Principles of Magnetic Resonance Imaging by Nishimura

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磁共振成像原理，stanford大学的Nishimura教授写作的
o Copyright 2010 by Dwight G. Nishimura All Rights Reserved dition 1.1 Printedbywwwulu.com Contents Contents v Acknowledgements I1 Magnetic Resonance imaging 1.2 Historical nole 1.3 Organization 2 Preliminaries 2.1 Complex numbers 5 2 Fourier transfor 2.2.1 One diner 2.2.2 Two Dimensional 2.3 Projections and the Central section Theorem 10 LO 3.2 The Central Section theoren 2.4 Sampling,,.,,,,,,, 12 5 Relation malri 2.6 Basic proba 17 2.6.1R 2.6.2 Random Proccsses 21 3 Overyiew 23 tion of mr 3.1.1 Main Field B 3.1.2 Radiofrequency field Bl 3.1.3 Linear gradient Fields gr 26 3.1.4 Bloch equ .2 entional MR Imaging methe CONTENTS CONTENTS 3.2.1 Selective Excitation 6 Excitation 3.2.2 2D Imaging Methods 3⊥ 6.1 Basic Excitation princip 6.1.1 General Formula 3.3 Other Approaches to MR imaging 3.4 Imaging Considerations 5.1.2 Rotating frame 6.1. 3 Graphical and Ar 3.5 Instrumentation 6.2 Selective excitation 3.6 problems 6.2.1G 4 Physics 45 6.2.2 Small Tip-Angle 4.1 Nuclear Spins ,.,45 6.2.3 Refocusing 6.2.4 General selective 4.2 Interaction with a Static magnelic Field 4 6.3 Excitation k-Space Inter 4.21 Magnetizaton 4 42.2 Precession 47 6.4 Selective Excitation Pull 6.5 Problerns 4.3 Interaction with a Radiofrequency Field 48 4.4 rclaxation 7 Imaging Considerations 4.4.1 Longitudinal Relaxation 50 7.1 off-Resonance sources 4.4.2 Transverse relaxation 50 7.1.1 Off-Resonance Ei 4.4.3 Relaxation Times of Biological Tissue 5⊥ 7.1.2 Off-Resonance el 4.5 Bloch Equation ..., .1 3 Echoes 4.6 Chemical shift 53 7.1.4 Spin echoes in ir 4.7 Problems 56 7.2 12 Relaxation 7. 3 Other nonidealities 5 Imaging Principies 57 7.3.1 Radiofrequency 3.1 Bloch equation 7.3.2 Gradient Field C 5.1.1 Homogeneous Object, Uniform Field ,,,58 7. 4 Image Contrast 5.1.2 Inhomogeneous Object, Nonuniform Field 61 7.4.1 Saturation recov 5.2 Signal equation 7.4.2 Inversion recove 5,2.1 Other Considerations of the Signal T: quation 7.5 Noise Considerations 3 Alternate Derivation: Signal equalion 69 7.5.1 Physical and ins 5.4 Fouricr Interprctation of the Signal Equation 6.2 Imaging Sequenc 7. 6 Problems 5.3 Ilustration of the k' -Space Perspective 5.6 Basic 2D Irrlaging Methoc 8 Volumetric Imaging and Fa 5.6.1 2D Projection Reconstruction 4 8.1 Volumetric Imaging 5.6.2 2D Fourier transform 8.1.I 3D Imaging by M ..7 Sampling Requirements: 2DFT Imaging 81 8.1.2 3D k- Space acqu 5.7.1 Field of view .,,,,84 8.⊥.3 Projective imagin 5.7.2 Spatial resolution 89 8.2 Fast Imaging .... 5.7.3 DFT image Reconstruction :·93 8.2.1 k-Space trajecto 5.8 Problems 。94 8.2.2 Signal-Generatio CONTENTS CONTENTS g Spectroscopic imaging 193 Preface 9.1 Effect of Chemical Shift on the Signal Equation ,,,..193 9.2 Spectroscopic Imaging Methods .194 This book gives an introduction to magnetic resonance(Mr)imaging, a re- 9.2.1 3D Fourier transform Spectroscopic imaging 19 markably rich field that touches on many disciplines in medicine, science, 9.2.2 Spectroscopic Imaging with Time-Varying Gradients . 198 and engineering. Because of this richness, this book makes no attempt to 9.2.3 N-Component Chemical Shift Imaging 199 provide a comprehensive treatment. Inslead, the goal is to present a man- ageable introduction to MR imaging by focusing on the basic principles of 10 Flow Imaging 203 image formation, image content, and performance considerations. Greater 10.1 FLOw Elects 2目3 emphasis is given to the signal processing elements than to the physics 10.1, 1 TOF Effccts a翟 and life-science elements of MR imaging 10.1.2 Phase effecls 205 This book grew out of notes prepared for a graduate-level course i 10.2 Flow Imaging Methods ..208 electrical engineering at Stanford. A background in Fourier transforms 10.2.1 TOF-Based methods 8 serves as the primary prerequisite. Elegant Fourier descriptions pervade 102.2 Phasc Bascd mcthods 。,210 the analysis of MR. Likewise, MR serves as a beautiful physical example of Fourier transforms 11夏 nstrumentation 213 11.1 Main field .Ithough structured for electrical engineering students, the material 11.2 Radiotrequency hieb 213 should be accessible to those from other technical fields. In a 10-week 11. 3 Gradicnt Ficlds 17 quarter(approximately 27 lecture hours), it is rcasonable to cover Chapters 3-7, whichrepresent the bulk of the material. The latter chapters (Chapters A Useful Constants and Conversions 2l9 8-11)provide brief overviews of imaging extensions and selected topics,A scmester-long course could cover the entire book with time for additional Bibliography 22l topics such as image reconstruction and imaging applications. Prohlems at the end of the chapters are provided not only to revicw the concepts but index 224 to challenge the student lo apply these concepts in new situations. Finally, it is worth mentioning that this book has long been a work-in progress Much morc material could be included. However, i have received feedback that the book is usable in its current form so i have decided to ke it available not Dwight G. Nishimura CONTENTS Acknowledgements For over 25 years, I have had the pleasure and privilege of working in MR with many talented pcople at Stanford in the Magnetic Resonance Sys tems Research Lab(MRSRL. These people, too numerous to list (see www- mrsrl. stanford.edu for a partial list), have substantially shaped the conten of this book. i am particularly indebted to Albert Macovski who introduced apter me to medical imaging and whose profound insights are an inportant pal of this book. Special thanks go to Steven Conolly, Bob Hu, Craig Meyer, John Pauly, Greig Scott, and Graham Wright for many helpful discussions I also express my appreciation to Shuye Huan for her tireless handling of Introduction past printings of this book I wish to acknowledge ronald Bracewell and Joseph Goodman for their early guidance and inspiration. I also wish to thank the students and teach. ing assistants of EE 369B (formerly EE 492B)who endured early versions of this book, toiled diligently through the problems, and provided invaluable The discovery of x-rays by roe k ful capability to the practice o Finally, I am extremely grateful to my family and, most importantly, to human body without surgical Ann for their unwavering and unconditional support imaging modalities exist, inclu trasound. nuclear medicine, ane Dwight G. Nishimura 1.1 Magnetic Resona nis book is al netc re modality that has gained wides include dliagnostic studies of the tal system. MR imaging remain new medical applications may nuclear magnetic resonance(N ties, the imaging communily o negative connotations associa involves no ionizing radiation normal operating conditions to as "Mri"and occasionally the An important characteristc a wide range of physical para mental parameters to set for anatomic imaging, valuable m thereby opening the possibilitie ies. MR is often compared to x- imaging. Although physical C CHAPTER⊥, INTRODUCTION l3. ORGANIZA TTON cross-sections, MR images can be made of planes at an arbitrary orienta tion. Moreover, MR images possess comparable spaLial resolution and far superior soft-tissue contrast. Representative mr images in Fig. 1.l demon trate the exquisite anatomical detail possible with MR. Shortcomings of MR include the sophisticated and expensive hardware required, the center- piece of which is a large powerful magnet that can present siling challenges in a hospital. Also, conventional MR imaging methods may involve scan minutes, making nonstationary regions (e.g, abdomen) more difficult to image 1.2 Historical notes NMR in condensed matter was discovered independently by Felix Bloch [ B1o46] and Edward Purcell [PlP46] in 1946, a discovery for which they shared the Nobel Prize in physics. nmR has long been used in chem- Lry and physics for studying molecular structure and diffusion. In 1973 Paul Lauterbur reported the first MR image [Lau73 using linear gradient fields. During the 1970s, most of the work in MR imaging took place it acadcmia-primarily in the United Kingdom. In the 1980s however, indus ioined forces with universities, investing substantial resources to de velop imaging systems for clinical use in medical centers. As a result, in age quality improved dramatically and clinical imagers proliferated across the world. Ovcrall, this rapidly-growing field encompasses an array of dis ciplines and many new developments continue to emerge. In 2003, the Nobel Prize in Physiology or Medicine was awardedl to Lauterbur and Sir Peter Mansfield for their discoveries in MR imaging 1 3 Organization The emphasis of this textbook is on developing a description of mr imag ing from a systems perspective. Fourier-domain descriptions pervade MR Figure 1. 1.: Representative Cross-Sectional Brain Images: (a) Sagittal cross analyses; hence, the material in this book is founded largely on lourie section;(b) Coronal cross section; (c)Transaxial cross section. Sagittal transform and lincar systems thcory, the basic principles of which arc coronal, and transaxial (or simply axial) refer to the slice orientation as reviewed in Chapter 2. Those readers unfamiliar with multi-dimensional illustrated. The images are presented in a standard display format with Fourier transform theory may find Chapter 2 particularly useful. Chapter anatomical location labels: anterior(A), posterior(P), superior(S), inferior also contains other mathematical pre luminaries such as rotation matrices (I), left(L), and right (R). Note that the "anatomic right"appears on the and basic probability mage left"in the coronal and axial images Chapter 3 provides a general overview of the material on Mr imaging that will be more fully developed in the subsequent chapters. The goal of this early chapter is to foster a physical intuition of the fundamental CHAPTER I. INTRODUCTION concepts. Those who have studied MR imaging would probably agree that the underlying principles, while elegant, may lack intuitive appeal initially Therefore, Chapter 3 should be viewed as a gentle survey of the material Although the emphasis of this book is on the systems analysis of mr imaging, Chapter 4 gives hackground information on the basic physics of nuclear magnetic resonance. For the purposes of this book, a classical description of the physics will suffice Chapter 2 Conventional MR imaging sequences may be divided into two parts excitation anld reception. Nuclei are excited by a radiofrequency magnetic field and produce a time signal which is recorded and processed to form an image. Chapter 5 delves into the basic principles involved in the reception Preliminaries portion of imaging while Chapter6 cleals with the excitation portion. It will be easier and more convenient to discuss the reception portion first. In both chapters however, elegant Fourier-domain perspectives will emerge from the analyses While Chapter 5 describes imaging from an idealized perspective, Chap This chapter contains a review ter 7 considers the effects of system imperfections and physical constraints will be useful for understandin on imaging. Also considered are image contrast and noise, both of which chapters are important issues with respect to disease diagnosis. In Chapter 8, extel sions to the basic imaging sequences are examined, specifically volumetric ( 3D)imaging and fast imaging 2.1 Complex number Chapter g touches on another imaging extension, spectroscopic imag ing, which concerns the imaging of specific chemical species with subtle A complex number c can be exp frequency differences. Mapping of such species can provide useful phys iological information. Flow imaging is the topic of Chapter 10. Several approaches exist to make MR sensitive to flowing material, enabling direct where i=√-⊥. Hence the real imaging or measurement of flowing material. The final chapter, Chapter 11, gives a brief overview of Mr instrumentation issues Alternatively, c can be expresse C= Ae i中 where A, the magnitude, and cp A In MR, phase plays an extreme nced with respect to the x often the important quantity CHAPTER 2. PRET.IMINARIES 2. 2. FOURIER TRANSFORMS It is convenient to use the complex number c to describe a vector re cach other; hence if x is in mm, then kx is in cycles/inn. Also common is siding in a plane with axes, say, in x and y. Hence a and b correspond the use of reciprpeaI variables t and f, where t is in units of sec andfin to the x and y components respectively, while A and h pertain to the units of cycles/sec or Hertz (Hz). vector's length and direction. Often considered is the case of a vector of to recover f(x)fro F(kx), the inverse 1D Fourier transform is magnitude A rotating in the xy plane at some angular frequency w in the lockwise direction and some starting phase o. This time-varying vector can be expressed as -a (kxe+izrrkx x dkx (t)=Ae Ae-Lotel 2.2.2 Iwo Dimensional where the - a suggests clockwise rotation The Fourier integral expands into multiple dimensions by the inclusion of additional integrals and complex exponentials. For the case of a two 22 Fourier transforms dimensional(2D)Fourier transform, the integral becomes Ihis section surveys the basic principles and theorems of Fourier trans- FCIx, ky) forms. A more in-depth coverage can be found in [Bra00, Pap621 2.2。1 one dimensional \oaa,J(, ye -i2r(kxx+ky y) dxdy f(x, y)sin 2TT( Kxx+kyy) dx dy(2. 12) The one-dimensional (I))Fourier transform of a function f(x)is Because the variables x and y often correspond to spatial coordinates F(kx) f(xe-L2Trkxx dx when dealing with 2D functions, the typical unit for x and y is mm and thus the unit for kx and ky is cycles mm. Therefore kx and ky are va where F(kx)is commonly called the frequency spectrum of f(x). The bles representing spatial frequency (as opposed to temporal frequency as Fourier transform integral can be rewritten as n the ID case when the dimension of x is time). A spatial frequency com- ponent is a sinusoidal variation over 2D space. For example, when dealing F(kx)=f(x)cos 2nrkxx dx f(x)sin 2rrKxx dx(2 with images the amplitude that varies over space is the gray level The 2D Fourier transform decomposes the 2D function into its spatial frequency components. Figure 2.1 shows an example of a cosinusoidal The Fourier transform decomposes the function f(x) into its cosinusoidal component at a particular spatial frequency. Figure 2. 1 a depicts this com and sinusoidal components at all frequencies kx. If f(x)is real-valued ponent as an grayscale image while lig. 2.1b is a mesh plot representation then--from Eg 2.7 and the fact that f(x) is decomposable into the sum In two dimensions, a spatial frequency component is defined by two pa of an even function and an odd function-F(kx) possesses Hermitian syrm- rameters, Kx and ky, or, equivalently, its magnitude lki and direction 0 metry; that is The period L of the sinusoid is given by 1/ki F(-kx)=F(Kx) Two special cases of 2D functions are separable functions and circu Therefore the real part of F(kx)is an even function while the imaginary larly symmetric functions part of F(kx) is an odd function. Similarly, if f(x) is strictly imaginary valued. then separable Functions F(-kx)=-F*(x) A function f(x, y) is separable in Cartesian coordinates if it can be writ a symmetry referred to as anti-Hermitian. Table 2.1 lists some functions ten as fx(x)fr(y). For cxamplc, the box function 27(x, y)is separable and their Fourier transforms. The units of x and kx are the reciprocal of being equal ton(x)n(y). With separable 2D functions, their 2D Fourier 【实例截图】
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