Methods for the design, optimization, and image correction of local insert gradient system in magnetic resonance imaging

Developing faster Magnetic Resonance Imaging (MRI) gradient systems is desirable for fast imaging pulse sequences. Because of their large size, reduced efficiency, and higher inductance, system body gradient coils are limited in performance. This limitation can be reduced by designing proper local gradient coils. In this dissertation, various types of local insert gradient coil designs, as well as their optimization and image correction techniques are investigated. Conventional method of designing planar gradient systems that uses twodimensional stream functions can be more computationally intensive than optimizing the one dimensional stream functions required for cylindrical gradients. In this dissertation, a novel planar gradient design method, which simplifies the twodimensional planar gradient coil design problem to a faster and easier one-dimensional problem by using a conformal mapping technique. The simulated gradient filed maps prove that the proposed gradient design technique creates same homogeneous gradient and works well throughout the imaging volume.

i METHODS FOR THE DESIGN, OPTIMIZATION, AND IMAGE CORRECTION OF LOCAL INSERT GRADIENT SYSTEM IN MAGNETIC RESONANCE IMAGING by Sung Man Moon A dissertation submitted to the faculty of The University of Utah in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Electrical and Computer Engineering The University of Utah December 2010 i Copyright © Sung Man Moon 2010 All Rights Reserved The University of Utah Graduate School STATEMENT OF DISSERTATION APPROVAL The dissertation of has been approved by the following supervisory committee members: , Chair Date Approved , Member Date Approved , Member Date Approved , Member Date Approved , Member Date Approved and by , Chair of the Department of and by Charles A. Wight, Dean of The Graduate School. ABSTRACT Developing faster Magnetic Resonance Imaging (MRI) gradient systems is desirable for fast imaging pulse sequences. Because of their large size, reduced efficiency, and higher inductance, system body gradient coils are limited in performance. This limitation can be reduced by designing proper local gradient coils. In this dissertation, various types of local insert gradient coil designs, as well as their optimization and image correction techniques are investigated. Conventional method of designing planar gradient systems that uses two-dimensional stream functions can be more computationally intensive than optimizing the one dimensional stream functions required for cylindrical gradients. In this dissertation, a novel planar gradient design method, which simplifies the two-dimensional planar gradient coil design problem to a faster and easier one-dimensional problem by using a conformal mapping technique. The simulated gradient filed maps prove that the proposed gradient design technique creates same homogeneous gradient and works well throughout the imaging volume. Many MRI applications such as DCE-MRI of the breast require high spatial and temporal resolution. The planar gradient systems as currently designed cannot create an imaging volume large enough to accommodate both breasts. A new concept for designing planar gradient systems, which has a superellipse shape and the addition of a field-modifying layer, has been presented. Homogeneous gradient volumes (HGVs) of the proposed superelliptical coil are increased as much as 182 % (x-gradient) over those iv of the original planar system. In addition, adding an extra field-modifying layer allows the further enlargement of the HGVs by 214 % (z-gradient). Gradient homogeneity of the flat gradient systems is worse compared to cylindrical gradient systems, because the magnetic field and gradient strength drop nearly exponentially with distance from the coil surface. As a result, imaging with a planar insert gradient coil is challenging and prone to image distortions. A new concept of image de-warping has been demonstrated that calculates the nonlinear gradient field as an analytical function, and applies to de-warp collected image data. The simulated results indicate that the usable imaging region (in plane) has improved by 450% over the conventional linear gradient imaging. i To my parents, Gap Tae Moon and Young Gil Cheon, whose love and support made this work possible TABLE OF CONTENTS ABSTRACT iii ACKNOWLEDGEMENTS ix CHAPTERS 1. INTRODUCTION 1 1.1 Motivation 1 1.2 Overview of This Work 3 1.3 References 5 2. BACKGROUND I - BASICS OF MAGNETIC RESONANCE IMAGING 8 2.1 Magnetic Resonance 8 2.1.1 Net Magnetization 9 2.1.2 Precession and Larmor Frequency 11 2.1.3 Free Induction Decay and Relaxation 13 2.2 Basics of Imaging Techniques 18 2.2.1 Signal Localization 18 2.2.2 Signal Detection 19 2.2.3 Gradient Encoding 23 2.2.4 K-space and Image Reconstruction 24 2.3 System Hardware 26 2.3.1 Main Magnets 28 2.3.2 Gradient Arrays 32 2.3.3 RF Coils 34 2.3.4 Computer System 34 2.3.5 Operating Console 35 2.4 References 35 3. BACKGROUND II - GRADIENT ARRAYS 37 3.1 Criteria of the Gradient Systems 39 3.1.1 Efficiency 39 vii 3.1.2 Inductance and Slew Rate 39 3.1.3 Homogeneity 40 3.1.4 Homogeneous Gradient Volume 43 3.1.5 Other Parameters 43 3.2 Types of Gradient Coils 43 3.2.1 Gradient Systems by Shape/Geometry 44 3.2.1.1 Cylindrical Geometry 44 3.2.1.2 Planar Geometry 44 3.2.1.3 Other Geometries 46 3.2.2 Gradient Systems by Function 46 3.2.2.1 Whole Body Gradient Coil 46 3.2.2.2 Local Insert Gradient Coil 47 3.3 Designing Methods for Conventional Cylindrical Gradient Coils 47 3.3.1 Target Field Method 47 3.3.2 Stream Function Method 50 3.3.3 Z-intercept Method 51 3.3.4 Boundary Element Method 53 3.3.5 Other Methods 54 3.4 Optimization Process 54 3.4.1 Gradient Descent and Conjugate Gradient Methods 56 3.4.2 Simulated Annealing Method 56 3.4.3 Genetic Algorithm Method 57 3.4.4 Other Methods 59 3.5 Other Considerations for Gradient Systems 59 3.5.1 Eddy Currents 59 3.5.2 Gradient Shielding 60 3.5.3 Force and Torque Balance 60 3.5.4 Acoustic Noise 61 3.6 References 62 4. LOCAL BI-PLANAR GRADIENT ARRAY DESIGN USING CONFORMAL MAPPING AND SIMULATED ANNEALING 73 4.1 Abstract 73 4.2 Introduction 74 4.3 Theory 76 4.4 Methods 79 4.5 Results 88 4.6 Discussion 92 4.7 References 95 viii 5. SUPEERELLIPTICAL INSERT GRADIENT COIL WITH A FIELD MODIFYING LAYER FOR BREAST IMAGING 98 5.1 Abstract 98 5.2 Introduction 99 5.3 Theory 100 5.3.1 Wire Pattern Design 101 5.3.2 Transformation to Superelliptical Geometry 102 5.4 Methods 104 5.4.1 Field-Modification with an Extra Layer 104 5.4.2 Geometric Design 107 5.4.3 Simulation/Optimization Process 109 5.4.4 Force and Torque Balance 114 5.5 Results 115 5.6 Discussion 125 5.7 Acknowledgements 126 5.8 References 126 6. IMAGE CORRECTION METHOD USING ANALYTICAL MODELING OF NONLINEAR GRADIENT FIELDS IN MRI 129 6.1 Abstract 129 6.2 Introduction 130 6.2.1 Physical Phantom Based Correction Techniques 131 6.2.2 Image Based Correction Techniques 132 6.2.3 Other Correction Techniques 132 6.3 Methods 133 6.4 Results 135 6.5 Discussion 138 6.6 Acknowledgements 138 6.7 References 140 7. CONCLUSIONS 143 7.1 Motivation 143 7.2 Future Work 145 7.2.1 Future Work Towards Improving Gradient System Performance 146 7.2.2 Future Work Towards Improving Image Distortion 147 ACKNOWLEDGEMENTS I would like to thank my research advisor Dr. Dennis L. Parker for his constant encouragement, deliberate consideration, and patience throughout my Ph.D. study. His profound understanding and mastery of medical imaging science made my research work in this field possible and worthwhile. I would also like to thank academic advisor Dr. Cynthia Furse for her thoughtful advice during my Ph.D. study and for being my committee chair. I give special thanks to Craig, Rock, Adam, Sathya, and Nelly for their friendship, kindness, encouragement, and helpful advice. My gratitude also goes to all of my committee members as well as the faculty, staff, and student colleagues in the Utah Center for Advanced Imaging Research (UCAIR) for their friendship and advice. Finally, I would like to express my deepest appreciation to my family and Namkoong's family. Their endless love, selfless caring, warm encouragement and support led me through the hardest times and have become the soul of this dissertation. CHAPTER 1 INTRODUCTION 1.1 Motivation Magnetic resonance imaging (MRI) is one of the most widely used medical imaging techniques to produce high quality images of the inside of the human body. This unique imaging method is noninvasive and requires no radioactivity. In addition, MRI has superior soft tissue contrast over other imaging modalities, as it utilizes various physical parameters of tissues to achieve optimal contrast between normal and abnormal tissues. As a result, MRI has become the imaging modality of choice in many clinical and research studies especially when extremely high contrast between soft tissues is needed. MRI creates a strong magnetic field in the main bore of the system, where the patient is placed. This main magnetic field establishes net magnetization when the patient is placed within the bore. In order to generate an MRI image from a specific region of interest (ROI) of the body, extra magnetic fields are needed. The extra magnetic fields, which are superimposed on the main magnetic field, allow the spatial encoding of the nuclear magnetic resonance (NMR) signals, which in turn, form the image. Gradient coils, also known as gradient arrays, are the part of the MRI system that generates these extra magnetic fields. In spite of its crucial role for MR imaging (MRI), only a small number of 2 researchers made contributions to gradient coil development until the mid 1980's [1]. This is partly because MRI image acquisition times were much longer than they are now, and did not involve rapid switching gradients. Since then, more efficient, highly homogeneous, and faster switching gradient coils have been developed to meet the needs of faster image acquisition time. Among these, there are several outstanding developments such as the introduction of the stream function (SF) for cylindrical surface current density [2-4], the use of distributed current loops [5], the development of minimization methods to reduce power dissipation [6], and the improvement of fabrication methods and the introduction of high-current gradient drivers. Developing a faster gradient system is very demanding but is desirable for fast imaging pulse sequences. Body gradient systems are limited in performance due to the large size, reduced efficiency, and high inductance. These limitations can be at least partly overcome using local gradient coils to improve image quality and speed. A local gradient system is smaller than conventional whole body gradient systems and is used as an insert to fit in the magnet bore. Local gradient coils have many advantages over body gradient systems including high efficiency and low inductance (high slew rate), which result in higher spatial and temporal resolution. Also, less patient body volume is required inside the highly changing magnetic field (dB/dt) resulting in less peripheral nerve stimulation (PNS) [7, 8]. Therefore, local gradient systems can be used very effectively when specific areas of the body such as head, neck, breasts or knee are imaged. In this dissertation, various types of local insert gradient coil designs, as well as their optimization and image correction techniques, are investigated. 3 1.2 Overview of This Work This dissertation is composed of seven chapters, including this introductory chapter. In Chapter 2, the basic principles of MR imaging are reviewed including a brief description of data acquisition and reconstruction techniques, as well as the hardware of MRI. In Chapter 3, details of the gradient coil development method is described. The concepts, performance parameters, and various types of gradient coils are covered followed by design and optimization methods, and practical considerations for the gradient coil. Chapter 4 presents a novel planar gradient design method using conformal mapping and simulated annealing optimization. The improved gradient performance required to achieve high spatial and temporal resolution in MRI may be achieved by using local gradient coils such as planar gradient inserts. Although the wire patterns for planar gradients can be designed using two-dimensional stream functions [9-11], optimization of the two-dimensional stream functions can be much more computationally intensive and time consuming than optimizing the one-dimensional stream functions required for cylindrical gradients. To address this problem, a simple and rapid method for designing planar gradient inserts to produce a high strength local gradient field and a reasonably uniform imaging region has been developed. This method reduces the two-dimensional planar gradient coil design problem to a faster and easier one-dimensional problem using conformal mapping. This work has been published [12, 13]. Many MRI applications such as DCE-MRI of the breast require high spatial and temporal resolution, and can benefit from improved gradient performance (increased 4 gradient strength and reduced gradient rise time) especially when they are designed for a target anatomy such as breasts. Chapter 5 describes a new concept for designing planar gradient systems, which consists of transformation of cylindrical gradients to a superellipse shape and the addition of a field-modifying layer. Even though there have been a few attempts to design flat gradient systems for breast imaging [14], these systems have not created an imaging volume large enough to accommodate both breasts. Furthermore, the gradient field produced is not homogeneous, dropping rapidly with distance from the gradient coil surface. To attain an imaging volume adequate for breast MRI, a conventional local planar gradient design has been transformed into a segment of a superellipse shape to create homogeneous gradient volumes (HGVs) that are bigger than those of the original planar local gradient system. In addition, adding an extra field-modifying (FM) layer also allowed the further enlargement of the homogeneous gradient volume near the gradient coil surface compared to the already enlarged HGVs of the superelliptical gradients. This work has been published as a conference publication and submitted to Magnetic Resonance in Medicine [15, 16]. In Chapter 6, a new concept of imaging planar insert gradient system is presented. Insert gradient coils can generate high strength gradient fields over a small HGV because of their small size. With higher gradient strength and slew rate, the insert gradient coils can achieve high spatial and temporal resolution with less peripheral nerve stimulation (PNS) [7]. However, the insert gradient coils typically suffer from relatively larger deviations in gradient strength over their smaller HGVs. Gradient homogeneity is especially bad with flat insert gradients because the magnetic field and gradient strength drop nearly exponentially with distance from the coil surface. As a result, imaging with a planar insert gradient coil is more challenging and prone to image 5 distortions. Calibration and image correction methods such as Gradwarp can de-warp field distortion from the images [17], but require a cumbersome and time-consuming pre-scan calibration. In this chapter, a new concept of image de-warping is presented that calculates the nonlinear gradient field of the planar insert gradient coil as an analytical function, and use this to de-warp collected image data. Chapter 7 is a summary describing the accomplishments achieved over the course of this dissertation. Some directions of future work are also suggested. 1.3 References [1] R. Turner, "Gradient coil design: a review of methods," Magn Reson Imaging, vol. 11, pp. 903-920, 1993. [2] W. A. Edelstein and J. F. Schenck, "Current streamline method for coil construction," US Patent 4,840,700, July 13 1987. [3] R. J. Sutherland, "Selective excitation in NMR and considerations for its application in three-demensional imaging," University of Aberdeen, vol. Ph.D. Thesis, 1980. [4] J. F. Schenck, M. A. Hussein, and W. A. Edelstein, "Transverse gradient field coils for nuclear magnetic resonance imaging," US Patent 4,646,024, November 2 1983. [5] M. W. Garrett, "Axially symmetric systems for generating and measuring magnetic field," J Appl Phys, vol. Part 1, pp. 1091-1171, 1951. [6] R. Turner, "Minimum Inductance Coils," J Phys E:Sci Instrum, vol. 23, pp. 948- 952, 1988. [7] B. Zhang, Y. F. Yen, B. A. Chronik, B. C. McKinnon, D. J. Schaefer, and B. K. Rutt, "Peripheral nerve stimulation properties of head and body gradient coils of 6 various sizes," Magn Reson Med, vol. 50, pp. 50-58, 2003. [8] R. E. Feldman, C. J. Hardy, B. Aksel, J. Schenck, and B. A. Chronik, "Experimental determination of human peripheral nerve stimulation thresholds in a 3-axis planar gradient system," Magn Reson Med, vol. 62, pp. 763-770, 2009. [9] M. A. Martens, L. S. Petropoulos, R. W. Brown, and J. H. Andrews, "Insertable biplanar gradient coils for magnetic resonance imaging," Rev Sci Instrum, vol. 62, pp. 2639-2645, 1991. [10] K. Yoda, "Analytical design of self-shielded planar coils," J Appl Phys, vol. 67, pp. 4349-4353, 1990. [11] E. C. Caparelli, D. Tomasi, and H. Panepucci, "Shielded biplanar gradient coil design," J Magn Reson Imaing, vol. 9, pp. 725-731, 1999. [12] S. M. Moon, K. C. Goodrich, J. R. Hadley, and D. L. Parker, "Local uni-planar gradient array design using conformal mapping and simulated annealing," Concepts Magn Reson B, vol. 35B, pp. 23-31, Feb. 2009 2009. [13] S. M. Moon, K. C. Goodrich, J. R. Hadley, and D. L. Parker, "Local uni-planar gradient array design using conformal mapping and simulated annealing," 16th ISMRM Proceedings, p. 1168, 2008. [14] C. F. Maier, H. N. Nikolov, K. C. Chu, B. A. Chronik, and B. K. Rutt, "Practical design of a high-strength breast gradient coil," Magn Reson Med, vol. 39, pp. 392-401, 1998. [15] S. M. Moon, K. C. Goodrich, J. R. Hadley, G. L. Zeng, G. R. Morrell, M. McAlpine, B. Chronik, and D. L. Parker, "Superelliptical insert gradient coil with a field modifying layer for breast imaging," Magn Reson Med, vol. in review, 2009. [16] S. M. Moon, K. C. Goodrich, J. R. Hadley, G. L. Zeng, G. Morrell, M. A. McAlpine, B. A. Chronik, and D. L. Parker, "Superelliptical insert gradient coil 7 with field modifying layers for breast MRI," 18th ISMRM Proceedings, Stockholm, Sweden, 2010. [17] G. H. Glover and N. J. Pelc, "Method for correcting image distortion due to gradient nonuniformity," US Patent 4,591,789, 1986. CHAPTER 2 BACKGROUND I - BASICS OF MAGNETIC RESONANCE IMAGING Magnetic resonance imaging (MRI) is a 2.1 Magnetic Resonance medical imaging technique that is the most commonly used in radiology to visualize detailed internal organs, structures, and tissues of the human body based on nuclear resonance of atoms within the body. In addition to MRI's noninvasive and nonionizing properties, its unparalleled soft tissue contrast makes it especially useful in medical imaging. The quantum phenomenon that is known as nuclear magnetic resonance (NMR) was first observed in 1937 by Columbia University professor, Isidor I. Rabi [1]. Since then, many researchers have played important roles in MRI research. A few distinctive researchers are Felix Bloch (Stanford University) [2] and Edward Purcell (Harvard University) [3] who developed instruments that can measure the nuclear magnetic resonance in bulk material such as liquids and solids, Paul Lauterbur (State University of New York at Stony Brook) who first introduced the 2D NMR image [4], and Peter Mansfield (University Nottingham, England) who further developed and mathematically proved the utilization of gradients in the magnetic field [5]. In this chapter, the general MRI concepts and systems hardware will be discussed. 9 2.1.1 Net Magnetization Hydrogen is the most abundant source of signal for magnetic resonance imaging (MRI). The hydrogen nuclei in the patient, specifically the protons, are like tiny magnets. Each individual proton has properties of spin and charge, which in turn, result in a magnetic moment. The magnetic moments are randomly variable in orientation in the body under normal circumstances (Figure 2.1(a)). The net magnetization of magnetic moments in the body is normally zero, because each magnetic moment points in a random direction, and the sum of these magnetic moments cancels each other. When the human body is positioned in a strong magnetic field, the magnetic moments tend to align either along or against the external magnetic field. To be more specific, for the nuclei such as 1H, 13C, 19F, 31P, which have spin quantum number of half (1/2), the magnetic moments will align in the direction of the external magnetic field (parallel, lower energy state) or opposite direction of the external magnetic field (anti-parallel, higher energy state). The amount of magnetic moments that align along the external magnetic field and against the external magnetic field is given by the Boltzmann distribution, and different at different magnetic field strength. There will be a slight excess of spins in the lower energy state (aligned with the external field), and this phenomenon establishes the net magnetization for MRI. In MRI this external magnetic field, called the main magnetic field (B0 field), is created by a very strong magnet, usually a superconducting magnet. This external magnetic field polarize the magnetization within the patient, and a net magnetization is established (Figure 2.1(b)). For clarification, the axes of the MRI system need to be defined. The z-axis, the direction of the main magnetic field, is defined as the toe-to- 10 Figure 2.1. Magnetic moments and their response to the external magnetic field. (a) In regular circumstances the net magnetization is zero because each magnetic moment points randomly, canceling each other's magnetization. (b) In the presence of the external magnetic field (e.g., the main magnet of the MRI system), a net magnetization is established along the external magnetic field (z-direction). Z Z (a) (b) B0 field 11 head or inferior-to-superior direction of the patient, the x-axis is defined as the left-to-right direction of the patient, and the y-axis is defined as posterior-to-anterior direction of the patient when the patient enters the magnet head first on their back (head-first supine position). 2.1.2 Precession and Larmor Frequency Another important concept to understand in MRI is the precession of the proton. When protons are in the strong external magnetic field, the magnetic moments of the proton will not only spin, but will also rotate around the direction of the external magnetic field. This is called precession (Figure 2.2). This motion can be described mathematically as follows: dM M B dt = ×γ (2.1)   x y z x y z i j k M M M B B B = γ  (2.2)   ( ) ( ) ( ) z y y z z x x z y x x y i B M B M j B M B M k B M B M γ  −    = − −     −   (2.3) where B represents the total external magnetic field, M represents the net magnetization vector, and γ represents the gyromagnetic ratio. The frequency of the precession is called the Larmor frequency, and the Larmor equation establishes the relationship between precession and the magnetic field strength as shown below: 0 0 ω =γ ⋅ B (2.4) 12 Figure 2.2. Precession of magnetic moments. A total magnetic moment (M) precesses along the external magnetic fields. 13 where B0 is the external magnetic field, and γ represents the gyromagnetic ratio. For example, the gyromagnetic ratio of the proton (1H) is 42.575 MHz/T, and 11.262 MHz/T for 23Na [6], where T is Tesla, the SI unit of magnetic field (B), where 1 T is equivalent to 104 Gauss (G). 2.1.3 Free Induction Decay and Relaxation As described earlier, when the patient is placed in the main magnet, a net magnetization is established in the direction of the main magnetic field (z-direction). For the magnetization, the z direction is the direction of the main magnetic field, and the x and y directions are orthogonal to this field direction. The net magnetization precesses around the main magnetic field is usually called the longitudinal magnetization, and is symbolized as Mz, where M means net magnetization and z means it is along the main field direction. During the imaging process, radio-frequency (RF) pulses that have the same frequency as the resonant frequency of the precessing nuclei are transmitted to the patient in the transverse plane (i.e., x/y plane). This RF energy tips some of the magnetization away from the direction of the main magnetic field into the x/y plane resulting in transverse magnetization (Mxy). Typical angles of RF pulses are 90 degrees for full excitation, 180 degrees for refocusing, and less than 90 for steady state excitation. After the RF pulse is turned off, while continuing its precession, the net magnetization relaxes back to its original state, which is a stable lower energy state. Felix Bloch [7] was the first to describe the behavior of the phenomenon of the magnetization vector, M=[Mx My Mz]: 14   0 2 1 x y ( z ) dM M B M i M j M M k dt T T γ + + = × − −  , (2.5) where i , j , k are unit vectors in the x, y, and z directions, and 0 M is the thermal-equilibrium magnetization. In general, the Bloch equation (Equation 2.5) is considered in the rotating frame of reference instead of the laboratory frame, which is stationary. The rotating frame is a frame of reference that is rotating along the z-axis of the laboratory frame with a frequency of 0 ω . In the rotating frame, the three unit vectors can be related to those of the stationary frame as    ˆ ' cos( ) sin( ) ˆ ' sin( ) cos( ) ˆ ' i t i t j j t i t j k k ω ω ω ω  = −  = +  =    (2.6) Now, the Bloch equation in the rotating frame can be described as  ' ' ' 0 2 1 x' y ' ( z )ˆ' eff dM M B M i M j M M k dt T T γ + + = × − −  , (2.7) with eff 0 B =B+ω γ , (2.8) where 0 0 k ω =ω and M and B both have their components expressed in the rotating frame. The original state is called the equilibrium state, because protons are stable in this lower energy state. The precession of the net magnetization induces electrical signal/current in nearby coils (by Faraday's law of induction). This oscillating signal is called free induction decay (FID), and FID decreases as the net magnetization returns to 15 its equilibrium after excitation into the transverse plane (Figure 2.3). The time for net magnetization to recover to its original equilibrium state is called relaxation time, and there are two kinds of relaxation time constants, T1, and T2. T1 relaxation happens when the energy from the RF pulse returns to its surroundings, which is known as the lattice, and the protons go back from their higher to the lower state of energy. This is why the T1 relaxation process is also called spin-lattice relaxation and longitudinal relaxation. The time that it takes for the longitudinal magnetization to go back to its original value is described by the longitudinal relaxation time and is different from tissue to tissue as the lattice (the surroundings of each tissue) is different. The longitudinal magnetization versus time after the RF pulse is shown in Figure 2.4a. When a patient is placed inside of the MR Imager, the spins in the body precess with slightly different frequencies, which are determined by the magnetic field strength that they are in, because the field inside of the MRI is not perfectly homogeneous. In addition, each spin is influenced by microscopic magnetic fields from the neighboring nuclei, which are different from tissue to tissue. As a result, when the RF pulse is off, the spins precess with different frequencies, and the spins soon become out of phase. The time it takes for the net magnetization to reach 1/e (37%) of its original magnetization as a result of getting out of phase (de-phasing) is called the T2 relaxation time or spin-spin relaxation time, shown in Figure 2.4b. These T1 and T2 relaxation processes were discovered by Nicolaas Bloembergen in 1948, along with Purcell and Pound [8]. They are different for different tissues/cells of the human body, resulting in different signal intensity and allowing us to detect different signal strength at a given time. 16 Figure 2.3. Free induction decay (FID) is an oscillating signal of the net magnetization with the precession frequency (Larmor frequency). RF signal intensity Time 17 Figure 2.4. Plots of the longitudinal and transversal magnetization after RF pulse is switched off. (a) T1 relaxation over time, (b) T2 relaxation over time Mz (longitudinal magnetization) Time Mxy (transverse magnetization) Time (a) (b) 18 2.2 Basics of Imaging Techniques The previous section described how magnetic fields interact with the molecules in the body. In this section, how to use these effects to create an image will be explained. 2.2.1 Signal Localization When a patient is placed in the strong and homogeneous magnetic field inside of the main magnet bore of the MRI system, each part of the patient's body within the homogeneous part of the magnetic will see the same magnetic field strength, resulting in the same Larmor frequency for every position in the body throughout this volume. This means that the signal from throughout that volume is the same when the RF pulse is applied. There is no way to tell where a certain signal came from in order to produce an image. In order to distinguish specific signals for different positions, the field must change as a function of position. This is done by adding an extra magnetic field that is called a "gradient field," and was applied for the first time by Paul Lauterbur [4]. The magnetic field of the gradient may point in many different directions, but since the main magnetic field is so strong, only the component of the gradient field in the same direction as the main field (Bz) has any appreciable effect. To spatially encode signals from the body in three-dimensional, gradient fields from three different axes are required. For this purpose, three different gradient coils are needed, which are referred to as the x-gradient coil, y-gradient coil, and z-gradient coil. For conventional two-dimensional Fourier MR imaging, these three gradients are defined as a slice selection gradient, frequency encoding gradient, and phase encoding gradient. For many imaging sequences, the slice selection gradient (Gz ) is turned on 19 while an RF pulse with a certain bandwidth is applied to spatially localize the excitation to a certain slice position ( z0 ) with thickness ( Δz ) as follows: RF BW = γ ⋅Gz ⋅ Δz . (2.9) Then the RF pulse is centered at the resonance frequency of spins at that location, 0 z , and the slice thickness (TH) will be determined as ( ) ( ) RF TH Δz = BW Gz ⋅γ . (2.10) Figure 2.5 shows how the slice selection is done. The main magnetic field (B0) is shown in Figure 2.5a, and the gradient field (Gz) varies its strength as a function of position (Figure 2.5(b)). These two different magnetic fields are superimposed (Figure 2.5c) resulting in the main magnetic field being modified to have different strengths at different positions. Now protons in different positions along the z-axis have different precession frequencies and can be excited separately by using different frequencies of RF pulses. The gradient fields and detail imaging processes such as gradient encoding will be discussed later in this chapter. Note that the gradient fields are produced by gradient coils, the main topic of this dissertation. 2.2.2 Signal Detection During an application of the RF pulse, with both the main magnetic field (B0) and a rotating RF field (B1(t)) are present, a rotating frame will see a stationary B1 field of B1 =B1(0)i, and an effective total magnetic field can be described as,    0 1' rf eff k B B k B i ω γ = + − (2.11)  1 0 B i' B rf kˆ' ω γ   = + −    . (2.12) 20 Figure 2.5. Slice selection strategy in MRI. (a) the main magnetic field, (b) the gradient magnetic field with linearly varying magnetic field strength in position along the z-axis, (c) the resulting modified magnetic field after superimposition of the main magnetic field and gradient field. Z (c) Z B0 field (a) (b) Gradient field strength Z B0 (Main magnetic field) Gz (Gradient field strength) Gz Total Field Strength(Bz(z))= B0 +Gz⋅z 21 In case rf 0 ω =ω , it is referred to as an on-resonant condition. When the on-resonance condition is not satisfied, the effective magnetic field will experience an off-resonance condition (Figure 2.6), which is described as follows,  1 ' ˆ' eff B B i k ω γ Δ  = +    . (2.13) After a short period of the RF pulse application, the B1 field will disappear, and the Bloch equation (Equation 2.7) can be simplified to ' 0 ' 1 ' ' ' ' 2 z z x y x y dM M M dt T dM M dt T −  =  = −  (2.14) where x'y' M is the magnetization in the transverse plane with x'y' x' y' M = M + iM . The solution to Equation 2.13 is 1 1 2 / / ' ' 0 / ' ' ' ' ( ) (0) (1 ) ( ) (0) t T t T z z t T x y x y M t e M e M M t e M − − −  = + −   = (2.15) This result confirms the exponential decay of the transverse magnetization and the exponential recovery of the longitudinal magnetization as described earlier (Figure 2.4). By Faraday's law of induction, a changing magnetic field (precession of the magnetic moments) can induce a current in the coil. The resulting electromotive force (emf) induced in the coil is expressed as 3 ( , ) receive( ) sample emf d d rM r t B r dt =− ∫ ⋅ [6]. (2.16) This equation can be written in terms of the individual components as 3 [ ( , ) receive( ) ( , ) receive( ) ( , ) receive( )] x x y y z z signal d d r M r t B r M r t B r M r t B r dt ∝ − ∫ ⋅ + ⋅ + ⋅ . (2.17) 22 Figure 2.6. Off-resonance excitation in the rotating frame. The bulk magnetization precesses around the direction of the effective magnetic field ( ). y' x' z 0 B ω γ    −    1 B eff B eff B 23 By substituting Equation 2.15 in Equation 2.17, the transverse magnetization signal that is detected by the receiver coil can be derived as 3 / 2( ) ( ) ( ) ( ( ) ( )) 0 0 0 t T r ,0 sin xy xy B signal∝ω ∫ d re− M r B r ω t+θ r −φ r (2.18) where 0φ (r) is the phase, and Bxy represents the coil sensitivity. More generally, Equation 2.18 is replaced by the demodulated form ( ) ( ) ( ( ) ( )) 2 0 0 3 / ( ) ( ) 0 ( ) ,0 t T r i t r Br xy xy s t d re M r B r e ω φ θ ω ∝ ∫ − Ω− + − (2.19) which is the multiplication of the signal by a sinusoid or cosinusoid with a frequency at or near 0 ω . 2.2.3 Gradient Encoding The gradient field is a linearly varying magnetic field that is superimposed on the main magnetic field. As described in Figure 2.5, the resultant magnetic field, made up of both the main field and the gradient field, varies spatially as 0 z B =B +G⋅r (2.20) with the local Larmor frequency of 0 ω =ω +γG⋅r. (2.21) From Equation 2.19, the signal with the frequency of Equation 2.21 can be rewritten as ( ) 3 ( ) i r Gt x y s t = ∫ ∫ d rρ r e− γ ⋅ (2.22) The corresponding Fourier pair is defined as ( ) 3 ( ) i2k r x y s k = ∫ ∫ d rρ r e− π ⋅ (2.23) or ( , , ) ( , , ) ( 2 ( )) x y z x y z s k k k =∫∫∫ dxdydzρ x y z ⋅exp −iπ k x+k y+k z (2.24) 24 where 0 k(t) =γ ∫tdτG(τ ). (2.25) 2.2.4 K-space and Image Reconstruction The detected signals, which are spatially encoded with the gradient magnetic fields, are stored as a raw data matrix in a complex space called the k-space. For two-dimensional imaging, k-space has two axes. Conventionally, the vertical axis (Ky) represents the phase axis, and the horizontal axis (perpendicular to the phase axis) represents the frequency axis (Kx) as shown in Figure 2.7(a). The raw signals received by the receiver coil (Equation 2.24) are filled in k-space as horizontal lines during each repetition time (TR). These lines are then shifted vertically (upward/downward) to fill the next horizontal line. For many imaging techniques, one horizontal line of k-space (readout) is acquired with each RF excitation. Before the next readout using a constant Gx, a pulse of Gy is used to generate a single encoding in the phase direction (vertical direction) as shown in Figure 2.7(b). From the mathematical point of view, sampling position in the k-space means integration of the gradient as shown previously in Equation 2.25 0 ( ) ( ) t x k t =γ ∫ Gx τ dτ (2.26) 0 ( ) ( ) t y k t =γ ∫ Gy τ dτ (2.27) where Gx is the gradient waveform along the x-axis, and Gy is the gradient waveform along the y-axis. Then the sampled complex raw signal data in k-space is used for image reconstruction by Fourier transformation. Note that the center part of the k-space 25 Figure 2.7. Illustration of the k-space. (a) Ky is the vertical phase encoding axis and the Kx axis is the frequency encoding (readout) axis, (b) an example k-space filling scheme. Ky Kx (a) (b) ΔKy FOVy ΔKx FOVx 26 determines the image contrast and the peripheral area determines image resolution and fine details. The units of Kx and Ky are in spatial-frequency, such as cycles/cm [9]. Figure 2.8 shows example waveforms of the gradient and RF and the k-space. For many conventional imaging techniques, one line of k-space is acquired with each RF excitation, which is conventionally called readout (Gx) encoding. Before readout encoding using a constant Gx, a pulse of Gy is used to generate a single encoding in the phase direction, which is called phase encoding. The time required to play out the pulse sequence (see Figure 2.8a) and acquire a line of k-space is called the repetition time (TR). Typically this time includes an excitation RF pulse and associated gradient, a readout gradient, a phase encoding gradient pulse as well as a gradient associated with the data acquisition window. The pulse sequence is repeated until sufficient data is acquired to reconstruct an image or a slice (or images for a volume). Magnetic Resonance Imaging (MRI) hardware consists of five components: (1) the main magnet system; (2) the gradient array system; (3) RF coil system; (4) computer system; and (5) operating console. [10] The main magnet provides an essential environment for nuclear spins to be aligned along the main magnetic field by creating a very strong homogeneous magnetic field inside the magnet bore (gantry), where the patient is positioned. The gradient array is a set of coils that provide tomographic imaging. In other words, they provide the means to choose image slices of the body by spatially varying the magnetic field. The RF coil allows us to excite certain parts and/or slice of the body, and also detect the signals from the body as a result of excitation. The computer system reconstructs images and synchronizes all the hardware. The operating 2.3 System Hardware 27 Ky Kx (b) Ny Nx Δt τy Figure 2.8. Example gradient waveforms and the k-space. (a) RF and gradient waveforms for Gx, Gx, and Gy as time progress, (b) corresponding k-space. (a) Gz Gx Gy RF Nx Δt Ny τy BWRF 28 console provides an input/output graphical-user-interface (GUI) for the operator. Each and every component of the system is essential and will be discussed in the following sections. Figure 2.9 shows a block diagram of a typical magnetic resonance imaging system. 2.3.1 Main Magnets The main magnet creates a strong homogeneous magnetic field inside the magnet bore. For the main magnets, high strength and high field homogeneity within the desired field-of-view (FOV) are essential. The field homogeneity is defined as 0,max 0,min 0,mean B B Field Homogeneity B − = [11]. (2.28) There are three types of magnets that are used for clinical MRI scanners; a permanent magnet, a resistive electromagnet, and a superconducting electromagnet. A permanent magnet creates a magnetic field between the two poles of the magnet from the permanently ferromagnetic materials. The advantages of the permanent magnet include no requirement for additional electrical power or cooling, very limited magnetic fringe field, and minimal missile effect. However, due to weight considerations, permanent magnets are usually limited to have maximum field strengths of 0.4 T [12]. Also, the main disadvantages of a permanent magnet are the supporting structures of the heavy magnet, and the less homogeneous main magnetic field; as a result, the field homogeneity of the permanent magnet MRI system is worse than the other systems and can be problematic. Generally, resistive magnet MRI systems have better field homogeneity and higher field strength than the permanent magnet MRI systems; however, they typically 29 Figure 2.9. Block diagram of the MRI scanner hardware. This dissertation focuses on the gradient coil. 30 require large current values and significant cooling of the magnet coils. The resistive magnet does not require cryogens, but needs a constant power supply to maintain a homogenous magnetic field, and can be quite expensive to maintain. Nowadays, manufacturers of MRI systems focus more towards the permanent magnet systems than resistive magnet systems, because permanent magnets can create magnetic fields that are as strong as resistive magnet systems. Due to its extremely high strength and field homogeneity, the most common type of magnet used in MRI systems is the cylindrical superconducting magnet. Because high field strength means better signal-to-noise-ratio (SNR) and better (i.e. higher) resolution, 1.5 T ~ 7.0 T superconducting magnet systems are predominant nowadays despite the high cost of the system and maintenance. As a superconductor needs to be kept below a certain temperature to attain superconducting properties, so the main magnet system requires a complex apparatus (cryostat) and frequent maintenance. Usually, Niobium-Titanium (NbTi) coils are used as a superconductor for MRI systems, and they are contained in liquid helium and liquid nitrogen at 4.2º K [13]. Figure 2.10 shows a cooling system of the superconducting magnet MRI. Some systems such as Siemens Tim Trio (Siemens Medical Solutions, Erlangen, Germany) only use helium cooling for the whole system without additional liquid nitrogen cooling. To achieve high field homogeneity, manufacturers use additional coils, called shim coils, to compensate the main magnetic field to create an even more homogeneous main field. A set of superconducting shim coils allows fine tuning of the magnetic field with homogeneity up to ±1 ppm in a 40 or 50 cm spherical volume. Another concern of the superconducting MRI system is the fringe field, which is an undesired magnetic field outside of the magnet bore. This fringe field can become 31 Figure 2.10. A cross section (axial) of the cryostat for MRI systems. Superconducting wires are enclosed in a liquid helium (LHe) casing, which is also enclosed in a liquid nitrogen (LN2) casing with vacuum layers in between. : Superconducting wire (Ni-Ti) : Liquid Nitrogen Cooling (< 77° K) : Liquid Helium Cooling (< 4° K) : Vacuum 32 strong as the main field strength increases. Therefore, the surrounding hardware as well as other medical devices can be affected by this fringe field and can malfunction. To overcome this problem and minimize the fringe field, magnetic field shielding techniques are used. 2.3.2 Gradient Arrays The gradient array (also called the gradient system or gradient coil) is a set of coils that produces a varying magnetic field within the magnet bore. There are three gradient coils that create changes in the main magnetic field (Bz) along three different perpendicular axes, which are generally known as the x-gradient, y-gradient, and z-gradient. In fact, these coils also produce By and Bx magnetic field components, however, these non-Bz components are ignored because they are perpendicular to the strong main field (B0), and have negligible effect for high-field MRI. Generally, the gradient array consists of multiple layers of copper wire patterns including layers of three orthogonal axes (x-, y-, and z-axis), a cooling layer, and gradient shielding layers. The important parameters of the gradient coil include its linearity, strength, slew rate, and inductance. Since the gradient strength affects image resolution, high gradient strength is desirable as shown in Figure 2.11. The slew rate is the maximum achievable rate of rise to its maximum gradient amplitude and it is expressed in the units of T/m/s or mT/m/ms. More detail regarding the gradient array such as criteria, types, design method, and wire patterns will follow in the next chapter. 33 Figure 2.11. An illustration shows the relationship between the gradient strength and the slice thickness ( Δz ); while the same bandwidth of RF pulse is applied, the higher strength gradient (dotted line, Gz') allows narrower slice selection than lower strength gradient (solid line, Gz). The ability to define a fine slice results in higher resolution of the image. Z-direction (slice position) Frequency (f0) 0 ω (RF pulse) Gz' Gz Magnetic field (B0) BW 0 Bz(z)=B +Gz' ⋅z 0 Bz(z)= B +Gz⋅z Δz 34 2.3.3 RF Coils The RF coils have two essential roles for MRI: (1) to induce spin precession by generating and applying pulses near the Larmor frequency, and (2) to detect resultant NMR signals. An RF coil can function as a transmitter, receiver, or both transmitter and receiver. During the transmission state, energy in the form of a radiofrequency (RF) pulse is transmitted throughout the volume of the RF coil. If the RF pulse is at the resonant frequency for a part of the human body, it can cause the magnetization to tip from the main magnetic field direction. The tipped magnetization rotates coherently (Equation 2.12) to the transverse magnetization (Mxy). As soon as the RF pulse is turned off, the rotating transverse magnetization (Mxy) induces current in the receiver coil as described in Equation 2.18. Although there are many types and variations of RF coils, RF coils can be categorized as two basic types: volume coils and surface coils. Volume coils are designed to surround the body part that is being imaged, while surface coils are designed to be placed in close proximity on the surface (top, side, or bottom) of the body part that is being imaged. Volume coils have a more uniform field distribution within the body than surface coils, as well as more uniform receiving sensitivity than surface coils. However, surface coils have the advantage of very high sensitivity near the coil surface. 2.3.4 Computer System The main function of the computer system is to synchronize hardware such as gradient coils and RF coils with a prescribed pulse sequence, and to reconstruct an image from the received signal as shown in Figure 2.9. 35 For image reconstruction, the computer systems of the current scanners have array processors, which allow very fast image reconstruction (20~50 images per second) [14]. A few concerns regarding the computer system are the storage spaces, and the multiuser/multitasking ability. 2.3.5 Operating Console To operate the MRI system, the operator has to input/select the scanning protocol, the pulse sequence. The operating console provides a graphical user interface (GUI) not only for selecting pulse sequences, but also for reviewing reconstructed images. A few more functions of the operating console include retrieving scanned raw data and reconstructed images, image assessment, and later access to the data. [1] I. I. Rabi, J. R. Zacharias, S. Millman, and P. Kusch, "A new method of measuring nuclear magnetic moments," Phys. Rev., vol. 53, p. 318, 1937. 2.4 References [2] F. Bloch, W. W. Hansen, and M. Packard, "The nuclaer induction experiment," Phys. Rev., vol. 70, pp. 474-485, 1946. [3] E. M. Purcell, H. C. Torrey, and R. V. Pound, "Resonance absorption by nuclear magnetic moments in a solid," Phys. Rev., vol. 69, pp. 37-38, 1946. [4] P. C. Lauterbur, "Image formation by induced local interactions: examples employing nuclear magnetic resonance," Nature, vol. 242, pp. 190-191, 1973. [5] P. Mansfield and P. K. Grannell, "NMR 'diffraction' in solids?," J. Phys. C: Solid State Phys., vol. 6, pp. L422-L426, 1973. [6] E. M. Haacke, R. W. brown, M. R. Thompson, and R. Venkatesan, Magnetic Resonance Imaging: Physical Principles and Sequence Design. New York, NY: 36 A John Wiley & Sons, Inc. Publication, 1999. [7] F. Bloch, "Nuclear Induction," Phys. Rev., vol. 70, pp. 460-474, 1946. [8] N. Bloembergen, E. M. Purcell, and R. V. Pound, "Relaxation effects in nuclear magnetic resonance absorption," Phys. Rev., vol. 73, pp. 679-712, 1948. [9] D. G. Nishimura, Principles of Magnetic Resonance Imagin: Stanford Press, 1996. [10] S. C. Bushong, Magnetic resonance imaging: physical and biological principles, Third ed. St. Louis, MO: Mosby, 2003. [11] G. B. Chavhan, MRI made easy. Kent, UK: Anshan, 2007. [12] J. L. Prince and J. M. Limks, Medical Imaging: Signals and Systems. Upper Saddle River, NJ: Pearson Prentice Hall, 2006. [13] V. Lee, Cardiovascular MR Imaging: Physical Principles to Practical Protocols, 1st ed. New York, NY: Lippincott Williams & Wilkins, 2005. [14] Z. P. Liang and P. C. Lauterbur, Principles of Magnetic Resonance Imaging: A Signal Processing Perspective. Piscataway, NJ: IEEE Press, 2000. CHAPTER 3 BACKGROUND II - GRADIENT ARRAYS In order to determine spatial position of protons in an image it is necessary to vary the magnetic field strength in a controlled manner over the imaged volume. Since the frequency of precession is determined by the magnetic field, spatial information can be derived easily by imposing magnetic field gradients. Current implementations of these spatially localizing gradients are achieved by varying the main component of the magnetic field (e.g., Bz), which is the same direction to the main magnetic field (B0), linearly along three axes (e.g., x-, y-, and z-axis, which are typically left/right, anterior/posterior, and superior/inferior directions respectively, and not necessarily perpendicular to each other). Note that other components (e.g., Bx and By) of the magnetic field than the component along with the main magnetic field (B0) are ignored because those components are very small compared to the main magnetic field. These gradient fields are turned on and off during the imaging process synchronized by prescribed pulse sequences. Achieving high gradient field homogeneity and linearity over large volumes, however, is challenging due to conflict between performance parameters such as efficiency, inductance, and homogeneity. Despite the crucial role of gradients in MR imaging, there were surprisingly few researchers contributing to gradient coil development until the mid 1980s [1]. Since then, more efficient, highly homogeneous, and faster switching gradient coils have been 38 developed to meet the needs of faster image acquisition time. There are several outstanding developments which include: an introduction of the stream function (SF) to specify current density [2-4], use of distributed current loops [5], minimization methods for power dissipation [6], an improvement in fabrication methods and high-current gradient power amplifiers [7-9]. As technology for other aspects of MRI develops such as high field main magnets, high-performance power amplifiers, and fast pulse sequences, faster gradient systems are needed more than ever because gradient coil performance is the main factor that determines imaging time and resolution. Fast-imaging sequences such as echo planar imaging (EPI) [10-12], would benefit enormously from improved gradient capabilities. Although up-to-date gradient systems achieve high gradient amplitudes and faster switching gradient fields with high voltage gradient amplifiers, MRI systems have reached the point that increases in gradient performance will also increase the risk of peripheral nerve stimulation (PNS), which results in patient discomfort and/or pain [13- 15]. The nerve stimulation can be reduced by using gradient coils with reduced volume for specific applications. For example, local head insert gradient systems with reduced volume, which have high gradient performance have been used [16, 17] and uniplanar or biplanar flat [18-24] insert gradient coils have been built to improve gradient performance for other applications. In summary, faster gradient systems are very demanding and desirable for their high efficiency for fast-imaging pulse sequences, where the system whole body gradient system is not enough. One of the main motivations for this dissertation is to develop various geometries of gradient coil systems with high-performance parameters and less PNS. 39 This chapter first defines gradient system performance measures and design criteria followed by types of existing gradient coils. The design and optimization methods for gradient systems are also described, as well as state-of-the-art systems and other considerations for the gradient system. 3.1 Criteria of the Gradient Systems This section defines the important performance measures for gradient coil systems including efficiency, inductance, homogeneity, and homogeneous gradient volume. 3.1.1 Efficiency Gradient efficiency is defined as gradient strength per unit current. Higher efficiency is attained by increasing wire density around the gradient system surface. 3.1.2 Inductance and Slew Rate An ideal gradient coil has a very high slew rate, which means short rising/switching time. This parameter is directly related to gradient amplifier voltage and gradient coil inductance as high inductance results in slower switching time for the same voltage. Thus, low inductance is desired. Local insert gradient systems have lower inductance than conventional whole body system gradients because of the reduced size. When inductance decreases, efficiency increases, and homogeneity is reduced. This is why insert gradient systems may be preferable to the system body gradient coil for some anatomy-specific imaging tasks that require fast imaging sequences such as EPI and gradient stringent imaging 40 tasks such as DWI (diffusion weighted imaging). 3.1.3 Homogeneity Homogeneity is important because it measures the uniformity of the resulting gradient field, which in turn, describes how much the resulting image will be distorted. A simple way to define homogeneity is shown in Figure 3.1. The desired magnetic field gradient within the defined field-of-view (FOV) is compared to the actual magnetic field generated. The root-mean-square (RMS) difference between the desired gradient and the actual gradient are integrated over the FOV along each axis to determine homogeneity for that axis. To represent homogeneity for rapid visual analysis, gradient fields are commonly displayed as a gradient field map. A gradient field map shows how much the generated magnetic field differs from the desired magnetic field of the gradient, and contour lines are used as an indication of deviation. In other words, each contour line of the gradient field map represents a deviation from the desired magnetic field of the gradient. As an example of the x-gradient, , , , ( ) ( ( )) / , x x z desired z true x x z x FOV Deviation B B xG B x G Δ = − = − (3.1) where x G is the gradient along the x-axis, and x z B is the z component of the magnetic field from the x-gradient ( / z dB dx ). Note that gradient field maps depend only on the gradient, and not on the actual magnetic field (B0). Examples of gradient field maps are shown in Figure 3.2. 41 Figure 3.1. Defining homogeneity for the x-gradient. The actual generated gradient field is represented by the dotted line, and the theoretical desired gradient is represented with the solid line. The RMS values of the difference in every x-position are used for homogeneity calculations. x z B is the z component of the magnetic field from the x-gradient). x BZ X Desired Gradient Actual Gradient 42 (a) (b) Figure 3.2. Example gradient field maps of the conventional cylindrical gradient. (a) x/y-axis view of the y-gradient field, and (b) x/z-axis view of the x-gradient field. y x z x 43 3.1.4 Homogeneous Gradient Volume Homogeneous gradient volume (HGV) describes the size of the imageable volume within a given gradient system. HGV is the volume where the gradient field is within a specified deviation. During the design/optimization process, this volume is maximized because this is the actual region where the distortion is considered acceptable. DSV (Diameter of homogeneous Spherical Volume) is another, similar, way to definition the size of the imageable homogeneous region. 3.1.5 Other Parameters There are other parameters to be considered when designing gradient coil systems such as low power dissipation, force and torque balance, cooling, total weight of the coil, and less acoustic noise. These parameters should not be neglected during the design process, because they are as important as the criteria described earlier, and will be discussed later in this chapter. 3.2 Types of Gradient Coils Gradient coils can be categorized in two ways. First, the gradient coils can be categorized by shape such as a cylindrical gradient coil, a planar gradient coil, and a hemi-spherical gradient coil. Second, they can be classified by function such as a whole body gradient coil, and a local insert gradient. 44 3.2.1 Gradient Systems by Shape/Geometry 3.2.1.1 Cylindrical Geometry Cylindrical geometry is the most available and standard geometry for the clinical gradient systems, and there have been numerous designs and technical developments for this geometry [1, 4, 6, 25-36]. The cylindrical geometry provides magnetic fields with the best homogeneity and the largest homogeneous gradient volume (HGV). (see Figure 3.3(a)). 3.2.1.2 Planar Geometry As the name implies, planar gradient systems (Figure 3.3(b)) use multiple layers of planes to create gradient fields in all x- y-, and z-axes rather than using a cylindrical form as in the cylindrical gradient system. There are two types of planar gradient systems, uni-planar and bi-planar systems. Bi-planar gradient systems [19, 21, 23, 24, 37-41] have two parallel plates, one above and one below the patient, and uni-planar gradient system [18, 22, 42-48] consists of one plane, usually placed under the patient. While a uni-planar gradient set gives better patient access and allows more room for the gradient system and RF coil hardware, bi-planar gradient coils give better homogeneity and a larger homogeneous imaging volume. A common application of planar gradient geometries are open MRI systems. Open MRI systems are designed for patients who are too big to fit in the cylindrical main magnet bore and/or the patients who have claustrophobia. The useable gradient fields of the planar gradient systems are smaller than those of cylindrical geometry gradient coils with less homogeneity. However, planar gradient systems create very strong gradient fields near the surface of the gradient coil, which yields high efficiency, 45 Figure 3.3. Various shape gradient coil systems. The lines representing wire patterns for each geometry. (a) cylindrical gradient coil, (b) planar gradient coils, (c) elliptical gradient coil (a) z z x (b) 46 allowing high resolution imaging. They can be placed very close to the patient. 3.2.1.3 Other Geometries Although cylindrical and planar geometry is the most common geometrical choices for the gradient systems, gradient coils can be designed on arbitrary surfaces in order to fit any specific anatomy. However, they become harder to design when the shape of the coil is not cylindrical, because it is complicated to derive surface current density equations for the arbitrary surface. So far, various geometries have been used for gradient system design. Cylindrical shaped gradient coils modified to have an elliptical cross-section are thought to better fit the human body/torso (Figure 3.3(c)). There have been a few studies about elliptical gradient systems [49-51]. Spherical/hemi-spherical shaped gradient coils have been developed because the geometry seems to fit the human head well [52, 53]. 3.2.2 Gradient Systems by Function 3.2.2.1 Whole Body Gradient Coil The whole body gradient system is the conventional gradient system for clinical scanners, which is integrated into structure of the MR system. Although whole body gradients have lower efficiency and higher inductance compared to local insert gradient systems, they have the advantage of a larger homogeneous imaging volume to cover a greater volume of the patient body without repositioning. Currently, most clinical imaging is done with conventional cylindrical gradient coils. 47 3.2.2.2 Local Insert Gradient Coil Local insert gradient systems are relatively smaller gradient systems than conventional whole body system gradient coils, and are used as an insert to fit in the magnet bore. The insert gradient systems are usually designed for specific applications such as head/neck [16, 17, 51], breast [39, 54-56] or knee imaging [57]. Despite disadvantages such as smaller homogeneous imaging volume, insert gradient systems can be very desirable for a number of reasons. These advantages include high efficiency for the equal driving current and low inductance, which results in higher slew rate. In addition, PNS issues can be reduced with the use of insert gradients that are designed for a reduced imaging volume, which helps to minimize the extent of the associated magnetic field excursions and the resulting electric fields and currents induced in the patient [14, 58]. In conclusion, the local insert gradient systems can be used very effectively for imaging with specific anatomy. 3.3 Design Methods for Conventional Cylindrical Gradient Coils This section explains the basics of gradient coil design and the underlying physics. Gradient coil design is basically deciding the position and shape of the wire on the chosen form shape (such as cylindrical). This wire position/shape is simply called a wire pattern, winding pattern, or fingerprint. Several design and optimization techniques that are commonly used are covered in this section. 48 3.3.1 Target Field Method Of several methods to design gradient coils that have been developed, the target field method is one of the most commonly used [1, 6, 35, 59-64]. The target field method was developed by Robert Turner in 1986 [35]. This method calculates the required surface currents directly from the specification of the desired field, referred to as the target field, by inverting Ampere's law. The advantages of the target field method include fast simulation speed and ease of implementation, because they use analytical equations, and there are many existing fast Fourier transform program modules already available. The target field method starts by assuming a current density and defining the Fourier transform of the surface current density (J) as ( ) 1 ( , ) 2 m im ikz z z j k d e dke J z π ϕ π ϕ ϕ π − ∞ − − −∞ = ∫ ∫ (3.2) ( ) 1 ( , ) 2 jm k d e im dke ikz J z π ϕ ϕ π ϕ ϕ ϕ π − ∞ − − −∞ = ∫ ∫ , (3.3) so that these equations satisfy the continuity equation ∇J =0 (3.4) (i.e. m( ) m( ) z j k ka j k m ϕ = − ). (3.5) The next step is to expand the Green's function in the cylindrical geometry, as shown below: ( , ') 1 1 ( ') ( ') ( ) ( ) ' im ik z z m m m G r r dke e I k K k r r ϕ ϕ ρ ρ π ∞ ∞ − − − − < > −∞ =−∞ = = − Σ ∫ , (3.6) where ρ<(ρ >) is the lesser (greater) of ρ and ρ ' , and m I and m K are the first and second kind of modified Bessel function as follows: 49 ( ) 1 ( / 2)( 1/ ) 1 2 z k k m m I z dke k π i ∞ + − − −∞ = ∫ (3.7) ( ) ( ) ( ) 2 sin( ) m m n K z I z I z n π π − − = (3.8) The vector potential (A) in cylindrical coordinates can be calculated as: ( , , ) 0 4 ' A z Jdv r r μ ρ ϕ π = − ∫ , (3.9) ( i.e. 0 ' ( ') 'sin( ') 4 ' J r dv A r r ϕ ρ μ ϕ ϕ π − = − ∫ , (3.10) 0 ' ( ') 'cos( ') 4 ' J r dv A r r ϕ ϕ μ ϕ ϕ π − = − ∫ (3.11) 0 ( ') ' 4 ' z z A J r dv r r μ π = − ∫ ) (3.12) And after further derivation with the Green's function expansion, the vector potential equations become: 0 1 1 1 1 1 1 1 1 exp( ) exp( )[ ( ( ) ( ) ( ) ( )) ( ) 4 ( ( ) ( ) ( ) ( )) ( )] m m m m m m m m m m m A i dk im ikz b I kb K k I kb K k j k a I ka K k I ka K k J k ρ ϕ ϕ μ ϕ ρ ρ π ρ ρ ∞ ∞ −∞ − − + + =−∞ − − + + − = − + − Σ ∫ (3.13) 0 1 1 1 1 1 1 1 1 exp( ) exp( )[ ( ( ) ( ) ( ) ( )) ( ) 4 ( ( ) ( ) ( ) ( )) ( )] m m m m m m m m m m m A dk im ikz b I kb K k I kb K k j k a I ka K k I ka K k J k ϕ ϕ ϕ μ ϕ ρ ρ π ρ ρ ∞ ∞ −∞ − − + + =−∞ − − + + = + + + Σ ∫ (3.14) 0 exp( ) exp( ) ( )( ( ) ( ) ( ) ( )) 2 m m z m m z m z m A dk im ikz K k bI kb j k aI ka J k μ ϕ ρ π ∞ ∞ −∞ =−∞ = Σ ∫ + (3.15) Finally, we can derive the magnetic field equation with these vector potential equations above as: ( , , ) 0 ( ) ( ) ' ( ) 2 im ikz m z m m m B r z a dke ϕ e k j k I kr K ka ϕ μ ϕ π ∞ ∞ < > −∞ =−∞ =− Σ ∫ ⋅ . (3.16) Note that even though the magnetic field is a vector field with components from all 50 three axes (i.e. [Bx, By, Bz]), the only field we are interested in for magnetic resonance imaging is Bz, which is in the direction of the main magnetic field. The target field method require a fairly large number of target field points for better accuracy, and is also less realistic in the manufacturing point of view since there is no direct restriction on the overall length of the coil. 3.3.2 Stream Function Method The stream function (SF) method, which was first introduced by Shenck and Sutherland in 1983 [3] is another very popular gradient coil design method [25, 32, 34, 37, 41]. Originally, stream functions were developed and used in fluid dynamics and are still being used by many researchers for gradient coil design. The stream function specifies the current densities, which in turn are used to specify the wire pattern to most closely approximate the desired current densities. Once the current density is specified, a wire is placed at every (or several) contour representing a specific current density. To determine the wire patterns with these stream functions, first we define a surface current density (J) on the surface of cylinder J = ∇ ×[S(ϕ,z)⋅ar]. (3.17) The S in Equation (3.17) is the stream function and is an arbitrary function initially. With this stream function we define the z (axial) component and ϕ (radial) component of J as shown below in the case of cylindrical coordinates 1 ( , ) z j S z r ϕ ϕ ∂ = − ∂ (3.18) j S( ,z) z ϕ ϕ ∂ = ∂ . (3.19) 51 By inverting these equations, we can derive the equation of S(φ,z) as ( , ) ( ) im ( )cos( ) m m Sϕz h z v e ϕ h z mϕ ∞ =−∞ = Σ ≈ , (3.20) where m=0 for the axial (z) gradient coil and ±1 for the transverse (x and y) gradient coil design. Once the stream function is defined, the wire positions are obtained according to the axial component, z, and azimuthal component,ϕ . However, without optimization, the stream function can be poorly chosen and produce a small HGV. Thus this method requires some type of optimization process. The optimization methods for gradient design will be discussed later in this chapter. 3.3.3 Z-intercept Method One group of researchers introduced a Z-intercept method [65-68] for cylindrical gradient coil design and the supershielding (SUSHI) method [69-72] for gradient shielding design. These two methods share the same basics. These researchers noticed that the turns of coil wire on the primary gradient coil are all perpendicular to the z-axis along the lines at ϕ =0 and ϕ =π in the conventional gradient wire patterns, and also the spacing between wire turns along these lines determines the allowed conductor width and positions. So, they have concluded that all characteristics of a transverse gradient coil can be expressed through the set of z-intercept points (Figure 3.4), where the coil current patterns intersect the cardinal axis ϕ =0 for the x-gradient coil design, and ϕ =pi for the y-gradient coil design. These z-intercept points are varied, monitored, and optimized until the resulting gradient field meets design criteria such as minimum coil inductance, gradient field homogeneity, and gradient strength. 52 Figure 3.4. Illustration of z-intercept points. Intercept points along the ϕ =0 line (z-axis) are called z-intercept points, which are varied and optimized for best gradient coil performance. 53 The Z-intercept method is somewhat similar to Turner's conventional method [1], ( ) arccos ( 1/ 2) n z n d Nz ϕ =  +    , (3.21) where N is a total number of wire turns, n is the index of each individual turn, z is the position along the z-axis, and d is the distance to the eye of the wire pattern. In the Z-intercept method, the ϕ along the z-axis is defined analytically as / ( ) arccos 1/ 2 ( ) / ( ) n eye P S z n Q z Q z N  −  Φ =     (3.22) where / ( ) ( ) ( ) ( ) 1 ( ) ( ) ( ) ( ) ( ) 2 2 2 2 NP S in in f f n n n n n Q Z erf z Z erf z Z erf z Z erf z Z = β β β β =  − + + − − − +    Σ ,(3.23) Zn are the z-intercept points, z is the position along the z-axis, β is a small parameter for smoothness of the current centroid, and Zeye is the root of ( ) 0 eye Q′ z = . The advantages of the Z-intercept method are that it allows smooth wire patterns for easy construction and relatively faster optimization processes due to the small number of parameters to be varied. However, this method does not explore all possible solutions of wire pattern design, and also requires a complicated inductance minimization process. 3.3.4 Boundary Element Method In recent years, there have been developments in gradient coil design for arbitrary geometries. Conventional methods such as the target field method only allow gradient designs for certain geometries where the solution exists and is simple in form (such as the cylindrical geometry). The stream function method allows more flexible geometries such as planar gradient geometry, however, the solutions become more 54 complicated and computation time increases significantly for the optimization process. The boundary element method (BEM), which is similar to the finite element method, defines the gradient coil geometry on an arbitrary surface. This surface consists of a large number of small triangular elements, and can be any shape (see Figure 3.5). The main disadvantage of this method is the calculation time. Since BEM involves a large number of mesh elements, the accuracy of the parameters obtained from the simulation results such as inductance, homogeneity, and efficiency, are heavily dependent on the number of mesh elements. Of course, a large number of mesh elements are desirable; however, calculation time will be increased exponentially. 3.3.5 Other Methods There have been other methods such as the matrix inversion method and the method of moments. The matrix inversion method [1, 73, 74] is related to the finite element method, and copes with any shape of coil geometry, like the BEM. However, it has relatively slow computation time, and gives less accurate field simulation results. The method of moments [75-78] is the prior version of FDTD/BEM, which uses the same triangular element. 3.4 Optimization Process Optimization of the gradient coil design is essential and very challenging not only because there are many parameters to be optimized, but also because of the conflict between the parameters. In other words, there are trade-offs between efficiency, inductance, and homogeneity. Therefore, finding an optimal operating point with high efficiency, good homogeneity, low inductance as well as with other performance 55 Figure 3.5. Illustration of boundary element method (BEM). Mesh grid of a top half of the superellipse is displayed showing that the BEM is capable of handling more flexible geometry than previous conventional gradient designing methods. 56 measures is not a simple task. Theoretically, any kind of optimization methods can be used for gradient coil design. However, optimization can take a long time and not every method can yield good results, because the design solution requires the optimization of many parameters and has many local minima. Various optimization methods such as the conjugate gradient method, genetic algorithm, simulated annealing method, and frequency optimization method have been used, and they are discussed in this section. 3.4.1 Gradient Descent and Conjugate Gradient Methods The gradient descent method and conjugate gradient methods, which are relatively simple and fast, have been used in a few studies [79-82]. However, these methods are often prone to being trapped in local minima, and therefore are not effective in solving multidimensional problems with many local minima. 3.4.2 Simulated Annealing Method Simulated annealing (SA) is one of the most popular methods for designing a gradient system, because this method can handle optimization problems with many parameters and local minima. Since Metropolis et al. introduced simulated annealing [83] in 1953, simulated annealing has been used for gradient coil system optimization [26, 27, 31, 40, 49, 84-86], and its modified technique called the fast simulated annealing has been introduced [34, 37, 41]. In medical imaging, simulated annealing has been applied to X-ray computed tomography (CT) image reconstruction [87], selective pulse design for MRI [88], gradient waveform design [89], and spectral curve fitting and parameter estimation for MR spectroscopy [90]. 57 The simulated annealing method mimics the annealing process of metallurgy. The simulated annealing represents the cost function as a temperature (T) and occasionally allows the solution to jump to a new parameter state to avoid being trapped in local minima. The occasional jump accepts some solutions that are worse than previous solutions with a probability that depends on Boltzmann statistics ( exp(−ΔE /(K ⋅T)) ). During each step of the optimization process, the SA algorithm replaces the current solution by a random nearby solution if the new solution is better than the previous solution, and guarantees that the optimized solution will not be stuck in local minima. The probability of acceptance of a new-worse solution decreases as the temperature decreases during the optimization process until it reaches a freezing state, which is the end of the simulation. Figure 3.6 illustrates the process of an SA optimization. In addition to this, by saving all iterated solutions, the simulated annealing algorithm can explore a wide range of possible solutions including an optimum solution as well as near-optimum solutions in the feature space. Hence, it is possible to review trade-offs between the simulation parameters as well as to evaluate other factors that might be important [28, 29]. 3.4.3 Genetic Algorithm Method The genetic algorithm (GA), which was invented by John Holland in 1975 [91], is another algorithm that is less susceptible to getting stuck in local optima than gradient search methods. Although GAs tend to be computationally expensive, they can handle the multivariable optimization problem and have been used to design gradient arrays [21]. 58 Figure 3.6. The temperature feature in the simulated annealing (SA) method avoids possible local minima trapping. (a) Example feature dimension of the system to be optimized (b) High system temperature; when the system temperature is high, SA process allows an occasional jump to any possible position (c) Medium system temperature; as SA progresses, the system temperature cools down, limiting its possibility to jump upward in the feature space (d) Low system temperature; as the system cools down close to a frozen state, which is the end of the process, the solution is trapped in the global minimum, theoretically. (a) (b) (c) (d) Feature dimension FOM (cost function) Local minima Feature dimension FOM Temp. Feature dimension FOM Temp. Feature dimension FOM Temp. 59 In genetic algorithm methods, the optimization process (which is called the evolution), usually starts from a population of randomly generated parameters (called individuals). In each generation as optimization progresses, the fitness (FOM) of every individual in the population is evaluated. Multiple individuals are stochastically selected from the current population based on their fitness, and the previous population is modified to form a new population. Then, the newly generated population is used for evaluation in the next iteration. The modification process includes paring, mating, mutation, and crossover. The algorithm finishes its optimization process when a satisfactory fitness level has been reached, or when a maximum number of generations has been executed [92]. 3.4.4 Other Methods There have been other attempts [43, 93-95] to design and optimize gradient coil systems, however, these other methods are less appreciated or are no longer used. 3.5 Other Considerations for Gradient Systems 3.5.1 Eddy Currents As the technology improves, MRI systems that have higher slew rate gradient coils are becoming available. This causes a new problem. Faster switching of the gradient fields (i.e., higher slew rate) causes a circulating flow of currents (eddy currents) in any conductor near the coils including patients and the MRI system hardware (cryostat and shim coils). In turn, these circulating eddies of current create induced magnetic fields, which oppose the change of the gradient fields due to Lenz's law. 60 As a result, eddy currents can cause degradation of the gradient wave form, resulting in image artifacts [17, 96]. Common techniques to reduce the influence of eddy currents on gradient fields are eddy current compensation, which apply pre-emphasis of current on the gradient amplifiers [97, 98] and gradient coil shielding. 3.5.2 Gradient Shielding To reduce eddy currents and to reduce interaction with the superconducting coils, gradient coil shielding is necessary. The external fringe magnetic field created by the primary gradient coil can be canceled by putting a secondary layer of winding pattern outside of the primary gradient coil layer [19, 47, 61, 63, 69, 70]. The current in the shielding layer designed to flows in the opposite direction to the primary coil. As a result, magnetic fields from the primary coil and the secondary shielding layer cancel outside the imaging volume [1, 34]. Even though adding shielding layer reduces the internal magnetic field, it can be compensated by applying increased current. Note that extra shielding layer increases inductance, which in turn increases slew rates. 3.5.3 Force and Torque Balance Because gradient systems generate magnetic fields, they can experience forces and torques in the main magnetic field. Balancing the force and torque is important for all types of gradient systems. For local insert gradient systems, especially, the force and torque estimation has to be made more carefully since the local gradient systems often designed as asymmetric in shape and off-positioned from the iso-center of the main magnetic field. Unbalanced force and torque of the insert gradient system can cause 61 serious patient endangerment during scan. Local insert gradient coil systems are usually very heavy (i.e., high density) compared to their size, and can weigh more than several hundred pounds. Moreover, in the case of asymmetrical gradient systems such as a uni-planar insert gradient, miscalculation of the force and torque balance can cause the gradient system to move suddenly, resulting in serious patient injuries. The net force calculation of the insert gradient coil is usually done by dividing the wire pattern into finite segments and using the Lorentz force equation [99] 1( ) N i i F I dl B = = Σ ×    . (3.24) The torque on the insert gradient coil due to the external Lorentz force from the main magnetic field can be calculated by [99] 1 ( ( )) N i i i τ I r dl B = = Σ × ×     . (3.25) Both equation 3.24 and equation 3.25 involve a numerical integration over the entire wire pattern. Each wire segment of the insert gradient will contribute to the net force, and the force on each wire segment will contribute to the torque. 3.5.4 Acoustic Noise Thumping noises caused by the gradient system are one of the most uncomfortable factors for patients. This noise occurs during the rapid alterations of currents within the gradient coils. These currents, in the presence of a strong static magnetic field of the MR system, produce significant Lorentz forces. The Lorentz force causes motion or vibration of the gradient coils which makes them impact their mounting hardware, which also flexes and vibrates. 62 The level of noise depends on a prescribed pulse sequence. 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Jiang, "A new eddy-curent compensation method in MRI," PIERS Online, vol. 3, pp. 874-879, 2007. 72 [98] M. Terpstra, P. M. Andersen, and R. Gruetter, "Localized eddy current compensation using quantutative field mapping," Journal of Magnetic Resonance, vol. 131, pp. 139-143, 1998. [99] H. D. Young and R. A. Freedman, University Physics, 11th ed.: Addison Wesley, 2003. [100] FDA(US), "Guidance for Industry: Guidance for the Submission Of Premarket Notifications for Magnetic Resonance Diagnostic Devices," 2003. CHAPTER 4 LOCAL BI-PLANAR GRADIENT ARRAY DESIGN USING CONFORMAL MAPPING AND SIMULATED ANNEALING 4.1 Abstract Many magnetic resonance imaging applications require high spatial and temporal resolution. The improved gradient performance required to achieve high spatial and temporal resolution may be achieved by using local gradient coils such as planar gradient inserts. The planar gradient set provides higher gradient performance because it is placed inside of the imaging bore of the magnet (within the body gradients) in close proximity to the imaging region. Although the wire patterns for planar gradients can be designed using two-dimensional stream functions and simulated annealing, optimization of the two-dimensional stream functions can be much more computationally intensive and time consuming than optimizing the one-dimensional stream functions required for cylindrical gradients. To address this problem, we have developed a simple and rapid method for the design of planar gradient inserts to produce a high strength local gradient field and a reasonably uniform imaging region. By using conformal mapping, the two-dimensional problem can be simplified to a faster and more easily calculated one dimensional problem. The mapping transforms the magnetic field and wire patterns in the cylindrical system into a magnetic field and wire patterns in the bi-planar geometry providing a tool for bi-planar gradient coil design 74 using a one-dimensional stream function. 4.2 Introduction Magnetic resonance imaging (MRI) systems are continually improving with increases in the main magnetic field strength, RF coil techniques, and gradient system performance. Many clinical MRI applications would benefit from further increases in spatial and temporal resolution. In general, MRI image acquisition speed and resolution are limited by gradient performance factors such as gradient imaging volume, homogeneity, efficiency, and inductance. Efficiency and inductance confine gradient amplitude and slew rate. In general, gradient coil performance is also limited by the potential for peripheral nerve stimulation (PNS) [1]. The potential for PNS can be reduced by using gradient coils with reduced imaging volume, which helps to minimize the extent of associated magnetic field excursions and the resulting electric fields and currents induced in the patient. One way to reduce the gradient imaging volume and increase gradient performance is to develop anatomy specific local gradient coils, such as planar gradient inserts. For example, in the case of dynamic contrast enhanced (DCE) MRI for breast lesion characterization, a planar gradient insert can achieve high gradient performance in the breast close to the planar surface. Planar gradient coils, where the currents are constrained to planar surfaces, create magnetic field gradients that are strong near the surface of the planar coils and decrease rapidly as distance from the surface increases. Bi-planar gradient coils give a more homogeneous gradient field than single plane gradient coils. The homogeneous 75 imaging volume for planar gradients is smaller than that of the whole body cylindrical gradients. However, the smaller imaging volume allows us to design systems using stronger and more rapidly changing gradient fields for improved resolution, efficiency, and achieving shorter imaging times. Higher performance coils can yield shorter echo-times, resulting in increased signal-to-noise ratio. Typical planar gradient systems are either bi-planar having two parallel plates, one above and one below the patient [2], or uni-planar, usually to be placed under the patient [3]. While a uni-planar gradient set gives better patient access and allows more room for the gradient system hardware, bi-planar gradient coils give better homogeneity and a larger homogeneous imaging volume. In this work, we present a simple and rapid method to design bi-planar gradient coil inserts to produce high strength gradient fields that are reasonably uniform over a wide imaging region. For cylindrical gradients, a one-dimensional stream function is used to specify currents on the cylinder surface [4, 5]. For planar gradients, however, the problem becomes two-dimensional, requiring two-dimensional stream functions [6-9]. Because of the larger number of parameters required to define two-dimensional stream functions, adjusting the parameters to find an optimal stream function becomes a more complicated and time consuming problem. With a proper transformation, a uniform field in the cylindrical system can be transformed into a uniform field in the bi-planar geometry. With this transformation, a cylindrical system stream function can be adjusted while minimizing a figure-of-merit (FOM) based upon inductance, gradient efficiency and homogeneity in the corresponding field of the bi-planar gradients by using simulated annealing (SA). Since 76 introduction by Metropolis et al. [10], simulated annealing has been used for various gradient coil optimization studies [11-15]. Stream function optimization using simulated annealing has advantages over conventional optimization processes such as the conjugate gradient descent method [16] because it allows occasional jumps upward with a probability given by the Boltzmann distribution and therefore is less likely to be trapped in local minima [17]. Genetic algorithms can handle multi variable optimization problems and have been used previously for gradient coil optimization processes [2, 18]. After a stream function has been selected for the cylindrical geometry, it is possible to transform the cylindrical system wire pattern to create a wire pattern for the bi-planar gradients. This process should reduce calculation time, simplify the optimization process, and overcome the complication of a two-dimensional stream function for planar gradient design. In addition, the stream function algorithm supports exploring a wide range of possible solutions including optimum solution as well as near-optimum solutions in the feature space. Hence, it is possible to review trade-offs of the simulation parameters between solutions, and evaluate other factors, which might be important [15, 17]. The detailed optimization processes are described in the following section. 4.3 Theory Our stream function algorithm uses conformal mapping to transform the magnetic field in the cylindrical coordinate into a field map in bi-planar geometry. For the algorithm, a magnetic field map in the three-dimensional volume of cylindrical coordinates is calculated from the current densities specified by a specific stream function along the z-axis. In the design of gradient coils, the currents (and ultimately the 77 wires) are usually constrained to a surface in order that the divergence of the surface current density equals zero [5]. The wire patterns can be calculated from the surface current densities, which can in turn be calculated from the derivatives of a stream function [4]. Given the surface current density function on a cylindrical surface, analytic expressions exist to calculate the magnetic field map as a function of position throughout the cylinder volume. For an unshielded transverse gradient, the magnetic field can be expressed as [19, 20]: 0 0 1 2 0 ( , ) ( ) ( )( ( ) ( )) 4 ikz z B z a dk j k e k I k K k K ka ϕ μ ρ ρ π ∞ −∞ = ∫ ⋅ ⋅ + (4.1) where a is the coil radius, z is the position along the z-axis, ρ is the radius of the field measurements, I and K are the modified Bessel functions, and jm (k) ϕ , m ( ) z j k are the components of the Fourier transform of the current density on the cylinder. Next, each slice along the z-axis in cylindrical coordinates is mapped into the bi-planar coordinates. Figure 4.1 shows a conformal mapping of magnetic field map from cylinder to bi-planar geometry with a transformation: w=Log [(r+1) /(r−1)], (4.2) where w and r are complex variables. Figure 4.1b is an axial slice of the magnetic field map in the cylinder calculated from the wire pattern on the cylinder (see Figure 4.1a), and Figure 4.1c is the field obtained with conformal mapping into the bi-planar coordinates. Because the conformal mapping transformation is surjective there are points which are not filled in after point-to-point mapping to the bi-planar geometry. To avoid empty pixels all points are determined by back-conformal-mapping: Every point of the conformal mapped image is back-mapped into the cylindrical coordinates using the inverse of the conformal transformation. Bilinear interpolation in the cylinder field 78 Figure 4.1. Wire patterns and simulated fields. a-c: wire pattern on the cylinder and its magnetic field map in the cylinder. d-e: wire pattern and its field map on the bi-plane. (All units in mm) a b c d e 79 is then used to obtain the value for the point in the bi-planar coordinates. This process is continued for all points until a fully-filled image is completed. Figure 4.1c illustrates a field map after calculating all points with reverse mapping technique. For comparison, conformal mapping was also used to transform the wire pattern on the cylinder surface to the bi-planar surfaces; see Figure 4.1d, and Figure 4.1e shows a plane of the field map calculated from this direct planar wire pattern using the Biot-Savart law. All the field maps from the wire patterns were generated by using the Biot-Savart law [20]: 0 3 (x) J(x ') x x ' x' 4 x x ' B d μ π − = × − ∫ (4.3) where J(x ') is the current density, x and x ' are the coordinate vectors from the element of length to an observation point. Thus, the conformal mapped field and the field calculated from the conformal mapped wire patterns are very similar. The detailed method for applying the theory described above will be described in the following section. 4.4 Methods In this work, we demonstrate a bi-planar gradient coil design that consisted of two planes of 42 cm in width and 60 cm in length above and below the patient (see Figure 4.2). Our target FOV was 15 cm (left/right), 16 cm (superior/inferior), and 20 cm (anterior/posterior). The FOV can be varied based upon the size and geometry of the gradient system, MRI scanner geometry, and application of interest as long as the gradient system hardware can fit into the magnet bore. In each iteration, simulated annealing starts with randomly chosen control parameters, which are used to determine an initial stream function, which in turn 80 determines the surface current density for a cylindrical geometry. Theoretically, there could be numerous ways to create the stream function, but a simple piecewise power stream function has been used for this study as expressed below: ( ) ( , ) ( ) cos( ), 1 ( / 1) , 0 ( ) ( ) 1, ( ) 1 ( ) (1 ) , 1 ( ) . S z h z z a if z a region I where h z if a z b region II z b b if b z region III α β ϕ = ϕ − − < <  = < <    − − − < <  (4.4) The piecewise power function has the advantage of yielding smoother wire patterns resulting in easier construction and increased gradient coil efficiency as the design eliminates wire jogs. Figure 4.3 shows the stream function plot using the piecewise power function method, and its fixed control points (circled points), which determine the shape of the stream function. An additional advantage of the piecewise power function is there are very few control points, resulting in simulated annealing easier and faster [11, 21]. However, the functional form of these stream function can limit the range of potential solutions. In general, a reduced number of SA parameters results in less flexible stream functions. Next, the analytical equations for the current density on the cylindrical surface are used to calculate the magnetic field over a three-dimensional volume within the cylinder. After the magnetic field for the specified FOV in three-dimension is calculated, conformal mapping by Eq. (4.2) is applied in several closely spaced axial planes (5 planes which are positioned in the center, 3 cm and 7.5 cm apart from the center toward each end) orthogonal to the z-axis of the cylinder to map the magnetic field in the 81 Figure 4.2. Bi-planar gradient system geometry. a: view through the magnet bore. b: view from above. 42cm x y 26cm 60cm 42cm x a b z 82 Figure 4.3. Stream function (SF) plot generated with piecewise power function. Circles show control points to determine SF. (α =2.15, β =1.8, a=0.55, b=0.568) 83 cylinder to the bi-planar geometry. The mapping is shown in Figure 4.4. The numbers shown in the figure indicate corresponding points in the two coordinate systems under the transformation. In this mapping, there are singular points that map to infinity, and they appear near each end on the x axis and y ≈ 0 (i.e., number 8 and 9). Every conformal mapping with an equation that includes division will have singular points when the divisor equals zero. In fact, this phenomenon causes the divergence of the wire pattern while mapping the wire patterns from the cylinder onto the plane if there are wires that are too close to the singular points. The simulation program avoids this problem by removing the outer loop of the winding pattern if the wire loop is not physically realizable in the available area of the plane. For the y-gradient, these singular points are trivial for a typical gradient coil design as they are near null points of the surface current density where the surface current density is zero. This means that at these points the magnetic field is zero ( Bz ≈ 0), having a minimal effect on the net desired gradient field. The next step is to calculate the cost function from the resulting conformal mapped field in the bi-planar geometry as shown below [22]: ( ) 2 5/ 6 cost function max rms = σ L Bo η, (4.5) which is a function of the parameters homogeneity, rms σ , efficiency, η , inductance, L , and max 5/ 6 Bo or the maximum strength of the magnetic field on the plane at a distance of 5/6 of the plane's half-width, respectively. This 5/ 6 max Bo parameter is useful to constrain large excursions of the total magnetic field external to the imaging volume, which primarily occurs in the case of transverse gradients. 84 x          y x y         -∞  ∞ (b) Figure 4.4. Point to point representation before (a) and after (b) the mapping. (a) 85 The homogeneity ( rms σ ), efficiency (η ), and inductance ( L ) are defined as [22]: 1 _ 2 rms GFOV z GFOV B r z dxdydz V σ η =   − ⋅  ∫ , (4.6) _ 2 GFOV z GFOV η = zB  r dxdydz z dxdydz   ∫ ∫ , (4.7) and 0 2 0 2 2 2 0 2 0 ( ) [( ( ) ( ))( ( ) ( ))] 4 L a dk j k I ka I ka K ka K ka I ϕ μ ∞ −∞ = ∫ ⋅ + + , (4.8) where GFOV V is the volume of the imaging FOV allowed by the gradient design (GFOV), and we define GFOV as the desired imaging volume in three-dimension. FOVx, FOVy, and FOVz are defined as the width, height, and length of the GFOV respectively. For inductance, a is radius, I, K are modified Bessel functions, and j (k) ϕ is the components of the Fourier transform of the current density on the cylinder as described in the Eq. (4.1). Note that the inductance is calculated in the cylindrical geometry rather than bi-planar geometry for simulation speed since inductance in the cylindrical geometry and inductance in the bi-planar geometry has a monotonic relationship as shown in Figure 4.5. The FOM parameters of the final cost function for the gradient volume are obtained by averaging over slices along the z-axis as described previously. For the x-gradient, a bi-planar wire pattern cannot be computed directly from the cylindrical x-gradient wire pattern because the current densities are not zero at the singular points of the conformal map. To overcome this problem, the stream function is modified to be S(ϕ , z) = h(z) cos(2ϕ ) instead of S(ϕ , z) = h(z) cos(ϕ ) as in the y-gradient (see Figure 4.6). This creates current nulls at the points of singularity in the 86 Figure 4.5. Inductance comparison between cylindrical and bi-planar geometry. Figure shows its monotonic relationship between inductance in the cylindrical and the bi-planar geometry through the simulation process. Inductance is calculated by using analytical equations and units are normalized. 87 a b Figure 4.6. X-gradient coil wire patterns. a: wire pattern on the cylindrical surface acquired with S(ϕ , z) = h(z) cos(ϕ ). b: wire pattern acquired with S(ϕ , z) = h(z) cos(2ϕ ) to create current nulls at the points of singularity in the conformal mapping for the x-gradient. 88 conformal mapping for the x-gradient. In other words, the wire pattern is doubled up for the x-gradient because cos(2ϕ ) causes the wire pattern to be wrapped two times around the circumference of the cylinder. The gradient coil design simulation program was written in the C programming language for performance. 4.5 Results Figure 4.7 shows simulation results in feature space. Each dot represents solutions visited, and the chosen optimum operating solution is shown as a diamond symbol. Dots are converging toward the bottom right corner which is the area of higher efficiency and lower rms σ value. The diamond symbol indicates the operating point that was chosen to obtain the simulated wire patterns and magnetic fields. The comparison between the gradient field transformed from the cylindrical coordinates and the field directly calculated from the bi-planar wire pattern transformed from the cylindrical coordinates is shown in Figure 4.8. The results show a great deal of similarity between the two approaches as expected, confirming the principal of the new design technique. Table 4.1 compares the efficiency and inductance for the cylindrical and bi-planar x- and y- gradient arrays for the optimally chosen solutions. It is clear that the bi-planar gradient shows high efficiency in both x and y gradient cases. The inductance of the x-gradient coil set for both cylindrical and bi-planar case is higher because x-gradient wire patterns are created by S(ϕ , z) = h(z) cos(2ϕ ) as described in the previous section (see Figure 4.6b). The inductance values shown in Table 4.1 are calculated using Grover's method [23] for different cylindrical and bi-planar wire 89 Figure 4.7. FOM plot from the simulation results for a y-gradient. (Diamond mark indicates the chosen solution) 90 Figure 4.8. Gradient homogeneity plot (y-gradient, Gy) on the x/y plane. a: gradient field map from cylinder-to-biplanar conformal mapped magnetic field. b: gradient map computed directly from the bi-planar wire pattern which is acquired from the same stream function. The dashed rectangle on both figures is same size. b x y x y a 91 Table 4.1. Efficiency and inductance of cylindrical and bi-planar gradients. The x-gradient inductance is higher than y-gradient because it is created using S(ϕ , z) = h(z) cos(2ϕ ) instead of S(ϕ , z) = h(z) cos(ϕ ) . X-gradient Y-gradient Cylindrical Bi-planar Cylindrical Bi-planar Efficiency 0.20mT/m/A 0.28mT/m/A 0.19mT/m/A 0.36mT/m/A Inductance 237.0μH 246.5μH 120.1μH 166.8μH 92 patterns. Figure 4.9 shows additional results for the x- and y-gradients in different axial views in order to characterize the imaging volume. The comparison of the conformal mapped fields and field maps obtained from direct wire patterns for the y-gradient in several different positions along the z-axis is shown in Figure 4.10. This demonstrates that the conformal mapping technique works well throughout the imaging volume. However, Figure 4.10e shows that the conformal mapped gradient field is slightly less homogeneous than the gradient field obtained directly from planar wire patterns. Note that the simulated annealing processing time varies considerably depending on the simulation parameters, including the number of iterations, the initial temperature, the cooling ratio, and the number of iterations to reach equilibrium. The calculation time varies between 10 to 12 hours to obtain reasonably optimized solutions using a 32-bit machine (Intel Centrino, 2.0 GHz). 4.6 Discussion In this work, a conformal mapping technique has been used for bi-planar gradient array design. The results after the simulation show a great deal of similarity between the optimized field and the field which is calculated using the Biot-Savart Law applied to the bi-planar wire pattern. The conformal mapping technique for the bi-planar gradient coil design has two distinct advantages. First, conformal mapping effectively results in planar gradients using a one-dimensional stream function. As a result, the simulated annealing algorithm process has been simplified to a small number of parameters to save computation time dramatically compared to approaches using a two-dimensional stream function, which 93 Figure 4.9. Gradient maps (5% contours) for a: x-gradient and b: y-gradients. Both figures show Axial, Coronal, and Sagittal slices through the volume. All units are expressed in mm. a b 94 Figure 4.10. Slice comparison along the z-axis (Gy field). (Left column) conformal mapped fields from cylinder to biplane (Right column) fields simulated from planar wire patterns. Figure shows homogeneous region extends 7.8 cm along the z-axis on each side, which gives us about 16 cm of FOVz. a. slices at the center b. slices at 2.0 cm from the center c. slices at 4.0 cm from the center d. slices at 6.0 cm from the center e. slices at 7.8 cm from the center 95 would require more control parameters. Second, the method of conformal mapping can be applied to various other geometries to design elliptical, hemispherical, or uni-planar gradient systems [24]. In principle, conformal mapping can be used for any arbitrary geometry where appropriate transformations can be derived. It appears that the bi-planar geometry is very suitable for this design procedure since the analytic magnetic fields are calculated assuming the cylindrical surface current density equation is defined on the full cylinder, 0 ≤ϕ < 2π . In other words, the bi-planar geometry preserves the upper and lower half ( 0 ≤ϕ <π , π ≤ϕ < 2π , respectively) of the surface current on each bi-plane. In conclusion, a new local insert gradient coil design method using conformal mapping technique and simulated annealing has advantage of faster and simpler optimization procedure and can be used for noncylindrical geometry. 4.7 References [1] B. Zhang, Y. F. Yen, B. A. Chronik, B. C. McKinnon, D. J. Schaefer, and B. K. 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CHAPTER 5 SUPERELLIPTICAL INSERT GRADIENT COIL WITH A FIELD MODIFYING LAYER FOR BREAST IMAGING 5.1 Abstract Many MRI applications such as DCE-MRI of the breast require high spatial and temporal resolution, and can benefit from improved gradient performance, e.g., increased gradient strength, and reduced gradient rise time. The improved gradient performance required to achieve high spatial and temporal resolution may be achieved by using local insert gradients specifically designed for a target anatomy. Today flat gradient systems cannot create an imaging volume large enough to accommodate both breasts. Further, the gradient field is not homogeneous, dropping rapidly with distance from the gradient coil surface. To attain an imaging volume adequate for breast MRI, we converted a planar local gradient system into a segment of a superellipse to create homogeneous gradient volumes (HGVs) that are 182% (Gx), 57% (Gy), and 75% (Gz) wider (left/right direction) than those of the original planar local gradient system. By adding an extra field-modifying (FM) layer, the homogeneous gradient volume near the gradient coil surface was enlarged by 67%, 89%, and 214% for Gx, Gy, and Gz respectively, over the already enlarged HGVs of the superelliptical gradients. 99 5.2 Introduction Many imaging tasks, such as dynamic contrast enhanced (DCE) MRI for breast lesion characterization, can benefit from high spatial and temporal resolution. The improved gradient performance required for high spatial and temporal resolution may be achieved by local gradient coils such as planar insert gradients. With higher gradient strength and slew rate, the local planar gradient can attain higher spatial and temporal resolution than the body gradients. In general, local gradient coils are designed to produce a smaller homogeneous gradient volume (HGV) compared to that of the MRI system whole-body gradient coils. For local planar gradient coils, the x-gradient, which requires four fingerprint patterns per plane rather than two fingerprints as in the y-gradient, results in the most limited imaging volume. As a