulse dimension of nullspace equals to N [34]. To exactly encode information in the S, only number of basis vectors equal to the

42 27 rank (ç N) are necessary. Hence, the RROD methods, in theory, reveal a minimum number of basic vectors or encodes necessary for exactly encoding information in the FOV. The RROD methods can be classified as one-sided or two-sided decomposition. We refer to the two-sided decomposition as the complete decomposition. The RRQR [51],[52] is an example of the one-sided RROD method. The RRQR transforms a matrix into a column permutation and an upper triangular. The RRQR is not a unique decomposition due to the defined tolerance criteria in its computation. The decomposition, however, takes a relatively short time to compute when compared to the SVD [34] method, which is a complete RROD. The SVD is a prominent example of an RRCOD method. The SVD transforms a matrix into three matrices: 2 matrices that contain, as column vectors, orthonormal basis sets, and a diagonal matrix. The SVD is the 2D separable orthogonal transform when written as an outer product. Its truncated form is the best leastsquare approximation of a matrix. The computation of the SVD is, however, expensive and is not easily parallelizable. Recently, there are other non-unique decompositions in this class aimed to be parallelizable, but involve tolerance error. Examples of such non-unique decompositions form a class of UTV decomposition [53]. In the KS methods, the underlying theory and algorithm to generate the basis sets are distinctly different from that of the RROD methods [34]. The KS methods are separated into symmetric and unsymmetric matrix decomposition. The Arnoldi method [54] is an example of the unsymmetric KS method. The symmetric KS method, on the other hand, is more feasible in practice due to the numerical stability of the method. The symmetric Lanczos tridiagonalzation algorithm is an example of this method. The advantage of KS, compared to RROD methods, is the

43 28 simplicity and parallelizability of the algorithms, which greatly increases computational speed. General Analysis In the non-adaptive technique, a 2D separable transform is analyzed in the 2D function form. The same transform can also be presented in the matrix form as F = U A V t (2.38) where A is a matrix to be transformed, U and V are the orthogonal matrices operating on the columns and on the rows of A, respectively, and F is a transform matrix. The 1D Fourier, RF impulse, and Hadamard transforms, as described in the previous section, are some examples of such orthogonal matrices (i.e. the U and V ). A general complete OD or separable linear transformation on a 2D data matrix A, whose dimension is N æ N, may be present in the form A = U æ V t (2.39) where U and V are orthonormal matrices ( U t U = V t V = I; I is the identity matrix), and t indicates conjugate transpose. Solving for æ we obtain æ=u t AV (2.40) As compared to eq. 2.38, æ is a transform matrix of A. U t and V transform the columns and rows of A, respectively. Notice that in the OD method the U and V transform matrices depend on the value of A, as opposed to the transform matrices used in the non-adaptive technique, which depend only on the indices of A.

44 29 If we present the orthonormal matrices U and V in the form U = ëu 1 u 2 æææu N ë (2.41) V = ëv 1 v 2 æææv N ë where each u i and v i consists of a set of linearly independent column vectors of U and V, respectively, then the data matrix A can be written as A =ëu 1 u 2 æææu N ëæëv 1 v 2 æææv N ë t (2.42) If the matrix æ is presented as a sum of each component in the matrix (æ ij ), then A can be defined as A = NX NX i=1 j=1 æ ij u i v t j (2.43) From eq A is transformed into the sum of the rank one matrices (the outer product u i v t j) weighted by the appropriate æ ij. This rank one matrix is also called the eigenimage. Singular Value Decomposition Singular Value Decomposition (SVD) is the complete RROD (RRCOD). It transforms a matrix A into the product: A = U æ V t (2.44) where U and V are the row (left) and column (right) singular vectors of A, respectively, and æ is a diagonal matrix whose diagonal components are the singular values (ç) of A. The SVD is thus a 2D separable orthonormal transform whose

45 30 row and column transforms can be represented separately as A A t = U æ 2 U t (2.45) and A t A = V æ 2 V t (2.46) where the columns of U and V are the singular vectors of A A t and A t A, respectively. The SVD can also be represented in vector outer product form as A = RX i=1 ç i u i v t i (2.47) where R ç N, R and N are the rank and dimension of A, respectively, u i and v i are the singular vectors, and ç i is the singular value of A. Eq states that the SVD is an expansion of the matrix A into a sum of eigenimages (i.e., the separable rank one matrices). If the singular values of the SVD are placed in monotonic decreasing order: ç 1 ç ç 2 ç æææçç R (2.48) and we attempt to approximate the matrix A by truncating the series expansion of the SVD as A K = KX i=1 ç i u i v t i (2.49) with K ç R, the truncation error norm from the approximation is represented by ka, A K k = RX i=k+1 where the matrix norm is the Euclidean measure defined as ç i (2.50) kak 2 = trèa A t è (2.51)

46 31 where tr is the trace or the sum of the diagonal elements of A A t. Zientara et al [7] suggests two possible decision criteria to truncate the series expansion of the SVD used for spatial encoding in MRI at the j th eigenimage: ç j ç 0 ç C 1 (2.52) or P K i=j+1 ç i P K i=0 ç i ç C 2 (2.53) In summary, the SVD, whose singular values are placed in descending order, is the 2D separable orthonormal transform whose truncation approximation is the most efficient least-square representation of the matrix. 2.3 SVD Encoded MRI Several properties of complete rank reveal orthogonal decompositions, such as SVD, are highly advantageous for efficient adaptive MRI encoding. First, digital images are often able to be described by a weighted sum of a relatively small set of eigenimages computed from the SVD. Second, the singular vectors (when ordered with respect to decreasing singular value) encode bands of low to high spatial frequency information. Therefore, large (small) regions of constant intensity in an FOV are encoded by singular vectors with corresponding large (small) singular values. Image noise is therefore encoded in eigenimages associated with small singular values. The approximation of the SVD by truncating the number of eigenimages, associated with small singular values, therefore primarily reduces image noise (i.e. by noise reduction) rather than affects the encoded objects in the image FOV. The process of SVD encoded MRI starts by acquiring base line data using any

47 32 non-adaptive basis set. The SVD is then used to decompose the base line data to obtain vertical (U) and horizontal (V ) encoding matrices, as well as the singular values. Using the results of the SVD computation, we derive the K orthonormal set of singular vectors (K MRI encoding steps) from a truncation criterion. The k th column of U, u k, represents a vertical encoding vector. On the other hand, the k th column of V, v k, represents a horizontal encoding vector. In any single acquisition step, either u k or v k can be used to acquire data (1D non-fourier encoding) in combination with the readout direction gradient (1D Fourier encoding) during the sampling period. Therefore, the K encoding steps necessary for a new image can be performed either by using all vertical encoding or by using all horizontal encoding or by a combination of both. An image reconstruction from the K SVD encoding steps is as follows. In each excitation, the acquired data represents a row (D row ) or column (D col ) matrix depending on whether the encoding employs u k or v k, respectively. The k-space data is reconstructed after complete K encoding acquisition by pre- and postmultiplying the data by the appropriate column of U or row of V : S = XKu k=0 u k D row èkè + XKv k=0 D col èkèv t k (2.54) where K = K u + K v. The image data F is obtained by performing 2D Fourier transform on S: F = C S R (2.55) where C and R are the row and column 1D inverse Fourier transform matrices.

48 33 Chapter 3 Methods and Implementations 3.1 System Overview The non-fourier encoded MRI experimental methods (non-real-time) have been implemented on a 1.5 Tesla GE SIGNA MR System available at the Division of Magnetic Resonance, Department of Radiology, Brigham and Women s Hospital, Boston, MA. The associated external processing has been implemented on SUN workstation computers available at the Surgical Planning Laboratory at that hospital. The system hardware can be separated into two parts: the SIGNA MR system, and the workstation system, as illustrated in Fig. 3æ1. The SIGNA system consists of three parts: the MR excitation unit, the Transceiver Processing and Storage (TPS) chassis, and the Host computer. The MR excitation unit consists of the x, y, and z MR gradient magnet coils, as well as RF transmitter and receiver coils, and all necessary auxiliary units needed to support these coils. The TPS chassis consists of an Integrated Pulse Generator (IPG) real time computer, RF transmitter and receiver, and memory boards. The TPS unit controls pulses to all MR gradient magnet coils, as well as RF coils, and receives (acquires and stores) MR data according to the pulse sequence software. The Host computer is the other computer system on the SIGNA system. It serves as the user interface, image database, and communication gateway for the system console. The workstation unit is another computer system connected to the SIGNA

49 34 SIGNA MR System MR Excitation Unit TPS Chassis Host Computer and Console TCP/IP SUN Workstation Figure 3æ1: The hardware system of the non-fourier encoded MRI method, implemented on a commercial 1.5 Tesla GE SIGNA MR system which is connected to an external SUN workstation computer via an Ethernet network using the TCP/IP network protocol. computer system via a switched Ethernet network using the TCP/IP protocol. The workstation unit serves as an external processing unit to compute RF excitation vectors and to reconstruct images according to the non-fourier encoding algorithm. 3.2 Pulse Sequence Implementation We implemented all pulse sequences using the Environment for Pulse programming In C (EPIC) software which is a part of the GE Software Pulse Sequence Developing Tools [37]. The EPIC software provides an integrated C language software environment for pulse sequence development. The EPIC compiler generates two object files from a source code. One object file (the Host Program) is executed on the host computer and the second file (the IPG Program) is to execute on the IPG board in the TPS chassis, as shown in Fig. 3æ2.

50 35 SIGNA MR System MR Excitation Unit Pre-Processing Workstation System IPG Program K-Space Mapping Orthonormal Basis Sets Calculation Image Reconstruction Host Program RF Vectors Converting Display Figure 3æ2: The software diagram of the non-fourier encoding method consists of the pulse sequence programs (the IPG and Host programs) on the SIGNA system as well as five processes on the SUN workstation for RF vector generation and image reconstruction. Three non-fourier encoding pulse sequence types were developed, spin echo, gradient-recalled echo, and multi-slice gradient-recalled echo pulse sequences Spin Echo The spin echo non-fourier encoding (SE nf ) pulse sequence is shown in Fig. 3æ3. This is a modification of the line scan approach [23] originally proposed by Weaver and Healy [24] for Wavelet encoding as an alternative to conventional Fourier encoding techniques. The pulse sequence had previously been implemented on non-commercial MR system by Panych et al [2] for Wavelet encoded MRI. It was also used later on that system by Zientara et al [7] to study the SVD encoding method. In this study, we implemented that pulse sequence on the commercial GE SIGNA MR system. The major difference between the SE con pulse sequence that employed Fourier

51 36 G fre G nf G slice RF mag RF phase Figure 3æ3: A spin echo non-fourier encoding pulse sequence diagram. (phase) encoding and our SE nf pulse sequence concerns the phase encoding function described in Chapter 2. With the pulse sequence as we have implemented it for encoding with the non-fourier basis set, spatial encoding is performed by applying a set of small-flip angle RF pulses during the play out of gradient along the non-fourier encoding direction (G nf ) (the small-flip angle RF excitation described in Chapter 2). A single slice is selected by using a spatially selective 180 æ pulse. The data are sampled during the frequency gradient (G fre ) play out in the same fashion as with the conventional frequency encoding method. The flip angle of the non-fourier RF excitation is calculated from the RF excitation area of a rectangular unit box pulse whose definition is described in Chapter

52 37 4, referenced to an area of the General Electric (GE) RF excitation standard pulse (i.e. a rectangular pulse with a 1 msec pulsewidth at the maximum magnitude is equal to a 180 æ flip angle). The flip angle of the RF excitation can be adjusted by scaling the magnitude or the pulsewidth of the rectangular unit box pulse. To minimize the chemical shift artifacts due to the RF excitation used in our encoding method, the pulsewidth of the rectangular unit box pulse was designed to be as short as possible. This is accomplished by keeping the magnitude of the pulsewidth at the maximum value and then adjusting the pulsewidth according to the flip angle value. For example, in our implementation, the pulsewidth of the rectangular box pulse (t rf ) is calculated from: t rf = A ref æ ç rf æ t ref A rf æ ç ref (3.1) where A ref and t ref are the amplitude (i.e. 1) and pulsewidth (i.e. 1msec) of the GE rectangular standard pulse, respectively, ç ref is the flip angle (i.e. 180 æ ) of the standard pulse. A rf is the amplitude of the rectangular box pulse which kept constant at maximum value (i.e. 1), and ç rf is the flip angle of the encoding method. We will refer to this flip angle as calculated flip angle (ç cal ) through out this thesis. The constant magnitude gradient spoiling method was employed in this pulse sequence to reduce artifact signals due to the remaining longitudinal magnetization following each excitation. The method applies constant gradient fields after read out of the echo in each TR interval, as shown in Fig. 3æ3. The spoilers can reduce artifact signals by dephasing the remaining transverse magnetization so that the net magnetization is decreased.

53 38 Figure 3æ4: The RF pulses for the two-shot slice selection using the phase cycling method. The first shot ([RF]I) uses a,90 æ hard pulse and 90 æ soft pulse. The second shot ([RF]II) employs a,90 æ soft pulse (i.e. 180 æ phase shift from the first shot) with the same hard pulse Gradient-Recalled Echo The volume gradient-recalled echo pulse sequence for non-fourier encoding had been implemented earlier by Panych et al [8] to demonstrate the feasibility of using a fast pulse sequence, for the non-fourier encoding for a dynamic MR imaging study. In this study, we further implemented an addition of slice selection techniques for a single-slice and single-slab selections on this pulse sequence. We have investigated one-shot slice selection techniques such as self-refocusing pulses [25]- [28] and optimized slice selection [29]-[32], as well as a two-shot slice selection technique using the phase cycling method [95].

54 39 Inside Slice B o z Outside Slice B o z (A) x x y y B o z B o z (B) x x y y B o z B o z (C) x x y y Figure 3æ5: The magnetization inside and outside the selected slice from the first (A) and second (B) shots of the two-shot slice selection technique using the phase cycling method. The subtraction of these two measurement data is shown in (C).

55 40 In the phase cycling method, a hard pulse (i.e. non spatial selective RF excitation using the rectangular pulse shape) and a soft pulse (i.e. a spatial selective RF excitation using the sinc pulse shape) are used to saturate the outside slice magnetization (i.e. flip the magnetization outside the slice into the transverse plane). The method requires two shots for each encoding. At the first shot, a,90 æ hard pulse and 90 æ soft pulse, as shown in Fig. 3æ4 ([RF]I), were used. The inside and outside slice magnetization after the first shot are illustrated in Fig. 3æ5A. At the second shot, a,90 æ soft pulse (i.e. 180 æ phase shift from the first shot) with the same hard pulse, as shown in Fig. 3æ4 ([RF]II), were used. The magnetization in the different volumes after this shot is depicted in Fig. 3æ5B. In practice, the magnetization outside the slice will never be completely saturated on each shot due to the variations of the MR equipment and tissues affecting RF deposition, as shown in Fig. 3æ5A and B. The difference of these two sampled data sets will, however, as best as possible minimizes the magnetization outside the slice, as shown in Fig. 3æ5C. Fig. 3æ6 illustrates a single-slice gradient-recalled echo (GRE) pulse sequence diagram using the phase cycling presaturation technique for a slice selection. At the first shot, the outside slice magnetization is presaturated using a non-selective,90 æ hard pulse and a slice-selective 90 æ soft pulse. Following the postsaturation phase, magnetization within the slice is excited by RF pulses formed from the orthonormal basis sets. Standard frequency encoding and data acquisition is performed in the same manner as with the SE nf encoding method. At the second shot, the same experiment is repeated by using a slice selective,90 æ soft pulse (i.e. 180 æ phase shift from the first shot). The subtraction of the data from the two shots is required in this technique prior to the image reconstruction process.

56 41 G fre G nf G slice RF mag 90 ±90 RF phase Figure 3æ6: A single-slice gradient-recalled echo non-fourier encoding pulse sequence diagram Multi-Slice Gradient-Recalled Echo For this study we implemented a multi-slice gradient-recalled echo non-fourier encoding technique (multi-slice GRE) for multiple slice acquisition within a slab. Fig. 3æ7 depicts the pulse sequence diagram resulting from the method, up to the end of the third echo. The phase cycling presaturation technique was also used in this pulse sequence for slab selection. The RF encoding method is performed in the same way as the single-slice gradient-recalled echo pulse sequence. Standard frequency encoding is performed along the G fre direction with data acquired only during the positive lobe of the G fre. The slab magnetization is further encoded

57 42 G fre G nf G slice RF mag 90 ±90 RF phase Figure 3æ7: A multi-slices gradient-recalled echo non-fourier encoding pulse sequence diagram is shown up to the end of the third echo. using normal phase encoding within the echo train to provide the multiple slices within the slab. 3.3 Software Processes on the Workstation The workstation unit serves as an external processing unit for the RF vector generation and image reconstruction of our non-fourier encoding method. The functions of the workstation unit can be separated into five parts: preprocessing, k- space mapping, image reconstruction, orthonormal basis sets calculation, and RF vector conversion and transfer, as shown in the right side of Fig. 3æ2. These pro-

58 43 cesses and their user interface were developed by using MATLAB software package (The MATHWORKS Inc., Natick, MA). The EXPECT software [40] were used to manage data transfer between the SIGNA system and the workstation Preprocessing The preprocessing phase concerns preprocessing of the output MR data (Y ) before the k-space mapping process occurs. The process differs with each pulse sequence. With the SE nf pulse sequence, the process performs conjugation on the Y data to reverse the conjugation effect on the Y data due to the 180 æ slice selection pulse, as shown in Fig. 3æ8. In the case of the GRE pulse sequence, which uses the phase cycling presaturation technique, the processing involves subtraction of the Y data from two of the slice-selection excitations whose slice-selective soft pulses differ by 180 æ (i.e. 90 æ and,90 æ pulses), as described in Section For the multi-slice GRE pulse sequence, the processing involves reorganization of the Y data into two sets of volume data (stacks of MR data slices) according to the flip angles of a slice-selective soft pulse (i.e. 90 æ or,90 æ pulses). Next, a 1D Fourier transform is performed to decode the phase encoding along the stack slice encoding direction of both volume data sets. Finally, the data subtractions on each slice of the sets data volumes are performed in the same fashion as those for the GRE pulse sequence K-Space Mapping K-space mapping is a procedure that maps the Y data into the k-space Fourier domain (S). It can be viewed as a process to decode or invert the encoding function of the RF excitation inputs. The MR data can be mapped onto the k-space matrix

59 44 B o z B o z y y x x B o Figure 3æ8: The conjugate effect on the non-fourier encoding data due to 180 æ RF pulse. (S) by multiplying the output matrix (Y ) with the inverse matrix of RF excitation inputs (P,1 )as S = P,1 Y (3.2) The detailed theory of this procedure is explained in Chapter Image Reconstruction Image reconstruction converts the complex-valued k-space data into a MR magnitude image. The process involves performing a 2D Fourier transformation on the k-space data, as described in Eq. 2.27, and transfers the magnitude data from the Fourier transform (MR image) to a display on the workstation s monitor Orthonormal Basis Set Calculation To increase encoding efficiency, the adaptive technique computes the orthonormal basis set from the contents of the FOV. The computation of such basis functions is

60 45 based on the Orthogonal Decomposition method, as described in Section Calculation of the orthonormal basis set entails generation of basis sets or RF vectors for the non-fourier encoding. For this study, we implemented the calculations of the orthonormal basis sets using MATLAB software. The calculation can be separated into two different schemes. One scheme uses predefined basis sets such as the RF impulse and Hadamard basis sets (the non-adaptive technique), while the other scheme calculates the basis set from the k-space data using matrix decomposition methods such as SVD and Lanczos (the adaptive technique). The SVD routine included in MATLAB [33] is used for the SVD calculation. The symmetric Lanczos algorithm using the normal forms of the matrix [34] was implemented in MATLAB for the Lanczos calculation. We also implemented a fast SVD computation using the Jacobi-SVD algorithm [35],[36] on a massively parallel computer, the CM-200 available at the Surgical Planning Laboratory, Brigham and Women s Hospital, Boston, MA. SVD Computation Using the Jacobi Algorithm There are different numerically stable methods for computing SVD [34], for example, the Jacobi methods [79], the Golub-Kahan-Reinsch method [71] and the QR methods [82]. Due to the high computational complexity of SVD, we have developed and implemented the algorithm in parallel to reduce the computational time. For parallel implementations, the Jacobi methods are far superior in terms of regularity, simplicity and local communications. The SVD computation employed on the CM-200 in this study is based on the Jacobi-SVD algorithm which diagonalizes the matrix by a series of matrix multiplications. The Jacobi-SVD method is suitable for parallel implementation using 2D plane rotations to diagonalize a general matrix. For any N æ N matrix, we may solve

61 46 N 2 problems simultaneously at each step by using an odd-even scheme [36]. Offdiagonal elements are successively zeroed until the sum of the off-diagonal elements is smaller than a desired value. We denoted a Jacobi rotation matrix of the angle æ in the èp; qè plane by T èp; q; æè where péq. The matrix T is the same as the identity matrix except regarding four elements: T èp; q; æè ç T pp T pq T qp T qq = cos æ sin æ,sin æ cos æ (3.3) The fundamental operation of the Jacobi rotation method is the diagonalization of a 2 æ 2 matrix by the rotation J èçè and Kèçè such that: Jèçè a b c d Kèçè = 4 x 0 0 y (3.4) Once the rotation angles ç and ç are found, we multiply the result and singular vector matrices in order to accommodate the effects of the rotation: æèl +1è = Jèçèæèlè Kèçè; U t èl +1è = JèçèU t èlè; Vèl+1è = VèlèKèçè where æè0è = a decomposed matrix èa N æn è, U t è0è and V è0è = identity matrix èiè, and l is rotation number. The computation was designed to rotate the N odd rows, and then the N even 2 2 rows in each rotation number (l). The termination criterion is chosen to be either a fixed number of rotations, or a test for the sum of the off-diagonal elements. The final result gives a diagonal matrix (æ) with positive diagonal elements (the

62 47 singular value), the left (U), and right (V t ) singular vector matrices Converting and Transferring RF Vectors In this phase, the vectors from the orthonormal basis set calculation are converted according to the requirements of the SIGNA system before being transferred to the system, as shown in Fig. 3æ9. Each vector s magnitude and phase found from the orthonormal basis set calculation is scaled according to the SIGNA hardware specifications [37]. The scaled vectors are then converted into the GE external wave form file format (i.e. 5X format) [37] and stored in a file on the hard disk of the SIGNA s Host computer. At the start of acquisition, the files containing the RF vectors are read into the IPG user s memory via the Host program while the download of the pulse sequence programs occurs. Each RF vector is then transferred to the pluse generator memory of the IPG program for play out, prior to the beginning of each RF excitation period as illustrated in Fig. 3æ Software Interface Our software was implemented in both research and user modes. For the research mode the user interface was written using the MATLAB software, a flexible environment to test and modify any new techniques found during investigation. Fig. 3æ10 depicts the user interface menu of the research mode, which displays the magnitude and phase of a RF vector, as well as its spatial profile (i.e. the Fourier transform of the RF vector). The user selects the desired type of the pulse sequence (SE nf (SE), GRE (GRE), or multi-slice GRE (MGRE)), the resolution of the image (128 or 256), and the encoding direction (horizontal or vertical). Then, the user selects from the types of the orthonormal basis set calculations. There are four orthonormal basis set calculation options; RF impulse, Hadamard, SVD, and

63 48 IPG Program V1 V2 V3 MR SIGNA System Host Program Workstation System Vectors from Orthonormal Basis Calculation Scaling Vn V MR Excitation Unit N e t w o r k N e t w o r k Format Converting Figure 3æ9: A block diagram of the RF vector conversion and transfer process. The vectors are first scaled according to the SIGNA hardware specifications and then converted into the GE file format, before being sent to the SIGNA s IPG computer. Each RF vector is sent to the pulse generator memory of the IPG program for play out prior to the beginning of each RF excitation period. Lanczos. Following calculation of the orthonormal basis sets the user can send the RF vectors to the SIGNA system with the Save&Transfer button. After the user activates the button, the program automatically establishes a network communication, and upon successful connection, transfers the data. To reconstruct an image a user can use the Recon Imp button for RF impulse encoding or the Recon NF button for cases of non-fourier encoding. The user mode interface was written with the UNIX tcsh file program [39], running on the Host computer of the SIGNA system. Thus, the user can work

64 49 SE GRE MGRE 256 res 128 res Horizontal Vertical Save& Transfer RF Impulse Hadamard SVD Lanczos Recon Imp Recon N-F Figure 3æ10: The research mode user interface of the non-fourier encoding process, which was implemented on the SUN workstation using the MATLAB software package. with the pulse sequence at the console of the SIGNA Host computer as if working with a conventional pulse sequence. Fig. 3æ11 depicts an example of an option menu for a multi-slice GRE pulse sequence using the SVD encoding method. The user has the options to: generate RF impulse vectors (1), generate SVD vectors (2), reconstruct RF impulse encoding images (3), or reconstruct SVD images (4). In the user mode the program will automatically transfer data between the SIGNA system and the workstation.

65 50 ***************************************************************** * MENU OPTIONS FOR THE MGE PULSE SEQUENCE ***************************************************************** 0 Exit 1 Generate RF Impulse Vectors 2 Generate SVD Vectors 3 Recon RF Impulse Images 4 Recon SVD Images Enter Option Number: Figure 3æ11: The menu options of the multi-slice GRE pulse sequence user mode, employing the SVD encoding implemented on the SIGNA s Host computer. 3.5 Non-Adaptive and Dynamic-Adaptive Algorithms From our implementation of the non-fourier encoding, non-adaptive and dynamicadaptive algorithms can be performed to study dynamic MRI applications. In the non-adaptive algorithm, the process starts by converting and transferring predefined basis set - such as RF impulse or Hadamard - to the IPG user s memory of the SIGNA system, as shown in Fig 3æ12. Dynamic images are acquired using the same RF vectors, generated from the predefined basis set, through out the application, as shown in Fig. 3æ13. The dynamic-adaptive algorithm, on the other hand, starts by acquiring a base line image using a non-adaptive technique, as shown in Fig. 3æ14. For the subsequent dynamic image, the measurement data is used to generated an adaptive basis set, such as the SVD. The calculated basis set is, then, converted and transferred to the SIGNA system to acquire the next image, as shown in Fig. 3æ2. The RF vectors are dynamically updated, according to the basis set calculated from the recent measurement data, through out the applications.

66 51 SIGNA MR System Workstation System MR Excitation Unit IPG Program Predefined Basis set Host Program RF Vectors Converting Figure 3æ12: The initial process of the non-adaptive algorithm. SIGNA MR System MR Excitation Unit Pre-Processing Workstation System IPG Program K-Space Mapping Image Reconstruction Host Program Display Figure 3æ13: The non-adaptive algorithm for dynamic MRI application.

67 52 SIGNA MR System MR Excitation Unit Pre-Processing Workstation System IPG Program K-Space Mapping Predefined Basis Set Image Reconstruction Host Program RF Vectors Converting Display Figure 3æ14: The initial process of the dynamic-adaptive algorithm

68 53 Chapter 4 Analysis of Non-Fourier Encoded MRI Using the Linear System Approach 4.1 Introduction The magnetization behavior that occurs during a non-fourier encoded MRI experiment can be described by the Bloch equation, using the small flip angle approximation described in Chapter 2. An alternative analysis method involves the use of the linear input-output (IO) system model [8], which also provides a useful description of the response of an MR system to RF excitation inputs. The linear system approach is a powerful tool because it yields both a mathematical and a physical model to analyze the behavior of an MR system. The IO system model, however, describes the response of an MR system to RF excitation inputs by assuming that the system output depends only on the input. In other words, it dictates that the system magnetization is at the equilibrium state (i.e. zero state) before the RF excitation input is applied. This assumption is valid only when the input repetition time (TR) is much longer than the system recovery time (T1) (i.e. TR é 5 æ T1). Such an assumption is, however, violated in fast MR acquisitions, as the TR of such an acquisition is typically not much longer than T1. The system output in this case thus does not depend only on the RF excitation input, but also on the non-equilibrium magnetization (i.e. the system state) generated by previous RF excitation inputs.

69 54 To describe the behavior of the MR system in such fast MR acquisitions, the linear input-state-output (ISO) system model [98], [99], which is an extension of the IO system model, is introduced. The ISO system has been used as a physical system model in various engineering fields [98]-[101]. For example, common linear electrical circuits, which consist of resistors, capacitors, or inductors, have a linear relationship between the electrical energy input and output response as long as their components are not forced to operate outside their normal range. If there is energy stored in the circuit (i.e. charges in the capacitors), the ISO system model can be used to analyze the output response to the electrical input and to the initial energy state of the circuit. On the other hand, if there is no energy stored in the circuit, only the IO system model is sufficient for analyzing the relationship between the electrical input and output response. In this study, the ISO system model is used for describing the MR system output generated by the combination of the RF excitation input and the non-equilibrium magnetization state of the system. The theory of the ISO system model will be reviewed in this chapter, followed by experimental results, with the aim of characterizing the non-fourier encoding MR experiment in terms of an analysis using the IO and ISO system models. 4.2 Theory The Linear Input-State-Output System For a diagnostic MRI, an MR system can be defined as a multivariable system, as shown in Fig. 4æ1, used for measurement of the proton density function sè~rè (where ~r = ëx; y; zë) of the object in the FOV. The system shown in Fig. 4æ1 is assumed to have n input, m output, and m state terminals. The inputs are de-

70 55 noted by p 1 ; p 2 ;:::; p n, as shown in Fig. 4æ1A, or by an n æ 1 column vector p =ëp 1 p 2 æææ p n ë t, as shown in Fig. 4æ1B which uses a thick line to represent a vector. The states and output are denoted by an m æ 1 column vector l =ël 1 l 2 æææ l n ë t and an m æ 1 column vector y =ëy 1 y 2 æææ y n ë t, respectively. (A) l 1 l2 lm (ktr) (ktr) (ktr) p (ktr) 1 p (ktr) 2 p (ktr) n Multivariable ISO System y ((ktr)+te) 1 y ((ktr)+te) 2 y ((ktr)+te) m (B) l(k) p(k) Multivariable ISO System y(k) Figure 4æ1: A multivariable system (A) represents an MR system with n input(p), m output(y), and m state(l) terminals. The inputs are excited repeatedly at time interval TR (repetition time). The outputs are sampled at the same time interval as the inputs but delayed by TE (echo time). The system diagram can be simplified (B) by omitting TE and TR, and representing its variable in vector form. For non-fourier encoded MRI, the system uses RF excitations as the inputs (p) and provide, as outputs (y), the complex-value data which can be mapped onto

71 56 the k-space data. The system also represents the non-equilibrium magnetization states (l) generated by the RF inputs. The measurement system includes all the physical hardware used for measurement (i.e. the main magnet, the gradient coils, and the RF transmitter and receiver, etc.), as well as the pulse sequence, except for TE (echo time) and TR (repetition time). The variable k in Fig. 4æ1 represents the encoding number, where k = 0 æææk, and K is the total number of encodes, where K ç n. The system inputs are repeatedly applied at the time interval TR, as shown in Fig. 4æ1A. The k th encoding input, therefore, excites the system at time t = k æ TR. The outputs are also sampled at the same time interval as the input but delayed by the TE period. The k th output is, thus, sampled at time t =èkætrè+te. The input and output timing diagram of the model in Fig. 4æ1A is illustrated in Fig. 4æ2A. In a typical MRI application, TE and TR were kept constant throughout the experiment. The system input, state, and output at the k th encoding number thus can be presented, as shown in Fig. 4æ1B, by omitting the time constants as pèkè; lèkè, and yèkè, respectively. The input and output diagram of this model (Fig. 4æ1B) is presented in Fig. 4æ2B. The system input (p) is comprised of the magnitude and phase of RF excitations, or the encoding function as described in Section 2.2. The encoding functions are obtained from orthonormal basis sets, such as Fourier, Hadamard, or SVD. To avoid an encoding artifact due to operation outside the linear range, the weighing function (the flip angle) of these basis sets should be lower than 30 æ [8]. The MR system is a shift variant system [8], thus multiple inputs (i.e. K inputs) are required to obtain the necessary responses from the system. The system output (y) can be mapped onto the k-space data using a process of decoding or inverting the encoding function. This process is described further

72 57 p(0) p(1) (A) p(k) y(0) y(1) y(k) 0 TE 1TR KTR t (B) y(0) p(0) y(1) p(1) y(k) p(k) 0 1 K k Figure 4æ2: Timing (A) and encoding (B) diagrams of the input (p) and output (y), according to the models in Fig. 4æ1A and B, respectively. in the following section. Once the system output is mapped onto the k-space (Fourier) domain, the MR image (i.e. the desired data) can be obtained (i.e. the MR image is reconstructed by performing the Fourier transform on the k-space data). In this model, all input and output terminals are assumed to be played out and sampled instantaneously. This model ignores the off-resonance effect and T2 relaxation time on the RF exciting and sampling. In other word, this model assumes that the RF exciting and sampling dwell time are short enough so that these effects can be ignored [8]. The non-equilibrium magnetization states (l) are generated by the RF inputs. If TR is much longer than T1, the non-equilibrium magnetization l can be neglected. The system model in this case is thus reduced to the IO system model. However,

73 58 if TR is not much longer than T1, the complete response should be analyzed only by using the ISO system model. In the ISO system model, the MR system can be considered as a linear system when a small flip angle (é 30 æ ) of the RF excitations are used [8] and the system output is the linear combination of the responses from the RF excitation input (p) and the non-equilibrium magnetization state (l) [98], [100]. In other words, the system output (the complete response) can be represented as: yèkè =y zs èkè +y zi èkè (4.1) where y zs èkè is the zero state response, which is the system output due only to the input when the system is at the zero state (i.e. the magnetization is at the equilibrium state prior to the play out of the RF excitation input). y zi èkè is the zero input response, which is the system output due only to the system state (the nonequilibrium magnetization state) when there is no input applied to the system. For the MR system, y zs èkè, which is the response due to the RF input, is the desired response while y zi èkè, which is the transient response, contributes the encoding artifact. Thus, the elimination or reduction of y zi èkè from the system output is a necessary procedure to improve the quality of the system output. In the following sections, y zs èkè and y zi èkè are further characterized using impulse and transient response analysis, respectively. Impulse Response Analysis For a zero state response (y zs èkè), whose output depends only on the input, the principle of superposition states that the theoretical response to all possible combinations of impulse (hard-pulse) can be computed if the response of an individual impulse is known [98], [100]. For example, in a two-input system (i.e. p =ëp 1 p 2 ë),

74 59 this statement can be expressed as: Sfapè1è + bpè2èg = asfpè1èg + bsfpè2èg (4.2) where pè1è and pè2è are system inputs, a and b represent any constant values, and S is the system s response. Eq. 4.2 indicates that if the responses to pè1è and pè2è, or Sfpè1èg and Sfpè2èg, are known, then the response to any linear combination of these inputs (Sfapè1è + bpè2èg) can be found, as defined on the right hand side of Eq This analysis, however, assumes that the system output depends only on the input (the IO system model) and that it is operated in a linear range. In the MR system, this assumption is valid only when TR is much longer than T1 and a small flip angle of the inputs were used. The superposition theory is a very useful tool for an analysis of the non-fourier encoding method because RF excitations generated by orthogonal basis sets, such as Hadamard or SVD, can be represented as the linear combination of hard-pulse trains generated by an RF impulse basis set. If the response to each hard-pulse is known, then the response to any basis set can be found in theory from the linear combination of the hard-pulses comprising that basis set. In the MR system, each RF excitation input is a stepwise function (i.e. a digital wave form). Each can be represented as a rectangular box function whose width and height are equal to the resolution (æt p ), and to the magnitude of the stepwise function, respectively. The rectangular box function can be further approximated as a hard-pulse train. This is illustrated in Fig. 4æ3, which is an example of such an approximation of the sinc pulse shape. The hard-pulse train, p H èkè, can be represented by p H èkè =ëp 1 p 2 æææ p n ë (4.3) where p i is a complex number representing both the magnitude and phase of the

75 Figure 4æ3: A sinc-pulse shaped RF input derived from the stepwise function (on the left) can be approximated as a rectangular box function (in the center), and also as a hard-pulse train function (on the right). i th hard-pulse train. For an MR system, the hard-pulse train approximation states that the magnetization is flipped instantaneously at time t = k æ TR, and then undergoes free precession during a time period of interval æ p. The pulsewidth of the waveform input (æ p ) is typically much less than the T1 relaxation time constant of the measured object. The free precession during the time interval (æ p ) thus has little effect on the longitudinal magnetization. The zero-state output of an MR system can be expressed in matrix form as Y zs = P R (4.4) Output = Input æ System Response (4.5) where P is a matrix, the rows of which contain RF excitation inputs (p). Y zs signifies a matrix whose rows hold the output data of the system (y zs ), which is the response to the coinciding rows of P. R is the impulse (hard-pulse) response matrix of the system. Once the system response (R) is known, the zero-state system output of any basis set input can be found theoretically using the superposition property as in Eq. 4.5.

76 61 The response to an individual hard-pulse (i.e. a row of R) can be obtained by using an RF unit box pulse (i.e. an RF impulse) that has a constant non-zero magnitude at only one step of the pulse train, æ p, and a zero magnitude elsewhere. By shifting the position of the non-zero magnitude of an RF unit box pulse on each excitation input, and repeating the excitation until all the positions of each step are excited, the response to all individual hard-pulses (R) can be found. The set of unit box pulse inputs can also be represented as identity matrix I. The impulse response matrix (R), obtained from the RF impulse input, has a direct map to the k-space Fourier domain data (S), acquired by using the Fourier (phase) encoding method, as described in Chapter 2 and also suggested in [8]. This mapping is only an approximation [8] because in the RF impulse method there are effects of relaxation and evolution of the magnetization during the excitation. In the Fourier (phase) encoding method there are, however, no such effects. In general, the zero-state system output matrix (Y zs ) can be mapped onto the k-space matrix by multiplying the output matrix with the inverse matrix of the RF excitation inputs, S = R = P,1 Y zs (4.6) In order to obtain the k-space data (S) it is therefore necessary that the RF excitation input matrix (P ) be an invertible matrix, or that the encoding function be an orthonormal basis set. Transient Response Analysis The zero input response (y zi èkè), also known as the transient response, is caused by a system state when there is no input applied to the system. In the MR system, when TR is much longer than T1, the transient response is negligible. For fast MR imaging, a typical TR may be shorter than T1. There is, therefore, a need

77 62 to analyze the transient response which contributes the encoding artifact in the complete response (yèkè). By eliminating or minimizing this artifact, the system output will approximately close to the zero state response (y zs èkè), which is the desired output (i.e. free from the encoding artifact). In this study, the transient response is analyzed by using the system controllability theory [98], [100], and by using the unit step input method [100]. The system is said to be controllable [100] if it is possible to transfer the system from the initial state (lè0è = 0) to some desired state lèkè = l d by using the appropriate sequence of control inputs pè0è; pè1è;:::; pèk,1è, where K is a finite number. On the other hand, if K can not be a finite number, the system is said to be uncontrollable. If the system is controllable, then the zero input response can be found by generating the state of this response using a sequence of control inputs. In other words, a zero input response can be obtained if the history of the sequence of inputs is known. For example, to find the zero input response at the K th encoding input, the sequence of inputs pè0è; pè1è;:::; pèk,1è are used to excite the system. The K th zero input response is obtained by sampling data at the K th step as the zero input is applied (i.e. pèkè = 0). Once the zero input responses of all the encodes are obtained, the desired response (Y zs ) can be obtained by subtracting the transient responses (Y zi ) from the system responses (Y ). The unit step input is another method that can be used for characterizing the temporal response of the MRI system. The transient response is analyzed from the deviation of a response to a unit-step input when the system is at rest or at equilibrium to a response of the same input when the system is at a predefined initial state. Fig. 4æ4 illustrates the transient response model and an analysis of the MRI system. This analysis assumes that the major transient signal is caused by only the previous input. For the unit-step-input transient analysis, each RF

78 63 p(1) p(1) y(1) y(1) p(1) p(2) p(2) Multivariable ISO System y(1) y(2) y(2) p(2) y(2) p(k) X y(k) RF-Pulse Input Avg. Signal y(1) Response #1 y(2) Response #2 System-Response Output X Figure 4æ4: The unit-step-input transient response model and an analysis of the MRI system showing the RF excitation pulse inputs (p) and the system response outputs (y). The average magnitudes of the system response depict a transient response for a typical TR (TR é T1) in fast imaging (the solid line) and the response for a long TR (T R éé T 1)(the dotted line). encoding input is used to excite the system repeatedly (i.e. 20 times). The average magnitude of the system response from such encoding inputs is investigated by comparing them to the results from the input-output response (zero state response), as shown in Fig. 4æ4. The average magnitude plot in Fig. 4æ4 depicts a system response for a typical TR (TR é T1) in fast imaging (the solid line), and the zero state response for a long TR (T R éé T 1) (the dotted line). The transient response, which creates encoding artifacts, is the deviation of the system response

79 64 from the zero state response. This analysis provides a method for measurement of the transient signal, not for elimination of the transient response, as in the previous analysis. For example, this measurement can be used to investigate the effect of the ordering of non- Fourier encoding vectors to the transient signal. By reordering the encoding inputs such that the effect of the transient signal on the system output is optimized, the desired output can thus be obtained with a minimum of artifacts. 4.3 Experimental Results and Analysis We present the experimental results in this section, with the aim of characterizing the non-fourier encoding MR experiment in terms of an analysis using the linear input-output (IO) and input-state-output (ISO) system models. The IO system model was employed for studying the system response to the non-fourier RF pulse inputs. By using this model, the amplitude variations (i.e. flip angles) of such inputs were investigated in order to reduce encoding artifacts and to increase the signal-to-noise ratio (SNR). To fulfill a necessary condition for the IO system model, as described in Section 4.2.1, a long TR (i.e. TR é 5 æ T1) was imposed in the experiments. The ISO system model was, on the other hand, employed for studying the transient response caused by the non-equilibrium magnetization (i.e. the system state) when TR was not much longer than T1. To study such a response, a short TR (i.e. TR ç T1) was employed in the experiments. The experiments in this section were performed on a 1.5 GE SIGNA MR system using the single slice spin-echo non-fourier encoding (SE nf ) pulse sequence, following the implementation described in Chapter 3. In all experiments in this thesis, images were acquired with NEX=1, FOV=24cm, and matrix=256x256 (except for multi-slice GRE images which is 128x128), using a standard quadrature

80 65 birdcage head coil for transmiting and receiving RF signal System Response Superposition Theory An experiment was performed, using a phantom, applying sets of RF unit box pulses at different flip angles. The objective of this experiment was to validate of the superposition theory in the linear range of magnetization flip angles, which is an important property of the linear system analysis. RF magnitude, C1 RF magnitude, C2 RF magnitude, C3 = (C1+C2)/2 RF magnitude, C4 = (C1 C2)/2 Figure 4æ5: A representation of the four types of RF unit box pulses used in the superposition experiment; the first and second patterns, C 1 and C 2, are basis RF unit box pulses, while C 3 and C 4 are linear combinations of patterns of C 1 and C 2. The experiment was performed on a doped-water phantom using a 1 second TR (i.e. TR é 5æT1) with four RF impulse patterns at two different flip angles. Four types of RF unit box pulses, used to excite the system, are illustrated in Fig.

81 (A) (B) Signal Magnitude k-space position Figure 4æ6: A comparison of the experimental excitation results (represented by the solid line) with the calculated-superposition results (the dashed line) of the C 3 RF input pattern acquired at 20 æ (a) and 50 æ (b) flip angles. 4æ5. The patterns were aimed to represent a linear addition and subtraction of two RF unit box pulses. The first and second patterns, C 1 and C 2, were basis RF unit box pulses. The third pattern (C 3 ) was a combination sum of half C 1 and C 2 (C 3 = C 1+C 2 2 ). The fourth pattern(c 4 ) was a different combination of half of C 1 and C 2 (C 4 = C 1,C 2 ). The constant factor (i.e. 1) in the C and C 4 patterns was imposed in order to maintain the same flip angle (i.e. a magnitude of the Fourier transform of the pulse) as in the unit box pulses. Figs. 4æ6 and 4æ7 compare the actual results of the excitations with the calculatedsuperposition results at two different flip angles, one flip angle in the linear range and the other in the non-linear range. The MR signals excited by patterns C 3 and

82 (A) (B) Signal Magnitude k-space position Figure 4æ7: A comparison of the experimental excitation results (represented by the solid line) with the calculated-superposition results (the dashed line) of the C 4 RF input pattern acquired at 20 æ (a) and 50 æ (b) flip angles. C 4 are illustrated in Figs. 4æ6 and 4æ7, respectively. Figs. 4æ6A and B show the MR signal of the C 3 excitation input resulting from the actual excitation (represented by the solid line), and from the calculation of C 1 and C 2 using the superposition theory (the dashed line) at the flip angles of 20 æ (linear range) and 50 æ (non-linear range), respectively. Figs. 4æ7A and B show the MR signal of the C 4 excitation input in the same type of comparison as that constructed in Fig. 4æ6. As seen from Figs. 4æ6 and 4æ7, the actual and calculated results are almost identical when the excitation flip angle is in the linear range, while there is a distinguishable difference if the flip angle is in the non-linear range. These experimental results support the basic superposition theory, provided that the flip angle input is in the linear

83 68 range. System Response to RF Excitation Input A series of experiments were performed with the DW phantom, using a 1 second TR (i.e. TR é 5 æt1), employing different types of non-fourier encoding. With these experiments we attempted to characterize the response of the system to the distribution of flip angles of the RF excitation input generated by the different orthonormal basis sets. Three different RF excitation encoding methods were used: the RF impulse, the Hadamard, and the SVD. The SVD encoding input was calculated with the k-space data (i.e. the impulse response data) acquired from the RF impulse input. Figure 4æ8: RF impulse images (A-F) acquired at flip angles of 10 æ ; 15 æ ; 20 æ ; 30 æ ; 40 æ, and 50 æ, respectively.

84 69 Figure 4æ9: Hadamard encoding images (A-F) acquired at flip angles of 10 æ ; 15 æ ; 20 æ ; 30 æ ; 40 æ, and 50 æ, respectively. The images acquired with different mean flip angles using the RF impulse, Hadamard and SVD encoding inputs are shown in Figs. 4æ8, 4æ9, and 4æ10, respectively. Each Figure shows 6 images, A-F, acquired at flip angles of 10 æ ; 15 æ ; 20 æ ; 30 æ ; 40 æ ; and 50 æ, respectively. The non-fourier and frequency encoding directions are along the horizontal and vertical directions of the images, respectively. The images acquired by the use of the RF impulse input (Fig. 4æ8), which is equivalent to the Fourier (phase) encoding method, show no encoding artifacts, as expected. The Hadamard (Fig. 4æ9) and SVD images (Fig. 4æ10), however, display vertical line artifacts, especially when a flip angle in the non-linear range is employed. To analyze the different encoding artifacts resulting from these encoding methods, we examined the RF spatial and phase profiles of each method. For the low-

85 70 Figure 4æ10: SVD encoding images (A-F) acquired at flip angles of 10 æ ; 15 æ ; 20 æ ; 30 æ ; 40 æ, and 50 æ, respectively. flip-angle excitation, the magnitude of the RF spatial profile is equal to the flip angle profile because the excited profile of magnetization is the Fourier transform of the RF encoding vectors, as analyzed in Section Figs. 4æ11A, B, and C show the magnitude and phase of RF excitation inputs generated from the RF impulse, Hadamard, and SVD methods, respectively. Fig. 4æ12 contains the flip angle and phase profiles associated with the RF input represented in Fig. 4æ11. The flip angle profiles (A ç è~rè, where ~r is a spatial direction) of Fig. 4æ12 are depicted in the scaled magnitude units of Fourier transform from Fig. 4æ11. The actual flip angle ç act è~rè is, then, equal to ç cal æ A ç è~rè, as described in Chapter 3. The ç cal is obtained from the area of RF magnitude input related to the standard area, as also described in Chapter 3. For example, if ç cal is 10 æ, ç act è~rè of

86 71 (A) Magnitude encode number 1 64 Phase k space direction (B) (C) Figure 4æ11: Three RF magnitude and phase inputs of the RF impulse encoding (a), Hadamard encoding (b), and SVD encoding (c) methods used in the system response studies. the RF impulse and Hadamard encoding methods will be equal to 10 æ throughout the FOV because from Figs. 4æ12A and B, the RF spatial profiles (A ç è~rè) of both methods are uniformly distributed with a unit one magnitude. The ç act è~rè in the SVD method, however, varies across the FOV due to a non-uniform distribution of A ç è~rè, as shown in Fig. 4æ12C. Fig. 4æ12 shows that the flip angle profiles are uniformly distributed for the RF impulse and Hadamard encoding methods inside the FOV and these methods use phase profiles to spatially encode. By contrast, the SVD encoding method uses the

87 72 Magnitude encode number 1 (A) space direction Phase (B) (C) Figure 4æ12: Magnitude and phase profiles of the RF impulse encoding (a), Hadamard encoding (b), and SVD encoding (c) methods. These profiles were generated by performing a Fourier transformation on the RF input illustrated in Fig. 4æ11. spatial, as well as the phase, profile to encode the object inside the FOV. The RF impulse method does not display any encoding artifact (Fig. 4æ8) because it does not require linear combinations of the RF hard-pulses. In other words, the method does not require the assumption or application of superposition. The Hadamard and SVD methods, on the other hand, use orthonormal basis sets which are linear combinations of the hard-pulse trains. These two encoding methods, therefore, are subject to encoding artifacts when the RF input flip angles

88 73 are in the non-linear range, since this violates the superposition assumption. The SNR and artifact-to-noise ratio (ANR) of these encoding methods will be further addressed, quantitatively, in the following sections. The Distribution and Cumulative Distribution Functions of the SVD Flip Angle Profile The actual flip angle profile ç act è~rè in the SVD encoding method is not uniformly distributed, as illustrated in the previous experiment (Fig. 4æ12C). This is because the SVD encoding method uses flip angle profiles that are dependent on the contents of the FOV. To avoid encoding artifacts due to the violation of the superposition assumption, the magnitude of ç act è~rè is limited only in the linear range. For the quantitative analysis of ç act è~rè, there is, thus, a need to use statistical parameters to represent this profile. The following analysis was designed to demonstrate a method to characterize the SVD flip angle by using the distribution and cumulative distribution functions. The flip angle distribution can be characterized from its histogram distribution such that the quantitative parameters - such as the mean and standard deviation - can be used to represent such distribution. The cumulative distribution function (CDF) is formally defined as the integral of the distribution function P èxè from 0 to x 0 such that, as x 0 goes to infinity, the CDF is equal to one. The CDF can be used to ascertain how much of the flip angle distribution is inside or outside the linear region. For example, the percentage of the flip angle population which is not exceeding the x 0 flip angle (i.e. 30 æ ), is equal to CDFèx 0 è æ 100 % of the total population. The SVD flip angle distribution can be characterized by the mean and standard deviations calculated from the distribution of the actual flip angle profile (ç act è~rè), as shown in Figs. 4æ13 and 4æ14. These are the same flip angles that were used

89 74 in the previous experiment. Figs. 4æ13A, B, and C show the distributions and CDF of the SVD RF spatial profiles at ç cal 10 æ ; 15 æ, and 20 æ, respectively. Figs. 4æ14A, B, and C depict the distribution and the CDF of the SVD spatial profile at ç cal 30 æ ; 40 æ, and 50 æ, respectively. The mean and standard deviations of each flip angle are shown with the corresponding distribution. Flip Angle=10deg Flip Angle=15deg Flip Angle=20deg (A) (B) (C) 600 mean= mean= mean=25.8 std=3.65 std=5.47 std=7.3 # of flip angle flip angle (deg) CDF (30)=99.38% CDF (30)=93.01% CDF (30)=82.92% CDF flip angle (deg) Figure 4æ13: The distribution and cumulative distribution functions for the SVD flip angles of 10 æ (a), 15 æ (b), and 20 æ (c), which were used in the system response studies. At the same RF excitation pulsewidth, the SVD image has a higher signal output than the uniform flip angle profile image because the mean flip angle of the

90 75 Flip Angle=30deg Flip Angle=40deg Flip Angle=50deg (A) (B) (C) # of flip angle mean=38.7 std= flip angle (deg) CDF (30)=63.89% mean=51.6 std= CDF (30)=50.51% mean=64.5 std= CDF (30)=41.66% CDF flip angle (deg) Figure 4æ14: The distribution and cumulative distribution functions for the SVD flip angles of 30 æ (a), 40 æ (b), and 50 æ (c), which were used in the system response studies. SVD spatial profile is greater than the ç cal used by the RF impulse and Hadamard methods. From the previous experiment, the artifacts caused by the SVD method are more prominent than those caused by the Hadamard method at the same flip angle due to this difference in mean flip angle. For the SVD method the distribution of flip angles is also graphically described by using the CDF, as shown in Figs. 4æ13 and 4æ14. This function can describe how much of the flip angle distribution is within the linear region (ç 30 æ ). For example,

91 76 about 83% of the flip angle population (Fig. 4æ13C) is inside the linear region when the mean SVD flip angle is 25 æ, while only about 64% of the flip angle population (Fig. 4æ14A) is inside the linear range when the mean SVD flip angle is 38 æ. Thus, when the SVD method is used, the CDF is useful for the characterization of the flip angles employed. To further investigate the characteristic of the distribution of SVD flip angle profile, an analysis was performed on different shape phantoms. The hypothesis is that if there is no significant difference of the flip angle profile relative to the FOV contents. The mean of SVD flip angle profile calculated from the base line (i.e. initial) image can be used, in general, to respresent the SVD flip angle profile of images during dynamic MRI applications. Fig. 4æ15 depicts the flip angle distributions, used with SVD encoding, found in the case of the circular, rectangular, and brain-shaped-gel [38] phantoms, calculated at the ç cal of 20 æ. The SVD flip angle distribution of these three objects does not show any statistically significant difference. The SVD human brain image, acquired at the same ç cal as used by the phantoms, is shown in Fig. 4æ16. Its flip angle distribution is also illustrated in the same figure. The SVD flip angle distribution of the human brain image also does not shown any significant difference relative to the distribution of the phantom images. This supports the hypothesis that the structure of the FOV contents is not a critical factor for the SVD flip angle profile. Thus the statitical parameters calculated from the base line image can be used, in genearal, to represent the SVD flip angle profile of images in dynamic MRI applications. In this section, we demonstrate a method to represent the SVD flip angle profile by using the distribution and cumulative distribution functions. The SVD flip angle distribution can be quantitatively represented by the mean and standard

92 77 Figure 4æ15: The SVD images and the accompanying flip angle distributions resulting from doped-water (represented in the top section), water-fat (middle section), and brain-shaped gel (the bottom section) phantoms deviation found from the histogram of the SVD RF flip angle profile. The CDF can be used to quantitatively represent the percentage of the flip angle population which is not exceeding the linear range. Optimal Flip Angle for SVD Encoding For an MRI, a high signal-to-noise ratio (SNR) is the preferred output for high diagnostic image quality. SNR is increased by using a high flip angle excitation. In the case of SVD encoding, however, only a relatively small flip angle can be

93 78 Figure 4æ16: The SVD images and the accompanying flip angle distributions resulting from saggital view human brain. employed due to the limit imposed by the superposition property. An experiment was performed, therefore, aimed at finding the optimal mean flip angle for SVD encoding which would provide the highest possible SNR at the lowest possible encoding artifact-to-noise ratio (ANR). A series of experiments was performed with the DW phantom using different flip angles and types of non-fourier encoding. The SNR plots of the RF impulse, Hadamard, and SVD encoding methods are shown in Fig. 4æ17. The SNR is the ratio of the average MR signal inside the object to the average MR signal of the background noise. The higher flip angle used, the higher the SNR obtained, as expected. The RF impulse and Hadamard methods result in the same SNR from each flip angle due to the use of a common uniform flip angle profile (i.e. ç act = ç cal ). The SVD method results, however, display a higher SNR when compared to the other methods at the same flip angle due to the higher mean flip angle used in SVD encoding, as described in the previous experiment. Fig. 4æ18 shows the ANR plots for the three encoding methods. The ANR is

94 RF Impulse +. Hadamard SVD 20 SNR flip angle (deg) Figure 4æ17: The signal to noise ratio (SNR) of images acquired by the use of RF impulse ( ), Hadamard (+.), and SVD ( ) encoding methods. the ratio of a standard deviation calculated from the MR signal inside the object to the mean MR signal of the same area. The ANR plots for the RF impulse and Hadamard methods do not show any significant differences as the flip angles are changed. The SVD method, however, displays a significant change in the ANR as the mean flip angle is varied. Fig 4æ18 indicates that the optimum ANR for the SVD method occurs with a mean flip angle of approximately 15 æ, 20 æ. The high ANR of the SVD method when the mean flip angle is less than 15 æ is most likely due to the low MR signal formed by the low flip angle excitation (i.e. a

95 RF Impulse +. Hadamard SVD ANR flip angle (deg) Figure 4æ18: The artifact to noise ratio (ANR) of images acquired by using the RF impulse ( ), Hadamard (+.), and SVD ( ) encoding methods. small amount of measured magnetizing contribution). The high ANR that results when the mean flip angle is greater than 20 æ is most likely due to the non-linear artifact when the superposition principle underlying the use of the SVD RF pulses is violated. The image quality of the SVD encoding method is limited by this non-linear encoding artifact. The ANR and SNR plots suggest that the optimal image quality when using the SVD method can be achieved by using a mean flip angle of approximately 15 æ, 20 æ.

96 The Transient Response In the ISO system model, the system output (i.e. the complete response) consists of the response due to the RF input (i.e. the system response) and the response due to the system state (i.e. the transient response) caused by non-equilibrium magnetization, as described in Section For a linear system, the complete response is a linear combination of the system and transient responses. When TR is much longer than T1 (i.e. TR é 5 æ T1), the transient response is negligible. The complete response in this situation, then, is approximately equal to the system response. The system response analysis using the IO system model, as was investigated in the previous section, is thus sufficient to characterize the system. For fast MR imaging a typical TR is, however, not much longer than T1. There is, therefore, a need to analyze the transient response. By eliminating or minimizing the transient response from the complete response, the system output will depend only on the RF input (i.e. free from an encoding artifact). To investigate the transient response, the experiments were performed first to analyze such a response using the linear ISO system model, and then to demonstrate a possible method of reducing the encoding artifact caused by the transient signal. Zero-Input Response Analysis A series of experiments was performed with different phantoms employing a specially arranged series of non-fourier encodes. The objective of these experiments were, first, the characterization of the zero-input response (i.e. the transient response) and then the demonstration of a method for eliminating such a response from the system output (i.e. the complete response). The analysis in this section, which is based on the ISO system model, can be used for any spatial selective encoding technique as long as the system is operated in the linear range. With

97 82 the objective of studying the adaptive encoding method, only SVD encoding was employed for the demonstration in this section. Effect of Transient Signal In the first set of experiments, a doped-water (DW) phantom, whose T1 was about 100 msec, was used to demonstrate the effect of TR on the zero-input response. The experiments were performed using 100 (i.e. TR ç T1) and 1000 (i.e. TR é 5æT1) msec TRs. The SVD encoding input was calculated with the k-space data (i.e. impulse response data) acquired from the RF impulse inputs. A total of 34 encoding inputs were used in each experiment (i.e. for each selected TR). The first 30 inputs were defined from 30 SVD vectors along the horizontal direction with highest weights (i.e. highest associated singular values). To illustrate the highest possible zero-input response, the SVD pulse inputs were arranged in an order of lowest to highest significance (i.e. lowest to highest signal output). The four remaining inputs were generated from a null vector (i.e. zero pulse input). From this input pattern, the signal outputs from the 31st to 34th encoding inputs were the zero-input responses caused by the non-equilibrium magnetization state generated by the SVD pulse inputs. The MR signals, acquired at 100 and 1000 msec TRs, from the 30th (the last SVD pulse) to the 34th input are shown in Figs. 4æ19A and B, respectively. According to the encoding inputs in this experiment, any signals after the 30th input (i.e. the second row in Fig. 4æ19) are the zero-input response (i.e. the transient response). In Fig. 4æ19, the system outputs, acquired by using a short TR, (Fig. 4æ19A), show the transient responses while there are no such responses when a long TR is used (Fig. 4æ19B). The experimental results clearly demonstrate transient responses caused by non-equilibrium magnetization when the input repetition time (TR) is not much longer than the system recovery time (T1).

98 83 (A) Signal Magnitude Response (y) (B) k space position 64 Signal Magnitude Response (y) k space position 64 Figure 4æ19: The zero-input responses (i.e. transient responses) of the DW phantom, acquired at 100 (TR ç T1) (A) and 1000 (TR é 5æT1) (B) msec TRs, starting from the 31st encoding input (the second row). To investigate the effectiveness of constant gradient spoiling, as described in Chapter 3, on reducing the transient signal, an experiment was performed on the same phantom using the same encoding inputs at 100 msec TR with three different spoiler pulse widths of 0 msec (no spoiler), 5 msec (typical choice of spoiler [103]) and 10 msec (twice that of normal) at the same gradient amplitude (i.e. 1 G/cm). Fig. 4æ20 shows the signals at the 31st encoding input (i.e. the first transient signal), obtained with no spoiler (the solid line), a normal spoiler (i.e. the +- line), and twice the duration of a normal spoiler (the dashed line). In Fig. 4æ20, there is

99 Signal Magnitude k space position Figure 4æ20: The transient signals from the zero-input response experiment using the SE nf pulse sequence with spoiler pulse widths of 0 msec (the solid line), 5 msec (the +- line) and 10 msec (the dashed line) at the same gradient amplitude (i.e. 1 G/cm). no significant difference among transient signals obtained from these three different durations of spoilers. The result of this experiment clearly suggests that the spoiler has no significant effect on reducing the transient signal. In other words, this result implies that the transient signal is not due to the remaining transverse magnetization from the previous encoding input per se because, if the transient signal were caused by such magnetization, the spoiler would dephase such a signal and be able to reduce it.

100 Signal Magnitude Response (y) k space position 64 Figure 4æ21: The zero-input responses (i.e. transient responses) of the DW phantom acquired by using the volume projection GRE pulse sequence with TR ç T1. The response starts from the 31st encoding input (the second row). A possible cause of this transient signal might be the 180 æ RF slice selection pulses employed in the SE nf pulse sequence. This hypothesis was confirmed by performing an experiment in the same way as in the spoiler experiment, except for using a volume projection non-fourier GRE pulse sequence (i.e. a GRE pulse sequence with no slice selection). The MR signals of this experiment, using no spoiler, from the 30th to 34th encoding inputs, are shown in Fig. 4æ21, which shows no significant transient signal. The result implies that the RF slice selection pulses should be the cause of the transient signal. This finding agrees with the

101 86 β1+β2=180 p(1) p(2) Transient Signal RF input TR RF input data acquisition t Figure 4æ22: The timing sequence of the pulse sequence diagram shows a possible cause of the transient signal from the combinations of imperfect 180 æ RF refocusing pulses from the recent (æ 2 ) and previous (æ 1 ) excitations. result from a fast SVD imaging experiment conducted by [8] using this volume pulse sequence because there was also no report of the transient signal in that experiment. Another possible explanation of the cause of the transient signal is that the transient signal is an echo signal generated by a 180 æ refocus pulse formed from a combination of imperfect 180 æ RF slice selection pulses, as shown in Fig. 4æ22. For a perfect RF slice selection pulse, as shown in Fig 4æ23A, the RF spatial profile has a flip angle of 180 æ inside the slice selection region and 0 æ outside that region. To achieve this perfect profile, the RF excitation profile needs to be a sinc pulse shape played out with an infinite time. In a practical situation, this sinc pulse is truncated (i.e. played out with a finite time), as shown in Fig. 4æ23B. The RF spatial profile in this case is therefore not perfect, and has a distribution of flip angles from 0 æ -180 æ (æ) around the leading and trailing edges. The transient signal can be generated if the combination of flip angles formed from these imperfect flip angles at the recent (æ 2 ) and previous (æ 1 ) excitations is equal to 180 æ (i.e. æ 1 + æ 2 = 180 æ ). The echo signal generated by the combination of RF pulses,

102 87 (A) (B) 180 β Figure 4æ23: The perfect (A) and imperfect (B) slice selection flip angle profiles generated from the RF sinc pulse shape without and with truncation, respectively. In the imperfect case, there is a distribution of flip angle from 0 æ, 180 æ (æ) around the leading and trailing edges. like the transient signal, is also called the stimulated echo signal [102]. Further investigation regarding the cause of the transient signal found in this study should focus on the effect of these RF slice-selection pulses, in the same manner as in the study of the stimulated echo [104]-[106]. Elimination of the Zero-Input Response From the linear ISO system theory, if the zero-input responses (i.e. the transient responses) of all the encoding inputs are known, the zero-state responses (i.e. the desired responses) can be obtained by subtracting the zero-input responses (i.e the transient responses) from the complete responses (i.e. the output responses). To demonstrate this method, an experiment was performed on two different phantoms, a doped-water (DW) and brain-shaped-gel (BG) phantom, which had short (ç 100 msec) and long (ç 700 msec) relaxation time constants, respectively. The

103 88 (A) first SVD set second SVD set t (B) TR 1 2 (n-1) 1 2 n t TR 5*T1 = RF input = zero pulse input = data acquisition Figure 4æ24: The experimental diagrams to acquire the complete (A), and the n th and (n+1) th zero-input (B), responses. TRs in this experiment were selected to be equal to the T1s of the objects. Two sets of the same 30 SVD encoding inputs were used in this experiment. The SVD encoding sets were generated by using the same criteria as in the previous experiment. The first set of SVD pulse inputs was used for generating a non-equilibrium magnetization state. The second set of SVD inputs was used for demonstrating the encoding artifact caused by the transient signal from the first SVD pulse set. The complete responses were obtained by playing out all the SVD pulse inputs (i.e. 60 encoding inputs) continuously, and acquiring data from the 31st to 60th encoding inputs, as shown in Fig. 4æ24A. Each zero-input response of the

104 (A1) 4000 (B1) (A2) 4000 (B2) Magnitude Signal Magnitude Signal k space position k space position Figure 4æ25: The complete (the first row) and zero-state (the second row) responses from the DW (A) and BG (B) phantoms are shown as the solid line. The ideal input response, obtained from the superposition theory of that pulse input, is displayed as the dashed line. second SVD input set was obtained by using the same criteria as in the previous experiment. All zero-input responses were obtained by repeating the experiment for each possible zero-input response. To let the system return to an equilibrium state, a delay time of 5 æ T 1 was maintained between each experiment, as shown in Fig. 4æ24B. For example, to obtain the n th zero-input response (Fig. 4æ24B), the 1st to (n-1) th encoding inputs were generated from the SVD pulse inputs and the n th input was generated from a null vector. The data was acquired at the n th

105 90 Figure 4æ26: The SVD images from the DW (A) and BG (B) phantoms reconstructed from the complete (1), zero-state (2), and ideal input (3) responses. encoding input. After a 5 æ T 1 delay time, the next experiment, for obtaining the (n+1) th zero-input response, was performed. Figs. 4æ25A and B show the complete and zero-state responses from the DW and BG phantoms, respectively. The responses were obtained from the 31st encoding input, which demonstrated the highest transient signal in this experiment. Figs. 4æ25A1 and A2 depict the complete and zero-state responses from the DW phantom (represented by the solid line), respectively. The calculated-superposition result (i.e. the ideal system response) was displayed as the dashed line. Figs. 4æ25B1 and B2 depict the responses from the BG phantom in the same manner as Fig. 4æ25A depicted the DW phantom. As shown in Figs. 4æ25A1 and B1, the output response is clearly different from the ideal system response. This is due to the transient response superimposed on the system response. However, when

106 91 the transient signal is eliminated from the output signal, the zero-state response shows no significant difference from the ideal system response, as shown in Figs. 4æ25A2 and B2. These results clearly demonstrate a successful method from the ISO system theory for eliminating the effect of a transient response. The SVD images reconstructed from the second SVD input set are shown in Fig. 4æ26. Figs. 4æ26A and B show the images from the DW and BG phantoms, respectively. Figs. 4æ26A1 and B1 are the images reconstructed from the complete response, which clearly show a vertical lines artifact caused by the transient response. Figs. 4æ26A2 and B2 illustrate the images reconstructed from the zerostate response which was obtained by subtracting the transient response from the complete response. The zero-state images (Figs. 4æ26A2 and B2) clearly show an elimination of the transient artifacts, which make the image closer to the images reconstructed from the ideal system responses (Figs. 4æ26A3 and B3). The results of this experiment clearly demonstrate a method, using the linear ISO system theory, for eliminating the transient artifacts. The method is, however, not suitable for practical applications because of the long acquisition time necessary for obtaining all of the zero-input responses. In a practical situation, the effect of the transient artifact might only be optimized by minimizing the transient signal or by minimizing the effect of its signal on the output signal. To minimize the transient signal, further investigation needs to focus on the effect of an imperfect RF slice selective pulse. The following sections are aimed at demonstrating a method for minimizing the effect of the transient signal on the output signal. The Transient Artifact in Static MR Imaging When using Fourier spatial encoding, the ordering of phase encodes can have a significant effect on image quality (e.g. in the fast acquisition interleaved spin-

107 92 echo method). In a dynamic MRI, the ordering of phase encodes are also employed to reduce motion artifacts [93]. The ordering of non-fourier encoding vectors (i.e. the ordering of RF excitations) also has profound effects on image quality, especially when shorter TRs are used. This ordering is a necessary method for understanding how the acquisition data can be obtained with a minimum of artifacts. The objective of this investigation is to find the RF excitation orderings that minimize undesirable transient effects on non-fourier encoded images. A series of experiments was performed, with the DW phantom, employing different reorderings of the non-fourier encodes. This experiment attempted to characterize the encoding artifact, due to the transient signal, in a static MR image and the effects of RF excitation orderings on such an artifact. Figure 4æ27 shows the static MR image of the DW phantom acquired using the Hadamard and SVD encoding methods at different TRs. Figs. 4æ27A, B, and C are Hadamard encoding images acquired at TR = 100, 300, and 1000 msec, respectively. Figs. 4æ27D, E, and F are SVD encoding images acquired at the same TRs. Both encoding methods clearly show transient signal artifacts when a short TR is used, as observed in Figs. 4æ27A and D. Transient artifacts can be minimized if the MR image is acquired after the magnetization reaches the steady state [14]. This is achieved by applying preexcitation RF pulses before the actual encoding RF excitation. However, this method requires an overhead of pre-excitation pulses which may not be suitable for fast MR imaging. Another way of minimizing the artifacts involves a reordering of the RF excitation inputs. That is a reordering of the encoding pulses. The encoding vectors can be reordered from the most to least significant encoding data (HTL), or from the least to most significant (LTH). In the HTL encoding order, the RF impulse

108 93 Figure 4æ27: Static MR images of the doped-water phantom using the Hadamard (A-C) and SVD encoding (D-F) methods acquired at TR = 100, 300, and 1000 msec, respectively. Note transient artifacts are visible when TR = 100 msec is used (A and D). and Hadamard encoding vectors are reordered such that the k-space data are acquired from the center to the periphery (i.e. from the lowest to highest spatial frequency components). The opposite acquisition order, from the periphery to the center of the k-space data, is employed in the LTH encoding order in both encoding methods. In the SVD method, the encoding vectors are reordered according to their associated singular values. For the HTL encoding order, the encoding vectors whose singular values are from the highest to lowest are employed (i.e. from the highest to lowest weighted eigenimages), and the opposite encoding order is used for the LTH.

109 94 Figure 4æ28: Hadamard (A and B) and SVD (C and D) encoding images acquired by using the HTL and LTH encoding orders, respectively. Note reduced transient artifacts for the LTH encoding order (B and D) when compared to the HTL order (A and C). Images acquired from two different orderings of Hadamard and SVD encodes are shown in Fig. 4æ28. Figs. 4æ28A and B show the Hadamard encoding images that were acquired from the HTL and LTH encoding orders, respectively. Figs. 4æ28C and D illustrate SVD images acquired from the same encoding order. The LTH encoding order, when used with both the Hadamard and SVD encoding vectors (Figs. 4æ28B and D), results in less transient artifacts than the HTL encoding order (Figs. 4æ28A and C). The HTL encoding order implies a method of encoding the MR system that starts with the encodes containing a majority of low spatial frequency components, and continues to those containing mostly high spatial fre-

110 95 quency components. The transient signal then imposes a non-ideal (artifact) signal onto the low spatial frequency informations of the image. Hence, the transient artifacts cause a negative visual effect in the image result. The LTH encoding order, however, is the reverse of HTL encode ordering. For LTH arranged encoding, the transient signal is only modulated around the least significant MR signal; therefore its effect is reduced. In other words, the transient signal in this case only imposes itself on the high frequency informations of the images which are not significant. The images, therefore, show less transient artifacts than in the HTL encoding order scheme. The experimental results in this section clearly demonstrate that the ordering of the RF inputs in the non-fourier encoding method can reduce the effect of the transient signal on the output response. The Transient Artifact in Dynamic MR Imaging An experiment was performed, using the DW phantom, employing different non- Fourier encoding ordering functions. The unit step analysis was employed to analyze the effects of transient artifacts in this experiment. The experiment was designed to characterize the transient artifacts in dynamic MR imaging, and to demonstrate that reordering the RF encoding inputs can reduce that artifacts. The same hypothesis, as with the transient artifacts in the static MR imaging experiment, was used to minimize the transient artifacts. The SVD vectors were computed from baseline k-space data, and 25 SVD vectors, with the highest associated singular values, which encode along the horizontal direction, were used to define RF pulses for encoding. Three different methods of ordering the RF excitations were examined. The first ordering of RF pulse excitations used SVD vectors with the highest to lowest associated singular values (i.e.

111 96 SVD encoding from the lowest to highest eigenimages). Each of the 25 pulses was repeated for 20 consecutive excitations, and the full set of excitations was repeated twice. The second pattern of excitations used SVD vectors ordering from the lowest to highest singular values (i.e. the opposite of the first pattern). The third pattern of excitations used a combination of the first and second orderings. In the third pattern, the first ordering was employed for one full set of excitations, and then the second ordering was employed for the second set. Since each RF pulse was used to excite repeatedly (20 times), we can study the transient response of the system (i.e. a unit step input analysis) under different excitation conditions ,,æ 2 2 1,,æ ,,æ 2 Figure 4æ29: The average magnitudes of response to each of the three excitation ordering methods, shown with the applicable encoding numbers. Arrows indicate the most significant transient effects on each pattern. The results shown in Fig. 4æ29 display the average magnitude of the responses from each of the three methods of excitation ordering during the time interval whose midpoint is the end of the first set of excitations. These results are the most demonstrative of the effects of excitation ordering on the transient response since the maximum change occurs in excited spatial frequency components of the FOV.

112 97 Figure 4æ30: Three sets of four dynamic MR images of a doped-water phantom. The images illustrate the use of reordering patterns one (A-D), two (E-H), and three (I-L). Note the presence of low frequency transient artifacts in patterns one and three (A-D and I-L). High frequency transient artifacts are clearly visible in pattern two (E-H). For example, in Fig. 4æ29 the first two sections of the first row show the responses when using the 24th and 25th RF pulses, and the following two sections show the results when using the 1st and 2nd RF pulses (the first reordering pattern). Row 2 shows the result of excitations by the 2nd, 1st, 25th, and 24th RF pulses (the second reordering pattern). Row 3 shows the result of excitation by the 2nd, 1st, 1st, and 2nd RF pulses (the third reordering pattern). The arrows indicate the major transient effects on each pattern. These results show that the image generated by the first and third ordering

113 98 methods should have low frequency noise artifacts because of the effect of the transient signal from the 1st encoding step on the 2nd encoding step. The ordering method of the second pattern, on the other hand, should show high frequency noise artifacts due to the large transient signal from the 1st encoding step strongly affecting the 25th encoding step. The hypothesis of this transient analysis was confirmed in a dynamic MR imaging experiment, shown in Fig. 4æ30, where each of the three ordering schemes was used to encode a doped-water phantom image. Each row shows four consecutive dynamic images obtained using the respective ordering schemes. The transient artifacts in the SVD encoded MRI method, demonstrated in this study, are caused by the object s magnetization not fully returning to the equilibrium state before the excitation of the next RF encoding pulse; this happens in the fast imaging case. The transient system analysis shows clearly that these artifacts are caused by the initial state of the MRI system inherited from the previous RF encoding inputs. The experiments in this study have shown that, by reordering the SVD encoding pulses according to their associated singular values and investigating their transient responses, these encoding artifacts can be minimized. 4.4 Summary and Conclusions The linear IO and ISO system models were used to analyze the non-fourier encoded MRI method. The linear MR system was defined as a black-box measurement system whose input was RF excitation and whose output was complexvalue data which could be mapped onto the k-space data. The system also represents the non-equilibrium magnetization states generated by the RF inputs. The impulse and transient responses were described theoretically and demonstrated experimentally in this Chapter.

114 Impulse Response Analysis The system impulse responses, which directly map onto the k-space data, were obtained by using a set of RF unit box pulses (i.e. RF impulses). Once these system impulse responses were obtained, the response to any orthonormal basis set could be theoretically computed using the superposition property. This analysis, however, assumes that the system output depended only on the input (the IO system model), and that it was operated in the linear range. In the MR system, this assumption is valid only when TR is much longer than T1 (i.e. TR é 5æT1) and a small flip angle of the inputs is used. The impulse response analysis focused on identifying experimental parameters and limitations of non-fourier encoding, particularly those that depended on, and were revealed by, the superposition assumption. The experimental results indicated that the superposition property is valid as long as the flip angle input is in the linear range (i.e. less than 30 æ ). Three different RF excitation encoding methods - RF impulse, Hadamard, and SVD - were used to characterize the output behavior of the system at different flip angles. The RF impulse encoding method did not show any encoding artifacts in the experimental results, because in theory it is physically equivalent to the Fourier (phase) encoding method. This is most likely the case because this method does not require the assumption of the superposition property in its theory or practical implementation. However, the Hadamard and SVD encoding methods, whose orthonormal basis sets are linear combinations of hard-pulse trains, are subject to encoding artifacts when the flip angle input is in the non-linear range. This is attributed to the violation of the principle of superposition. The SVD flip angle, whose profile depends on the object inside the FOV, can be characterized by its distribution and CDF. The experimental results indicate

115 100 that the mean SVD flip angle is normally higher than the flip angles of both the RF impulse and Hadamard methods at the same RF pulse width. The CDF can be used to find the percentage of the SVD flip angle population that is inside the linear range. The image quality of the SVD encoding method is limited by the encoding artifact that results from the violation of the superposition property. The SNR and ANR results of the experiments with phantoms suggest that optimal image quality can be achieved by using a mean flip angle of approximately 15 æ - 20 æ Transient Response Analysis For a linear system, the complete response is a linear combination of the system (zero-state) and transient (zero-input) responses. When TR is much longer than T1, the transient response is negligible. For fast MR imaging a typical TR is, however, not much longer than T1. There is, therefore, a need to analyze the transient response. In this study, the transient response is analyzed by using the system controllability theory and the unit-step input method. From the zero-input response analysis, the experimental results clearly demonstrate a transient response caused by non-equilibrium magnetization when TR is not much longer than T1. Moreover, the experiment showed that the spoiler has no significant effect on reducing this transient signal. This result implies that the transient signal is not due to the remaining transverse magnetization from the previous encoding input per se. A possible cause of this transient signal, therefore, might be the imperfect 180 æ RF slice selection pulses employed in the SE nf pulse sequence. This hypothesis was confirmed by the experiment using the volume projection non-fourier pulse sequence. Further investigation regarding the transient signal found in this study should focus on these imperfect RF slice selection

116 101 pulses in the same manner as in the study of the stimulated echo. A method for eliminating the transient response, using the controllability theory, was demonstrated experimentally. The experiment result clearly showed that this method can be used successfully to eliminate the transient responses from the complete responses. The method is, however, not suitable for fast MRI applications because of the long acquisition time necessary for obtaining all of the zero-input responses. In a practical situation, the effect of the transient artifact might only be optimized by minimizing its effect on the output signal. As the static and dynamic MR images show, the transient artifact can be reduced by reordering the encoding functions to minimize the transient effects on the low frequency encode. In the case of static MR imaging, the encode ordering from lowest to highest significance results in less transient artifacts than the encode ordering from highest to lowest. This is because the transient signal modulates the less significant MR signal in the first encoding order, but affects the most significant MR signal in the second order. In the case of dynamic MR imaging, the unit-step input was shown to be useful in analyzing the reordering scheme. By reordering the encoding function according to the result suggested by the unit-step analysis, we have shown a method for reducing the transient artifacts in dynamic MR imaging when using the non- Fourier encoding method.

117 102 Chapter 5 Encoding Efficiency of the Non-Fourier Encoded MRI Methods The encoding efficiency of non-fourier encoded MRI methods is the major topic of this chapter. First, the method of analyzing this efficiency in the eigenimages (the encoding factor) and in the encoding vectors (the principal angle factor) is introduced, followed by the demonstrations and discussions of the experimental and simulated results regarding the encoding efficiency. 5.1 Background Encoding Factor As described in Chapter 2, non-fourier encoding methods can be separated into two classes: non-adaptive and adaptive techniques. Non-adaptive techniques use a predefined basis set which is independent of the contents of the FOV. Adaptive basis techniques, on the other hand, use a basis set calculated from information of the contents in the FOV. The latter technique, which has the possibility of generating a near-optimal basis set, that is, the possibility of encoding more information from the FOV from the same number of samplings than the former technique. The adaptive technique thus has the possibility of providing greater encoding efficiency than the non-adaptive basis technique, depending upon the rates of change and the types of changes in the contents of the FOV.

118 103 To analyze the encoding efficiency of different encoding methods, the encoding factor (EF), which is a standard deviation measurement of the difference between the complete encoding image and each eigenimage, is calculated. The EF is defined as EFèrè= stdëæi rèx; yèë æ 100 ; r ç M (5.1) stdëi M èx; yèë where, std is the standard deviation, M is a dimension (N) or rank (R) of the image matrix depending on the encoding technique, and æi r èx; yè is the difference of signal intensity of the complete image, I M èx; yè, to the sum of r eigenimages: æi r èx; yè =æ M i=1 E ièx; yè, æ r i=1 E rèx; yè (5.2) where E i is the i 0 th eigenimage, æ M i=1 E ièx; yè = I M èx; yè, and x; y are the image coordinates. From linear algebra, only the number of eigenimages equal to the rank of matrix (R) are necessary for exactly encoding the image [108]. In the adaptive method, the rank of the matrix is determined from a rank revealing orthogonal decomposition such as the SVD. Therefore, only R encoding vectors, where R ç N, are necessary to obtain the complete encoding image in the method. In the nonadaptive method, however, there is no information about the rank of the image. N encoding vectors are, then, used to obtain the complete encoding image in the non-adaptive method. The EF serves as a measurement of the convergence of the sum eigenimages to the complete image. For example, the sum eigenimages from the SVD method can be represented in vector outer product form, described in Section 2.2.3, as æ r i=1 E i = rx i=1 ç i u i v t i (5.3)

119 104 If the singular values of the SVD are placed in monotonic decreasing order, the sum eigenimages will also monotonicly convert in decreasing fashion. The measurement, therefore, depends on the order of the encoding vectors employed to obtain the image. For comparisons in this study, the order of encoding vectors starts from the most significant, and continues to the least significant data (i.e. from the center to the periphery of k-space in the Fourier method, or from the high to low associated singular values (ç) in the SVD method). The measurement can be employed to quantitatively analyze the efficiency of different encoding methods. The encoding efficiency of different methods is compared by scaling the EF value obtained from the different encoding methods referring to the same image, to the same maximum value (i.e. 100è). This comparison is valid only when the encoding methods under comparison yield the same results from the complete sum of eigenimages (i.e. the methods encode the same image contents). Two possible methods exist in which to use the EF measurement. One is to compare the EF value obtained from different encoding methods using the same number of encodes. In this fashion, EF is a measurement of the encoding efficiency, with a lower value of EF indicating better encoding efficiency. To illustrate, Fig. 5æ1 shows the EF measurement of two different encoding methods ( X and Y) applied to the same image. The EF measurement of method X is shown by the dashed line and by the solid line in method Y. As shown in Fig. 5æ1, encoding method X has a lower EF value than method Y for the same number of encodes. For instance, as shown in Fig. 5æ1A, the EF value of method X at the 60th encode is about 13è while it is about 52è in method Y. This suggests that the sum of the first 60 eigenimages of method X is only about 13è different than the complete result as compared to about 52è different than the complete result in method Y. Method

120 105 (A) (B) Figure 5æ1: Two comparisons of the EF measurements from encoding methods X (the dashed line) and Y (the solid line) used in the same image. The first method (A) compares the EF value of these two encoding methods at the same encoding number (i.e. 60). The second method (B) compares the encoding methods at the same EF value (i.e. 27%). X is thus judged to be more efficient than method Y based upon this measurement. A second method is to compare encoding methods at the same EF value. This is a comparison of how many encodes are required by different encoding methods to acquire similar information in the FOV (i.e. the same EF value). For example, in Fig. 5æ1B, method X requires only about 29 encodes to obtain similar information as compared to about 128 encodes required in method Y. Method X is thus about four times more efficient than method Y. For the quantitative analysis of the adaptive encoding method in this chapter, the efficiency from this comparison will be referred to as the encoding efficiency factor (EEF), which is the ratio of the

121 106 encodes required by the Fourier encoding method to the adaptive (SVD) encoding methods to obtain a similar image quality (measured by having identical EF values) Principal Angle Factor The major advantage of adaptive non-fourier encoding is the possibility of optimizing the number of encodes using current FOV information during dynamic MRI. For example, the SVD encoding method provides an orthonormal basis set whose truncated form is the best least square approximation of the contents in the FOV, as described in Section SVD encoding therefore has the possibility of providing greater encoding efficiency than a non-adaptive technique, such as the Fourier encoding method, whose truncation approximation is the band limit approximation of the spatial frequency components in the FOV. Unlike in the non-adaptive technique, whose encoding efficiency is not altered by a change in the FOV, the encoding efficiency in the adaptive technique is affected by changes occurring in the FOV causing its contents to differ from the baseline image used to determine the encodes. This is due to a specific encoding basis set employed for the contents in the FOV. To use the adaptive encoding technique in dynamic MRI applications, the variation in the efficiency of the use of a particular basis set due to the changes in the FOV needs to be considered. In the adaptive encoding method, like the SVD, basis sets for encoding are derived from vertical (U) or horizontal (V ) encoding matrices obtained by calculating the image matrix decomposition of the baseline data. These basis sets after truncation, based upon user-desired criterion, as described in Section 2.2.3, are then employed for acquisition and reconstruction of the contents in the FOV. The basis set of the estimate is frequently calculated from using information from

122 107 newly acquired data, and then employed in the next acquisition and reconstruction. For example, let fu i g and fv i g be vertical and horizontal basis sets computed from baseline data, or an image estimate (S), and let S 0 denote an ideal image acquired from the FOV. According to the acquisition and reconstruction method in [108], a subset of fu i g or fv i g basis sets is chosen, based upon a user-truncation criterion, and then employed for vertical, U t S 0, or horizontal, S 0 V, acquisitions of S 0. Image reconstruction is, then, accomplished by using UèU t S 0 è and ès 0 V èv t. These acquisition and reconstruction formulae for the vertical and horizontal directions can be rearranged as P u S 0 and èp v S 0t è t, where P u = UU t and P v = VV t, respectively. From a linear algebra point of view, P u and P v are the orthogonal projections onto the ranges of S and S t [34], respectively. The acquisition and reconstruction method in [108] can be stated, in theory, as the projections of the vector subspaces that encode the ideal image onto the vector subspaces spanned by the reduced encoding basis sets computed from the image estimate [108]. Images acquired using this method are then the estimate of the idea image, S 0, which can be presented as [108] S 0 = S 0 UV + E0 (5.4) where S 0 UV is the component of S 0 which has projections onto the vector subspaces spanned by fu i g and fv i g, and E 0 represents information that is not acquired. E 0 contains components due to truncation, as well as projection errors. The truncation error depends on user-defined criterion while the projection error is due to a non-ideal projection of the FOV onto the subspaces spanned by the encoding basis sets of the estimate. The maximum efficiency of this encoding method will be accomplished when the matrix norm of E 0, ke 0 k, is minimized. In other words,

123 108 the quality and artifact in images acquired using the adaptive encoding depends on the projections of one vector subspace onto another. These projections, or how similar or close the basis sets are that encode S and S 0, are reflected in the principal angles [108] between the vector subspaces of S and S 0. The principal angle is defined [34] as: Let V and W be two subspaces of C m and assume that p = dimèv è ç dimèw è= q ç1. The principal angles ç 1 ;:::;ç q 2 ë0; ç ë, between V and W, are recursively 2 defined by cos ç k = max v2v subject to the constraints max w2w v T w = v T k w k; kvk = kwk =1 (5.5) v? v i ; w? w i ; i =1;:::;k,1 (5.6) If the subspace V and W have the same dimension (i.e. p = q), the distance between these two subspaces is defined to be distèv; Wè =jsinèç p èj (5.7) where ç p is the largest principal angle. The definition states that the k th principal angle represents the minimum angle existing between a vector in one subspace, and a second vector in an other subspace, with the restriction that neither of these two vectors contributed to the formation of the first (k-1) st principal angles. The maximum of the sines of the principal angles is normally defined and used by mathematicians as the distance between two vector subspaces. The definition is, however, less useful in our case, since this maximum value is typically unity. More appropriate for our analysis of the encoding efficiency is the distribution of

124 109 the sines of the principal angle between two vector subspaces. sin(principal Angle(k)) (A) Z(FOV,white noise) Y(FOV,some changes in FOV) X(FOV,no change in FOV) k (B) Y =0.0028*k Z =0.0070*k Z Y X k Figure 5æ2: Three distributions of the sines of the principal angles represent three different scenarios (A). The first case (X) illustrates the ideal encoding vector (i.e. no change in the FOV). The second case (Y) shows a typical case when there are some changes in the FOV. The final case (Z) represents the worst case scenario (i.e. the basis set was calculated from the white noise contents in the FOV). The first 60 principal angles (the box at the bottom left of (A)) are used for the estimation of the distribution of principal angles using first order regression approximation as shown and plotted as the dotted line in (B). In MRI, ideal encoding is defined as the case in which all principal angles are zero. Ideal encoding means that the basis set obtained from the estimate image, S, can be used perfectly to encode the ideal image, S 0. In practice, however, noise in the FOV prevents ideal encoding. When encoding is non-ideal, the distribution of the sines of principal angles is typically sigmoidal, with a range from 0 to 1 [108]. For example, three different distributions of the sines of principal angles

125 110 are shown in Fig. 5æ2A, as the lines X, Y, and Z, respectively. The distribution of X refers to the case when no change occurs in the FOV (i.e. the ideal encoding) while the distribution of Z was computed from an ad hoc chosen worst case scenario (i.e. the basis set was calculated from the white noise contents in the FOV). The distribution of Y demonstrates a typical case when there are some changes of contents in the FOV. In this study, the method described in [97] is employed to calculate the principal angle. For quantitative analysis, the slope of a linear approximation (i.e. the first order regression approximation) to the portion of the distribution of sines of the first 60 principal angles is used for estimating the distribution of the principal angles. Figure 5æ2B shows examples of this approximation compared to the distribution of Y and Z. From Fig. 5æ2B, the first order regression approximation of Y and Z (Y and Z ) were computed from the first 60 principal angles, as shown in the box at the bottom left of Fig. 5æ2A, and shown as the dashed lines in Fig. 5æ2B. To describe quantitatively the estimate of the distribution of sines of the principal angle, the first order term of the approximation is further scaled by the value calculated from the worst case non-ideal encoding basis. We choose a basis set computed from an image containing white noise for this non-ideal encoding basis (i.e. the distribution of Z). We call this measurement, after the scaling, the principal angle factor (PAF). For example, the PAF of Y s distribution can be defined as PAFèèè = SL Y SL worst æ 100 (5.8) where SL Y is the slope of a linear approximation estimated from Y s distribution (i.e. the first term of Y in Fig. 5æ2B), SL worst is the same approximation of the white noise contents in the FOV (i.e. the first term of the Z in Fig. 5æ2B). From this definition, the PAF value varies from 0è in the perfect encoding (i.e. the ideal

126 111 case or the most efficient encoding), as in X s distribution, to 100è in the worst case non-ideal encoding basis, as in Z s distribution. In this example, the PAF of Y s distribution, which demonstrates some changes in the FOV, is about 40è. 5.2 Quantitative Analysis of Encoding Efficiency The analysis of the MR images in this section is aimed at characterizing the efficiency of the non-fourier encoding method. First, the encoding efficiency of different encoding methods is quantitatively compared using the EF analysis. Such comparison demonstrates the highest efficiency of the adaptive encoding technique when using encoding basis sets that ideally encode. Next, the encoding efficiency of the adaptive encoding methods at various SNR is investigated. This investigation is aimed at analyzing the effect of noise in the FOV on the efficiency of the encoding methods because such noise prevents the ideal encoding condition. Finally, the effects of geometric changes in the FOV contents are investigated with regarding to encoding efficiency. These effects represent the worst case scenario of changes in the FOV during the dynamic process. The experiments were performed on a 1.5 GE SIGNA MR system using the SE nf pulse sequence described in Chapter 3. The simulation were performed on a SUN workstation using MATLAB software tools Encoding Efficiency in Cases of Ideal Encoding A Comparison of Different Encoding Methods The EF analysis is performed on the brain-shaped-gel phantom acquired at long TR (i.e. 3 seconds), using different encoding basis sets. The objective of the analysis is to compare quantitatively the encoding efficiency of the non-adaptive and

127 112 adaptive techniques in the non-fourier encoding method to that of the Fourier encoding method. In this analysis, the basis sets in the adaptive encoding methods differ from those of ideal encoding in that the only change in the FOV is due to noise which had been reduced by acquiring the image at long TR. The encoding efficiency of the adaptive encoding technique in this analysis should, therefore, represent the best case scenario, the maximum encoding efficiency as compared to the non-adaptive technique. Three different encoding methods - RF impulse, Hadamard, and SVD - are used to encode the phantom. The RF impulse method is used to represent the conventional Fourier encoding method because these two methods can theoretically obtain the same encoding results as described in Chapter 2. Hadamard and SVD encoding are employed to represent the non-adaptive and adaptive techniques in the non-fourier encoding method, respectively. The sum of five different eigenimages of these three encoding methods are displayed in Fig. 5æ3. The image results of the RF impulse, the Hadamard, and the SVD encoding methods are shown in the first, second, and third columns, respectively. From top to bottom, each column illustrates five image results using 4, 8, 16, 32, and 256 encoding numbers, respectively. The magnitudes of the projection of these images along the spatial selective encoding direction (the horizontal direction of images) are also shown in Figs. 5æ4 and 5æ5. The projection obtained from the RF impulse, the Hadamard, and the SVD, are shown in Figs. 5æ4 and 5æ5, as the solid line, in the first, second, and third columns, respectively. The projection of the completely encoded image (i.e. the image acquired by using all encoding vectors) is also shown in Figs. 5æ4 and 5æ5 as the dotted line. Figs. 5æ4A and B show the projection magnitudes of these three encoding methods from 4 and 8 encodes, respectively, while Figs. 5æ5A and B display the projection magnitudes

128 113 Figure 5æ3: The comparison of the eigenimages from the RF impulse (the first column (A)), Hadamard (the second column (B)), and SVD (the third column (C)) encoding inputs. Each column from top to bottom (1-5) illustrates the 4th, 8th, 16th, 32nd, and 256th eigenimages, respectively. A brain-shaped gel phantom is used in this experiment.

129 114 (A1) Fourier Encoding, n=4 1 1 (A2) Hadamard Encoding 1 (A3) SVD Encoding scaled projection magnitude image space (B1) Fourier Encoding, n=8 1 1 (B2) Hadamard Encoding 1 (B3) SVD Encoding Figure 5æ4: The magnitude of the projection of the RF impulse, Hadamard, and SVD images along the spatial selective encoding direction (the horizontal direction of images in Fig. 5æ3) are illustrated in the first, second, and third columns, respectively. The first and second rows show the projections from 4 and 8 encodes, respectively. from 16 and 32 encodes, respectively. In the image results of these encoding methods (Fig. 5æ3), the visual quality of the images from the same number of encodes is significantly different. For the same encoding numbers (the same row in Fig. 5æ3), the SVD encoding method provides image results qualitatively closer to the complete image (the last row) than the RF impulse and Hadamard methods. The projection data from these encoding methods (Figs. 5æ4 and 5æ5) also show the same results as the image data. From the projection data, the RF impulse method provides a sinusoidal projection profile while a step-like projection profile is the character of the Hadamard

130 115 (A1) Fourier Encoding, n= (A2) Hadamard Encoding 1 (A3) SVD Encoding scaled projection magnitude image space (B1) Fourier Encoding, n= (B2) Hadamard Encoding 1 (B3) SVD Encoding Figure 5æ5: The magnitude of the projection of the RF impulse, Hadamard, and SVD images along the spatial selective encoding direction (the horizontal direction of images in Fig. 5æ3) are illustrated in the first, second, and third columns, respectively. The first and second rows show the projections from 16 and 32 encodes, respectively. method. The SVD method, however, has a projection profile which is similar to the object s projection profile. The RF impulse method shows the sinusoidal profile because a sinusoidal basis set is used in the method, as described in Section The step-like profile in the Hadamard method is due to the binary basis set employed. The SVD method, in contrast, has a profile similar to the object s profile because the SVD method employs a basis set which is calculated from the contents of the object. As shown in the projection data (Figs. 5æ4 and 5æ5) and image results (Fig. 5æ3), the SVD method can encode more information from the FOV than the RF impulse and Hadamard

131 Encoding Factor number of encodes Figure 5æ6: The EF analysis of the RF impulse (the solid line), Hadamard (the dotted line), and SVD (the dashed-dotted line) from the brain-gel phantom (Fig. 5æ3). methods at the same encoding number. This is due to a specific encoding basis set employed for the contents in the FOV by the SVD encoding method as compared to the predefined basis sets employed by the RF impulse and Hadamard methods. From the image results and projection data, all encoding methods provide the same contents of the FOV when all encoding vectors are used (i.e. the last row of Fig. 5æ3 and the dotted lines in Figs. 5æ4 and 5æ5). It is, therefore, valid to use the EF measurement to analyze quantitatively the encoding efficiency of the results of these encoding methods. The EF analysis of these three encoding methods is depicted in Fig. 5æ6. The EF measurements of the RF impulse, Hadamard, and SVD methods are plotted

132 117 as the solid, dotted, and dashed-dotted lines, respectively. In Fig. 5æ6, the EF measurements from the RF impulse and Hadamard encoding methods are quite similar. This implies that the encoding efficiencies of these two methods are compatible. The EF measurement of the SVD encoding method, however, is substantially lower than that of the other two methods. The SVD encoding method is thus judged to be more efficient than the RF impulse and Hadamard encoding methods according to this measurement. For instance, if 32 encodes are used to acquire the phantom in these three encoding methods (i.e. the fourth row of Fig. 5æ3), the image result of the SVD encoding method is only about 20è different than the result from 256 encoding numbers (i.e. the last row of Fig. 5æ3) while it is about 80è different if it is encoded by the RF impulse and Hadamard encoding methods. For the second method of EF comparison on this data, Table 5.1 shows four comparisons of different encoding numbers used by the SVD and RF impulse methods to obtain similar contents in the FOV. The table illustrates the value of the truncation criterion (C 1 ), the number of encodes used by the SVD method, the EF value, and the EEF value in the first, second, third, and fourth columns, respectively. The four rows of table 1 from top to bottom represent the quantitative analysis of the image results acquired by the SVD encoding method using 4, 8, 16, and 32 encodes, respectively. This comparison represents a quantitative analysis of the SVD image results shown in the first four rows (C1-C4) of Fig. 5æ3. For example, from the fourth row of table 5.1, the user desired to employ SVD encoding vectors whose associated singular values are above 2:49è of the maximum singular value (i.e. C 1 = 2:49è) by using only the encoding vectors whose associated singular values meet with this truncation criterion (i.e. 32 encodes). The vectors are then employed for the acquisition and reconstruction process. From the EF analysis, the image result from this process (Fig. 5æ3C4) will be, then,

133 118 C1(%) #encodes(svd) EF (%) EEF Table 5.1: The EEF analysis of the SVD method at 4, 8, 16, and 32 encodes on the brain-gel phantom (Fig.5æ3). about 23è (i.e. EF = 23è), different than the exact result from 256 encoding numbers (Fig. 5æ3C5). Furthermore, from the EEF analysis, if the RF impulse or Fourier encoding methods are used to obtain this image result, about 138 encoding numbers are required by such methods as compared to only about 32 encoding numbers required by the SVD method (i.e. EEF = 4:33). This quantitative analysis clearly demonstrates the possibility of the greater efficiency of the SVD encoding method over the RF impulse or Hadamard encoding methods. From the EF analysis in this section, the adaptive basis encoding technique (the SVD method) at an ideal encoding condition demonstrates the possibility of a greater encoding efficiency over the conventional Fourier encoding method (the RF impulse method) and the non-adaptive basis encoding technique (the Hadamard method). A Study of Encoding Efficiency in the SVD Method Due to Different Image Contents To investigate the variation of the encoding efficiency in an adaptive encoding technique, like the SVD, due to the different contents in the FOV, simulations of acquisition and reconstruction processes are performed on different MR images using the SVD and Fourier encoding methods. This simulation is aimed at demonstrating the variation of the encoding efficiencies in the SVD method at an ideal

134 119 basis encoding condition. Figure 5æ7: Four typical MR images obtained from the conventional pulse sequence used in the simulation. Four typical MR images are used in this simulation, as shown in Fig. 5æ7. The axial (ABR) (Fig. 5æ7A) and saggital (SBR) (Fig. 5æ7B) human brain images are obtained by using a conventional spin echo pulse sequence while the axial (ABO) (Fig. 5æ7C) and coronal (CBO) (Fig. 5æ7D) human body images are obtained by using a spoiled grass pulse sequence [37]. MATLAB software tools are used to simulate the acquisition and reconstruction process [108]. The simulation process starts by first transforming the images into the k-space

135 EF(%) number of encodes Figure 5æ8: The average of the EF measurement from the four typical MR images (Fig. 5æ7) acquired and reconstructed in simulation using the Fourier (solid line) and SVD (dashed-dotted line) encoding methods. domain using a 2D FFT operation. The identity matrix is used to represent the encoding basis sets for the Fourier encoding method. The basis sets from the SVD encoding method, on the other hand, are obtained by performing SVD calculation on the k-space data. Next, subsets of these basis sets are employed to simulate the acquisition and reconstruction process. In this simulation, the SVD basis set represents ideal encoding (i.e. there is no change in the FOV). Finally, the result from the process is transformed into the spatial domain (i.e. the image domain) using 2D inverse FFT. The EF and EEF measurements were then employed for a quantitative comparison of the encoding efficiency of these simulation results. Fig. 5æ8 shows the average of the EF measurement from the four typical MR

136 EEF number of encodes Figure 5æ9: The average of the EEF measurement of the SVD method from the four typical MR images (Fig. 5æ7). images (Fig. 5æ7) acquired and reconstructed in simulation using the Fourier (solid line) and SVD (dashed-dotted line) encoding methods. As shown in the figure, the EF measurement of the SVD encoding method implies more rapid convergence of the sum of eigenimages to the final result faster than the Fourier method. This implies that the SVD encoding method is more efficient than the Fourier method. From Fig. 5æ8, the EF measurement of the SVD method (B) can be separated into two parts. The first part, from the 1st to 75th encoding number, shows a rapid decrease of EF values, while the other part, from the 76th to 256th encoding number, shows no significant change in EF values. In this simulation, this is due to the singular vector ordering with respect to decreasing singular value. That is the method encodes a band of low to high spatial frequency information. Therefore,

137 122 large (small) regions of constant intensity in an FOV are encoded by singular vectors with corresponding large (small) singular values. The range of rapid decrease (large slope) of EF values in the SVD method case (i.e. from 1 to 75 encodes) relates to the encoding of information of the image s contents in the FOV while the range of no change in EF values (smaller slope) (i.e. from 76 to 256 encodes) relates to the encoding of the image noise (high frequency contents). From Fig. 5æ8 it is apparent that the SVD encoding method approximation, which truncates the number of eigenimages associated with small singular values, primarily reduces image noise rather than affects the encoded large objects in the FOV. The truncation approximation of the Fourier encoding method is the band limit approximation of the spatial frequency components in the FOV and in use can eliminate mid and high spatial frequency information including FOV object contents. The SVD method, whose truncation is the best least square approximation of the contents in the FOV, therefore, provides a better truncating approximation than the Fourier method when they are compared at the same number of encodes. Fig. 5æ9 shows the average of the EEF measurement calculated from the EF values in Fig. 5æ8. From the figure, the average EEF values vary from 10 at the first encode to 1 at the last encode. This suggests that the more encoding numbers employed by the SVD method, the less efficient the method will be, compared to the Fourier encoding method. The EEF measurement also shows the same character as the EF measurement in that the first encoding part (i.e. from 1 to 75) shows a rapid change of EEF values while the second part (i.e. from 76 to 256) does not show much change at all. This measurement agrees with the EF measurement that only a subset of the SVD basis set is necessary to obtain most of the contents of the image in the FOV, and any other additional encoding vectors employed will primarily encode noise. That is why the EEF values initially change rapidly and

138 123 do not vary appreciably as the second half of the encodes are included. From the EF and EEF measurements in the simulation results in this section, an adaptive encoding method like the SVD shows more efficient encoding than the conventional Fourier encoding method. This is due to the encoding basis set employed by the SVD method specifically for the contents in the FOV. Furthermore, the SVD method can provide a better truncating approximation than the Fourier method when the methods are compared at the same encoding numbers. This is due to a least square truncating approximation in the SVD method as compared to a band limit spatial frequency approximation in the Fourier method. The measurements in this simulation, however, represent the best case scenario (i.e. the greatest efficiency) of the SVD method, which is due to the ideal encoding used in the simulations of this section Encoding Efficiency of the SVD Method at a Non-Ideal Encoding Condition The EF analysis is performed on the brain-shaped-gel phantom acquired at different TRs (i.e. different SNRs) using the SVD encoding method. The objective of the analysis is to investigate quantitatively the effect of a non-ideal basis encoding condition due to noise in the FOV. This analysis represents the encoding efficiency of the SVD encoding method in practical situations when the only changes in the FOV are due to the noise (i.e. there is no change due to dynamic processing in the FOV). The EF analysis is performed on the simulated and experimental data. The simulated data represent the encoding result from an ideal encoding condition while the experimental data represent the results from a non-ideal encoding condition caused by the noise. The analysis is performed on the SVD images acquired

139 124 at 100, 500, and 2000 msec TRs. The baseline images are obtained by using the RF impulse basis set. The SVD images are obtained using basis sets from the SVD computed from the baseline data. For each TR data set, the simulated results are obtained by using the SVD basis set to simulate the acquisition and reconstruction process, where acquisition is simulated from an FOV given by the baseline data set. The same method described in the previous section is used in this study. The experimental data is obtained by using the same SVD basis sets to acquire and reconstruct the experimental images. The simulation results, therefore, represent ideal encoding while the experimental data represent the case of non-ideal encoding due to the presence of the noise (A) SNR= (B) SNR= (C) SNR=36.77 EF (%) number of encodes Figure 5æ10: The EF measurement of the SVD results at low (A), medium (B), and high (C) SNRs. The measurements show the SVD results at the ideal (the dotted line) and non-ideal (the solid line) encoding conditions as well as the measurement of the Fourier result (the dashed-dotted line).

140 125 Figs. 5æ10A-C show the EF measurement from the simulated and experimental results at low (3.19), medium (17.0), and high (36.77) SNRs, respectively. The data were obtained by using 100, 500, and 2000 msec TRs. The SNR is calculated from the ratio of the average MR signal inside the object to the average MR signal of the background noise. In Fig. 5æ10, the simulated and experimental results as well as the result from Fourier encoding method are shown as the dotted, solid, and dashed-dotted lines, respectively. As shown, the acquisition in the non-ideal encoding case (the solid line) is less efficient than that from the ideal encoding case (the dotted line), but it is more efficient than from the Fourier method. The difference between the non-ideal and ideal results is more prominent when more encodes are considered. This is because, in this investigation, the singular vectors are ordered by decreasing associated singular values. The encoding method therefore sequentially encodes bands of low to high spatial frequency information as a greater number of encodes are used. The high spatial frequency components of noise is therefore the predominant contribution to that encoded by the SVD method at the high encoding numbers. The difference between non-ideal encoding caused by noise, and the ideal encoding, is thus prominent at the high encoding numbers while both produce similar results for fewer encodes (ç 60). The difference at the high encoding numbers between the non-ideal and ideal conditions, however, will have a less significant effect on practical encoding efficiency because the SVD method is approximated by truncating the number of eigenimages, associated with small singular values (i.e. at high encoding numbers as in the encoding order of this study). For example, if only the first 40 encodes are employed for the acquisition and reconstruction process at a medium SNR (Fig. 5æ10B), then from the resulting EF measurement we will see no significant difference in encoding efficiency between the non-ideal and ideal encoding conditions.

141 126 From the EF measurement of the simulated results in this section, the nonideal encoding condition caused by noise in the FOV has a more significant effect at the greater rather than fewer numbers of encodes when the encoding vectors are ordered from high to low associated singular values. The effect, however, can be reduced in practice because in the SVD method the number of eigenimages is truncated causing a filtering of the high frequency noise components Encoding Efficiency of the SVD Method in the Worst-case Scenario The major advantage of an adaptive encoding technique, such as the SVD, is its ability to optimize encoding in a dynamic MRI series. The basis set from this technique is calculated based upon the contents in the FOV. Changes in the FOV thus affect the encoding efficiency of this technique. Changes in the FOV during dynamic MRI applications can be separated into geometric (i.e. transitional or rotational), appearance and disappearance, and intensity changes which can happen partially or completely in the FOV. Geometric changes in the FOV can be due to movement of the base line image during the application, such as voluntary and involuntary physiological motions. Intensity changes, on the other hand, can be due to the procedure during the application, such as applied heat energy during the laser ablation treatment. Also during interventional procedures, objects or anatomy of patient and physician can enter or leave the FOV. The SVD method has been employed [8] to demonstrate the encoding efficiency in a dynamic MRI application in which intensity changes occurred. In this section, the effect of geometric changes on encoding efficiency is investigated. PAF analysis is performed on the horizontal and vertical encoding vectors of the SVD calculated from a reference meat phantom to vectors calculated from

142 127 the simulated change in geometry of the same phantom. The objective of this simulation is to demonstrate the effect of geometric changes on encoding efficiency. This demonstration represents the worst-case scenario for encoding efficiency in the SVD method because the change in this simulation happens everywhere in the FOV. In other words, this demonstration represents the highest possible projection error that can happen due to change in the FOV. This projection error can also be viewed as the error caused by non-ideal projections of vector subspaces due to change in the FOV onto the vector subspaces spanned by the encoding basis sets of the ideal projection image (i.e. the baseline image). Geometric change is separated into translation and rotation effects. The translation of image S is accomplished by transforming the signal intensity from the (x; y) coordinate to the (x 0 ;y 0 ) coordinate using the following definition: x 0 = x, h y 0 = y, v (5.9) where h and v are horizontal and vertical translation variables in a pixel unit. In this study, the horizontal and vertical translations have been investigated separately to reveal the effect of such changes on both directions of the SVD encoding vectors. The rotation of the image S is defined by transforming the signal intensity from the (x; y) coordinate to the (x 0 ;y 0 ) coordinate using the following equation: x0 y = cos æ,sin æ sin æ cos æ where æ is the rotation angle in the radius unit x y (5.10) PAF analysis of these two geometry changes are analyzed on the vertical and horizontal encoding vectors.

143 128 Effects of Translation on Encoding In this simulation, we start with the hypothesis that translation of FOV contents in the same direction as the SVD encoding should show a projection error, while translation in the orthogonal direction to that of SVD encoding would show an error of lesser magnitude. Here, we assumes that the image contents do not leave the FOV. That is only 2D in-plane motion is simulated. The projection errors differ since the SVD method employs a spatial selective encoding technique in one direction, and Fourier encoding in the other. If the translation is along the Fourier encoding direction, which is a non-spatial selective encoding technique, the effect of such a translation is minimal. On the other hand, if the translation is along the SVD encoding direction, which is a spatial selective encoding technique, the effect of that translation would greater upon the acquired and reconstructed result. Figure 5æ11: The effect of the horizontal translation on the image encoded by the vertical (the first row (A)) and the horizontal (the second row (B)) encoding vectors from the SVD method. Each row shows four horizontal translations (1-4) from left to right. The reference image for the PAF analysis is in the second column. Fig. 5æ11 illustrates the effect of the horizontal translation on the images using

144 PAF(%) Shift Position (pixel) Figure 5æ12: The PAF analysis of the horizontal translation on the vertical (solid line) and horizontal (dashed line) encoding vectors. Figure 5æ13: The effect of the vertical translation on the image encoded by the vertical (the first row (A)) and the horizontal (the second row (B)) encoding vectors from the SVD method. Each row shows four horizontal translations (1-4) from left to right. The reference image for the PAF analysis is in the second column.

145 PAF(%) Shift Position (pixel) Figure 5æ14: The PAF analysis of the vertical translation on the vertical (the solid line) and horizontal (the dashed line) encoding vectors. the horizontal and vertical encoding vectors. The images from the horizontal encoding vector are shown in the second row (Fig. 5æ11B) while the first row (Fig. 5æ11A) shows the images from the vertical encoding vectors. Each row in Fig. 5æ11 shows four horizontal translations in a left to right direction. The reference image for the PAF analysis is in the second column. As shown in Fig. 5æ11, the vertical encoding images do not demonstrate any significant encoding artifacts while there are encoding artifacts clearly visible in the horizontal encoding images (the second row of Fig. 5æ11). The PAF analysis of this horizontal translation experiment is depicted in Fig. 5æ12. The PAF of the vertical encoding images is shown by the solid line, and the PAF of the horizontal encoding images is shown by the dashed-line. The PAF of the horizontal encoding images is about 30è while it is

146 131 only about 9è for the vertical encoding images. The PAF values of these encoding images also agree with the image quality in Fig 5æ11. The effect of the vertical translation on the images is depicted in Fig. 5æ13. The first row of Fig. 5æ13 (A) shows images from the vertical encoding vectors while the horizontal encoding images are shown in the second row (B). Each row in Fig. 5æ13 shows four vertical translations in a top to bottom direction. As opposed to the results of the previous simulation, the horizontal encoding images do not show encoding artifacts while the vertical encoding images clearly show encoding artifacts. The PAF of the vertical (solid line) and vertical (dashed line) encoding images are illustrated in Fig. 5æ14. The PAF of the vertical encoding images in this translation is about 85è while it is only about 5è for the horizontal encoding images. The simulation results from these two translations suggest that the encoding efficiency will have a minimal effect when the translation occurs along the Fourierencoded direction and will have a non-negligible effect when the translation occurs along the SVD encoding direction. Effects of Rotation on Encoding In this simulation, we start with the hypothesis that the rotation of the baseline image relative to the FOV contents should show a projection error since it may affect the projection profile along the SVD encoding direction. The rotation effect on the vertical and horizontal encoding images is shown in Figs. 5æ15 and 5æ16, respectively. Each figure shows eight images which are rotated by 0 æ ; 45 æ ; 90 æ ; 135 æ ; 180 æ ; 225 æ ; 270 æ ; and 315 æ. The reference image is 0 æ rotation (the middle right image in Figs. 5æ15 and 5æ16). PAF analysis of the vertical and horizontal encoding images is illustrated in Figs. 5æ17A and B, respectively. The

147 132 Figure 5æ15: The rotation effect on the vertical encoding images using the SVD method. From the middle right image (the reference image) in a counterclockwise direction, eight different rotation angles: 0 æ ; 45 æ ; 90 æ ; 135 æ ; 180 æ ; 225 æ ; 270 æ ; and 315 æ, are depicted.

148 133 Figure 5æ16: The rotation effect on the horizontal encoding images using the SVD method. From the middle right image (the reference image) in a counterclockwise direction, eight different rotation angles: 0 æ ; 45 æ ; 90 æ ; 135 æ ; 180 æ ; 225 æ ; 270 æ ; and 315 æ, are depicted.

149 134 (A) 90 (B) Figure 5æ17: The angular plot of the PAF analysis of the rotation effect of the horizontal (A) and vertical (B) encoding vectors from the SVD method is illustrated as the thin-solid line. The PAF of the worst-case scenario is illustrated as the thickdotted line. PAF of the encoding images is shown by the thin-solid line while the PAF of the worst case is shown by the thick-dotted line. When we compare image quality in Figs. 5æ15 and 5æ16, the vertical encoding images demonstrate more encoding artifacts than the horizontal encoding images. This is most likely due to the different encoding profiles of the vertical and horizontal vectors. When the vertical encoding vectors are used to encode the rotated images, some parts of the images, especially around the edges, are lost. This is due to the fact that some parts of the FOV contents are outside the object s contents of the encoding profile calculated from the reference image. The horizontal encoding vectors, on the other hand, are able to encode all the contents of the images, due to the change in the rotation, which are still inside the object s contents of the encoding profile. For example, the vertical encoding profile and the projection of a 45 æ rotated baseline image along the vertical direction are shown as the dotted and solid lines

150 135 2 x 106 (A) x 106 (B) Magnitude Signal image space Figure 5æ18: The SVD encoding (dashed line) and image (solid line) projection profiles along the vertical (A) and horizontal (B) directions. The rotated 45 æ image in Fig. 5æ16 was used to generate the image profiles in Fig. 5æ18A, respectively. The horizontal encoding profile, and the projection of the same image as in Fig. 5æ18A, but along the horizontal direction are also shown as the dotted and solid lines in Fig. 5æ18B. As shown in Fig. 5æ18, the object s contents of the vertical encoding profile is only a part of the rotated profile of the contents along the vertical direction while the horizontal encoding profile can cover all of the rotated profile along the horizontal direction. Some parts of the rotated vertical profile, especially around the edges of the image contents, therefore, can not be encoded by the vertical encoding vectors. In other words, some parts of the vector subspaces of the rotated vertical profile can not be projected onto the vector subspaces spanned by the vertical encoding basis set. Those parts, there-

151 136 fore, cannot be encoded by the encoding vectors. The horizontal encoding basis set, on the other hand, can encode all parts of the rotated horizontal profiles because such a basis set can span the vector subspaces to project onto all parts of the vector subspaces of the rotated horizontal profiles. This explains why, in this case, the vertical encoding demonstrates more encoding artifacts than the horizontal encoding. The simulated results in this section demonstrate the effect of rotational change on encoding efficiency in the contents of an example image. The degree of effect depends on the difference between the object s contents of the encoding profiles and the profiles of the FOV; if all the FOV contents are inside the object s contents of the encoding profiles, a weaker effect is expected than when some parts of the FOV contents are outside the object s contents of the encoding profiles. When using an adaptive encoding technique in dynamic MR applications, this effect needs to be considered because of the significant reduction in encoding efficiency Summary and Conclusions The efficiency of non-fourier encoding methods has been investigated. The encoding factor (EF) and principal angle factor (PAF), which are used in the analysis of such efficiency on the eigenimages and on the encoding vectors, respectively, have been employed to analyze encoding efficiency quantitatively. The experimental and simulated results regarding encoding efficiency have been presented and discussed. The EF analysis demonstrates the higher encoding efficiency of an adaptive encoding technique, such as the SVD, in case of ideal and non-ideal encoding when compared to non-adaptive techniques like the RF impulse or Hadamard methods. This is due to the encoding basis set employed by the SVD method being computed specifically for the contents in the FOV es-

152 137 timate. Furthermore, the SVD method can provide a better truncated encoding approximation than the conventional Fourier method when they are compared at the same encoding numbers. This is due to the encodes of the SVD method contains bands of information in the spatial frequency domain compared to the single frequency encoded per encode in the Fourier method. From the EF measurement in the simulated results of the SVD method in the non-ideal encoding condition caused by noise in the FOV, the difference in encoding efficiency between the non-ideal and ideal encoding cases involves eigenimages acquired by the SVD encoding vectors with small associated singular values (i.e. the high spatial frequency components), which are dominated by noise. The difference, however, can be reduced since these encodes are most often truncated in practice. The effects of geometric change on the encoding efficiency in the adaptive encoding technique have been investigated. Encoding efficiency will be minimally affected when the translation of the image occurs along the Fourier encoding direction, and will be affected more significantly when the translation occurs along the spatial selective encoding direction. Rotation of the image also has a significant effect on the encoding efficiency. The degree of the effect depends on the difference between the object s projection along each axis and the profiles of the FOV; if all the FOV contents are inside the object s x and y axis projections, a weaker effect is expected than when some parts of the FOV contents are outside the object s x and y axis projections in the estimate. When using the adaptive encoding technique in dynamic MR applications, the effects due to geometric change need to be considered because of the significant reduction in encoding efficiency. Synthetic data can be added to the image estimate k-space data to accomodate this eventuality.

153 138 Chapter 6 Applications of the Non-Fourier Encoded MRI in Dynamic MRI The experiments in this chapter were aimed at demonstrating the encoding efficiency of using the non-fourier encoding methods, especially the adaptive technique, as well as to demonstrate the feasibility of using such encoding methods to acquire multiple slice images within a slab for dynamic MRI applications. 6.1 Single-Slice Spin-Echo SVD Encoded MRI A series of experiments was performed in non-real time with a fruit phantom, employing different types of non-fourier encoding, and using non-adaptive and adaptive algorithms. The experiments were performed on the fruit phantom using the SE nf pulse sequence with a 25 æ flip angle, TE = 30 msec, and TR = 1000msec. Three different basis sets - the RF impulse, Hadamard, and SVD - were employed to encode the phantom. To demonstrate the change in the FOV during the dynamic MRI applications, a biopsy needle was inserted in, and withdrawn from, the phantom in non-real time fashion. Twenty-seven dynamic images were acquired from each basis set input. The RF impulse and Hadamard encoding inputs were pre-defined and did not change throughout the experiments (i.e. the non-adaptive algorithm). The SVD encoding input, on the other hand, changed with each acquisition im-

154 139 age using the dynamically adaptive algorithm [7] which calculates the new encoding input from recent measurement data (i.e. the k-space data). The keyhole algorithm [92], which combines recent acquisition data with base line data in the reconstruction process, was also investigated for these three encoding inputs Dynamically Adaptive Algorithm The dynamic MRI images using the RF impulse, Hadamard, and SVD encoding inputs are illustrated in Fig. 6æ1, 6æ2, and 6æ3, respectively. The non-adaptive algorithm was used in the RF impulse and Hadamard methods while the dynamically adaptive algorithm was applied in the SVD method. Each figure shows 27 dynamic images starting from the top left corner, progressing from left to right, and ending at the bottom right. The biopsy needle was inserted into the phantom at the top-left part. Each image was reconstructed from the 32 most significant encodes (i.e. the 32 encodes around the center of k-space in the RF impulse method, or the 32 encodes from the 32 highest singular vectors in the SVD method). From the image results of these three encoding methods, the RF impulse and Hadamard images show lower image resolution than the SVD images, as expected. The intensity changes due to the biopsy needle are clearly visible in the SVD images, but are only slightly visible in the RF impulse and Hadamard images. The superior image quality in the SVD method over the RF impulse or Hadamard methods is due to the higher encoding efficiency in the SVD method. This higher encoding efficiency can be achieved because of the near optimal encoding vectors generated specifically for the contents of this FOV. The quantitative analysis of the encoding efficiency in these three encoding methods, is illustrated in Fig. 6æ4A and B. Fig. 6æ4A shows the EF measurement from the RF impulse input (the dashed line) and from the SVD method (the solid

155 140 line) at the 1st dynamic image (i.e. the base line image). Fig. 6æ4B shows the EF from the same encoding methods but the SVD image is at the 14th dynamic image (i.e. the midpoint of the dynamic image series). Due to similar results of EF measurement from the RF impulse method, the EF measurement of the Hadamard encoding results is not plotted here. As shown in Fig. 6æ4, the EF measurement from the SVD method is significantly lower than from the RF impulse method. The EF measurement from the SVD method at the midpoint of the dynamic series is slightly higher than at the beginning of the dynamic series. This is probably due to the change generated by the biopsy needle and background noise. For example, using 32 encodes, the EF value of the RF impulse input is about 40è while it is only about 3è in the SVD method at the 1st dymanic image and only about 6è at the 14th dynamic image. The slight change in encoding efficiency in the SVD method is also demonstrated in the PAF analysis. Fig. 6æ5 illustrates the PAF value of each dynamic image calculated from the experimental and ideal encoding vectors of that image. The ideal encoding vectors were generated by applying the SVD computation to the reconstructed image. In this experiment, the PAF value is less than 15è throughout the dynamic series. This implies that the experimental encoding vectors in the SVD method are just slightly less efficient than in the ideal case. This experiment demonstrates the efficiency of the SVD encoding method in this dynamic MRI application. A higher image resolution can be achieved in the SVD method than at the same encoding numbers in the RF impulse and Hadamard methods. The quantitative analysis in this experiment shows that only about 32 encoding numbers were required for the SVD encoding method to achieve almost the same contents in the FOV (i.e. less than 10% difference) as compared to 128 encoding steps required for the non-adaptive basis technique. The encoding effi-

156 141 ciency of the SVD method in this experiment is thus about four times higher than that of the conventional method The Keyhole Reconstruction Algorithm The keyhole reconstruction algorithm [92] was employed in the dynamic MRI adaptive and non-adaptive methods from the previous experiment. The images of this reconstruction algorithm from the RF impulse, Hadamard, and SVD methods are depicted in Fig. 6æ6, 6æ7,and 6æ8, respectively. Each figure shows the same 27 dynamic images as the previous experiment except that each image is reconstructed by using a combination of the 16 most significant encodes from the recent acquisition (i.e. only half of the acquisition data as the previous reconstruction images) and another 112 encodes from the base line data. Comparing the images from these three encoding methods, the RF impulse and Hadamard images show a very similar image resolution of the phantom to the SVD images. This is because of the additional encoding data in the keyhole reconstruction process from the base line image. The intensity change due to the biopsy needle, on the other hand, is significantly less visible in the RF impulse and Hadamard methods than in the SVD method. This is due to the small number of acquisition steps (i.e. 16) used to update the dynamic images. The SVD images, however, still provide a reasonable contrast where the biopsy needle is inserted. This is because of the higher encoding efficiency in the SVD method. This result suggests that the keyhole reconstruction algorithm can be used to substantially reduce the encodes in the SVD method (i.e. by about 50% in this experiment). The keyhole method yields a good result only, as demonstrated in this section, when there is no geometric change occurring in the contents in the FOV during the duration of imaging.

157 Multi-Slice Gradient-Recalled-Echo SVD Encoded MRI Zientara et. al. [7] first introduced the SVD encoded MRI in order to reduce the redundancy of spatial encoding during dynamic MRI. Panych et. al. [8] developed and implemented an SVD encoding method using a fast gradient-echo sequence which can update a 2D dynamic MR image in a sub-second interval and later introduced a 3D interleaved EPI technique for SVD encoding [107]. Here, we demonstrate for the first time an implementation of multi-slice gradient-recalledecho SVD encoded MRI that has been applied to image a human brain. It provides efficiency encoding of multiple 2D images within a slab, that is useful for dynamic MRI applications. The experiment was performed on a 1.5 GE SIGNA MR system using the multi-slice GRE pulse sequence, as described in Section The pulse sequence is employed as follows. Before the dynamic series begins, base-line image data (S) are obtained by Fourier encoding. Data from each echo train is stored in each column of S. Next, the SVD of the auto correlation matrix, S t S, is computed. The desired SVD vectors, according to the truncation criterion in [7], are then converted to RF pulses for encoding as described in [8]. Reconstruction entails performing a 2D inverse Fourier transform along the frequency and phase encoding direction and performing SVD reconstruction as described in [7] along the SVD encoding direction. For example, to encode K sections within a slab, the multi-slice GRE pulse sequence is used with K gradient echoes. For each of N shots, a new SVD encoding pulse is applied for a total of N encodes along the SVD direction. Fig. 6æ9 shows six out of eight slices of the sagittal brain images of an healthy adult volunteer acquired by using 64 excitations of SVD encoding in the horizon-

158 143 tal direction with TE=30 msec and TR=1sec. These images are the proton density images. To reduce the artifact due to the imperfections of the slab-selection technique, as described in Section 3.2.3, we subtract k-space data from two excitations with a 180 æ variation (i.e. +90 æ and,90 æ ) of the slab-selective soft pulse on each excitation. Fig. 6æ10 shows the same images as in Fig. 6æ9 acquired from the same pulse sequence by using 128 excitations of the equivalent Fourier encoding (the RF impulse method). The anatomy within the slices is shown in the image in Fig. 6æ11, acquired using a conventional gradient-echo pulse sequence with 128 pulse encodes and the same TE and TR as used for the images in Fig. 6æ9. The result in Fig. 6æ9 clearly demonstrates the feasibility of our SVD encoding method to acquire multiple slice images for dynamic MRI applications. Our implementation, however, still requires two excitations for each data set due to the slab-presaturation artifact. Further study should be focused on investigating other single excitation slab-presaturation techniques. 6.3 Summary The results from the dynamic non-real time MRI experiment clearly demonstrate the efficiency of the SVD encoding method as compared to the non-adaptive basis set technique. The quantitative analysis in this experiment shows that the encoding efficiency of the SVD method is about four times higher than that of the conventional method. The keyhole reconstruction algorithm can also be used to substantially reduce the encoding numbers necessary in the SVD method. The image results acquired by using multi-slice GRE pulse sequence clearly demonstrate the feasibility of the SVD encoding method to acquire multiple slice images for dynamic MRI applications.

159 144 Figure 6æ1: The dynamic MRI experimental results using the RF impulse encoding method. Twenty-seven dynamic images are depicted starting from the top left corner progressing from left to right and ending at the bottom right. Each image is reconstructed from the 32 most significant encodes.

160 145 Figure 6æ2: The dynamic MRI experimental results using the Hadamard encoding method. Twenty-seven dynamic images are depicted starting from the top left corner progressing from left to right and ending at the bottom right. Each image is reconstructed from the 32 most significant encodes.

161 146 Figure 6æ3: The dynamic MRI experimental results using the SVD encoding method. Twenty-seven dynamic images are depicted starting from the top left corner progressing from left to right and ending at the bottom right. Each image is reconstructed from the 32 most significant encodes.

162 EF(%) Encoding Number 10 2 EF(%) Encoding Number Figure 6æ4: The comparison of the EF measurements from the RF impulse method (dashed line) and the SVD method (solid line) at the 1st (i.e. the starting image) (A) and 14th (i.e. the midpoint of the dynamic image series) (B) dynamic images PAF(%) Image Number Figure 6æ5: The PAF analysis from the SVD encoding method on the dynamic imaging experiment.

163 148 Figure 6æ6: The dynamic MRI experimental results using the RF impulse encoding method. Twenty-seven dynamic images are depicted starting from the top left corner progressing from left to right and ending at the bottom right. Each image is reconstructed using a combination of the 16 most significant encodes from the recent acquisition and of another 112 encodes from the base line data (i.e. the keyhole algorithm).

164 149 Figure 6æ7: The dynamic MRI experimental results using the Hadamard encoding method. Twenty-seven dynamic images are depicted starting from the top left corner progressing from left to right and ending at the bottom right. Each image is reconstructed using a combination of the 16 most significant encodes from the recent acquisition and of another 112 encodes from the base line data (i.e. the keyhole algorithm).

165 150 Figure 6æ8: The dynamic MRI experimental results using the SVD encoding method. Twenty-seven dynamic images are depicted starting from the top left corner progressing from left to right and ending at the bottom right. Each image is reconstructed using a combination of the 16 most significant encodes from the recent acquisition and of another 112 encodes from the base line data (i.e. the keyhole algorithm).

166 151 Figure 6æ9: Six out of eight slices of the sagittal brain images of an healthy adult volunteer acquired by using 64 excitations of SVD encoding in the horizontal direction with TE=30msec and TR=1sec.

167 152 Figure 6æ10: The same images as in Fig. 6æ9 acquired from the same pulse sequence by using 128 excitations of the equivalent Fourier encoding (the RF impulse method).

168 153 Figure 6æ11: The anatomy images within the slices acquired using a conventional gradient-echo pulse sequence with 128 pulse encodes and the same TE and TR as using for the images in Fig. 6æ9.