Magnetic Resonance Imaging: Physical Principles and Sequence Design Second Edition Robert W. Brown, Ph.D. Institute Professor and Distinguished University Professor Case Western Reserve University, Cleveland, Ohio, USA 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 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. 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Mark Haacke ... [et al.]. c1999. Includes bibliographical references and index. ISBN 978-0-471-72085-0 (cloth) I. Cheng, Yu-Chung N., author. II. Haacke, E. Mark., author. III. Thompson, Michael R., author. IV. Venkatesan, Ramesh., author. V. Title. [DNLM: 1. Magnetic Resonance Imaging. 2. Physical Phenomena. WN 185] RC78.7.N83 616.07′548–dc23 2014000051 Cover design by Wiley Printed in 10 9 8 7 6 5 4 3 2 1 Contents 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 1.3.1 Fundamental Interaction of a Proton Spin with the Magnetic Field . 3 1.3.2 Equilibrium Alignment of Spin . . . . . . . . . . . . . . . . . . . . . 4 1.3.3 Detecting the Magnetization of the System . . . . . . . . . . . . . . . 5 1.3.4 Magnetic Resonance Spectroscopy . . . . . . . . . . . . . . . . . . . . 7 1.3.5 Magnetic Resonance Imaging . . . . . . . . . . . . . . . . . . . . . . 7 1.3.6 Relaxation Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3.7 Resolution and Contrast . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3.8 Magnetic Field Strength . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3.9 Key Developments in Magnetic Resonance . . . . . . . . . . . . . . . 12 2 Classical Response of a Single Nucleus to a Magnetic Field 19 2.1 Magnetic Moment in the Presence of a Magnetic Field . . . . . . . . . . . . 20 2.1.1 Torque on a Current Loop in a Magnetic Field . . . . . . . . . . . . . 20 2.1.2 Magnet Toy Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.2 Magnetic Moment with Spin: Equation of Motion . . . . . . . . . . . . . . . 25 2.2.1 Torque and Angular Momentum . . . . . . . . . . . . . . . . . . . . . 25 2.2.2 Angular Momentum of the Proton. . . . . . . . . . . . . . . . . . . . 26 2.2.3 Electrons and Other Elements . . . . . . . . . . . . . . . . . . . . . . 27 2.2.4 Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.3 Precession Solution: Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.3.1 Precession via the Gyroscope Analogy . . . . . . . . . . . . . . . . . 29 2.3.2 Geometrical Representation . . . . . . . . . . . . . . . . . . . . . . . 31 2.3.3 Cartesian Representation . . . . . . . . . . . . . . . . . . . . . . . . . 32 iii iv Contents 2.3.4 Matrix Representation . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.3.5 Complex Representations and Phase . . . . . . . . . . . . . . . . . . 34 3 Rotating Reference Frames and Resonance 37 3.1 Rotating Reference Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.2 The Rotating Frame for an RF Field . . . . . . . . . . . . . . . . . . . . . . 41 3.2.1 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.2.2 Quadrature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.3 Resonance Condition and the RF Pulse . . . . . . . . . . . . . . . . . . . . . 44 3.3.1 Flip-Angle Formula and Illustration . . . . . . . . . . . . . . . . . . . 45 3.3.2 RF Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.3.3 Different Polarization Bases . . . . . . . . . . . . . . . . . . . . . . . 47 3.3.4 Laboratory Angle of Precession . . . . . . . . . . . . . . . . . . . . . 49 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 ~ 4.5.1 Bloch Equation for B = B zˆ+B xˆ . . . . . . . . . . . . . . . . . 62 ext 0 1 ′ 4.5.2 Short-Lived RF Pulses . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.5.3 Long-Lived RF Pulses . . . . . . . . . . . . . . . . . . . . . . . . . . 64 5 The Quantum Mechanical Basis of Precession and Excitation 67 5.1 Discrete Angular Momentum and Energy . . . . . . . . . . . . . . . . . . . . 68 5.2 Quantum Operators and the Schr¨odinger Equation . . . . . . . . . . . . . . 72 5.2.1 Wave Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.2.2 Momentum and Angular Momentum Operators . . . . . . . . . . . . 74 5.2.3 Spin Solutions for Constant Fields . . . . . . . . . . . . . . . . . . . . 76 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 Contents v 7.3.1 General Expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 7.3.2 Spatial Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 7.3.3 Signal Demodulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 7.3.4 Dependent Channels and Independent Coils . . . . . . . . . . . . . . 107 7.4 Dependence on System Parameters . . . . . . . . . . . . . . . . . . . . . . . 107 7.4.1 Homogeneous Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 7.4.2 Relative Signal Strength . . . . . . . . . . . . . . . . . . . . . . . . . 108 7.4.3 Radiofrequency Field Effects . . . . . . . . . . . . . . . . . . . . . . . 110 8 Introductory Signal Acquisition Methods: Free Induction Decay, Spin Echoes, Inversion Recovery, and Spectroscopy 113 8.1 Free Induction Decay and T . . . . . . . . . . . . . . . . . . . . . . . . . . 114 2∗ 8.1.1 FID Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 8.1.2 Phase Behavior and Phase Conventions . . . . . . . . . . . . . . . . . 115 8.1.3 T Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 2∗ 8.1.4 The FID Sequence Diagram and Sampling . . . . . . . . . . . . . . . 119 8.2 The Spin Echo and T Measurements . . . . . . . . . . . . . . . . . . . . . . 120 2 8.2.1 The Spin Echo Method . . . . . . . . . . . . . . . . . . . . . . . . . . 120 8.2.2 Spin Echo Envelopes . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 8.2.3 Limitations of the Spin Echo . . . . . . . . . . . . . . . . . . . . . . . 124 8.2.4 Spin Echo Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 8.2.5 Multiple Spin Echo Experiments . . . . . . . . . . . . . . . . . . . . . 125 8.3 Repeated RF Pulse Structures . . . . . . . . . . . . . . . . . . . . . . . . . 126 8.3.1 The FID Signal from Repeated RF Pulse Structures . . . . . . . . . . 127 8.3.2 The Spin Echo Signal from Repeated RF Pulse Structures . . . . . . 129 8.4 Inversion Recovery and T Measurements . . . . . . . . . . . . . . . . . . . 131 1 8.4.1 T Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 1 8.4.2 Repeated Inversion Recovery . . . . . . . . . . . . . . . . . . . . . . . 134 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 9.1.1 Complex Demodulated Signal . . . . . . . . . . . . . . . . . . . . . . 142 9.1.2 Magnetization and Effective Spin Density . . . . . . . . . . . . . . . . 143 9.2 Frequency Encoding and the Fourier Transform . . . . . . . . . . . . . . . . 144 9.2.1 Frequency Encoding of the Spin Position . . . . . . . . . . . . . . . . 144 9.2.2 The 1D Imaging Equation and the Fourier Transform . . . . . . . . . 145 9.2.3 The Coverage of k-Space . . . . . . . . . . . . . . . . . . . . . . . . . 146 9.2.4 Rect and Sinc Functions . . . . . . . . . . . . . . . . . . . . . . . . . 147 9.3 Simple Two-Spin Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 9.3.1 Dirac Delta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 9.3.2 Imaging Sequence Diagrams Revisited . . . . . . . . . . . . . . . . . 151 9.4 Gradient Echo and k-Space Diagrams . . . . . . . . . . . . . . . . . . . . . . 151 9.4.1 The Gradient Echo . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 9.4.2 General Spin Echo Imaging . . . . . . . . . . . . . . . . . . . . . . . 156 vi Contents 9.4.3 Image Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 9.5 Gradient Directionality and Nonlinearity . . . . . . . . . . . . . . . . . . . . 162 9.5.1 Frequency Encoding in an Arbitrary Direction . . . . . . . . . . . . . 162 9.5.2 Nonlinear Gradients . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 10 Multi-Dimensional Fourier Imaging and Slice Excitation 165 10.1 Imaging in More Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 10.1.1 The Imaging Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 166 10.1.2 Single Excitation Traversal of k-Space . . . . . . . . . . . . . . . . . 169 10.1.3 Time Constraints and Collecting Data over Multiple Cycles. . . . . . 171 10.1.4 Variations in k-Space Coverage . . . . . . . . . . . . . . . . . . . . . 174 10.2 Slice Selection with Boxcar Excitations . . . . . . . . . . . . . . . . . . . . . 175 10.2.1 Slice Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 10.2.2 Gradient Rephasing After Slice Selection . . . . . . . . . . . . . . . . 178 10.2.3 Arbitrary Slice Orientation . . . . . . . . . . . . . . . . . . . . . . . . 180 10.3 2D Imaging and k-Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 10.3.1 Gradient Echo Example . . . . . . . . . . . . . . . . . . . . . . . . . 184 10.3.2 Spin Echo Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 10.4 3D Volume Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 10.4.1 Short-T 3D Gradient Echo Imaging . . . . . . . . . . . . . . . . . . 194 R 10.4.2 Multi-Slice 2D Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . 195 10.5 Chemical Shift Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 10.5.1 A 2D-Spatial 1D-Spectral Method . . . . . . . . . . . . . . . . . . . . 200 10.5.2 A 3D-Spatial, 1D-Spectral Method . . . . . . . . . . . . . . . . . . . 204 11 The Continuous and Discrete Fourier Transforms 207 11.1 The Continuous Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . 208 11.2 Continuous Transform Properties and Phase Imaging . . . . . . . . . . . . . 209 11.2.1 Complexity of the Reconstructed Image. . . . . . . . . . . . . . . . . 211 11.2.2 The Shift Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 11.2.3 Phase Imaging and Phase Aliasing . . . . . . . . . . . . . . . . . . . 212 11.2.4 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 11.2.5 Convolution Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 11.2.6 Convolution Associativity . . . . . . . . . . . . . . . . . . . . . . . . 218 11.2.7 Derivative Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 11.2.8 Fourier Transform Symmetries . . . . . . . . . . . . . . . . . . . . . . 220 11.2.9 Summary of Continuous Fourier Transform Properties. . . . . . . . . 220 11.3 Fourier Transform Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 11.3.1 Heaviside Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 11.3.2 Lorentzian Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 11.3.3 The Sampling Function . . . . . . . . . . . . . . . . . . . . . . . . . . 223 11.4 The Discrete Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . 223 11.5 Discrete Transform Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 225 11.5.1 The Discrete Convolution Theorem . . . . . . . . . . . . . . . . . . . 226 11.5.2 Summary of Discrete Fourier Transform Properties . . . . . . . . . . 227 Contents vii 12 Sampling and Aliasing in Image Reconstruction 229 12.1 Infinite Sampling, Aliasing, and the Nyquist Criterion . . . . . . . . . . . . . 230 12.1.1 Infinite Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 12.1.2 Nyquist Sampling Criterion . . . . . . . . . . . . . . . . . . . . . . . 232 12.2 Finite Sampling, Image Reconstruction, and the Discrete Fourier Transform. 237 12.2.1 Finite Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 12.2.2 Reconstructed Spin Density . . . . . . . . . . . . . . . . . . . . . . . 239 12.2.3 Discrete and Truncated Sampling of ρˆ(x): Resolution . . . . . . . . . 240 12.2.4 Discrete Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . 242 12.2.5 Practical Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 12.3 RF Coils, Noise, and Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . 245 12.3.1 RF Field-of-View Considerations . . . . . . . . . . . . . . . . . . . . 245 12.3.2 Analog Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 12.3.3 Avoiding Aliasing in 3D Imaging . . . . . . . . . . . . . . . . . . . . 250 12.4 Nonuniform Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 12.4.1 Aliasing from Interleaved Sampling . . . . . . . . . . . . . . . . . . . 250 12.4.2 Aliasing from Digital-to-Analog Error in the Gradient Specification . 258 13 Filtering and Resolution in Fourier Transform Image Reconstruction 261 13.1 Review of Fourier Transform Image Reconstruction . . . . . . . . . . . . . . 262 13.1.1 Fourier Encoding and Fourier Inversion . . . . . . . . . . . . . . . . 262 13.1.2 Infinite Sampling and Fourier Series . . . . . . . . . . . . . . . . . . 263 13.1.3 Limited-Fourier Imaging and Aliasing . . . . . . . . . . . . . . . . . 263 13.1.4 Signal Series and Spatial Resolution . . . . . . . . . . . . . . . . . . 264 13.2 Filters and Point Spread Functions . . . . . . . . . . . . . . . . . . . . . . . 264 13.2.1 Point Spread Due to Truncation . . . . . . . . . . . . . . . . . . . . 265 13.2.2 Point Spread for Truncated and Sampled Data . . . . . . . . . . . . 266 13.2.3 Point Spread for Additional Filters . . . . . . . . . . . . . . . . . . . 267 13.3 Gibbs Ringing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 13.3.1 Gibbs Overshoot and Undershoot . . . . . . . . . . . . . . . . . . . . 267 13.3.2 Gibbs Oscillation Frequency . . . . . . . . . . . . . . . . . . . . . . . 269 13.3.3 Reducing Gibbs Ringing by Filtering . . . . . . . . . . . . . . . . . . 270 13.4 Spatial Resolution in MRI . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 13.4.1 Resolution after Additional Filtering of the Data . . . . . . . . . . . 277 13.4.2 Other Measures of Resolution . . . . . . . . . . . . . . . . . . . . . . 278 13.5 Hanning Filter and T Decay Effects . . . . . . . . . . . . . . . . . . . . . . 281 2∗ 13.5.1 Resolution Due to the Hanning Filter . . . . . . . . . . . . . . . . . . 281 13.5.2 Partial Fourier T Reconstruction Effects . . . . . . . . . . . . . . . . 281 2∗ 13.6 Zero Filled Interpolation, Sub-Voxel Fourier Transform Shift Concepts, and Point Spread Function Effects . . . . . . . . . . . . . . . . . . . . . . . 283 13.6.1 Zero Padding and the Fast Fourier Transform . . . . . . . . . . . . . 283 13.6.2 Equivalence of Zero Filled Image and the Sub-Voxel Shifted Image . 284 viii Contents 13.6.3 Point Spread Effects on the Image Based on the Object Position Relative to the Reconstructed Voxels . . . . . . . . . . . . . . . . . . 285 13.7 Partial Fourier Imaging and Reconstruction . . . . . . . . . . . . . . . . . . 286 13.7.1 Forcing Conjugate Symmetry on Complex Objects . . . . . . . . . . . 290 13.7.2 Iterative Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . 290 13.7.3 Some Implementation Issues . . . . . . . . . . . . . . . . . . . . . . 292 13.8 Digital Truncation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 14 Projection Reconstruction of Images 297 14.1 Radial k-Space Coverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 14.1.1 Coverage of k-Space at Different Angles . . . . . . . . . . . . . . . . 299 14.1.2 Two Radial Fourier Transform Examples . . . . . . . . . . . . . . . . 300 14.1.3 Inversion for Image Reconstruction . . . . . . . . . . . . . . . . . . . 301 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 14.4.1 X-Ray Analog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 14.4.2 Back-Projection Method . . . . . . . . . . . . . . . . . . . . . . . . . 311 14.4.3 Projection Slice Theorem and the Fourier Reconstruction Method . . 313 14.4.4 Filtered Back-Projection Method . . . . . . . . . . . . . . . . . . . . 314 14.4.5 Reconstruction of MR Images from Radial Data . . . . . . . . . . . . 316 14.5 Three-Dimensional Radial k-Space Coverage . . . . . . . . . . . . . . . . . . 317 14.6 Radial Coverage Versus Cartesian k-Space Coverage . . . . . . . . . . . . . . 320 14.6.1 Image Distortion Due to Off-Resonance Effects: Cartesian Coverage Versus Radial Sampling . . . . . . . . . . . . . . . . . . . . . . . . . 321 14.6.2 Effects of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 14.6.3 Cartesian Sampling of Radially Collected Data . . . . . . . . . . . . . 323 15 Signal, Contrast, and Noise 325 15.1 Signal and Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326 15.1.1 The Voxel Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326 15.1.2 The Noise in MRI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 15.1.3 Dependence of the Noise on Imaging Parameters . . . . . . . . . . . . 328 15.1.4 Improving SNR by Averaging over Multiple Acquisitions . . . . . . . 331 15.1.5 Measurement of σ and Estimation of SNR . . . . . . . . . . . . . . . 333 0 15.2 SNR Dependence on Imaging Parameters . . . . . . . . . . . . . . . . . . . . 334 15.2.1 Generalized Dependence of SNR in 3D Imaging on Imaging Parameters 334 15.2.2 SNR Dependence on Read Direction Parameters . . . . . . . . . . . . 336 15.2.3 SNR Dependence on Phase Encoding Parameters . . . . . . . . . . . 340 15.2.4 SNR in 2D Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 15.2.5 Imaging Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 15.3 Contrast, Contrast-to-Noise, and Visibility . . . . . . . . . . . . . . . . . . . 342 15.3.1 Contrast and Contrast-to-Noise Ratio . . . . . . . . . . . . . . . . . . 343 15.3.2 Object Visibility and the Rose Criterion . . . . . . . . . . . . . . . . 343 15.4 Contrast Mechanisms in MRI and Contrast Maximization. . . . . . . . . . . 345