Robert W. Brown, Ph.D.
Institute Professor and Distinguished University Professor
Case Western Reserve University, Cleveland, Ohio, USA
His research group efforts have resulted in over 200 published papers and abstracts, and his former students hold at least 150 patents (eight co-authored by him) and he has done important work in radiation physics, MRI, PET, CT, electromagnetics, inverse methods, mechanical and thermal modeling, nonlinear dynamics, EEG, MEG, sensors, and physics education, as well as a professional-life-long involvement in elementary particle physics and cosmology.

Yu-Chung N. Cheng, Ph.D.
Associate Professor of Radiology
Wayne State University, Detroit, Michigan, USA

E. Mark Haacke, Ph.D.
Professor of Radiology, Wayne State University, Detroit, Michigan, USA
Professor of Physics, Case Western Reserve University, Cleveland, Ohio, USA
Adjunct Professor of Radiology, Loma Linda University, Loma Linda, California, USA
Adjunct Professor of Radiology, McMaster University, Hamilton, Ontario, Canada
Distinguished Foreign Professor, Northeastern University, Shenyang, Liaoning, China
Director of The Magnetic Resonance Imaging Institute for Biomedical Research and Professor of Radiology, Department of Biomedical Engineering, Wayne State University. Dr. Haacke has two decades of experience teaching courses in physics, mathematics and statistics.

Michael R. Thompson, Ph.D.
Principal Scientist, Toshiba Medical Research Institute,
Cleveland, Ohio, USA

Ramesh Venkatesan, D.Sc.
Manager, MR Applications Engineering
Wipro GE Healthcare Pvt. Ltd., Bangalore, Karnataka, India

Foreword to the Second Edition xvii

Foreword to the First ~ Edition xxi

Preface to the Second Edition xxvii

Preface to the First Edition xxix

Acknowledgements xxx

Acknowledgements to the First Edition xxxi

1 Magnetic Resonance Imaging: A Preview 1

1.1 Magnetic Resonance Imaging: The Name 1

1.2 The Origin of Magnetic Resonance Imaging 2

1.3 A Brief Overview of MRI Concepts 3

2 Classical Response of a Single Nucleus to a Magnetic Field 19

2.1 Magnetic Moment in the Presence of a Magnetic Field 20

2.2 Magnetic Moment with Spin: Equation of Motion 25

2.3 Precession Solution: Phase 29

3 Rotating Reference Frames and Resonance 37

3.1 Rotating Reference Frames 38

3.2 The Rotating Frame for an RF Field 41

3.3 Resonance Condition and the RF Pulse 44

4 Magnetization, Relaxation, and the Bloch Equation 53

4.1 Magnetization Vector 53

4.2 Spin-Lattice Interaction and Regrowth Solution 54

4.3 Spin-Spin Interaction and Transverse Decay 57

4.4 Bloch Equation and Static-Field Solutions 60

4.5 The Combination of Static and RF Fields 62

5 The Quantum Mechanical Basis of Precession and Excitation 67

5.1 Discrete Angular Momentum and Energy 68

5.2 Quantum Operators and the Schrödinger Equation 72

5.3 Quantum Derivation of Precession 77

5.4 Quantum Derivation of RF Spin Tipping 80

6 The Quantum Mechanical Basis of Thermal Equilibrium and Longitudinal Relaxation 85

6.1 Boltzmann Equilibrium Values 86

6.2 Quantum Basis of Longitudinal Relaxation  89

6.3 The RF Field 92

7 Signal Detection Concepts 95

7.1 Faraday Induction 96

7.2 The MRI Signal and the Principle of Reciprocity 99

7.3 Signal from Precessing Magnetization 101

7.4 Dependence on System Parameters 107

8 Introductory Signal Acquisition Methods: Free Induction Decay, Spin Echoes, Inversion Recovery, and Spectroscopy 113

8.1 Free Induction Decay and T∗ 2 114

8.2 The Spin Echo and T2 Measurements 120

8.3 Repeated RF Pulse Structures 126

8.4 Inversion Recovery and T1 Measurements 131

8.5 Spectroscopy and Chemical Shift 136

9 One-Dimensional Fourier Imaging, k-Space and Gradient Echoes 141

9.1 Signal and Effective Spin Density 142

9.2 Frequency Encoding and the Fourier Transform 144

9.3 Simple Two-Spin Example 147

9.4 Gradient Echo and k-Space Diagrams 151

9.5 Gradient Directionality and Nonlinearity 162

10 Multi-Dimensional Fourier Imaging and Slice Excitation 165

10.1 Imaging in More Dimensions 166

10.2 Slice Selection with Boxcar Excitations 175

10.3 2D Imaging and k-Space 184

10.4 3D Volume Imaging 194

10.5 Chemical Shift Imaging 197

11 The Continuous and Discrete Fourier Transforms 207

11.1 The Continuous Fourier Transform 208

11.2 Continuous Transform Properties and Phase Imaging 209

11.3 Fourier Transform Pairs 220

11.4 The Discrete Fourier Transform 223

11.5 Discrete Transform Properties 225

12 Sampling and Aliasing in Image Reconstruction 229

12.1 Infinite Sampling, Aliasing, and the Nyquist Criterion 230

12.2 Finite Sampling, Image Reconstruction, and the Discrete Fourier Transform 237

12.3 RF Coils, Noise, and Filtering 245

12.4 Nonuniform Sampling 250

13 Filtering and Resolution in Fourier Transform Image Reconstruction 261

13.1 Review of Fourier Transform Image Reconstruction 262

13.2 Filters and Point Spread Functions 264

13.3 Gibbs Ringing 267

13.4 Spatial Resolution in MRI 272

13.5 Hanning Filter and T∗2 Decay Effects 281

13.6 Zero Filled Interpolation, Sub-Voxel Fourier Transform Shift Concepts, and Point Spread Function Effects 283

13.7 Partial Fourier Imaging and Reconstruction 286

13.8 Digital Truncation 293

14 Projection Reconstruction of Images 297

14.1 Radial k-Space Coverage 298

14.2 Sampling Radial k-Space and Nyquist Limits 302

14.3 Projections and the Radon Transform 308

14.4 Methods of Projection Reconstruction with Radial Coverage 310

14.5 Three-Dimensional Radial k-Space Coverage 317

14.6 Radial Coverage Versus Cartesian k-Space Coverage 320

15 Signal, Contrast, and Noise 325

15.1 Signal and Noise 326

15.2 SNR Dependence on Imaging Parameters 334

15.3 Contrast, Contrast-to-Noise, and Visibility 342

15.4 Contrast Mechanisms in MRI and Contrast Maximization 345

15.5 Contrast Enhancement with T1-Shortening Agents 358

15.6 Partial Volume Effects, CNR, and Resolution 363

15.7 SNR in Magnitude and Phase Images 365

15.8 SNR as a Function of Field Strength 368

16 A Closer Look at Radiofrequency Pulses 375

16.1 Relating RF Fields and Measured Spin Density 376

16.2 Implementing Slice Selection 381

16.3 Calibrating the RF Field 383

16.4 Solutions of the Bloch Equations 387

16.5 Spatially Varying RF Excitation 393

16.6 RF Pulse Characteristics: Flip Angle and RF Power 400

16.7 Spin Tagging 405

17 Water/Fat Separation Techniques 413

17.1 The Effect of Chemical Shift in Imaging 413

17.2 Selective Excitation and Tissue Nulling 420

17.3 Multiple Point Water/Fat Separation Methods 428

18 Fast Imaging in the Steady State 447

18.1 Short-TR, Spoiled, Gradient Echo Imaging 448

18.2 Short-TR, Coherent, Gradient Echo Imaging 468

18.3 SSFP Signal Formation Mechanisms 481

18.4 Understanding Spoiling Mechanisms 498

19 Segmented k-Space and Echo Planar Imaging 511

19.1 Reducing Scan Times 512

19.2 Segmented k-Space: Phase Encoding Multiple k-Space Lines per RF Excitation for Gradient Echo Imaging 514

19.3 Echo Planar Imaging (EPI) 522

19.4 Alternate Forms of Conventional EPI 530

19.5 Artifacts and Phase Correction 543

19.6 Spiral Forms of EPI 549

19.7 An Overview of EPI Properties 556

19.8 Phase Encoding Between Spin Echoes and Segmented Acquisition 560

19.9 Mansfield 2D to 1D Transformation Insight 563

20 Magnetic Field Inhomogeneity Effects and T∗2 Dephasing 569

20.1 Image Distortion Due to Field Effects 570

20.2 Echo Shifting Due to Field Inhomogeneities in Gradient Echo Imaging 580

20.3 Methods for Minimizing Distortion and Echo Shifting Artifacts 587

20.4 Empirical T∗2 603

20.5 Predicting T∗2 for Random Susceptibility Producing Structures 611

20.6 Correcting Geometric Distortion 615

21 Random Walks, Relaxation, and Diffusion 619

21.1 Simple Model for Intrinsic T2 620

21.2 Simple Model for Diffusion 622

21.3 Carr-Purcell Mechanism 624

21.4 Meiboom-Gill Improvement 626

21.5 The Bloch-Torrey Equation 628

21.6 Some Practical Examples of Diffusion Imaging 632

22 Spin Density, T1 and T2 Quantification Methods in MR Imaging 637

22.1 Simplistic Estimates of ρ0, T1 T2 638

22.2 Estimating T1 and T2 from Signal Ratio Measurements 640

22.3 Estimating T1 and T2 from Multiple Signal Measurements 647

22.4 Other Methods for Spin Density and T1 Estimation 649

22.5 Practical Issues Related to T1 and T2 Measurements 657

22.6 Calibration Materials for Relaxation Time Measurements 665

23 Motion Artifacts and Flow Compensation 669

23.1 Effects on Spin Phase from Motion along the Read Direction 670

23.2 Velocity Compensation along the Read and Slice Select Directions 675

23.3 Ghosting Due to Periodic Motion 683

23.4 Velocity Compensation along Phase Encoding Directions 688

23.5 Maximum Intensity Projection 698

24 MR Angiography and Flow Quantification 701

24.1 Inflow or Time-of-Flight (TOF) Effects 702

24.2 TOF Contrast, Contrast Agents, and Spin Density/T∗2 -Weighting 711

24.3 Phase Contrast and Velocity Quantification 719

24.4 Flow Quantification 730

25 Magnetic Properties of Tissues: Theory and Measurement 739

25.1 Paramagnetism, Diamagnetism, and Ferromagnetism 740

25.2 Permeability and Susceptibility: The →H Field 744

25.3 Objects in External Fields: The Lorentz Sphere 745

25.4 Susceptibility Imaging 755

25.5 Brain Functional MRI and the BOLD Phenomenon 760

25.6 Signal Behavior in the Presence of Deoxygenated Blood 766

26 Sequence Design, Artifacts, and Nomenclature 779

26.1 Sequence Design and Imaging Parameters 780

26.2 Early Spin Echo Imaging Sequences 785

26.3 Fast Short TR Imaging Sequences 791

26.4 Imaging Tricks and Image Artifacts 798

26.5 Sequence Adjectives and Nomenclature 812

27 Introduction to MRI Coils and Magnets 823

27.1 The Circular Loop as an Example 824

27.2 The Main Magnet Coil 827

27.3 Linearly Varying Field Gradients 838

27.4 RF Transmit and Receive Coils 846

28 Parallel Imaging 859

28.1 Coil Signals, Their Images, and a One-Dimensional Test Case 860

28.2 Parallel Imaging with an x-Space Approach 865

28.3 Parallel Imaging with a k-Space Approach 873

28.4 Noise and the g-Factor 885

28.5 Additional Topics in Acquisition and Reconstruction 888

A Electromagnetic Principles: A Brief Overview 893

A.1 Maxwell's Equations 894

A.2 Faraday's Law of Induction 894

A.3 Electromagnetic Forces 895

A.4 Dipoles in an Electromagnetic Field 896

A.5 Formulas for Electromagnetic Energy 896

A.6 Static Magnetic Field Calculations 897

B Statistics 899

B.1 Accuracy Versus Precision 899

B.1.1 Mean and Standard Deviation 900

B.2 The Gaussian Probability Distribution 901

B.2.1 Probability Distribution 901

B.2.2 z-Score 901

B.2.3 Quoting Errors and Confidence Intervals 902

B.3 Type I and Type II Errors 902

B.4 Sum over Several Random Variables 904

B.4.1 Multiple Noise Sources 905

B.5 Rayleigh Distribution 906

B.6 Experimental Validation of Noise Distributions 907

B.6.1 Histogram Analysis 907

B.6.2 Mean and Standard Deviation 909

C Imaging Parameters to Accompany Figures 913

Index 923

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