Magnetic Resonance Elastography

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

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• Eletromagnetismo eletricidade, magnetismo e
• A ressonância magnética