Magnetic Resonance Elastography

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 xiiiForeword xvPreface xviiAcknowledgments xixNotation xxiList of Symbols xxiiiIntroduction 1Part I Magnetic Resonance Imaging 71 Nuclear Magnetic Resonance 91.1 Protons in a Magnetic Field 91.2 Precession of Magnetization 101.2.1 Quadrature Detection 111.3 Relaxation 131.4 Bloch Equations 141.5 Echoes 151.5.1 Spin Echoes 151.5.2 Gradient Echoes 171.6 Magnetic Resonance Imaging 171.6.1 Spatial Encoding 181.6.1.1 Slice Selection 191.6.1.2 Phase Encoding 191.6.1.3 Frequency Encoding 202 Imaging Concepts 232.1 k-Space 232.2 k-Space Sampling Strategies 262.2.1 Segmented Image Acquisition 272.2.1.1 Fast Low-Angle Shot (FLASH) 272.2.1.2 Balanced Steady-State Free Precession (bSSFP) 282.2.2 Echo-Planar Imaging (EPI) 302.2.3 Non-Cartesian Imaging 322.3 Fast Imaging 332.3.1 Fast Imaging Strategies 332.3.2 Partial Fourier Imaging 342.3.3 Parallel Imaging 352.3.3.1 Grappa 362.3.4 Impact of Fast Imaging on SNR and Scan Time 373 Motion Encoding and MRE Sequences 413.1 Motion Encoding 433.1.1 Gradient Moment Nulling 443.1.2 Encoding of Time-Harmonic Motion 463.1.3 Fractional Encoding 503.2 Intra-Voxel Phase Dispersion 513.3 Diffusion-Weighted MRE 523.4 MRE Sequences 533.4.1 Flash-mre 533.4.2 bSSFP-MRE 553.4.3 Epi-mre 57Part II Elasticity 614 Viscoelastic Theory 634.1 Strain 634.2 Stress 674.3 Invariants 684.4 Hooke’s Law 694.5 Strain-Energy Function 704.6 Symmetries 714.7 Engineering Constants 754.7.1 Young’s Modulus and Poisson’s Ratio 754.7.2 Shear Modulus and Lamé’s First Parameter 764.7.3 Compressibility and Bulk Modulus 774.7.4 Compliance and Elasticity Tensor for a Transversely Isotropic Material 794.8 Viscoelastic Models 804.8.1 Elastic Model: Spring 814.8.2 Viscous Model: Dashpot 824.8.3 Combinations of Elastic and Viscous Elements 834.8.4 Overview of Viscoelastic Models 894.9 Dynamic Deformation 924.9.1 Balance of Momentum 924.9.2 Mechanical Waves 964.9.2.1 Complex Moduli and Wave Speed 984.9.3 Navier–Stokes Equation 994.9.4 Compression Modulus and Oscillating Volumetric Strain 1004.9.5 Elastodynamic Green’s Function 1014.9.6 Boundary Conditions 1034.10 Waves in Anisotropic Media 1044.10.1 The Christoffel Equation 1054.10.2 Waves in a Transversely Isotropic Medium 1064.11 Energy Density and Flux 1104.11.1 Geometric Attenuation 1134.12 Shear Wave Scattering from Interfaces and Inclusions 1144.12.1 Plane Interfaces 1154.12.2 Spatial and Temporal Interfaces 1184.12.3 Wave Diffusion 1214.12.3.1 Green’s Function of Waves and Diffusion Phenomena 1254.12.3.2 Amplitudes and Intensities of Diffusive Waves 1265 Poroelasticity 1315.1 Navier’s Equation for Biphasic Media 1335.1.1 Pressure Waves in Poroelastic Media 1365.1.2 Shear Waves in Poroelastic Media 1405.2 Poroelastic Signal Equation 142Part III Technical Aspects and Data Processing 1456 MRE Hardware 1476.1 MRI Systems 1476.2 Actuators 1536.2.1 Technical Requirements 1536.2.2 Practicality 1536.2.3 Types of Mechanical Transducers 1547 MRE Protocols 1618 Numerical Methods and Postprocessing 1658.1 Noise and Denoising in MRE 1658.1.1 Denoising: An Overview 1658.1.2 Least Squares and Polynomial Fitting 1678.1.3 Frequency Domain (k-Space) Filtering 1688.1.3.1 Averaging 1688.1.3.2 LTI Filters in the Fourier Domain 1708.1.3.3 Band-Pass Filtering 1728.1.4 Wavelets and Multi-Resolution Analysis (MRA) 1728.1.5 FFT versus MRA in vivo 1748.1.6 Sparser Approximations and Performance Times 1758.2 Directional Filters 1768.3 Numerical Derivatives 1798.3.1 Matrix Representation of Derivative Operators 1828.3.2 Anderssen Gradients 1838.3.3 Frequency Response of Derivative Operators 1868.4 Finite Differences 1879 Phase Unwrapping 1919.1 Flynn’s Minimum Discontinuity Algorithm 1939.2 Gradient Unwrapping 1959.3 Laplacian Unwrapping 19610 Viscoelastic Parameter Reconstruction Methods 19910.1 Discretization and Noise 20110.2 Phase Gradient 20410.3 Algebraic Helmholtz Inversion 20510.3.1 Multiparameter Inversion 20710.3.2 Helmholtz Decomposition 20710.4 Local Frequency Estimation 20810.5 Multifrequency Inversion 21010.5.1 Reconstruction of φ 21110.5.2 Reconstruction of |G ∗ | 21310.6 k-MDEV 21410.7 Finite Element Method 21710.7.1 Weak Formulation of the One-Dimensional Wave Equation 21810.7.2 Discretization of the Problem Domain 21910.7.3 Basis Function in the Discretized Domain 22010.7.4 FE Formulation of the Wave Equation 22110.8 Direct Inversion for a Transverse Isotropic Medium 22410.9 Waveguide Elastography 22511 Multicomponent Acquisition 22912 Ultrasound Elastography 23312.1 Strain Imaging (SI) 23512.2 Strain Rate Imaging (SRI) 23512.3 Acoustic Radiation Force Impulse (ARFI) Imaging 23512.4 Vibro-Acoustography (VA) 23712.5 Vibration-Amplitude Sonoelastography (VA Sono) 23712.6 Cardiac Time-Harmonic Elastography (Cardiac THE) 23712.7 Vibration Phase Gradient (PG) Sonoelastography 23812.8 Time-Harmonic Elastography (1D/2D THE) 23812.9 Crawling Waves (CW) Sonoelastography 23812.10 Electromechanical Wave Imaging (EWI) 23912.11 Pulse Wave Imaging (PWI) 23912.12 Transient Elastography (TE) 24012.13 Point Shear Wave Elastography (pSWE) 24012.14 Shear Wave Elasticity Imaging (SWEI) 24012.15 Comb-Push Ultrasound Shear Elastography (CUSE) 24112.16 Supersonic Shear Imaging (SSI) 24112.17 Spatially Modulated Ultrasound Radiation Force (SMURF) 24112.18 Shear Wave Dispersion Ultrasound Vibrometry (SDUV) 24112.19 Harmonic Motion Imaging (HMI) 242Part IV Clinical Applications 24313 MRE of the Heart 24513.1 Normal Heart Physiology 24513.1.1 Cardiac Fiber Anatomy 24713.1.2 Wall Shear Modulus versus Cavity Pressure 24913.2 Clinical Motivation for Cardiac MRE 25013.2.1 Systolic Dysfunction versus Diastolic Dysfunction 25013.3 Cardiac Elastography 25213.3.1 Ex vivo SWI 25313.3.2 In vivo SDUV 25313.3.3 In vivo Cardiac MRE in Pigs 25413.3.4 In vivo Cardiac MRE in Humans 25613.3.4.1 Steady-State MRE (WAV-MRE) 25613.3.4.2 Wave Inversion Cardiac MRE 25913.3.5 MRE of the Aorta 26014 MRE of the Brain 26314.1 General Aspects of Brain MRE 26414.1.1 Objectives 26414.1.2 Determinants of Brain Stiffness 26414.1.3 Challenges for Cerebral MRE 26414.2 Technical Aspects of Brain MRE 26514.2.1 Clinical Setup for Cerebral MRE 26514.2.2 Choice of Vibration Frequency 26614.2.3 Driver-Free Cerebral MRE 26914.2.4 MRE in the Mouse Brain 27014.3 Findings 27114.3.1 Brain Stiffness Changes with Age 27214.3.2 Male Brains Are Softer than Female Brains 27314.3.3 Regional Variation in Brain Stiffness 27414.3.4 Anisotropic Properties of Brain Tissue 27414.3.5 The in vivo Brain Is Compressible 27614.3.6 Preliminary Findings of MRE with Functional Activation 27714.3.7 Demyelination and Inflammation Reduce Brain Stiffness 27714.3.8 Neurodegeneration Reduces Brain Stiffness 27914.3.9 The Number of Neurons Correlates with Brain Stiffness 28014.3.10 Preliminary Conclusions on MRE of the Brain 28015 MRE of Abdomen, Pelvis, and Intervertebral Disc 28315.1 Liver 28315.1.1 Epidemiology of Chronic Liver Diseases 28615.1.2 Liver Fibrosis 28715.1.2.1 Pathogenesis of Liver Fibrosis 28915.1.2.2 Staging of Liver Fibrosis 29115.1.2.3 Noninvasive Screening Methods for Liver Fibrosis 29215.1.2.4 Reversibility of Liver Fibrosis 29315.1.2.5 Biophysical Signs of Liver Fibrosis 29315.1.3 MRE of the Liver 29415.1.3.1 MRE in Animal Models of Hepatic Fibrosis and Liver Tissue Samples 29415.1.3.2 Early Clinical Studies and Further Developments 29515.1.3.3 MRE of Nonalcoholic Fatty Liver Disease 30315.1.3.4 Comparison with other Noninvasive Imaging and Serum Biomarkers 30415.1.3.5 MRE of the Liver for Assessing Portal Hypertension 30715.1.3.6 MRE in Liver Grafts 30915.1.3.7 Confounders 31015.2 Spleen 31115.2.1 MRE of the Spleen 31115.3 Pancreas 31415.3.1 MRE of the Pancreas 31515.4 Kidneys 31515.4.1 MRE of the Kidneys 31615.5 Uterus 31815.5.1 MRE of the Uterus 31815.6 Prostate 31915.6.1 MRE of the Prostate 32015.7 Intervertebral Disc 32115.7.1 MRE of the Intervertebral Disc 32216 MRE of Skeletal Muscle 32516.1 In vivo MRE of Healthy Muscles 32616.2 MRE in Muscle Diseases 33017 Elastography of Tumors 33317.1 Micromechanical Properties of Tumors 33317.2 Ultrasound Elastography of Tumors 33617.2.1 Ultrasound Elastography in Breast Tumors 33717.2.2 Ultrasound Elastography in Prostate Cancer 33817.3 MRE of Tumors 33917.3.1 MRE of Tumors in the Mouse 34017.3.2 MRE in Liver Tumors 34217.3.3 MRE of Prostate Cancer 34417.3.3.1 Ex Vivo Studies 34417.3.3.2 In Vivo Studies 34517.3.4 MRE of Breast Tumors 34517.3.4.1 In Vivo MRE of Breast Tumors 34617.3.5 MRE of Intracranial Tumors 347Part V Outlook 351Dimensionality 351Sparsity 352Heterogeneity 353Reproducibility 353A Simulating the Bloch Equations 355B Proof that Eq. (3.8) Is Sinusoidal 357C Proof for Eq. (4.1) 359D Wave Intensity Distributions 361D. 1 Calculation of Intensity Probabilities 361D. 2 Point Source in 3D 362D. 3 Classical Diffusion 363D. 4 Damped Plane Wave 365References 367Index 417