Preface

Magnetic Resonance Imaging (MRI) plays a fundamental role in medicine, particularly for the evaluation of brain, spine, heart, muscle, and other soft tissues. Indeed, modern MRI techniques and associated hardware have become increasingly sophisticated, resulting in the acquisition of large volumes of clinically invaluable information within ever-decreasing scan times. Unlike X-ray imaging, computed tomography (CT), and ultrasound (US), wherein an energy source and detector pair are used to form images, the signal for MRI originates intrinsically within the body through the manipulation of a fundamental property of matter - the nuclear spin. In addition, MR signals strongly depend not only on the physical properties of the targeted nuclei themselves but also on processes linking nuclei, the molecules containing them, and interactions with the surrounding environment, providing a rich variety of image contrasts not seen in other modalities. Via these interactions, in vivo MRI can obtain unique biological information vital to our understanding of health and disease. This textbook focuses on developing a physical and mathematical understanding of these in vivo processes and how they can be utilized by Magnetic Resonance Spectroscopy (MRS) to measure individual biochemicals in the body or by MRI to generate unique image contrast.

Although there are multiple excellent MRI textbooks, the material presented here addresses what we think is a largely unmet need. The material bridges the gap between the physics of magnetic resonance (MR) image formation and the in vivo processes that influence the detected signals, equipping the reader with the mathematical tools essential to study the spin interactions leading to various contrast mechanisms. Specifically, the material arises from the lecture notes for a Stanford University Department of Radiology course taught for over 15 years, typically taken by engineering and biomedical physics PhD students as their second graduate class in MRI. The first course, based on the classical description of MRI starts with the Bloch equations followed by discussions of radiofrequency (Rf) excitation in combination with the linear gradient fields used for image formation. Although immensely powerful, this traditional approach based on following the evolution of a bulk magnetization vector arising from the sum of a very large number of independent nuclear spins, does not fully explain many important in vivo processes. For example, T1 and T2 relaxation times are typically included as phenomenological constants, whereas the physics driving these processes provides important insights connecting in vivo data and underlying anatomy and physiology. Indeed, individual spins have no T1 or T2, as these relaxation parameters are only emergent properties of a large collection or ensemble of spins!

More generally, interactions between spins (e.g., J coupling, dipolar coupling, and chemical exchange) provide fundamental contributions to image contrast. This textbook aims to introduce the reader to the tools needed to analyze these spin interactions with a goal to answer many intriguing questions including the following:

• Do tissues behave more like liquids or solids (liquid-state versus solid-state Nuclear Magnetic Resonance [NMR] requires distinctly different analysis tools)?
• If gray matter, white matter, and cerebrospinal fluid (CSF) in brain have similar water content, why do these tissue types show such marked contrast differences depending on the chosen MRI sequence?
• Why is fat bright in Fast Spin Echo (FSE) imaging?
• How does the presence of oxy- versus deoxy-hemoglobin affect the MRI of blood?
• Why is the T2 (and not the T1) of tendons strongly dependent on the angular orientation of the tendon with respect to the main magnetic field B0?
• Why is the chemical shift of water, but not fat, temperature-dependent?
• Why is the MRI contrast agent gadolinium-diethylenetriamine penta-acetic acid (Gd-DTPA) used to shorten T1, whereas dysprosium-DTPA (Dy-DTPA) has almost no T1 shortening effects but is useful as a T2 agent?

To analyze spin-spin interactions, we have chosen to start from a quantum mechanical (QM) formulation. It is not the case that the QM derivation is any more rigorous than that from classical physics. Rather, both static and dynamic spin-spin interactions are most easily incorporated using a QM approach. An emphasis is placed on understanding the associated math and physics while maintaining the critical intuition needed to make such knowledge useful in practice, and topical questions for the reader have been added throughout the text to encourage students to think critically and develop a deeper understanding of the material. We have also included exercises (with solutions provided in a companion volume) and biographical sketches to best capture the individual contributions to the rich history of the development of MR theory and practice. The chapters are organized as described below.

• Chapter 1 introduces the subject of MR with some of the important historical developments.
• Chapter 2 contains a brief description of the source of nuclear magnetism followed by a classical description of MR starting from the Bloch equations and then covering the important topics of Rf excitation, signal reception, transverse and longitudinal magnetization, and spatial localization via gradients. This leads to the fundamental imaging equation of MRI.
• Chapter 3 outlines the physics and mathematical concepts underlying the aspects of quantum mechanics most relevant to analyzing MR. Descriptions of MR are then provided in Hilbert and Liouville spaces with a focus on maintaining the vector formulation of MR that provides much of the physical intuition underlying classical MRI. As part of this development, the density operator, the fundamental quantity underlying the QM description of a collection of spins, is introduced.
• Chapter 4 covers the primary terms in the nuclear spin Hamiltonian, i.e., the QM operator corresponding to the energy of a spin system. Internal and external interactions are separately described followed by a discussion of the key concepts of spin populations and phase coherences.
• Chapter 5 describes the Product Operator Formalism (POF) of MR, the key analysis tool allowing a robust, yet intuitive, description of MR pulse sequences via a series of rotations in a vector space. POF is the basic tool for analyzing modern MRS sequences that exploit both J coupling and multiple quantum coherences. In particular, a solid understanding of J coupling is fundamental to understanding in vivo MRS.
• Chapter 6 gives an overview of in vivo magnetic resonance spectroscopy (MRS) techniques and clinical applications. MRS is a key technique for measuring in vivo metabolism and, hence, very important in its own right. Furthermore, a robust understanding of J coupling is a key first step before moving on to the more difficult-to-analyze dipolar coupling phenomenon that is the primary driver of in vivo relaxation.
• Chapter 7 introduces the fundamental mechanisms dominating in vivo MR relaxation, including high-level descriptions of dipolar coupling and chemical exchange phenomena.
• Chapter 8 develops Redfield theory, a rigorous mathematic framework for deriving relaxation rates in liquids. This formulation is particularly relevant in that most MR-detectable in vivo tissues are well modeled as liquids, with some notable exceptions. Redfield theory also provides a convenient approach for deriving a third MR relaxation parameter beyond T1 and T2, namely, T1? (also known as spin-lattice relaxation in the rotating frame).
• Chapter 9 discusses the various types of MRI contrast agents and their associated physics, focusing on the clinical agents used to shorten T1, T2, and .
• Chapter 10 is the final chapter and focuses on a representative set of in vivo tissues, highlighting concepts introduced in the earlier chapters.

As we embark on this journey into the fascinating realm of in vivo magnetic resonance, it is with great pleasure and gratitude that we acknowledge the collective efforts and support that have shaped this textbook. We extend our sincere thanks to the myriad trainees and colleagues, especially those at the Stanford University Radiological Sciences Laboratory, whose contributions have been invaluable. While it's impractical to individually list everyone, we express our heartfelt gratitude to all. Thank you.

We also extend our appreciation to Dr. Martin Preuss for his expert guidance through the intricate publication process, Neena Ganjoo for skillfully managing publishing timelines, and a special commendation to Sindhuraj Kuttappan and Sakthivel Kandaswamy for their exceptional assistance in editing and manuscript preparation.

Finally, a project of this magnitude demands not only intellectual collaboration but also the unwavering support of our spouses and families. We are profoundly grateful for their infinite patience and enduring encouragement, without which this volume...