Magnetic Resonance Elastography Physical

Magnetic resonance elastography (MRE) is a medical imaging technique that combines magnetic resonance imaging (MRI) with mechanical vibrations to generate maps of viscoelastic properties of biological tissue. It serves as a non-invasive tool to detect and quantify mechanical changes in tissue structure, which can be symptoms or causes of various diseases. Clinical and research applications of MRE include staging of liver fibrosis, assessment of tumor stiffness and investigation of neurodegenerative diseases.
The first part of this book is dedicated to the physical and technological principles underlying MRE, with an introduction to MRI physics, viscoelasticity theory and classical waves, as well as vibration generation, image acquisition and viscoelastic parameter reconstruction.
The second part of the book focuses on clinical applications of MRE to various organs. Each section starts with a discussion of the specific properties of the organ, followed by an extensive overview of clinical and preclinical studies that have been performed, tabulating reference values from published literature. The book is completed by a chapter discussing technical aspects of elastography methods based on ultrasound.

Ingolf Sack is professor for Experimental Radiology and Elastography at Charité - Universitätsmedizin Berlin, Germany. He received a PhD in Chemistry at Freie Universität Berlin, Germany, for the development of methods in NMR spectroscopy. He worked at the Weizmann Institute in Rehovot, Israel, and at the Sunnybrook Hospital Toronto, Canada. Since 2003 he leads an interdisciplinary team of physicists, engineers, chemists and physicians which has pioneered pivotal developments in time-harmonic elastography of both MRI and ultrasound for many medical applications.

Sebastian Hirsch is a postdoctoral fellow in the Department of Radiology at the Charité - Universitätsmedizin Berlin, Germany. After studying physics at the University of Mainz, Germany, he joined Charité, where he works on pressure-sensitive MRE and the development of data acquisition strategies.

Jürgen Braun is an assistant professor at the Charité - Universitätsmedizin Berlin, Germany. He received his PhD degree in physical chemistry from Albert-Ludwigs-University in Freiburg, Germany, for the elucidation of reaction kinetics with liquid and solid state NMR. He possesses long standing professional experience in elastography, medical engineering, and image processing.

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 Lamé’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 𝜑 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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Magnetic Resonance Elastography Physical Background and Medical Applications

Summary

Author Biography

Table of Contents

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