# Fundamentals of In Vivo Magnetic Resonance

## Spin Physics, Relaxation Theory, and Contrast Mechanisms

1. Edition March 2024

288 Pages, Softcover*Wiley & Sons Ltd*

**978-1-394-23309-0**

Authoritative reference explaining why and how the most important, radiation-free technique for elucidating tissue properties in the body works

In Vivo Magnetic Resonance helps readers develop an understanding of the fundamental physical processes that take place inside the body that can be probed by magnetic resonance imaging (MRI) and magnetic resonance spectroscopy (MRS), uniquely bridging the gap between the physics of magnetic resonance (MR) image formation and the in vivo processes that influence the detected signals, thereby equipping the reader with the mathematical tools essential to study the spin interactions leading to various contrast mechanisms.

With a focus on clinical relevance, this book equips readers with practical knowledge that can be directly applied in medical settings, enabling informed decision-making and advancements in the field of medical imaging. The material arises from the lecture notes for a Stanford University Department of Radiology course taught for over 15 years.

Aided by clever illustrations, the book takes a step-by-step approach to explain complex concepts in a comprehensible manner. Readers can test their understanding by working on approximately 60 sample problems.

Written by two highly qualified authors with significant experience in the field, In Vivo Magnetic Resonance includes information on:

* The fundamental imaging equations of MRI

* Quantum elements of magnetic resonance, including linear vector spaces, Dirac notation, Hilbert Space, Liouville Space, and associated mathematical concepts

* Nuclear spins, covering external and internal interactions, chemical shifts, dipolar coupling, J-coupling, the spin density operator, and the product operator formalism

* In vivo MR spectroscopy methods

* MR relaxation theory and the underlying sources of image contrast accessible via modern clinical MR imaging techniques

With comprehensive yet accessible coverage of the subject and a wealth of learning resources included throughout, In Vivo Magnetic Resonance is an ideal text for graduate students in the fields of physics, biophysics, biomedical physics, and materials science, along with lecturers seeking classroom aids.

About the Companion Website xv

1 Introduction 1

1.1 A Brief History of MR 1

1.2 NMR versus MRI 3

1.3 The Roadmap 5

2 Classical Description of MR 11

2.1 Nuclear Magnetism 11

2.2 Net Magnetization and the Bloch Equations 13

2.3 Rf Excitation and Reception 14

2.4 Spatial Localization 15

2.5 The MRI Signal Equation 16

2.6 Summary 19

Exercises 20

Historical Notes 23

3 Quantum Mechanical Description of MR 27

3.1 Introduction 27

3.1.1 Why Quantum Mechanics for Magnetic Resonance? 27

3.1.2 Historical Developments 27

3.1.3 Wave Functions 29

3.2 Mathematics of QM 32

3.2.1 Linear Vector Spaces 32

3.2.2 Dirac Notation and Hilbert Space 33

3.2.3 Liouville Space 36

3.3 The Six Postulates of QM 38

3.3.1 Postulate 1 38

3.3.2 Postulate 2 38

3.3.3 Postulate 3 39

3.3.4 Postulate 4 39

3.3.5 Postulate 5 39

3.3.6 Postulate 6 40

3.4 MR in Hilbert Space 44

3.4.1 Review of Spin Operators 44

3.4.2 Single Spin in a Magnetic Field 44

3.4.3 Ensemble of Spins in a Magnetic Field 46

3.5 MR in Liouville Space 49

3.5.1 Statistical Mixture of Quantum States 50

3.5.2 The Density Operator 51

3.5.3 The Spin-lattice Disconnect 52

3.5.4 Hilbert Space versus Liouville Space 52

3.5.5 Observations About the Spin Density Operator 53

3.5.6 Solving the Liouville von Neuman Equation 55

3.6 Summary 57

Exercises 58

Historical Notes 61

4 Nuclear Spins 67

4.1 Review of the Spin Density Operator and the Hamiltonian 67

4.2 External Interactions 68

4.3 Internal Interactions 69

4.3.1 Chemical Shift 71

4.3.2 Dipolar Coupling 72

4.3.3 J Coupling 72

4.4 Summary 75

Exercises 75

Historical Notes 78

5 Product Operator Formalism 81

5.1 The Density Operator, Populations, and Coherences 81

5.1.1 Spin Systems and Associated Density Operators 81

5.1.2 Density Matrix Calculations 85

5.2 POF for Single-Spin Coherence Space 88

5.3 POF for Two-Spin Coherence Space 90

5.4 Branch Diagrams 94

5.5 Multiple Quantum Coherences and 2D NMR 97

5.6 Polarization Transfer 100

5.7 Spectral Editing 103

5.7.1 J-difference Editing 103

5.7.2 Multiple-quantum Filtering 104

5.8 Summary 105

Exercises 106

Historical Notes 111

6 In vivo MRS 113

6.1 ¯1H MRS 113

6.1.1 Acquisition Methods 113

6.1.2 Detectable Metabolites and Applications 120

6.2 ¯31P-MRS 126

6.3 ¯13C-MRS 127

6.3.1 Acquisition Methods 127

6.3.2 ¯13C Infusion Studies 132

6.3.3 Hyperpolarized 13 c 132

6.4 Deuterium Metabolic Imaging 138

6.5 ¯23Na-MRI 140

6.6 Summary 140

Exercises 141

7 Relaxation Fundamentals 145

7.1 Basic Principles 145

7.1.1 Molecular Motion 145

7.1.2 Stochastic Processes 147

7.1.3 A Simple Model of Relaxation 150

7.2 Dipolar Coupling 153

7.2.1 The Solomon Equations 153

7.2.2 Calculating Transition Rates 155

7.2.3 Nuclear Overhauser Effect 158

7.3 Chemical Exchange 160

7.3.1 Introduction 160

7.3.2 Effects on Longitudinal Magnetization 161

7.3.3 Effects on Transverse Magnetization 162

7.3.4 Examples 164

7.4 In Vivo Water 167

7.4.1 Hydration Layers 167

7.4.2 Tissue Relaxation Times 168

7.4.3 Magic Angle Effects 169

7.4.4 Magnetization Transfer Contrast (MTC) 170

7.4.5 Chemical Exchange Saturation Transfer (CEST) 172

7.4.5.1 Amide Proton (-NH) Transfer (APT) 173

7.4.5.2 Hydroxyl (-OH) CEST 173

7.4.5.3 Amine (-NH2) CEST 173

7.5 Summary 174

Exercises 174

Historical Notes 179

8 Redfield Theory of Relaxation 181

8.1 Perturbation Theory and the Interaction Frame of Reference 181

8.2 The Master Equation of NMR 182

8.3 Calculating Relaxation Times 185

8.4 Relaxation Mechanisms 187

8.4.1 Dipolar Coupling Revisited 187

8.4.2 Scalar Relaxation of the 1 st Kind and 2 nd Kind 189

8.4.3 Chemical Shift Anisotropy (CSA) 191

8.5 Relaxation in the Rotating Frame 191

8.5.1 Physics of T1rho 192

8.5.2 The Spin-Lock Experiment 194

8.5.3 Choosing the Optimum Spin-Lock Frequency 195

8.5.4 Rf Power Considerations 200

8.5.5 Adiabatic Spin-Lock 201

8.5.6 Applications 202

8.6 Illustrative Redfield Theory Examples 202

8.6.1 Hyperpolarized ¯13C-urea 202

8.6.2 Hyperpolarized ¯13C-Pyr 203

8.7 Summary 207

Exercises 208

Historical Notes 210

9 MRI Contrast Agents 213

9.1 Paramagnetic Relaxation Enhancement 213

9.1.1 Solomon-Bloembergen-Morgan Theory 215

9.1.2 Gd¯3+-Based T1 Contrast Agents 218

9.2 T2and T¯*2Contrast Agents 219

9.2.1 T2, Diffusion, and Outer-Sphere Relaxation 219

9.2.2 SPIOs and USPIOs 219

9.3 PARACEST Contrast Agents 220

9.4 Contrast Agents in the Clinic 221

9.4.1 Gd-Based Agents 222

9.4.2 Iron-Based Agents 223

9.5 Summary 225

Exercises 225

10 In vivo Examples 229

10.1 Relaxation Properties of the Brain 229

10.1.1 Morphological Imaging 229

10.1.2 Perfusion Imaging 229

10.1.3 Diffusion-weighted Imaging (DWI) 230

10.1.4 Imaging Myelin 232

10.1.5 Susceptibility-weighted Imaging (SWI) 232

10.2 Relaxation Properties of Blood 233

10.2.1 Hemoglobin and Red Blood Cells 233

10.2.2 MRI Blood Oximetry 235

10.2.3 Functional Magnetic Resonance Imaging (fMRI) 236

10.2.4 MRI of Hemorrhage 238

10.3 Relaxation Properties of Cartilage 241

10.3.1 T2Mapping 243

10.3.2 DWI 244

10.3.3 T1rho Mapping and Dispersion 244

10.3.4 gagCEST 245

10.3.5 dGEMRIC 245

10.3.6 Ultrashort TE (UTE) Imaging 246

10.3.7 Sodium MRI 246

10.3.8 Summary 248

10.4 Synopsis 248

Exercises 249

Further Readings 251

Quantum Mechanics 251

Spin Physics 251

Magnetic Resonance Imaging (MRI) 251

In vivo Magnetic Resonance Spectroscopy 251

Relaxation Theory 252

Clinical MRI 252

References 253

Index 265

Keshav Datta, PhD, is Vice President, Research & Development, at VIDA Diagnostics Inc., Coralville, IA, USA, a precision lung health company, accelerating therapies to patients through AI-powered lung intelligence. He is also a Consulting Research Scientist at Stanford University, Stanford, CA, USA.