About the Authors xiii
Foreword xv
Preface xvii
Acknowledgments xix
Notation xxi
List of Symbols xxiii
Introduction 1
Part I Magnetic Resonance Imaging 7
1 Nuclear Magnetic Resonance 9
1.1 Protons in a Magnetic Field 9
1.2 Precession of Magnetization 10
1.2.1 Quadrature Detection 11
1.3 Relaxation 13
1.4 Bloch Equations 14
1.5 Echoes 15
1.5.1 Spin Echoes 15
1.5.2 Gradient Echoes 17
1.6 Magnetic Resonance Imaging 17
1.6.1 Spatial Encoding 18
1.6.1.1 Slice Selection 19
1.6.1.2 Phase Encoding 19
1.6.1.3 Frequency Encoding 20
2 Imaging Concepts 23
2.1 k-Space 23
2.2 k-Space Sampling Strategies 26
2.2.1 Segmented Image Acquisition 27
2.2.1.1 Fast Low-Angle Shot (FLASH) 27
2.2.1.2 Balanced Steady-State Free Precession (bSSFP) 28
2.2.2 Echo-Planar Imaging (EPI) 30
2.2.3 Non-Cartesian Imaging 32
2.3 Fast Imaging 33
2.3.1 Fast Imaging Strategies 33
2.3.2 Partial Fourier Imaging 34
2.3.3 Parallel Imaging 35
2.3.3.1 GRAPPA 36
2.3.4 Impact of Fast Imaging on SNR and Scan Time 37
3 Motion Encoding and MRE Sequences 41
3.1 Motion Encoding 43
3.1.1 Gradient Moment Nulling 44
3.1.2 Encoding of Time-Harmonic Motion 46
3.1.3 Fractional Encoding 50
3.2 Intra-Voxel Phase Dispersion 51
3.3 Diffusion-Weighted MRE 52
3.4 MRE Sequences 53
3.4.1 FLASH-MRE 53
3.4.2 bSSFP-MRE 55
3.4.3 EPI-MRE 57
Part II Elasticity 61
4 Viscoelastic Theory 63
4.1 Strain 63
4.2 Stress 67
4.3 Invariants 68
4.4 Hooke's Law 69
4.5 Strain-Energy Function 70
4.6 Symmetries 71
4.7 Engineering Constants 75
4.7.1 Young's Modulus and Poisson's Ratio 75
4.7.2 Shear Modulus and Lame's First Parameter 76
4.7.3 Compressibility and Bulk Modulus 77
4.7.4 Compliance and Elasticity Tensor for a Transversely Isotropic Material 79
4.8 Viscoelastic Models 80
4.8.1 Elastic Model: Spring 81
4.8.2 Viscous Model: Dashpot 82
4.8.3 Combinations of Elastic and Viscous Elements 83
4.8.4 Overview of Viscoelastic Models 89
4.9 Dynamic Deformation 92
4.9.1 Balance of Momentum 92
4.9.2 MechanicalWaves 96
4.9.2.1 Complex Moduli andWave Speed 98
4.9.3 Navier-Stokes Equation 99
4.9.4 Compression Modulus and Oscillating Volumetric Strain 100
4.9.5 Elastodynamic Green's Function 101
4.9.6 Boundary Conditions 103
4.10 Waves in Anisotropic Media 104
4.10.1 The Christoffel Equation 105
4.10.2 Waves in a Transversely Isotropic Medium 106
4.11 Energy Density and Flux 110
4.11.1 Geometric Attenuation 113
4.12 ShearWave Scattering from Interfaces and Inclusions 114
4.12.1 Plane Interfaces 115
4.12.2 Spatial and Temporal Interfaces 118
4.12.3 Wave Diffusion 121
4.12.3.1 Green's Function ofWaves and Diffusion Phenomena 125
4.12.3.2 Amplitudes and Intensities of DiffusiveWaves 126
5 Poroelasticity 131
5.1 Navier's Equation for Biphasic Media 133
5.1.1 PressureWaves in Poroelastic Media 136
5.1.2 ShearWaves in Poroelastic Media 140
5.2 Poroelastic Signal Equation 142
Part III Technical Aspects and Data Processing 145
6 MRE Hardware 147
6.1 MRI Systems 147
6.2 Actuators 153
6.2.1 Technical Requirements 153
6.2.2 Practicality 153
6.2.3 Types of Mechanical Transducers 154
7 MRE Protocols 161
8 Numerical Methods and Postprocessing 165
8.1 Noise and Denoising in MRE 165
8.1.1 Denoising: An Overview 165
8.1.2 Least Squares and Polynomial Fitting 167
8.1.3 Frequency Domain (k-Space) Filtering 168
8.1.3.1 Averaging 168
8.1.3.2 LTI Filters in the Fourier Domain 170
8.1.3.3 Band-Pass Filtering 172
8.1.4 Wavelets and Multi-Resolution Analysis (MRA) 172
8.1.5 FFT versus MRA in vivo 174
8.1.6 Sparser Approximations and Performance Times 175
8.2 Directional Filters 176
8.3 Numerical Derivatives 179
8.3.1 Matrix Representation of Derivative Operators 182
8.3.2 Anderssen Gradients 183
8.3.3 Frequency Response of Derivative Operators 186
8.4 Finite Differences 187
9 Phase Unwrapping 191
9.1 Flynn's Minimum Discontinuity Algorithm 193
9.2 Gradient Unwrapping 195
9.3 Laplacian Unwrapping 196
10 Viscoelastic Parameter Reconstruction Methods 199
10.1 Discretization and Noise 201
10.2 Phase Gradient 204
10.3 Algebraic Helmholtz Inversion 205
10.3.1 Multiparameter Inversion 207
10.3.2 Helmholtz Decomposition 207
10.4 Local Frequency Estimation 208
10.5 Multifrequency Inversion 210
10.5.1 Reconstruction of 211
10.5.2 Reconstruction of |G?| 213
10.6 k-MDEV 214
10.7 Finite Element Method 217
10.7.1 Weak Formulation of the One-DimensionalWave Equation 218
10.7.2 Discretization of the Problem Domain 219
10.7.3 Basis Function in the Discretized Domain 220
10.7.4 FE Formulation of theWave Equation 221
10.8 Direct Inversion for a Transverse Isotropic Medium 224
10.9 Waveguide Elastography 225
11 Multicomponent Acquisition 229
12 Ultrasound Elastography 233
12.1 Strain Imaging (SI) 235
12.2 Strain Rate Imaging (SRI) 235
12.3 Acoustic Radiation Force Impulse (ARFI) Imaging 235
12.4 Vibro-Acoustography (VA) 237
12.5 Vibration-Amplitude Sonoelastography (VA Sono) 237
12.6 Cardiac Time-Harmonic Elastography (Cardiac THE) 237
12.7 Vibration Phase Gradient (PG) Sonoelastography 238
12.8 Time-Harmonic Elastography (1D/2D THE) 238
12.9 CrawlingWaves (CW) Sonoelastography 238
12.10 ElectromechanicalWave Imaging (EWI) 239
12.11 PulseWave Imaging (PWI) 239
12.12 Transient Elastography (TE) 240
12.13 Point ShearWave Elastography (pSWE) 240
12.14 ShearWave Elasticity Imaging (SWEI) 240
12.15 Comb-Push Ultrasound Shear Elastography (CUSE) 241
12.16 Supersonic Shear Imaging (SSI) 241
12.17 SpatiallyModulated Ultrasound Radiation Force (SMURF) 241
12.18 ShearWave Dispersion Ultrasound Vibrometry (SDUV) 241
12.19 Harmonic Motion Imaging (HMI) 242
Part IV Clinical Applications 243
13 MRE of the Heart 245
13.1 Normal Heart Physiology 245
13.1.1 Cardiac Fiber Anatomy 247
13.1.2 Wall Shear Modulus versus Cavity Pressure 249
13.2 Clinical Motivation for Cardiac MRE 250
13.2.1 Systolic Dysfunction versus Diastolic Dysfunction 250
13.3 Cardiac Elastography 252
13.3.1 Ex vivo SWI 253
13.3.2 In vivo SDUV 253
13.3.3 In vivo CardiacMRE in Pigs 254
13.3.4 In vivo CardiacMRE in Humans 256
13.3.4.1 Steady-State MRE (WAV-MRE) 256
13.3.4.2 Wave Inversion Cardiac MRE 259
13.3.5 MRE of the Aorta 260
14 MRE of the Brain 263
14.1 General Aspects of Brain MRE 264
14.1.1 Objectives 264
14.1.2 Determinants of Brain Stiffness 264
14.1.3 Challenges for Cerebral MRE 264
14.2 Technical Aspects of Brain MRE 265
14.2.1 Clinical Setup for Cerebral MRE 265
14.2.2 Choice of Vibration Frequency 266
14.2.3 Driver-Free Cerebral MRE 269
14.2.4 MRE in the Mouse Brain 270
14.3 Findings 271
14.3.1 Brain Stiffness Changes with Age 272
14.3.2 Male Brains Are Softer than Female Brains 273
14.3.3 Regional Variation in Brain Stiffness 274
14.3.4 Anisotropic Properties of Brain Tissue 274
14.3.5 The in vivo Brain Is Compressible 276
14.3.6 Preliminary Findings of MRE with Functional Activation 277
14.3.7 Demyelination and Inflammation Reduce Brain Stiffness 277
14.3.8 Neurodegeneration Reduces Brain Stiffness 279
14.3.9 The Number of Neurons Correlates with Brain Stiffness 280
14.3.10 Preliminary Conclusions on MRE of the Brain 280
15 MRE of Abdomen, Pelvis, and Intervertebral Disc 283
15.1 Liver 283
15.1.1 Epidemiology of Chronic Liver Diseases 286
15.1.2 Liver Fibrosis 287
15.1.2.1 Pathogenesis of Liver Fibrosis 289
15.1.2.2 Staging of Liver Fibrosis 291
15.1.2.3 Noninvasive Screening Methods for Liver Fibrosis 292
15.1.2.4 Reversibility of Liver Fibrosis 293
15.1.2.5 Biophysical Signs of Liver Fibrosis 293
15.1.3 MRE of the Liver 294
15.1.3.1 MRE in Animal Models of Hepatic Fibrosis and Liver Tissue Samples 294
15.1.3.2 Early Clinical Studies and Further Developments 295
15.1.3.3 MRE of Nonalcoholic Fatty Liver Disease 303
15.1.3.4 Comparison with other Noninvasive Imaging and Serum Biomarkers 304
15.1.3.5 MRE of the Liver for Assessing Portal Hypertension 307
15.1.3.6 MRE in Liver Grafts 309
15.1.3.7 Confounders 310
15.2 Spleen 311
15.2.1 MRE of the Spleen 311
15.3 Pancreas 314
15.3.1 MRE of the Pancreas 315
15.4 Kidneys 315
15.4.1 MRE of the Kidneys 316
15.5 Uterus 318
15.5.1 MRE of the Uterus 318
15.6 Prostate 319
15.6.1 MRE of the Prostate 320
15.7 Intervertebral Disc 321
15.7.1 MRE of the Intervertebral Disc 322
16 MRE of Skeletal Muscle 325
16.1 In vivo MRE of Healthy Muscles 326
16.2 MRE in Muscle Diseases 330
17 Elastography of Tumors 333
17.1 Micromechanical Properties of Tumors 333
17.2 Ultrasound Elastography of Tumors 336
17.2.1 Ultrasound Elastography in Breast Tumors 337
17.2.2 Ultrasound Elastography in Prostate Cancer 338
17.3 MRE of Tumors 339
17.3.1 MRE of Tumors in the Mouse 340
17.3.2 MRE in Liver Tumors 342
17.3.3 MRE of Prostate Cancer 344
17.3.3.1 Ex Vivo Studies 344
17.3.3.2 In Vivo Studies 345
17.3.4 MRE of Breast Tumors 345
17.3.4.1 In Vivo MRE of Breast Tumors 346
17.3.5 MRE of Intracranial Tumors 347
Part V Outlook 351
Dimensionality 351
Sparsity 352
Heterogeneity 353
Reproducibility 353
A Simulating the Bloch Equations 355
B Proof that Eq. (3.8) Is Sinusoidal 357
C Proof for Eq. (4.1) 359
D Wave Intensity Distributions 361
D.1 Calculation of Intensity Probabilities 361
D.2 Point Source in 3D 362
D.3 Classical Diffusion 363
D.4 Damped PlaneWave 365
References 367
Index 417
