Fundamentals of NMR and MRI: From Quantum Principles to Medical Applications 3031479750, 9783031479755

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Fatemeh Khashami

Fundamentals of NMR and MRI From Quantum Principles to Medical Applications

Fundamentals of NMR and MRI

Fatemeh Khashami

Fundamentals of NMR and MRI From Quantum Principles to Medical Applications

Fatemeh Khashami Advanced Imaging Research Center The University of Texas Southwestern Medical Center Dallas, TX, USA

ISBN 978-3-031-47975-5 ISBN 978-3-031-47976-2 https://doi.org/10.1007/978-3-031-47976-2

(eBook)

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

To my uncle, Abdolali, my lifelong inspiration To my niece, Sarina, with endless love

Preface

In this book, we will explore the fascinating applications of quantum principles in nuclear magnetic resonance (NMR) and magnetic resonance imaging (MRI) technologies, where the laws of quantum mechanics uncover the secrets of matter and magnetic phenomena, offering readers an in-depth understanding of the underlying concepts and applications of these techniques. The main goal of this book is to bridge the gap between the quantum foundation and medical applications of NMR and MRI technologies, with a transparent treatment of the quantum physics of NMR and MIR to make it accessible to a wide range of scientists such as biologists and medical scientists, chemists, and physicists. This book begins with a concise introduction to Quantum Mechanics in Chap. 1, where we discuss the Schrödinger equation, quantum formalism, and Hilbert space structure. We explore state vectors, operators, and the fascinating concept of spin in quantum mechanics. We further discuss the density matrix formalism that is important for the physics of spin systems in NMR and MRI. In Chap. 2, “Quantum Physics and NMR Foundations,” we discuss the essential elements of NMR systems and describe the Zeeman effect and the Boltzmann distribution. We analyze the behavior of magnetic fields on spin populations at thermal equilibrium and explore the dynamics of the macroscopic magnetization using the density matrix description. In Chap. 3, “Bloch Equation Description Without Relaxation Time,” we investigate the Bloch equation in both the laboratory frame and the rotating frame. We introduce the flip angle of the RF pulse and the geometric representation of RF pulses through density matrix formalism. The exploration of the Bloch equation continues in Chap. 4, “Bloch Equation Description with Relaxation Time,” where the relaxation time is also taken into account. We introduce longitudinal and transverse magnetization in the spin system. We explore how the Fourier transformation method and the density matrix approach offer a deeper understanding of magnetization dynamics. In Chap. 5, “Molecular Motion, Correlation, and Relaxation Time,” we examine the influence of dipole-dipole interactions and correlation time on relaxation processes. We introduce the Bloembergen-Pound-Purcell theory and the spectral vii

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density function, discussing the impact of molecular motion on relaxation time and introducing the Solomon-Bloembergen-Morgan theory. Chapter 6, “Chemical Shift, NMR Spectroscopy, and Beyond,” discusses NMR spectroscopy, starting with the discussion of electron density shielding. We explore the chemical shift concept and NMR sensitivity and receptivity for various nuclei. Proton-NMR, Carbon-NMR, Phosphorus-NMR, Nitrogen-NMR, Fluorine-NMR, and Xenon-NMR spectroscopy are discussed. In Chap. 7, “Spin-Echo and Spin-Lock Pulse Sequences in MRI System,” we shift our focus to the world of MRI. We analyze the principles behind spin-echo and spin-lock pulse sequences. The density matrix perspective of spin-echo is explored, providing a fundamental understanding of imaging techniques. Chapter 8, “Gradient-Echo Pulse Sequence in MRI System,” introduces the concept of gradient coils and the gradient-echo pulse sequence. We learn about time and frequency domain signals and explore echo-planar and spiral imaging sequences, maximizing signal quality for gradient echo imaging. Chapter 9, “Spin-Spin Coupling,” discusses two-spin systems, investigating the density matrix and Hamiltonian for such systems. We examine energy levels, transition energies, and free induction decay, differentiating between strong and weak spin-spin coupling and exploring quantum transitions. Chapter 10, “Hyperpolarized MRI Technique and Its Application in Medical Science,” introduces hyperpolarized MRI, examining the spin population and various hyperpolarization techniques. Applications in medical imaging using hyperpolarized carbon, helium, and xenon MRI are explored. Finally, in Chap. 11, “Some Specific NMR and MRI Techniques,” we explore some interesting techniques like superconducting quantum interference devices (SQUIDs), earth’s magnetic field NMR (EFNMR), and NMR cryoporometry (NMRC) for porous materials, offering a brief overview of these specialized techniques. I have included exercises with solutions at the end of each chapter to help you in your learning journey, where you might gain extra information about the materials provided in the chapters. Appendix sections provide additional mathematical support and calculations to enrich your comprehension. I sincerely hope this book will serve as an informative and helpful resource and inspire you to enrich your knowledge about this interesting subject. Dallas, TX, USA

Fatemeh Khashami

Acknowledgments

I am grateful to the individuals who have supported and inspired me in writing this book. First and foremost, I would like to thank Craig R. Malloy, who encouraged me throughout this journey. I would like to express my gratitude to Vlad G. Zaha for supporting me during my postdoctoral research at UT Southwestern Medical Center. I also thank my colleagues, A. Dean Sherry, Eunsook Jin, Anke Henning, Ivan E. Dimitrov, Maximilian Fuetterer, Zoltan Kovacs, Jae Mo Park, and Joseph A. Hill, for their valuable help in enriching various aspects of my knowledge in my study and research at UT Southwestern Medical Center. I am also grateful to Ashley White for proofreading parts of the manuscript. Special thanks go to Merry Stuber and her supportive team from Springer Nature Switzerland publisher for all the help during the publication of this book. Finally, I express my deepest gratitude to my family and friends. Your love, support, and encouragement have been the most valuable aid throughout this endeavor, without which this book would not have turned into a reality. Dallas, TX, USA

Fatemeh Khashami

ix

Introduction

Nuclear magnetic resonance (NMR) and magnetic resonance imaging (MRI) are two critical technologies that have revolutionized the way we study the human body and materials at the molecular level. MRI is widely used in medical science for diagnostic imaging, while NMR is an essential tool for chemists and physicists to investigate the properties of matter. These interesting technologies are byproducts of our understanding of the atomic world and the discovery of quantum theory. In fact, it is impossible to fully analyze the working principles of these technologies without referring to quantum physics and its properties. Therefore, to explain the concepts behind these technologies, we need to start with the story of quantum mechanics. The story of quantum mechanics is perhaps the most perplexing story developed so far in the history of science. Max Planck pioneered the quantum mechanical description of physical systems and introduced the quanta of energy in December 1900. He used this description to resolve the long-standing problem of black body radiation. The story of black-body radiation goes back to 1859 when the famous German physicist Kirchhoff formulated the problem of finding, for each temperature, the precise spectrum distribution of the emitted radiation. The long-standing problem of black-body radiation and the failure of classical physics to provide a correct physical description of the phenomenon in the ultraviolet spectrum was an important issue at the heart of physics known by the end of the nineteenth century. This failure was usually described as the “ultraviolet catastrophe.” There were also other unexplained phenomena, such as the photoelectric effect and the spectrum of hydrogen gases at the time. In 1905, Albert Einstein realized that a profound implication of black-body radiation is that light must be composed of particles called photons. Therefore, he argued that light is not just a continuous wave, as described by Maxwell equations, but has particle-like nature. This idea successfully explained the photoelectric effect. Ernest Rutherford’s experiment showed that most of the mass of an atom is located in its center, leading to the discovery of the atom’s nucleus in 1911. This observation brought up many issues which could not fit within the frame of classical physics. In 1913, Niels Bohr gave his atomic model to resolve these issues. Even

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though his model successfully resolved some of the problems, it was still insufficient to settle all the issues. Later, Louis de Broglie developed a formalism that assigned a wave nature to the particles like electrons in hydrogen atoms. This development, combined with Einstein’s photon concepts, enabled him to introduce the concept of wave-particle duality. A description that cannot be analyzed in classical physics formalism. The serious development of quantum mechanics came later in the 1920th, with the work of Werner Heisenberg, Max Born, Pascual Jordan, Erwin Schrödinger, Paul Dirac, Wolfgang Pauli, and many others. A game-changing step was taken by Heisenberg in 1925 by discovering the so-called matrix mechanics, which was soon laid in firmer ground by Born, Jordan, and Heisenberg himself. The other ground-breaking step was taken by Schrödinger who discovered the wave equation formalism of quantum mechanics, known as the Schrödinger equation. Since Schrödinger equation is a wave equation, this formalism was known as wave mechanics. We mentioned some important observations that led to formulating the quantum theory and abandoning classical physics. One of these interesting observations was the discovery of the angular-momentum quantization of quantum particles. This discovery was achieved by the famous Stern–Gerlach experiment in 1922. A few years later, in 1925, George Eugene Uhlenbeck and Samuel Abraham Goudsmit introduced electron spin, which attributes intrinsic angular momentum to the electron. In the same year, Wolfgang Pauli introduced the so-called Pauli exclusion principle. Finally, in 1928, Paul Dirac provided a theory to understand the spectrum of the hydrogen atom, which considers the spin of the quantum system. His formulation is known as the Dirac equation, a relativistic equation of spin-.1/2 systems such as electronics. The discovery of the spin in quantum systems provides the basis for the story of MRI and NMR techniques that we are going to explore in this book. In 1938, Israel Isaac Rabi extended the setup of the Stern-Gerlach experiment to measure the nuclear magnetic spin resonance. This was the first time such a phenomenon was observed in a molecular beam, and the principle of NMR was established. With the knowledge of this fundamental achievement, Felix Bloch and Edward Purcell developed NMR based on Rabi’s discovery and generated the first NMR signals from liquid water and solid paraffin in 1946. The central concept used in NMR and MRI techniques is the Zeeman effect due to the interaction of nuclear spins with some external magnetic field. The Zeeman effect was discovered by the Dutch physicist Pieter Zeeman in 1896 and played an essential role in developing quantum mechanics. The mathematical description of the dynamics in such a spin system is described by the so-called Bloch equation. The Bloch equation provides the details about the spin relaxation dynamics and MRI sequence process. The signal provided by the Bloch equation can be translated to the frequency domain using the Fourier transform (FT) method. The technique of converting the time domain NMR signal to the frequency domain signal is called the FT-NMR method, corresponding to the free induction decay (FID). Richard R. Ernst followed the FT-NMR method and

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developed the FT-NMR spectroscopy in 1991. After these discoveries, researchers enhanced the NMR signal and developed many useful techniques. Moreover, in the early twenty-first century, superconductive magnets opened a new window to NMR development. Combining the superconductive magnet with the FT-NMR method established new progress in having a powerful NMR machine. Interestingly, the fundamental physics that emerged from quantum mechanics gradually evolved and opened the way to medical imaging as a diagnostic device for investigating cancerous and normal tissues. We need to emphasize that NMR was the first generation of Rabi’s discovery, whereas MRI is the second generation of this phenomenon. Cutting-edge NMR and MRI technologies provide practical tools for understanding and describing various medical issues and provide valuable answers to questions that were difficult to address otherwise. In this book, we focus on the quantum mechanical foundations of NMR and MRI technologies and try to create a solid and comprehensible foundation for the quantum mechanical framework of the technologies. In order words, the aim of this book is to bridge the gap between the technological application of quantum mechanics and the fundamental principles based on which these technologies are built. We believe that learning the fundamental frameworks behind the technologies is beneficial for many reasons. One obvious but extremely important reason is that improving the existing technologies and building new and more valuable techniques usually require a good understanding of the fundamental principles of the systems and how they work. Such knowledge can also help us to have a good perspective of each step and element of the operation process and the imaging procedure. Understanding quantum mechanics can assist in managing specific system designs and manipulations, like spin-echo pulse sequences and spin dynamics procedures. Likewise, quantum mechanics helps us to understand individual energy levels and explore the system’s phase coherence and magnetization relaxation. The longitudinal and transverse magnetization processes can be successfully explained by quantum mechanics. Similarly, in quantum mechanics, we describe the spin system with different numbers of states in wave function formalism as a superposition of various eigenstates. We can calculate the eigenvalue and measure the expectation value of the system. In NMR and MRI, by defining an operator, we can measure the magnetization vector of spin along the x-axis, for instance. Therefore, we can observe the time average of an ensemble for an individual spin. These facts help us explain the atom’s structure, chemical properties, magnetization, relaxation time, and all surrounding atom’s environmental properties in NMR or MRI spectroscopy. Furthermore, NMR and MRI are among the most successful applications of quantum mechanics, and it is quite interesting to learn how the fundamental laws of the atomic world can also be extremely important in medical science. Understanding the practical procedures and the applications of the fundamental principles and the developments is important for designing advanced MRI techniques that can help researchers to study the complexities of biological tissues

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and diagnose various medical conditions. Such enhancements usually require to understand how the practical utilities of the technology operate, the limitations and desired properties on the one hand, and the fundamental physics governing the system and technology on the other.

Contents

1

Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The Schrödinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Elements of Quantum Formalism and Hilbert Space Structure . . . . 1.4 State Vectors and Operators in Quantum Formalism . . . . . . . . . . . . . . . 1.5 Uncertainty Relation Between Two Operators . . . . . . . . . . . . . . . . . . . . . . 1.6 Spin in Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Density Matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solution 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solution 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problem 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solution 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 7 9 14 15 23 27 27 27 28 29 29 30

2

Quantum Physics and NMR Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Spins and NMR Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The Zeeman Effect in NMR Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 The Boltzmann Distribution in NMR Systems . . . . . . . . . . . . . . . . . . . . . 2.5 The Effect of Magnetic Field on Spin Populations at Thermal Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 The Dynamics of the Macroscopic Magnetization in NMR Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 The Density Matrix Description of NMR Systems . . . . . . . . . . . . . . . . . 2.8 The Dynamics of the Density Matrix in NMR Spin Systems . . . . . . 2.9 Entropy and the Origin of the Boltzmann Distribution . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solution 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31 31 31 34 37 39 41 45 47 48 52 52 52 xv

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3

4

5

Contents

Problem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solution 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53 53

Bloch Equation Description Without Relaxation Time . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Bloch Equation in the Laboratory Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Bloch Equation in the Rotating Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Flip Angle of the RF Pulse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Geometric Representation of the RF Pulse by the Density Matrix Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solution 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solution 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55 55 55 58 62

Bloch Equation Description with Relaxation Time . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 General Formalism of the Bloch Equations . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 The Longitudinal Magnetization Behaviors in the Spin System . . . 4.4 The Transverse Magnetization Behaviors in the Spin System . . . . . 4.5 Tracking the Longitudinal and Transverse Magnetization . . . . . . . . . 4.6 The Fourier Transformation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 The Dynamics of Magnetization with the Density Matrix Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Insights from the Heisenberg Uncertainty Principle . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solution 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solution 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71 71 71 72 76 80 82

Molecular Motion, Correlation, and Relaxation Time . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Dipole-Dipole Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Correlation Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 The Bloembergen–Pound–Purcell Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 The Spectral Density Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 The Effect of Molecular Motion on the Relaxation Time . . . . . . . . . . 5.7 The BPP Theory and Relaxation Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 The Solomon–Bloembergen–Morgan Theory . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solution 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solution 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91 91 91 93 94 95 97 98 101 103 103 104 105 105

63 66 66 66 67 67

85 86 87 87 88 88 89

Contents

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Problem 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Solution 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 6

Chemical Shift, NMR Spectroscopy, and Beyond . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Electron Density Shielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 The Chemical Shifts of Fat and Water Molecules . . . . . . . . . . . . . . . . . . 6.4 Electron Shielding for Organic Chemicals . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 NMR Sensitivity and Receptivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Proton-NMR Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Carbon-NMR Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8 Phosphorus-NMR Spectroscopy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.9 Nitrogen-NMR Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.10 Fluorine-NMR Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.11 Xenon-NMR Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solution 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solution 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

107 107 107 110 112 113 115 117 117 118 118 119 119 119 119 120 120

7

Spin-Echo and Spin-Lock Pulse Sequences in MRI System . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 MRI Scanner. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Spin-Echo Pulse Sequence in MRI System . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Spin-Echo Pulse Sequence from the Density Matrix Perspective. . 7.5 Spin-Echo Pulse Sequence Parameters of Some Biological Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Spin-Lock Pulse Sequence in MRI System . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solution 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solution 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

121 121 121 124 127

Gradient-Echo Pulse Sequence in MRI System . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Gradient Coils’ Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Gradient-Echo Pulse Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Converting Time Domain FID Signal into a Frequency Domain Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Sampling Data Space in Gradient-Echo Pulse Sequence . . . . . . . . . . . 8.6 Maximizing the Signal Quality for Gradient-Echo Imaging . . . . . . . 8.7 Echo-Planar and Spiral Imaging Sequence . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

139 139 139 144

8

129 134 137 137 137 137 138

145 148 151 153 155 155

xviii

Contents

Solution 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 Problem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 Solution 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 9

Spin–Spin Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Density Matrix and Hamiltonian for the Two-Spin System . . . . . . . . 9.3 The Energy Levels and Transition Energy in the Two-Spin System 9.4 The Free Induction Decay for the Two-Spin System . . . . . . . . . . . . . . . 9.5 The Strong Spin–Spin Coupling in the System . . . . . . . . . . . . . . . . . . . . . 9.6 The Weak Spin–Spin Coupling in the System . . . . . . . . . . . . . . . . . . . . . . 9.7 Quantum Transition in the Two-Spin System . . . . . . . . . . . . . . . . . . . . . . . 9.8 The Homonuclear and Heteronuclear Systems . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solution 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solution 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problem 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solution 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

Hyperpolarized MRI Technique and Its Application in Medical Science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 The Spin Population in Hyperpolarized MRI Technique . . . . . . . . . . . 10.3 Different Hyperpolarization Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Hyperpolarized Carbon MRI and Its Applications in Medical Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Hyperpolarized Helium and Xenon MRI and Their Applications in Medical Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solution 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solution 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

Some Specific NMR and MRI Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Superconducting Quantum Interference Devices . . . . . . . . . . . . . . . . . . . 11.3 Earth’s Magnetic Field NMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 NMR Cryoporometry for Porous Materials . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 A Summary of Processing NMR and MRI Experiment . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solution 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solution 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

157 157 157 159 162 163 166 168 169 171 171 172 173 173 173 173 175 175 175 179 181 183 186 186 186 186 186 187 187 187 190 191 194 195 195 196 196 196

Contents

xix

A

Direct Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

B

The Density Matrix and Hamiltonian Elements for the Two-Spin System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

C

The Eigenstate and Eigenvalue of the Two-Spin System . . . . . . . . . . . . . . . 203

D

The Free Induction Decay Signal Elements for the Two-Spin System 207

E

The Fourier Transformation Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

Chapter 1

Quantum Mechanics

1.1 Introduction In this chapter, we discuss the quantum tools and techniques essential to understanding NMR and MRI technologies. The discussion will be brief but self-contained. However, a curious reader of quantum mechanics and people who want a more in-depth understanding of quantum theory may need to consult books that are specifically focused on quantum mechanics and its principles. We start with introducing the Schrödinger equation, which is central to the entire formulation of quantum mechanics. We show, in detail, how one can arrive at this equation starting from the standard wave equation. Then we introduce the Dirac notation and Hilbert space structure of quantum mechanics. We establish the Heisenberg equation for the dynamics of quantum systems as well as the Heisenberg uncertainty principles. We emphasize the two-level (spin-.1/2) quantum system and describe the physics of such a system in detail. There are two reasons for our emphasis on spin-.1/2 system. The first reason is that the two-level (spin-.1/2) quantum system is the simplest quantum system that one can consider. Thus, it can help explain the quantum mechanic’s frame without getting involved in some technically complicated settings. The second reason is that, as will become clear in the later chapters, spin-.1/2 systems are central for uncovering the physics behind the NMR and MRI techniques. We finally discuss the density matrix formulation of the quantum mechanics central to NMR and MRI physics.

1.2 The Schrödinger Equation The Schrödinger equation is one of the most central equations in physics and maybe one of the most important scientific findings. This equation was the formalism to © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Khashami, Fundamentals of NMR and MRI, https://doi.org/10.1007/978-3-031-47976-2_1

1

2

1 Quantum Mechanics

develop the wave theory of quantum mechanics [1–3]. This was important in the early days of quantum mechanics since physicists have been quite familiar with wave equations for a long time. This is not surprising since all electromagnetism was known to be described by Maxwell’s wave equations [4–6]. Also, the physics of sounds and fluid mechanics involve wave equations that were well-established at the time. A wave equation usually involves the second derivatives of both space and time. It can be expressed as .

∂ 2 ψ(x, t) 1 ∂ 2 ψ(x, t) = , ∂x 2 v2 ∂t 2

(1.1)

where the wave propagates by the velocity v. For example, we can show that electromagnetic waves can be expressed in this form, where the propagation velocity v is replaced by the speed of light c. Therefore, a solution of the wave equation can be a .sin or .cos function such as ψ(x, t) = A sin(kx − ωt),

.

(1.2)

where .ω = vk, and moreover .ω is the frequency at which the wave oscillates. Also, k = 2π/λ is the wave number, and .λ is wavelength. Furthermore, A is the amplitude of the wave. We note that the solution of the wave equation can be written as

.

ψ(x, t) = Aei(kx−ωt) .

.

(1.3)

In classical physics, however, the wave functions are real, as the physical measurements must be real-valued. Thus, this solution is not physical in this realm [7–9]. To introduce the Schrödinger equation, let us consider a particle subjected to a potential, such as an electron binding to a nucleus in an atom, and work out the wave function. Using de Broglie’s idea, the wave number can be expressed by the Planck constant .h¯ as k=

.

p 2π = , λ h¯

(1.4)

where p is a particle’s momentum that measures its motion [10–12]. On the other hand, if a particle behaves like a wave, we should be able to assign an angular frequency to it such that ω = 2πf =

.

E , h¯

(1.5)

1.2 The Schrödinger Equation

3

where f is the wave frequency, and .E is the energy related to the wave function. Moreover, the Planck constant is approximately .h¯ = 1.054 × 10−34 J.S in the international system of units (SI). Therefore, the particle wave function in the form of a .sin or .cos function from the above equations can be expressed as ( ψ(x, t) = A sin(kx − ωt) = A sin

.

px − Et h¯

) ,

(1.6)

,

(1.7)

and ( ψ(x, t) = A cos(kx − ωt) = A cos

.

px − Et h¯

)

respectively. Now, if we plug one of these solutions, for instance, the .sin solution, into Eq. (1.1), we obtain A

p2

.

h¯ 2

( sin

px − Et h¯

) =A

) ( 1 E2 px − Et . sin h¯ v 2 h¯ 2

(1.8)

This can hold if p2 =

.

1 2 E . v2

(1.9)

However, for non-relativistic particles, the kinetic energy is given by E=

.

p2 . 2m

(1.10)

Therefore, the relation .E2 = p2 v 2 cannot be correct. Hence, the wave equation of the classical wave does not hold for de Broglie’s wave theory. Then, what are the wave equation and the wave function in this case? To recover the correct expression for the energy, we note that the first derivative of the wave function in time gives one factor of .E, such that .

∂ψ = ∂t

(

) ( ) −E px − Et A cos , h¯ h¯

(1.11)

however, the second derivative of the wave function in space gives two factors of p as ) ) ( ( px − Et ∂ 2ψ −p2 . A sin . (1.12) = h¯ ∂x 2 h¯ 2

4

1 Quantum Mechanics

Therefore, the wave equation must be in the form of .

∂ψ ∂ 2ψ =α 2. ∂t ∂x

(1.13)

But still, there is a problem here. A .sin or a .cos function does not satisfy this equation. From, Eqs. (1.11) and (1.12), the left-hand side is a .cos function, while the right-hand side is a .sin function. However, the wave function is a complex function of the form of .A exp(i(px − Et))/h. ¯ This can be easily seen by plugging this wave function into Eq. (1.13) as A

.

−iE i(px − Et) i(px − Et) −p2 exp = Aα 2 exp . h¯ h¯ h¯ h¯

(1.14)

Since .E = p2 /2m, one can solve .α and get .α = i h/2m. Therefore, the wave ¯ equation can be found to be .

i h¯ ∂ 2 ψ ∂ψ = , ∂t 2m ∂x 2

(1.15)

h¯ 2 ∂ 2 ψ ∂ψ =− . 2m ∂x 2 ∂t

(1.16)

or, equivalently, i h¯

.

Now, if the electron is exposed to an external potential, .V (x), the equation must be written as ] [ −h¯ 2 ∂ 2 ∂ . + V (x) ψ = i h¯ ψ. (1.17) 2 ∂t 2m ∂x This is the celebrated Schrödinger equation. In the above equation the term is kinetic energy of the system. This can be easily seen from the fact that the momentum can be given as

−h¯ 2 ∂ 2 . 2m ∂x 2

p=

.

h¯ ∂ . i ∂x

(1.18)

Therefore, the sum of kinetic and potential energy gives the Hamiltonian of the system written as H=

.

p2 + V (x). 2m

(1.19)

1.2 The Schrödinger Equation

5

This provides the Schrödinger equation as i h¯

.

∂ψ = Hψ. ∂t

(1.20)

To consider this from a slightly different angle we can express the kinetic energy as E=

.

p2 h¯ 2 k 2 . = 2m 2m

(1.21)

On the other hand, if an electron behaves like a wave, we should be able to assign a frequency to it such that .E = hω. ¯ These two equations provide ω=

.

hk ¯ 2 . 2m

(1.22)

An important observation is that the frequency associated with the electron is proportional to .k 2 , not k as in .ω = vk that we introduced earlier. Therefore, the wave function of an electron must be written as .ψ = exp i(kx − ωt). Therefore, we developed a solution to the wave equation of the electron. Now, taking the second derivative in space is proportional to .k 2 as .

∂ 2ψ = −k 2 exp i(kx − ωt) = −k 2 ψ. ∂x 2

(1.23)

For the first time derivative, we have .

hk hk ∂ψ ¯ 2 ¯ 2 = −i exp i(kx − ωt) = −i ψ. ∂t 2m 2m

(1.24)

Therefore, the second derivative in space is proportional to .k 2 , while the first derivative in time has similar proportionality. Since our equation needs to be independent of k, we arrive at Eq. (1.16) from these two equations. Even though we have written this in one-dimensional form, it is straightforward to generalize it to three dimensions. In fact, the wave function of a particle at time t and position .r = (x, y, z) is presented by .ψ(r, t). For this system, the Hamiltonian of the system is defined as H=

.

p2 + V (r), 2m

(1.25)

such that we have h¯ ∇, .p = i

( where

∇=

) ∂ ∂ ∂ , , . ∂x ∂y ∂z

(1.26)

6

1 Quantum Mechanics

Therefore, the three-dimensional Schrödinger equation is i h¯

.

∂ ψ(r, t) = Hψ(r, t). ∂t

(1.27)

This is the time evolution equation of the wave function. Therefore, this equation is also called the time-dependent Schrödinger equation. One important issue that was not clear for the physicists in the first place was the meaning of the wave function. A wave function is a complex function in general. Therefore, it is not easy to attribute physical reality to it. Any measurement outcome is a real-valued quantity, and no complex number can be an outcome of a measurement in an experiment. It was left to Max Born to resolve this issue. In fact, according to him, a wave function represents the probability density of finding a particle at position r at time t. Thus, the probability of finding a particle between r and .r + dr, at time t, is given by |ψ(r, t)|2 d 3 r.

.

(1.28)

From the above equation, we attain the probability of finding the particle in the interval between .r1 and .r2 as f p=

r2

.

|ψ(r, t)|2 d 3 r.

(1.29)

r1

We immediately expect that f

+∞

.

−∞

|ψ(r, t)|2 d 3 r = 1,

(1.30)

which is called the normalization condition of the wave function. If the system is stationary, the energy .E of the system does not change over time. In this case, the wave function is given by ψ(r, t) = e−iEt/h¯ φ(r).

.

(1.31)

Inserting this solution into the Schrödinger equation yields Hφ(r) = Eφ(r),

.

(1.32)

that is called the time-independent Schrödinger equation. This, in fact, is the eigenfunction of the Hamiltonian of the system. We will discuss this later in more detail when we introduce the matrix representation of quantum states. It is also easy to see that

1.3 Elements of Quantum Formalism and Hilbert Space Structure

|ψ(r, t)|2 = |φ(r)|2 ,

.

7

(1.33)

which means that the probabilities, in this case, do not vary with time. Thus, they are referred to as stationary states.

1.3 Elements of Quantum Formalism and Hilbert Space Structure In matrix mechanics formalism, wave functions can be described by specific vector representations in the Hilbert spaces. In simple terms, a Hilbert space is a vector space equipped with an inner product [13]. In this framework, state vectors can represent wave functions using the Dirac notations. The Dirac notation is a relatively simple and helpful representation. For the wave function .ψ, we attribute the state vector .|ψ> in the Hilbert (vector) space. The representation .|.> is called a Ket representation [14]. Similarly, .ψ ∗ is replaced by the representative . and, similarly, the spin-down setting as .| ↓>. These, therefore, build the basis of a two-dimensional Hilbert (vector) space. The details of the physics of the spin system will be introduced later. However, here, we only need it to be a simple two-level system. To show more clearly how we can express these kets in the vector form, we can introduce the matrix (vector) forms of the basis as

8

1 Quantum Mechanics

Fig. 1.1 The description of the spin-.1/2 in terms of the state vector. The superposition of spin-up and down can take the general form of Eq. (1.38)

( ) 1 .| ↑> = , 0

and

( ) 0 | ↓> = . 1

(1.36)

Since these are the basis of a vector space, we have .

= = 0,

= = 1.

(1.37)

A general quantum state of a single spin on this basis can be described as |ψ> = α| ↑> + β| ↓>.

.

(1.38)

This means that before we measure the system, the electron can be expressed as the superposition of both up and down spin states. However, only one of the two possibilities can be observed if we measure the system, which will become clear soon. The superposition state is shown in Fig. 1.1. In this configuration, the spin-down state is associated with the quantum number of .−1/2, and similarly, the spin-up state is associated with the quantum number of .1/2. We can express the state above in a vector form such that ( ) ( ) ( ) 1 0 α .|ψ> = α +β = . 0 1 β

(1.39)

The corresponding bra of the spin state can be written as , are not unique for the representation of the quantum state. This is because the basis of vector space is not unequally defined. To see this, we can consider, for example, the new basis 1 |+> = √ (| ↑> + | ↓>), 2 . 1 |−> = √ (| ↑> − | ↓>). 2

(1.46)

10

1 Quantum Mechanics

This new basis also obeys the orthogonality rules as the old ones = = 0, .

(1.47)

= = 1.

The state .|ψ> = α| ↑> + β| ↓>, in the new basis, can be expressed as α−β α+β |ψ> = √ |+> + √ |−>. 2 2

(1.48)

.

One can easily confirm by plugging in the definition of the new basis in terms of the old basis, such that α−β α+β |ψ> = √ |+> + √ |−> 2 2 ( ) ( ) α+β 1 α−β 1 = √ √ (| ↑> + | ↓>) + √ √ (| ↑> − | ↓>) 2 2 2 2

.

= α| ↑> + β| ↓>.

(1.49)

With this simple example, it is now easy to formulate the concept in a more general setting. It is quite clear to see that, in this vector notation, the time evolution of Schrödinger equation takes the form ∂ ˆ H|ψ(t)> = i h¯ |ψ(t)>. ∂t

(1.50)

.

The state vector of a system can be expanded in its vector basis as the superposition of various eigenstates |ψ> =

∞ E

.

cn |an >,

(1.51)

n=1

where the set of .{|an >} are the basis of the Hilbert space. For example, in the spin example one can define .|a1 > = | ↑> and .|a2 > = | ↓> and take all others to be zero. Also, the coefficients .cn define the probability of finding the system in the state .|an > with .pn = ||2 = |cn |2 . If our Hilbert space is an N-dimensional space, we can assign a column vector (matrix) representation for the basics, such as ⎛ ⎞ 1 ⎜0⎟ . |a1 > = ⎜ ⎟ , ⎝·⎠ 0

⎛ ⎞ 0 ⎜1⎟ ⎟ |a2 > = ⎜ ⎝ · ⎠, 0

···

⎛ ⎞ 0 ⎜0⎟ ⎟ |aN > = ⎜ ⎝ · ⎠. 1

(1.52)

1.4 State Vectors and Operators in Quantum Formalism

11

Thus, the state can also be expressed in this representation as ⎛

⎞ ⎛

c1 ⎜ ⎟ ⎜ c2 ⎟=⎜ .|ψ> = ⎜ ⎝ ⎠ ⎝ · ·

⎞ ⎟ ⎟. ⎠

(1.53)

cN

The basis .{|an >} are the stationary states of the system that was addressed earlier, and we have Hˆ |an > = En |an > .

(1.54)

.

Therefore, stationary states are the eigenvectors of the Hamiltonian. This also indicates that ˆ

e− h¯ En t |an > = e− h¯ Ht |an > . i

i

.

Since .|ψ> = obtained as

E∞

n=1 cn |an >,

(1.55)

the dynamics of the general state vector can be

ˆ

|ψ(t)> = e− h¯ Ht i

.

∞ E

ˆ

cn |an > = e− h¯ Ht |ψ(0)>. i

(1.56)

n=1

In other words, the time evolution of the system can be determined by ˆ

|ψ(t)> = e− h¯ Ht |ψ(0)>. i

.

(1.57)

It is easy to check that this wave vector satisfies the Schrödinger equation, and it can easily be written as |ψ(t)> =

∞ E

.

i

cn e− h¯ En t |an > ,

(1.58)

n=1

where .exp(−iEn t/h) ¯ is a complex quantity. To find the inner product of two different states, in general, suppose that |ψ> =

d E

αi |i>,

i=1 .

|φ> =

d E i=1

(1.59) βi |i>.

12

1 Quantum Mechanics

The inner product of these two states can be given by =

( d E

⎞ )⎛ d E βi∗ ⎠ j =1

i=1

.

=

d d E E

βi∗ αj =

i=1 j =1

=

d E

d d E E

βi∗ αj δi,j

(1.60)

i=1 j =1

βi∗ αi ,

i=1

where .δi,j = 1 for .i = j . We have learned so far how quantum states are defined and how the time evolution of quantum wave functions are described. We proceed with discussing how we measure physical quantities in quantum systems. To clarify how the quantity measurement on an ensemble of systems works, let us consider the example of the spin-.1/2 system discussed earlier. We consider the state of the spin of the electrons to be .|ψ>. The spin-up and spindown measurements correspond to the outcome of the measurements .Sz = h/2 ¯ and h .Sz = .−h/2, respectively (this is just . times the two possible values of . ±1/2, which ¯ ¯ will be discussed in details soon). The expectation value for the z-component of the spin (average of spin after measurement in a large ensemble) is .

=

h¯ h¯ ||2 − ||2 , 2 2

(1.61)

where .||2 and .||2 are the probabilities of finding the electron in spin-up and spin-down, respectively. This is also equivalent to .

/ | | \ | | = ψ |Sˆz | ψ ,

(1.62)

with the spin operator h¯ Sˆz = [| ↑> as

.

ˆ = .

.

(1.64)

The operator can be expressed in the diagonal form on its basis such that ⎛

a1 ⎜ · ˆ=⎜ .A ⎝ · ·

· a2 · ·

⎞ · · · · ⎟ ⎟. · · ⎠ · aN

(1.65)

For instance, .| ↑> and .| ↓> are the two eigenstates of the spin operator .Sˆz , with the corresponding eigenvalues .+h/2 respectively. In other words, ¯ and .−h/2, ¯ h¯ Sˆz | ↑> = + | ↑>, 2 . h¯ Sˆz | ↓> = − | ↓>. 2

(1.66)

Therefore, in the diagonal form, the operator .Aˆ is given by Aˆ =

N E

.

ai |i>} are the eigenstates of the ˆ with eigenvalues .{ai }, since operator .A,

.

ˆ >= A|j

N E

ai |i> =

i=1

N E

ai |i>δi,j = aj |j >.

(1.68)

i=1

The operator representation of the physical observable has an important consequence in quantum formalism. In fact, two quantum operators do not commute in ˆ we can have general. In other words, given two operators .Aˆ and .B, ˆ Aˆ Bˆ /= Bˆ A.

(1.69)

.

This is due to the fact that matrices do not necessarily commute in general. For instance, if we take ( A=

.

) 01 , 10

( B=

) 1 0 , 0 −1

(1.70)

14

1 Quantum Mechanics

we can verify that ( AB =

.

) 0 −1 , 1 0

( BA =

) 0 1 , −1 0

(1.71)

ˆ The commutator of two operators .Aˆ and .B, ˆ denoted by obviously .Aˆ Bˆ /= Bˆ A. ˆ ˆ .[A, B], is defined by .

] [ ˆ Bˆ = Aˆ Bˆ − Bˆ A. ˆ A,

(1.72)

As an important example, the commutator of the x-components of the operators of the position and momentum, .xˆ and .pˆ x is given by .

[ ] x, ˆ pˆ x = i h. ¯

(1.73)

1.5 Uncertainty Relation Between Two Operators The other important quantity is the uncertainty relation between the two operators. ˆ and . ˆ are the ˆ where . To see this let us consider two operator .Aˆ and .B, ˆ ˆ expectation values of .A and .B, with respect to the state .|ψ>. In other words, we ˆ = ˆ ˆ = . ˆ have . and . Therefore, we can introduce the two ˆ ˆ operators .AA and .AB such that ˆ AAˆ = Aˆ − , .

ˆ ABˆ = Bˆ − .

(1.74)

The variance (uncertainty) of these operators is given with respect to the expectation values of their square, such that // \ // \ 2 ˆ ˆ 2, AA = (AA) = Aˆ 2 − .

// \ // \ ˆ 2 = Bˆ 2 − ˆ 2. AB = (AB)

(1.75)

It turns out that there is a relation between the product of the uncertainty of the two operators and their commutators such that [15] AAAB ≥

.

1 ˆ ˆ ||. 2

(1.76)

1.6 Spin in Quantum Mechanics

15

This is the uncertainty relation between two operators in quantum mechanics. As a fundamental example, for the position and the momentum operators, we have AxApx ≥

.

]>| 1 || or .| ↑> and .|z; −> or .| ↓>. The first case represents an electron with .Sz = +h/2, while the second case represents ¯ an electron with .Sz = −h/2. Mathematically, we can express this description as the ¯ eigenstates and eigenvalues equations, which are written as h¯ Sˆz |z; +> = + |z; +>, 2 . h¯ Sˆz |z; −> = − |z; −>. 2

(1.88)

We have also seen that the spin state basis can be written as column vectors, thus ( ) 1 |z : +> = |1> ←→ , 0 . ( ) 0 |z : −> = |2> ←→ . 1

(1.89)

By replacing the above relation in our definition, we can check (

)( ) h¯ 1 = + |z; +>, 0 2 . ( )( ) h¯ 1 0 h¯ 0 Sˆz |z; −> = + = − |z; −>. 1 2 0 −1 2 h¯ Sˆz |z; +> = + 2

1 0 0 −1

(1.90)

Of course, there is nothing special about the z axis. If we set up the Stern–Gerlach experiment in the x direction, we will detect the spin in that direction. This, of course, is true about any other arbitrary direction. Therefore, we need to define the

20

1 Quantum Mechanics

operators for the x and the y components, .Sˆx and .Sˆy as well. Therefore, .Sˆx and .Sˆy are expressed as (

) 01 , 10 . ( ) h¯ 0 −i . Sˆy = 2 i 0 h¯ Sˆx = 2

(1.91)

Moreover, these operators satisfy the commutation relations [ ] Sˆx , Sˆy = i h¯ Sˆz , [ ] ˆy , Sˆz = i h¯ Sˆx , . S [ ] Sˆz , Sˆx = i h¯ Sˆy .

(1.92)

This can be shown by directly calculating the communication relations, such as .

[ ] h2 (( 0 1 ) ( 0 −i ) ( 0 −i ) ( 0 1 )) ¯ Sˆx , Sˆy = − = i h¯ Sˆz . 10 i 0 i 0 10 4

(1.93)

In the spin algebra framework, other important operators are called the Pauli matrices .σi (for .i = x, y, z), which is given by h¯ Sˆi = σi . 2

(1.94)

.

Then the explicit form of the Pauli matrices are ( σx = σ1 =

.

) 01 , 10

( σy = σ2 =

0 −i i 0

(

) ,

σz = σ3 =

) 1 0 . 0 −1

(1.95)

The square of these operators is just the unity matrix .

(σ1 )2 = (σ2 )2 = (σ3 )2 = 1.

(1.96)

Also, the Pauli matrices have zero traces as .

Tr (σi ) = 0,

i = 1, 2, 3.

(1.97)

So far, we have considered the spin operator in the three orthogonal directions of the coordinate system. We can attain the spin operator in any other direction based on this. To do so, we consider an arbitrary direction specified by the unit vector .n as

1.6 Spin in Quantum Mechanics

21

) ( n = nx , ny , nz = (sin θ cos φ, sin θ sin φ, cos θ ),

.

(1.98)

where .θ and .φ are the polar and azimuthal angles. We can define the operators .S as the vector of the spin operators ) ( S = Sˆx , Sˆy , Sˆz .

(1.99)

.

Therefore, the spin operator in the direction of the unit vector can be expressed as h¯ Sˆn ≡ n · S ≡ nx Sˆx + ny Sˆy + nz Sˆz = n · σ , 2

.

(1.100)

where .S = (h/2)σ . We note that .Sˆn is simply an operator in)the direction of the ¯ ( vector .n. For example, when .n is along x, one has . nx , ny , nz = (1, 0, 0) and .Sˆn becomes .Sˆx . We can easily see that .

( ) ( ) h¯ Tr Sˆn = ni Tr Sˆi = ni Tr (σi ) = 0. 2

(1.101)

Also, we have ( .

Sˆn

)2

( )2 ( )2 h¯ h¯ 2 = (n · σ ) = . 2 2

(1.102)

The operator .Sˆn can be explicitly written as ( ( [ ( ) ) )] h¯ 01 0 −i 1 0 nx Sˆn = + ny + nz 10 i 0 0 −1 2 . ) ( h¯ nx − iny nz , = 2 nx + iny −nz

(1.103)

thus, we have .

h¯ Sˆn = 2

(

cos θ sin θ e−iφ sin θ eiφ − cos θ

) .

(1.104)

Solving the operator’s eigenvalues provide .±h/2, where the eigenstates can be ¯ denoted as .|n; ±>. Hence, we can write h¯ Sˆn |n; ±> = ± |n; ±>. 2

.

(1.105)

22

1 Quantum Mechanics

Fig. 1.5 (a) A general angular momentum vector in quantum mechanics. (b) The quantization of angular momentum vector for .l = 2 in quantum physics. In this case, five possible states exist for five different quantum numbers

The spin algebra, in fact, is similar to the angular momentum algebra defined through the operators .Lˆ x , Lˆ y , and .Lˆ z . These operators, similar to the spin-.1/2 operators above, as [ ] Lˆ x , Lˆ y = i h¯ Lˆ z , .

[ ] Lˆ y , Lˆ z = i h¯ Lˆ x , [ ] Lˆ z , Lˆ x = i h¯ Lˆ y .

(1.106)

These algebras determine the quantization beyond the two-state system of spin1/2 particles. For instance, for spin-1 particles, there will be three possible states. In general, for angular momentum (spin) l, there will be .2l + 1 different states. We illustrate this, for the case of .l = 2, in Fig. 1.5. We will consider the energy separation of spin in the following chapter. Interestingly, spin algebra obeys the same rules as angular momentum algebra in quantum mechanics. In angular momentum, a system rotates around a specific vector. Therefore, one might think that when we say, for instance, the electron has spin, which means an electron rotates (spins) around itself, similar to the earth’s

.

1.7 Density Matrix

23

rotation around itself (its center of mass). However, this is not true. In fact, spin is an intrinsic property of the quantum system, and no spinning or rotation is involved.

1.7 Density Matrix As was discussed earlier, a quantum state is denoted by a vector in a Hilbert space. To illustrate, if we measure the spin of an electron and find the pure state .| ↑>, then by applying a magnetic field in a different direction, we can rotate the spin into the different pure states. For example, it can point the spin system in the x direction. In general, the state can be described as a superposition of the two bases of the spin in the z direction, such as .|ψ> = α| ↑> + β| ↓>. This is a pure quantum state since it is denoted as a vector in a Hilbert space. In many realistic settings, however, we have no idea which pure state of the system is prepared. Imagine a mechanism that creates .| ↑> or .| ↓> state, each with .50% probability. Now, if we receive spins from such as ensemble, we have only a probabilistic description of the system. This is different from preparing a pure quantum state. In this case, the state of the system is mixed. To have a better insight into the difference between the pure and mixed states, suppose one prepares an ensemble of two different states .|ψ1 > and .|ψ2 >. Now, with probability .1/2, we get state .|ψ1 >, and with probability .1/2, we get the state .|ψ2 >. However, we are never told which state we are provided. How can we describe the system’s state in our possession once we receive a state? To do so, we must describe the system using a density matrix. The density matrix is not only an important concept in quantum mechanics but is also very helpful for investigating NMR and MRI systems. The density matrix deals with different types of spin systems, which can explain various parts of NMR and MRI. The density matrix for the states .|ψ1 > and .|ψ2 > that was described above can be written as ρ=

.

1 1 |ψ1 > , where each state is associated with probability .pi , then the density matrix describing the system is ρ=

n E

.

pi |ψi > is present in the mixture (.pi = 1 for some i), then we return to the pure state setting. In this case, the density matrix reduces to .ρ = |ψi > ≡

αl βl

) (1.109)

.

There are two eigenvalues .| ↑> and .| ↓> in two directions; therefore, the density matrix is given by .(2 × 2) matrix. From the definition of the density matrix, we have ( .

ρl = |ψl > Thus, we can define matrix elements by three parameters .l , σy l , l , where ) 1( ∗ βl αl + αl∗ βl . 2

(1.111)

< | | >2 ) < > 1( ∗ βl αl − αl∗ β . σy l = ψl |σy | ψl = 2

(1.112)

l = =

.

And, .

Also, .

l =

) 1( 2 |α| − |β|2 . 2

(1.113)

The matrix elements of Eq. (1.110) can be written as

.

< > l − i σy l = αl βl∗ , ) 1( 2 l = |αl | − |βl |2 . 2

(1.114)

Therefore, the density matrix of the subsystem l can be expressed as ( .

ρl =

< > ) l − i σy l + l < > . l + i σy l 21 − l 1 2

(1.115)

We can write the total density matrix as ρ=

E

.

l

pl |ψl > = (| ↑> + | ↓>)/ 2. For this pure state, the density matrix is given by ( ρˆ = |+> < by | the Schrödinger equation. From the density matrix .ρ = En is governed | | ψ ψ p j j , we can determine the dynamics using the time derivative of the j =1 i density matrix such that ⎞ ⎛ n E d ⎝ dρ = pi |ψj > / \ < > where .|a|2 + |b|2 = 1, calculate ., . Sy , ., . Sx2 , . Sy2 , . Sz2 .

Solution 1 The expectation value of .Sx : (

( ∗ ∗ ) h¯ . = = a , b 2 .

=

01 10

( ) )( ) h¯ ( ∗ ∗ ) b a a ,b = a b 2

) ) ( h¯ ( ∗ a b + b∗ a = h¯ Re ab∗ . 2

The expectation value of .Sy : > < | | > ( ) h¯ Sy = ψ |Sy | ψ = a ∗ , b∗ 2

< .

(

0 −i i 0

( )( ) ) h¯ ( ∗ ∗ ) −bi a a ,b = ai b 2

) > h¯ ( ) ( Sy = i −a ∗ b + b∗ a = −h¯ Im ab∗ . 2

< .

The expectation value of .Sz : .

( ) h¯ = = a ∗ , b∗ 2 .

=

(

1 0 0 −1

( )( ) ) h¯ ( ∗ ∗ ) a a a ,b = −b b 2

) ) h¯ ( 2 h¯ ( ∗ a a − b∗ b = |a| − |b|2 . 2 2

28

1 Quantum Mechanics

For .Si2 we have Sx2 =

.

Sy2 =

.

2 .Sz

h¯ 2

h¯ = 2

h¯ 2

(

(

(

)

01 10

)

0 −i i 0

1 0 0 −1

)

h¯ 2 h¯ 2 h¯ 2

(

(

(

01 10

)

0 −i i 0 1 0 0 −1

=

h¯ 2 4

) = )

(

(

h¯ 2 4

h¯ 2 = 4

10 01

(

)

10 01 10 01

=

h¯ 2 I 4

) =

h¯ 2 I 4

=

h¯ 2 I. 4

)

2

Therefore, .Sx2 = Sy2 = Sz2 = h¯4 I. For all three operators, we have ( ( ) )( ) \ / | | \ ( ) h¯ 2 1 0 h¯ 2 ( ∗ ∗ ) a a | | a ,b Si2 = ψ |Si2 | ψ = a ∗ , b∗ = b b 4 01 4

/ .

=

) h¯ 2 ( 2 |a| + |b|2 . 4

Since .|a|2 + |b|2 = 1, we have / .

\ / \ / \ h2 ¯ . Sx2 = Sy2 = Sz2 = 4

Problem 2 Consider ( |ψ> =

.

cos θ eiφ sin θ

where .φ and .θ are real parameters. (a) (b) (c) (d)

Find the density matrix .ρ = |ψ> orthogonal to .|ψ>.

) ,

Exercises

29

Solution 2 (a) We have ( ) to be ( ) p .|φ> = q for some complex p and q. Now, considering the orthogonality condition = 0, we can find that a solution for the state |φ> is ( |φ> =

.

−eiφ sin θ cos θ

) .

Problem 3 Suppose we are given the Hamilton related to the x component of the Pauli operator such that Hˆ = h¯ ωσx .

.

Find the solution of the Schrödinger equation i h¯

.

for the initial state

d ˆ |ψ> = H|ψ> dt

30

1 Quantum Mechanics

( ) 1 .|ψ(t = 0)> = . 0 Also, find the probability ||2

.

and find the time at which the state becomes orthogonal to the initial state.

Solution 3 The solution of the Schrödinger equation i h¯

.

d ˆ |ψ> = H|ψ> dt

gives the general dynamical formula ˆ

|ψ(t)> = e−i Ht/h¯ |ψ(t = 0)>.

.

This also can be written as ˆ

|ψ(t)> = U (t)|ψ(t = 0)> = e−i Ht/h¯ |ψ(t = 0)>,

.

where .U (t) is a unitary matrix. Since .σx2 = I we have ˆ

e−i Ht/h¯ ≡ U (t) =

.

(

) cos(ωt) −i sin(ωt) . −i sin(ωt) cos(ωt)

Thus |ψ(t)> = U (t)

.

( ) ( )( ) ( ) 1 cos(ωt) −i sin(ωt) 1 cos(ωt) = = . 0 −i sin(ωt) cos(ωt) 0 −i sin(ωt)

Also, we find the probability p = ||2 = cos(ωt).

.

The initial state evolves to the orthogonal state when p = ||2 = cos2 (ωt) = 0,

.

which gives .ωt = π/2, 3π/2, . . . .

Chapter 2

Quantum Physics and NMR Foundations

2.1 Introduction In this chapter, we provide a fundamental understanding of NMR and MRI spin systems from a quantum physics perspective. We introduce essential concepts such as the Zeeman effect, the Boltzmann distribution, nuclear magnetization, and spin populations in the thermal equilibrium state. Also, we consider the spin population and magnetization for different spin-.1/2 systems for deeper understanding. To frame the principles of NMR and MRI in thermal equilibrium, we briefly discuss the density matrix formalism for a spin-.1/2 system and investigate the general form of the density matrix for both thermal and non-thermal equilibrium conditions. We also discuss how thermal distribution emerges in the systems. This chapter, in fact, uses tools from quantum mechanics that was introduced in the previous chapter and some tools from the physics of thermodynamics to form the basis for understanding the NMR systems.

2.2 Spins and NMR Systems NMR is a powerful technique based on the fundamental principle of quantum mechanics. The nuclei of all atoms have spins that give rise to specific magnetic properties, denoted as I . The magnitude and direction of the nuclear spin rely on the specific atom and its regional surroundings. The spin I can be .I = 0, 1/2, 1, 3/2, · · · . In this book, we mainly focus on the spin .I = 1/2 system, which can provide the fundamental physics behind the NMR and MRI. A simple scenario of the NMR process can be understood by considering how 1 . H nucleus behaves. In the frame of quantum mechanics, the principles of NMR arise from the fact that nuclei have spins, which can provide signals in physical or biological samples when placed in an external magnetic field [17–20]. Therefore, © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Khashami, Fundamentals of NMR and MRI, https://doi.org/10.1007/978-3-031-47976-2_2

31

32

2 Quantum Physics and NMR Foundations

Fig. 2.1 (a) Schematic visualization of spin .I = 1/2. (b) The spin .I = 1/2 has two possible outcomes for .Iz with √ two different .mI = ±1/2. The total magnitude of the spin angular momentum vector is .|I-| = h¯ 3/4

from a quantum mechanics viewpoint, in the absence of an external magnetic field, - z gain arbitrary directions that are B-0 , the nuclei with magnetic dipole moments .μ out of phase in general. Thus, their spins cancel out each other such that their net spin and net magnetic moment are zero [1, 21, 22]. In the previous chapter, we discussed the physics of the spin-.1/2 system in the Stern-Gerlach experiment. In this experiment, we deal with the election spin in an atom. However, as was mentioned earlier, in a simple NMR setting, we have the nuclear spin degrees of freedom. Of course, the quantum description of the spin in both cases is similar. This facilitates the description of the NMR systems with .I = 1/2. We represent the spin angular momentum for .I = 1/2 along the .z-axis in Fig. 2.1a and b, which is given by .

h¯ Iz = mI h¯ = ± , 2

.

(2.1)

where .h¯ is the Planck’s constant that is equal to .h/2π = 1.054 × 10−34 J.S. The total magnitude of the spin angular momentum vector for spin .I = 1/2 is / / 3 .|I | = h . ¯ I (I + 1) = h¯ 4 The magnetic dipole moment of nuclei, .μ - z , with .mI = ±1/2, is defined as

(2.2)

2.2 Spins and NMR Systems

33

1 μ - z = mI h¯ γ = ± hγ ¯ , 2

.

(2.3)

where .γ is gyromagnetic ratio with units MHz .Tesla−1 or .rad Tesla−1 sec−1 [21, 23, 24]. The gyromagnetic ratio for the electron is given by γe = −ge

.

e , 2me

(2.4)

where e is the elementary charge of the electron, .me is the electron’s mass, and .ge is the electron’s gyromagnetic factor (g-factor), which is about 2, as was mentioned earlier. Similarly, the nucleon’s or proton gyromagnetic ratio is given by γP = gP

.

e , 2mP

(2.5)

where .mP is the proton’s mass, and .gP is the nucleon’s g-factor, which is about 5.58 [1, 18, 25]. These equations show that the gyromagnetic ratio is inversely proportional to the mass [23, 24]. From gyromagnetic ratios of electron and proton, - e , is defined by the Bohr magneton of the electron, .μ

.

μ -e =

.

eh¯ , 2me

(2.6)

and the nuclear magneton of the proton, .μ - P , is given by μ -P =

.

eh¯ . 2mP

(2.7)

Since .mP is almost 2000 times .me (.mP ≈ 2000 me ), .μ - P is much larger than μ - e (.μ - P >> μ - e ). Typically, .me = 9.1 × 10−31 kg, and .mP = 1.67 × 10−27 kg [1, 18, 25]. The gyromagnetic ratio corresponds to the magnetic moment .μ - z and the spin number .I for a specific nucleus (or electron) [26, 27] is expressed as

.

γ =

.

2π μ -z , hI

(2.8)

which can be positive or negative. Once .μ - z and .I are parallel (aligned in the same direction), .γ is positive value. An example of positive .γ is .1 H nucleus, which is −1 .42.58 MHz .Tesla [20, 28]. Alternatively, once .μ - z and .I are aligned in opposite directions, .γ is a negative value. An example of negative .γ is for a single electron, which is .−28024.95 MHz .Tesla−1 [20, 28]. Note that we usually represent the electron spin number by S and the nuclei spin number by I , however, in the above equation, I represents both.

34

2 Quantum Physics and NMR Foundations

2.3 The Zeeman Effect in NMR Systems The degeneracy of the spin can be removed by the so-called Zeeman effect, where an applied magnetic field can split the spin into different quantum levels. It was mentioned that the energy split of the nucleus spin for a system with spin I has .2I + 1 degenerate states, which results in energy splitting in a magnetic field. Note that I here has a similar role as the angular momentum l in the previous chapter, which obeys .2l + 1 splitting. The splitting of .2I + 1 energy levels is described by the spin quantum number, .mI . As mentioned before, spin .I = 1/2 is suitable for the NMR system. Therefore, we have two energy levels from the selection rule .2I + 1. Thus, spin-.1/2 has orientation in the direction up or down along the z-axis. The energy levels are usually denoted as .| ↑> or .|α> state and .| ↓> or .|β> state. The spin magnetic moment is orthogonal to the magnetic field and is written as a sum of spin-up and spin-down states [18, 19]. From the Zeeman effect, in the absence of an external magnetic field, .B-0 , all nuclei of a non-interacting system are randomly oriented. In Fig. 2.2a, we represent a collection of proton nuclei oriented randomly without a magnetic field. These nuclei can be attributed to .1 H atom, which is an essential element for many NMR systems. Therefore, the magnetic dipole moment of the atoms cancels out each other. Hence, their nuclei occupy a single energy level, which can be described by the total energy to be zero. On the other hand, by applying an external magnetic field, the nuclei precess with the magnetic field [29, 30]. We illustrate the concept of the Zeeman effect and energy splitting for an NMR system in Fig. 2.2b. Accordingly, the number of spins in the energy level .α is n.α and, similarly, in the energy level .β is n.β , where the sum of energy levels gives the total number of the spins in the sample. In other words, the nuclei along .B-0 direction are denoted as spin-up or .α state with .mI = +1/2. Alternatively, the nuclei against .B-0 direction are called spin-down or .β state with .mI = −1/2. Due to the Zeeman splitting, the interaction energy of a spin state with a magnetic field in the z-axis is directly proportional to the magnetic field strength. Thus, the energy of the system is described as [25, 28, 31, 32] E = −μ - · B-0 .

.

(2.9)

The energy along the z-axis is given by Ez = −μ - z · B-0 .

.

(2.10)

By replacing Eq. (2.3) in the above relation, we have Ez = −μ - z · B-0 = −(mI hγ ¯ )B0 .

.

(2.11)

2.3 The Zeeman Effect in NMR Systems

35

Fig. 2.2 Scheme of the Zeeman effect. (a) In the absence of the magnetic field .B-0 , .1 H atoms lay in random orientations, and their nuclei occupy a single energy level, which is 0. (b) Due to Zeeman splitting, when the strong magnetic field is applied, some nuclei align parallel to .B-0 (.α state) and others anti-parallel .B-0 (.β state). More of the nuclei occupy the lower energy level at thermal equilibrium. The number of spins in the lower and higher level is called .nα and .nβ , respectively

For spin .I = 1/2, we have the energy level from Eq. (2.11) as (

1 .Ez = − ± h ¯γ 2

) B0 .

(2.12)

For the two energy levels system, the energy difference is AE = Eβ − Eα = hγ ¯ B-0 = hω ¯ 0,

.

(2.13)

where .Eα and .Eβ are the energy level of .α and .β states, which express in the angular frequency of the system. Moreover, the NMR signal relies on the energy difference between the two-level state of the nucleus [23, 31]. To have a better understanding of the general principle of the Zeeman splitting, in Fig. 2.3, we illustrate the Zeeman energy level splitting for spin .I = 1 and .I = 2. For any other spin, the formalism is quite similar, in fact. We depict how the energy level changes in the presence of an external magnetic field. The energy level is

36

2 Quantum Physics and NMR Foundations

Fig. 2.3 Scheme of the Zeeman energy level splitting for spin .I = 1 and .I = 2 by selection rule .2I + 1 energy levels. For .I = 1, the spin system can be divided into three energy levels. For .I = 2, the spin system can be split into five energy levels in the presence of the magnetic field. The difference between the energy level .mI = 0 and .mI = +1 is .AE = h¯ γ B0 . Similarly, the difference between the energy level .mI = −1 and .mI = +1 is .AE = 2h¯ γ B0

divided into three levels for spin .I = 1 from the selection rule .2I + 1, where we have .AmI = 0, ±1. Similarly, the energy level is split into five for spin .I = 2, where we have .AmI = 0, ±1, ±2. Equation (2.13) is the Zeeman energy for the spin system. For instance, in Fig. 2.3, for .I = 1, the energy splitting is given by

.

E0 + hγ ¯ B0 E0 E0 − hγ ¯ B0

AmI = −1 AmI = 0 . AmI = +1

(2.14)

Thus, the energy difference between the two energy levels .mI = 0 and .mI = +1 is obtained as AE = h¯ γ B0 = hω ¯ 0.

.

(2.15)

The energy difference between .mI = +1 and .mI = −1 is written as AE = 2hγ ¯ B0 = 2hω ¯ 0.

.

(2.16)

2.4 The Boltzmann Distribution in NMR Systems

37

In the above relation, the nucleus spin precess at a specific frequency, .ω0 , known as the Larmor frequency, has the unit .rad sec−1 [31, 33, 34]. The Larmor frequency is the precession frequency of the spins around the axis of the magnetic field, which is defined by .ω0 = γ B0 [28, 35]. Corresponding to the larger mass of the proton and its smaller gyromagnetic ratio compared with the electron, the proton has a smaller Larmor frequency and results in lower signal frequency.

2.4 The Boltzmann Distribution in NMR Systems The enormous contrast between classical and quantum mechanics is related to energy quantization, where each quantum state i corresponds to the energy .Ei in quantum mechanics. To consider the system of NMR where spins can take different energy levels, we should note that the probability of the system is described by the Boltzmann distribution, which is written as .

exp (−Ei /kB T ) Ni , =E N i exp (−Ei /kB T )

(2.17)

E where .N = i=1 Ni , .Ni is the number of particles with the specific energy .Ei , and the sum of the exponential energy in the denominator is called the partition function. Moreover, .kB = 1.3805 × 10−23 J/K is the Boltzmann constant, and T is the temperature in Kelvin (K) [17–20]. Furthermore, we can define .1/kB T = β. The Boltzmann distribution describes the probability of nuclei of the spin-.1/2 system in the up or down states at temperature T , which represents the number of nuclei in each spin state [36]. The number of spins in the lower and higher levels is indicated by .nα and .nβ , respectively. At thermal equilibrium, the probability of spin populations for up and down states are given by .

pα =

1 nα = e−βEα , n0 Z

pβ =

nβ 1 = e−βEβ , n0 Z

(2.18)

where .n0 is the total number of nuclei per unit volume, which is nα + nβ = n0 .

.

(2.19)

Later in this chapter, we derive the Boltzmann distribution from more fundamental principles of statistical mechanics. The partition function for the system is usually denoted by .Z [37, 38]. The partition function for the spin-.1/2 system is given by Z = e−βEα + e−βEβ = e−βEα (1 + e−βAE ).

.

(2.20)

38

2 Quantum Physics and NMR Foundations

The sum of the probabilities of finding spin in each state should be conserved, as we prove in Eq. (1.45). Thus, we have pα + pβ = 1.

.

(2.21)

The above relation means that the NMR system state may have a probability .pα or .pβ . Additionally, the Boltzmann distribution indicates that NMR signal intensity is connected to the applied magnetic field and the inverse of temperature [39, 40]. The number of the higher-level spin populations or the energy separation between the lower and the higher states increases as the magnetic field increases. We also note that the number of spins in the lower state is slightly greater than in the higher energy at thermal equilibrium state. With these considerations, the internal energy can be expressed as E = nα Eα + nβ Eβ .

.

(2.22)

Also, the spin polarization is calculated from Eq. (2.18), which is written as pα − pβ β hω ¯ 0 , = tanh 2 p α + pβ

p=

.

(2.23)

where at thermal condition with high temperature we have .hω ¯ 0 = A↑ | ↑> + A↓ | ↓> = Aα |α> + Aβ |β>,

.

(2.43)

where .Aα and .Aβ are the probability amplitudes, which shows the probability of finding the system in states .|α> and .|β>, respectively. From Sect. 1.4, Hamiltonian .H is the operator energy of the system is expressed as E

H=

.

Ei |ψi > the other hand, the off-diagonal elements of .ρ, from Eq. (1.115), are [.l ± i σy l ]. The off-diagonal elements determine the coherence between the two states. The non-zero of off-diagonal elements indicates coherence between the two states. Therefore, the density matrix for hydrogen atoms at thermal equilibrium is written as [ −βE ] α 1 0 e .ρ = (2.50) . 0 e−βEβ e−βEα + e−βEβ By replacing the energy of each state, the density matrix for hydrogen atoms at thermal equilibrium can be approximated as [ ρ=

.

1 2

+

h¯ γ B 4kB T

0

0 1 2

−

h¯ γ B 4kB T

] ,

where we also can write ) [ ( ([ ]) ] hγ 1 1 hγ 10 ¯ B ¯ B 10 I+ σz . .ρ = = + 01 2kB T 0 1 2 2kB T 2

(2.51)

(2.52)

To make it more concise, we define .C = hγ ¯ B/2kB T and write ρ=

.

1 (I + Cσz ) . 2

(2.53)

The hydrogen density matrix shows that the off-diagonal elements of the hydrogen atom are zero at thermal equilibrium. This means that there is no coherence at thermal equilibrium. For further analysis of the system, we calculate from Eq. (1.123) the magnetization value along the x-, y-, and z-axis as

2.8 The Dynamics of the Density Matrix in NMR Spin Systems

47

= Tr(ρMz ) /= 0, .

= Tr (ρMx ) = 0, < > ( ) My = Tr ρMy = 0.

(2.54)

Note that the magnetization is proportional to the spin matrix in specific directions. Therefore, for convenience, we can take . = , . = , and . = to verify this relation. In particular, we have . = Tr(ρIz ) = hγ ¯ B/4kB T . The above relation shows that the magnetization value along the x- and y-axis is zero, which means that there is no magnetization population at thermal equilibrium along the x- and y-axis. Furthermore, we can calculate the diagonal element of the density matrix from Eq. (2.48), as ραα = = pα . .

ρββ = = pβ .

(2.55)

On the other hand, if we consider the spin system in a non-equilibrium state, we have the density matrix as [ ρ=

.

] pα c , c pβ

(2.56)

where .pα and .pβ are diagonal elements representing the spin population, and c is a real off-diagonal element dedicating the phase coherence, which is dependent on the resonance frequency. Therefore, the magnetization value is non-zero along the x-axis, where . = Tr (ρMx ) /= 0. Thus, we have net magnetization in the transverse plane at a non-equilibrium state.

2.8 The Dynamics of the Density Matrix in NMR Spin Systems In this part, we investigate how a spin system’s density matrix changes over time. To calculate that, we need to use the Liouville–von Neumann equation, as defined before. The dynamic of the density matrix is derived from Eq. (1.128), which is given by ρ(t) = e− h¯ Ht ρ(0)e h¯ Ht , i

.

i

(2.57)

where .ρ(0) is the density matrix at .t = 0, and .ρ(t) is the density matrix at time t, defined as the evolution of the density matrix over time. To calculate the off-diagonal elements of the density matrix, we have

48

2 Quantum Physics and NMR Foundations

= . i

.

i

(2.58)

Thus, we can write i

= e h¯ (Eβ −Eα )t .

.

(2.59)

By substituting the equations Eβ = hω ¯ β,

Eα = h¯ ωα ,

.

and

ω = ωβ − ωα ,

(2.60)

we have the off-diagonal element of the spin system as = eiωt .

.

(2.61)

Therefore, for a spin-.1/2 system, the matrix form is given by [

] pα c12 eiω12 t .ρ(t) = , c21 eiω21 t pβ

(2.62)

∗ , and .ω where .c12 and .c21 are parameters satisfying .c12 = c21 12 and .ω21 are the frequency difference between the two states, with .ω12 = −ω21 .

2.9 Entropy and the Origin of the Boltzmann Distribution The underlying physics of NMR is rooted in essential concepts of thermodynamics systems. Some critical questions behind the NMR systems are related to the Boltzmann distribution [62, 63]. As was demonstrated in this chapter, the Boltzmann distribution plays an important role in the physics of NMR. However, it was not discussed how the Boltzmann distribution emerges from the first principles. For example, why we define .pi = exp(−βEi )/Z? How we know .β = 1/kB T in E the Boltzmann distribution? Why the partition function is defined as .Z = i exp (−βEi )? Why .pi have an exponential form with respect to .Ei ? The Boltzmann distribution is the distribution at equilibrium conditions. A thermal system is in equilibrium when the entropy is maximum [64, 65]. Thus, we start with the entropy and find the maximum of the entropy. The entropy of the system is given by S=−

E

.

pi log pi ,

(2.63)

i

where suppose .kB = 1 for convenience. Here .pi is the probability distribution of the system. There are two constraints for .pi . The first constraint is that the sum of probability is 1. In other words, we can write

2.9 Entropy and the Origin of the Boltzmann Distribution

E .

49

pi = 1.

(2.64)

i

The second constraint is energy conservation, by which the average energy is fixed. In other words, we can express as E .

pi Ei = E,

(2.65)

i

where .E is the total energy of the system and is fixed. We use the Lagrange multipliers method to find the maximum entropy concerning the above relations [66, 67]. Thus, we define F =−

E

.

pi log pi − α

( E

i

) pi − 1 − β

( E

i

) pi Ei − E ,

(2.66)

i

where .−α and .−β are the Lagrange variables. Note that we deliberately took the minus signs for convenience. Now, we find the maximal of .Fj concerning .pj , as .

] [ ∂Fj = − log pj + 1 + α + βFj = 0. ∂pj

(2.67)

Then, we have .

log pj = −(1 + α) − βEj .

(2.68)

pj = e−(1+α) e−βEj ,

(2.69)

We obtain .pj as .

where .exp(−(1 + α)) is a Lagrange parameter. Thus, we rename it .exp(1 + α) ≡ Z. Hence, we find for the ith probability pi =

.

1 −βEi e . Z

(2.70)

This answers the question of why the probabilities take the exponential form. Note that .Z and .β are based on the Lagrange multipliers, and we still do not know what theyE are. These probabilities must satisfy the above constants. From the first constant, . i pi = 1, we have E .

i

pi =

1 E −βEi e = 1. Z i

(2.71)

50

2 Quantum Physics and NMR Foundations

Therefore, we arrive at .

Z=

E

e−βEi .

(2.72)

i

So, the partition function is the explicit form of .Z. Note that .Z is a function of β. E We still do not know what .β is. To find the answer, we use the constraint as . i pi Ei = E. Thus, we have .

E1 e−βEi Ei . Z

E=

.

(2.73)

i

Or we can write .

1 E −βEi e Ei = E. Z

(2.74)

i

On the other hand, from .Z = .β is expressed as

.

E i

exp (−βEi ), the derivative of .Z with respect to

E ∂Z =− Ei e−βEi . ∂β

(2.75)

i

Equivalently, we have E .

e−βEi Ei = −

i

∂Z . ∂β

(2.76)

Combining with the above equation for the energy constraint yields E=−

.

∂ log Z 1 ∂Z =− . Z ∂β ∂β

(2.77)

In fact, we can attain the energy of the system using the partition function based on this equation. Now, with all these analyses, we recalculate the entropy for the equilibrium system from Eqs. (2.63) and (2.70), as S=+

E1 e−βEi (+βEi + log Z) Z i

.

β E −βEi log Z E −βEi = e . e Ei + Z Z i

i

(2.78)

2.9 Entropy and the Origin of the Boltzmann Distribution

51

From Eq. (2.74), we get S = βE + log Z,

.

(2.79)

E where . i exp (−βEi ) /Z = 1. It is notable that the energy of the system is determined by the partition function. Therefore, the entropy of the system can also be determined by the partition function of the system. This demonstrated how powerful the partition function can be. In fact, we can calculate important thermodynamic quantities by using the partition function. Note that we still do not know the parameter .β. This should be determined in some way. To do so, we calculate the temperature of the system. According to the definition, the temperature of a system is determined by .

dE = T ⇒ dE = T dS. dS

(2.80)

1 dE = dS. T

(2.81)

Thus, we have .

On the other hand, from Eq. (2.79), we have ∂ log Z dβ ∂β ( ) ∂ log Z = βdE + E + dβ. ∂β

dS = βdE + Edβ + .

(2.82)

Therefore, from Eq. (2.77), we have ∂ log Z = 0. ∂β

E+

.

(2.83)

Thus, by considering the above relation, dS in Eq. (2.82) can be written as dS = βdE.

.

(2.84)

Hence, by comparing with the definition of the temperature, we can write β=

.

1 , T

where we suppose .kB = 1 in our calculation.

(2.85)

52

2 Quantum Physics and NMR Foundations

Exercises Problem 1 (a) Calculate nβ + nα from Eq. (2.26). (b) Calculate |nβ − nα | from Eq. (2.26). (c) Show that pα + pβ = 1.

Solution 1 (a) By substituting the expressions for nα and nβ into the sum nα + nβ , we have n0 .nβ + nα = 2

(

hω ¯ 0 1− 2kB T

)

n0 + 2

( ) hω ¯ 0 1+ 2kB T

= n0 .

(2.86)

We have presented the sum of nα , and nβ equals n0 , the number of spins at the thermal equilibrium. | | (b) To calculate |nα − nβ |, we can ( ) ( ) h¯ ω0 hω n0 ¯ 0 1− − 1+ | 2kB T 2 2kB T ( ) n0 h¯ ω0 . =2 2 2kB T

n0 .|nβ − nα | = | 2

= n0

h¯ ω0 . 2kB T

(2.87)

(c) To calculate pα + pβ , we have from Eq. (2.18) p α + pβ =

.

nβ nα + n0 n0

−Eβ α 1 −E [e kB T + e kB T ] Z −AE 1 −Eα = [e kB T (1 + e kB T )] Z −Eα −AE 1 [e kB T (1 + e kB T )]. = −Eα −AE e kB T (1 + e kB T )

=

= 1. Therefore, we have shown that pα + pβ = 1.

(2.88) (2.89)

Exercises

53

Problem 2 Find the entropy of a spin-.1/2 system that is at thermal equilibrium with temperature T .

Solution 2 The entropy for the equilibrium system is written as S = −kB

E

.

pi log pi .

(2.90)

i

For a two-level system (spin-.1/2), we have two possibilities as pα =

.

e−βEα , Z

pβ =

e−βEβ . Z

(2.91)

We know .pα + pβ = 1, thus, .pβ = 1 − pα . Therefore, the entropy is given by ( ) S = −kβ pα log pα + pβ log pβ

.

(2.92)

= −kβ (pα log pα + (1 − pα ) log(1 − pα )) . For the thermal equilibrium, we showed S = βE + log Z.

.

(2.93)

We also have E=

E

.

pi Ei = pα Eα + pβ Eβ =

i

) 1 ( −βEα e Eα + e−βEβ Eβ . Z

(2.94)

Thus, we have S=

.

) β ( −βEα e Eα + e−βEβ Eβ + log Z, Z

(2.95)

where Z = e−βEα + e−βEβ .

.

(2.96)

Chapter 3

Bloch Equation Description Without Relaxation Time

3.1 Introduction In this chapter, we discuss the physics behind the Bloch equations and provide the foundations for the dynamics of the magnetization vector. We investigate the basis for the magnetization vector in the laboratory and rotating frame under the influence of a magnetic field. We can attain valuable insights into the MRI sequences by understanding these essential principles. We consider the quantum mechanics perspective on the RF pulse and the effect of the rotating frame on the spin system. We investigate the Hamiltonian and the density matrix of the spin system in the rotating frame. We also provide a geometric perspective using the Bloch sphere representation and introduce rotation operators, which can help to visualize the spin system’s dynamics in NMR. The framework we develop in this chapter provides an important basis for understanding the later chapters. By considering the Bloch equations and the magnetization vector’s behavior, we can pave the way for more advanced MRI techniques that are practically used in the field of medical imaging.

3.2 Bloch Equation in the Laboratory Frame Once a spin system is placed in a magnetic field, the spin of the sample induces a magnetic dipole moment μ - = γ S,

.

(3.1)

resulting in a torque .τ-, which is defined as the change of the angular momentum, .S, with respect to the time, that can be determined by

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Khashami, Fundamentals of NMR and MRI, https://doi.org/10.1007/978-3-031-47976-2_3

55

56

3 Bloch Equation Description Without Relaxation Time

d S. dt

(3.2)

d S1 dμ . = γ dt dt

(3.3)

τ- =

.

Therefore, .τ- can be expressed as τ- =

.

On the other hand, the torque can be described in terms of the cross product of the magnetic dipole moment and the magnetic field, which is given by τ- = μ - × B.

.

(3.4)

From the above equations, the differential equation of the magnetic dipole moment is written as .

dμ - =μ = γ (μ - × B) - × γ B. dt

(3.5)

- for a spin system contains a large number of protons with The magnetization .M - n and is defined by the magnetic dipole moment .μ - = M

TotalE Proton

.

μ - n.

(3.6)

n=1

The first-order differential equation of the macroscopic magnetization, which describes the motion of a spin-.1/2 system in a magnetic field, is expressed by putting Eqs. (3.6)–(3.5)

.

d M(t) = M(t) × γ B(t), dt

(3.7)

where .ω = γ B(t) corresponds to a precession of the magnetization vector about the magnetic field, .B(t). As we noted, both magnetization and magnetic field can be time dependent. The differential magnetization equation is called the Bloch equation [18, 25, 32]. Considering each of the Bloch elements in the magnetic field .B(t) = Bx (t), .By (t), .Bz (t), we attain the differential equations of the magnetization in the laboratory frame from Eq. (3.7). Therefore, the above cross product can be written as .

( ) dMx (t) = γ My (t)Bz (t) − Mz (t)By (t) , . dt

(3.8)

3.2 Bloch Equation in the Laboratory Frame

57

Fig. 3.1 Scheme of the laboratory frame with coordinates .(x, y, z). The magnetization vector, .M, precesses around the main magnetic field .B-0 along the z-axis at Larmor frequency .ω0 . The rotating magnetic field .B-RF along the x-axis rotates clockwise with the same frequency as .ω0 in the .x − y plane. The blue arrow represents .B-RF . The yellow arrow represents .M

dMy (t) = γ (Mz (t)Bx (t) − Mx (t)Bz (t)) , . dt ( ) dMz (t) = γ Mx (t)By (t) − My (t)Bx (t) , dt

(3.9) (3.10)

where .M(t) = Mx (t), .My (t) and .Mz (t) are the elements of magnetization in the x, y and z directions. The magnetic field can be either a static magnetic field along the z-axis or a rotating magnetic field in the .x − y plane in the laboratory frame [68, 69]. We schematically express the dynamics of the Bloch equations in Fig. 3.1. In this figure, we display the frame of .x, y, and z coordinates, which are in the laboratory frame. Once we place the spin system in the static magnetic field along the z-axis (where we have .Bx (t) = 0, .By (t) = 0 and .Bz (t) = B0 ), by considering .ω0 = γ B0 , the rate of change of magnetization from the Bloch equation becomes .

dMx (t) = ω0 My (t), . dt dMy (t) = −ω0 Mx (t), . dt dMz (t) = 0. dt

(3.11) (3.12) (3.13)

58

3 Bloch Equation Description Without Relaxation Time

The rate of magnetization implies that the magnetization vector .M(t) rotates around .B0 along the z-axis at its Larmor frequency .ω0 . The solution of the above equation is written as .

Mx = A sin(ω0 t), .

(3.14)

My = A cos(ω0 t), .

(3.15)

Mz = b,

(3.16)

where A and b are positive constant. Moreover, Eq. (3.16) is a constant term due to the stationary magnetic field along the z-axis. The magnetization vector is along the z-axis, and the derivation of magnetization can be zero according to the above equation. Therefore, we must introduce another magnetic field to induce spin dynamics in the NMR system. To do so, we apply a rotating magnetic field in the .x − y plane, which is called the RF pulse denoted as .B-RF . The RF pulse can transmit the Larmor frequency .ωRF , which is close to the angular speed of .M(t) in the laboratory frame [28]. If x component of the rotating magnetic field vector is designated by .B-x,RF and y component is denoted by .B-y,RF , thus, we have .

ˆ . B-x,RF = B1 cos(ω0 t)i,

(3.17)

B-y,RF = B1 sin(ω0 t)jˆ, .

(3.18)

ˆ B-z,RF = B0 k,

(3.19)

where we suppose a two-dimension coordinate, e.g., .+x- and .+y-axes. Moreover, B1 is the amplitude of the RF pulse. From the above equation, .B1 carries the Larmor frequency .ω0 in the laboratory frame. Thus, we can track magnetization dynamics in the laboratory frame by replacing the rotating magnetic field in Eqs. (3.8)–(3.10). As one may notice, it is somewhat complicated if we consider both .M and .B-RF from the laboratory’s view. Therefore, we describe the spin system in a rotating frame to make it more convenient. The rotating frame simplifies description and enables more intuitive understanding.

.

3.3 Bloch Equation in the Rotating Frame As was mentioned earlier, evaluating the magnetization dynamics in a rotating frame is valuable. Describing the spin system in the rotating frame is helpful for simplifying the calculation of the related equations between the spin system and the magnetic field and constructing pulse sequence for the specific area of interest [70, 71].

3.3 Bloch Equation in the Rotating Frame

59

Fig. 3.2 Scheme of the rotating frame with new coordinates .(x ' , y ' , z' = z). The magnetization vector, .M, is gradually turned around the .B-RF along the .x ' -axis. The field .B-RF is weaker than .B-0 in the new frame; therefore, .M rotates slowly in the rotating frame. Furthermore, in the rotating frame, the reference frame can rotate with Larmor frequency .ωrot

When we place our spin system in the rotating frame, the frame rotates at a Larmor frequency .ωrot about the z-axis. As is shown in Fig. 3.2, the x- and yaxis rotate about the z-axis with the Larmor frequency .ωrot in the rotating frame. Moreover, the z-axis is the same in the laboratory and rotating frame. Therefore, the rotating frame has transferred to .x ' , y ' , and .z' = z. The coordinate axes are written as ⎛

⎞ ⎛ ⎞⎛ ⎞ iˆ' iˆ cos(ωrot t) − sin(ωrot t) 0 ' . ⎝ jˆ ⎠ = ⎝ sin(ωrot t) cos(ωrot t) 0 ⎠ ⎝ jˆ ⎠ . 0 0 1 kˆ ' kˆ

(3.20)

In an MRI or NMR system, the magnetization vector .M presses around .B-RF with a new Larmor frequency in the new frame. The magnetization vector rotates relatively slowly compared with the laboratory frame because .B-RF is weaker than .B-0 in the rotating frame. Therefore, the spin system senses a smaller Larmor frequency in the new frame. To simplify the Bloch calculation, we consider .B-RF in the rotating frame as -RF = (B1 , 0, 0), where the amplitude of applied magnetic field is .B1 along the .B x-axis. In Fig. 3.3, we schematically display the NMR instrument and present a scheme of the main magnetic field .B-0 and .B-RF . The diagram shows .B-0 along the z-axis and perpendicular to .B-RF . As depicted in Fig. 3.2, the magnetization vector is established at its equilibrium - The initial value initiates point along the z-axis in the rotating frame, denoted as .M. at its maximum magnetization point along the z-axis.

60

3 Bloch Equation Description Without Relaxation Time

Fig. 3.3 Scheme of NMR Instrument. The diagram represents the main magnetic field .B-0 and the RF pulse .B-RF . .B-0 is along the z-axis and .B-RF is perpendicular to .B-0 . Vacuum chamber (CV), Liquid nitrogen (LN), Liquid helium (LH). Note that an NMR machine usually uses RF pulses between .60–900 .MHz

On the other hand, the rotating frame system feels the influence of the rotation of the coordinate system, which is described by an extra magnetic field against the z-axis, known as the fictitious field .ωrot /γ as is shown in Fig. 3.4a. The fictitious field is generated in the rotating frame and can reduce the effect of .B-0 in the frame. We also maintain an effective magnetic field, denoted as .B-eff , as illustrated in Fig. 3.4b. This field generates the coordinate vector along the z-axis, given by ˆ and a coordinate vector along .x ' -axis represented as -eff (z) = (B0 − ωrot /γ )k, .B ' ' -eff (x ) = B1 iˆ [72]. Thus, .B-eff is written as .B ) ( ωrot ˆ k. B-eff = B1 iˆ' + B0 − γ

.

(3.21)

Note that an effective magnetic field along the z-axis can produce the resonance offset frequency, .ωoff . We have denoted as .ωoff = ω0 − ωrot in the rotating frame. We, likewise, can calculate the angle, .θ , between .B-eff and .x ' -axis, which is expressed as .

tan θ =

B0 − ωrot /γ . B1

(3.22)

3.3 Bloch Equation in the Rotating Frame

61

Fig. 3.4 Scheme of the effective magnetic field, .B-eff , in the rotating frame. (a) With the presence of .B-RF along the .x ' -axis and induced fictitious field (.ωrot /γ ) against .B-0 along the z-axis. (b) The point of .B-eff vector along the z-axis is .B0 − ωrot /γ , and along the .x ' -axis is .B1 . The angle between -eff and .x ' -axis is .θ .B

Thus, .θ is provided as θ = tan−1

.

(

B0 − ωrot /γ B1

) .

(3.23)

The magnitude of the effective magnetic field is defined by [

)2 ] 21 ( ω rot -eff | = B12 + B0 − = o/γ . .|B γ

(3.24)

Thus, the precession frequency in the rotating frame is written as ]1 [ 1 2 2 . o = [(γ B1 )2 + (ω0 − ωrot )2 ] 2 = (γ B1 )2 + ωoff

.

(3.25)

From the above equation, we can express the magnetization vector precesses about .B-eff in the rotating frame at frequency .o. If the RF pulse and the rotating frame do not resonate at the same frequency, one can say that they are off-resonant, where we have .ωoff /= 0. Hence, the fictitious field does not cancel with .B-0 , and a component of the effective field is initiated along the z direction. On another side, if we consider the rotating coordinates with .ωrot = γ B-0 = ˆ then we can ignore the impact of .B-0 and the induced magnetic field .ωrot /γ in ω0 k, Eq. (3.25). Thus, the spin system attains the same Larmor frequency as its natural frequency, .ω0 , which means they are resonant, where we obtain .ωoff = 0. Thus, the RF pulse can rotate the magnetization vector of the spin system from its longitudinal axis to its transverse plane and detect the maximum signal as frequency .o = γ B1 .

62

3 Bloch Equation Description Without Relaxation Time

On the other hand, if we assume the high power of the RF pulse in the system, the second part of Eq. (3.25) can be neglected, where we have .γ B1 >> ωoff . Therefore, -eff ≈ B1 iˆ' and we arrive at .o ≈ γ B1 . In contrast, once the RF pulse is turned off, .B the magnetization vector starts to precess with the offset frequency, where we have .o ≈ ωoff . Moreover, the Bloch equation in the rotating frame is noted by

.

- ' (t) dM - ' (t) × γ B-eff (t), =M dt

(3.26)

- ' (t) is the magnetization vector that rotates around .B-eff (t) in the rotating where .M frame. Note that both elements can be time dependent in the rotating frame.

3.4 Flip Angle of the RF Pulse In this part, we investigate the flip angle of the RF pulse as a practical parameter in NMR and MRI sequences. We suppose the spin system is not provided enough time to achieve equilibrium. By applying an RF pulse with a specific angle, the magnetization vector can stop in the middle of its way. Therefore, the angle of the RF pulse that can cover the duration time is known as the flip angle of the RF pulse, which is denoted as .α. We can handle the magnetization vector’s rotation by applying the RF pulse with a flip angle. The flip angle during the RF pulse can change its direction to correspond to the duration time of the RF pulse and the amplitude of the RF pulse. In this way, we can optimize the scan time. Therefore, the angle of the spin system, which is illustrated in Fig. 3.5a, is calculated by Flip Angle = Duration time of the RF pulse × Amplitude of the RF pulse. (3.27)

.

The amplitude of the RF pulse is the maximum of .B1 that is applied to the spin system, which is recorded in micro Tesla, .μT . Moreover, the duration time of the RF pulse, .t, is the time it takes for the magnetization .M to transfer to the new position, expressed in ms. Therefore, the RF pulse over the duration time changes the initial state of .M to its final state. To illustrate, we demonstrate a flip angle .α = 180◦ or .π in Fig. 3.5b, where the magnetization shifts from its initial state along the z-axis to the opposite direction (its final state) along the .−z-axis. Therefore, the spin is shifted .180◦ from its starting point, which is given by α = ot = γ B1 t,

.

(3.28)

3.5 Geometric Representation of the RF Pulse by the Density Matrix Formalism

63

Fig. 3.5 Scheme of the flip angle of the RF pulse. (a) Represent the amplitude and duration time of the RF pulse. Note that we plot a simple diagram for the RF pulse. In the sequence, the RF pulse is an envelope pulse shape. (b) The magnetization vector flips onto .x ' − y ' plane with the flip angle ◦ .α = 180

where .α is measured in degrees or radians. The duration time can be written as t=

.

α . γ B1

(3.29)

Moreover, the RF pulse can be time dependent, where we can write f α=

.

t

γ B1 (t ' )dt ' .

(3.30)

0

This relation indicates that the stronger the magnetic field, the shorter the duration time is recorded. In this way, the magnetization vector gets to its maximum point faster, resulting in a shorter scan time.

3.5 Geometric Representation of the RF Pulse by the Density Matrix Formalism Now, we consider the geometric representation of the RF pulse using the density matrix representation [17, 73]. To this aim, we define the general form of the Hamiltonian by considering the effect of the RF pulse, which is given by .H. The density operator by the RF pulse under the resonance condition can be described as ρ(t) = e− h¯ Ht ρ(0)e h¯ Ht . i

.

i

64

3 Bloch Equation Description Without Relaxation Time

If an RF pulse is applied along the x-axis, we can define .H as Hˆ = hγ ¯ B1 Iˆx ,

(3.31)

.

where .Iˆx is the Pauli matrix for a spin-.1/2 system along the x-axis. For .t = T , the time evolution of the density matrix can be written as ˆ

ˆ

ρ(T ) = e−iγ B1 T Ix ρ(0)eiγ B1 T Ix .

.

(3.32)

We define .α = γ B1 T , which leads ˆ

ˆ

ρ(α) = e−iα Ix ρ(0)eiα Ix ,

.

(3.33)

where .ρ(α) is the density matrix after rotating with angle .α. This, in fact, is the rotation along the x-axis, with the angle .α. For a general setting, if the density matrix .ρ(0) rotates around an arbitrary axis with the rotation angle .α, we can write ρ(α) = R(α)ρ(0)R(−α).

(3.34)

.

Comparing Eq. (3.34) with Eq. (3.33), we have ˆ

R(α) = Rαx = e−iα Ix ,

.

(3.35)

and ˆ

iα Ix R(−α) = R−α . x =e

.

(3.36)

In the above equations, .R(α) rotates the magnetization vector about the axis with the angle .α, and .R(−α) is an inverse rotation operator, such that it rotates the magnetization vector about the axis with an angle .−α. Considering the above equations, we can investigate the effect of the RF pulses on the magnetization vector for spin-.1/2 nuclei. For instance, in the case of a rotation angle .α = − π2 along the x-axis we have −π

+ π2

ρ(α) = Rx 2 ρ(0)Rx

.

π

π

= e+i 2 Ix ρ(0)e−i 2 Ix .

(3.37)

Thus, different rotation operators can be obtained depending on the flip angle parameters. Similarly, when the RF pulse is applied along the y-axis and z-axis, the rotation induced by the pulse is described by .Rαy = exp(−iα Iˆy ) and .Rαz = exp(−iα Iˆz ), respectively. Here, .Iy and .Iz are the Pauli matrices along the y- and z-axis, respectively. To visualize the rotation process, we express the magnetization motion on the Bloch sphere, which can help us to understand the rotation in the spin system [74–

3.5 Geometric Representation of the RF Pulse by the Density Matrix Formalism

65

Fig. 3.6 Scheme of the Bloch sphere representation of the magnetization vector rotation in the spin system. (a)–(b) Rotation of the vector about the x-axis, which rotates to the y direction with ±π

the rotation angle .α = ± π2 , denoted by .Rx 2 . (c) Rotation of the vector along the x-axis to the z direction by applying the rotation operator .Rπx . (d)–(e) Rotation of the vector about y-axis, which ±π

rotates to the x direction with rotation angle .α = ± π2 , denoted by .Ry 2 . (f) Rotation of the vector along the y-axis to the z direction by applying the rotation operator .Rπy . The yellow vector represents the magnetization vector on the Bloch sphere. The yellow dash arrows show where the vector is directed after rotating along the axes. The purple sphere is the Bloch sphere

76]. In Fig. 3.6, we represent the magnetization behavior on the Bloch sphere with rotation angle .α = ± π2 and .α = ±π along the x- and y-axis. - z along the z-axis. Then, In Fig. 3.6a and b, we have the magnetization vector .M we apply the inverse .90◦ RF pulse along the x-axis. The magnetization vector flips from the z-axis to the .+y-axis by clockwise rotation. Similarly, once we apply a .90◦ RF pulse along the x-axis, the magnetization vector tilts from its initial state to the .−y-axis by counterclockwise rotation. In Fig. 3.6c, we apply the .180◦ RF pulse along the x-axis, and the magnetization vector moves from its initial state in .z − x plane to .x − y plane by counterclockwise rotation. Note that the final direction is along the z-axis. We represent the spin rotation along the y-axis by applying the inverse .90◦ , the .90◦ , and the .180◦ RF pulse in Fig. 3.6d–f. To have more sense of the rotation in the matrix form, we provide related calculations behind the spin rotation in the Exercises 3.5.

66

3 Bloch Equation Description Without Relaxation Time

Exercises Problem 1 Consider the effect of the RF magnetic field (ranging of .1 × 10−5 − 3 × 10−5 T) on the duration time of the RF pulse, .t at .α = 180◦ , .α = 90◦ , .α = 10◦ , and .α = 5◦ for carbon and proton samples.

Solution 1 The plot represents the magnetic field’s effect on the RF pulse duration time, .t. In Fig. 3.7, we show that by increasing the magnetic field, .B-1 , a shorter duration time, .t, can be recorded. As we present in the plot, due to the smaller gyromagnetic ratio of the carbon (.10.71 MHz .Tesla−1 ) compared with the hydrogen atom (.42.58 MHz −1 .Tesla ), the carbon needs a longer time than the hydrogen atom to flip the spins onto the .x − y plane, which is .t ∝ 1/γ . For example, we calculate the duration time of the RF pulse for proton and carbon at .B-1 = 30 .μT, which is the amplitude of the RF pulse. For .α = 180◦ , the .1 H atom requires .t = 2.45 ms to flip the spins onto the .x − y plane. On the other hand, the .13 C atom with a smaller gyromagnetic ratio than .1 H atom needs .t = 9.77 ms to excite the spins. For .α = 90◦ , the .13 C atom has a higher duration time, .t = 4.88 ms, than the hydrogen atom, .t = 1.22 ms.

Fig. 3.7 Problem 1

Exercises

67

For .α = 10◦ , the .13 C atom has a longer duration time, .t = 0.54 ms, than the hydrogen atom, .t = 0.13 ms. For .α = 5◦ , the .13 C atom has a higher duration time, .t = 0.27 ms, than the hydrogen atom, .t = 0.068 ms.

Problem 2 Calculate the rotation operator for .I = 1/2 with rotation angle .α = ± π2 and .α = π . Apply the RF pulse to the density matrix from Eq. (2.51) [ ρ=

.

1 2

+

h¯ γ B 4kB T

0

0 1 2

−

h¯ γ B 4kB T

] .

Solution 2 The Pauli matrix representations for the simple .I = 1/2 case are given by [ ] 1 01 .Ix = 2 10

[ ] 1 0 −i Iy = 2 i 0

[ ] 1 1 0 Iz = . 2 0 −1

(3.38)

The rotation operators are expanded as a matrix form [60, 75] [

] cos α2 −i sin α2 , −i sin α2 cos α2 . ] [ cos α2 + sin α2 . Rαy = − sin α2 cos α2 Rαx =

(3.39)

The rotation matrices around x- and y-axis with rotation angle .α = ± π2 are written as ] ] [ √1 √ [ −i π cos π4 −i sin π4 2 2 2 = −i 1 , Rx = √ √ −i sin π4 cos π4 2 2 (3.40) . [ 1 i ] ] [ −π √ √ − π2 −i sin cos −π 2 2 4 4 = . Rx = √i √1 −i sin −π cos −π 4 4 2

2

68

3 Bloch Equation Description Without Relaxation Time

Also, [

cos π4 + sin π4 Ry = − sin π4 cos π4 π 2

−π Ry 2

=

[

.

[

]

sin −π cos −π 4 4 = −π − sin 4 cos −π 4

√1 2 − √1 2

[

] =

+ √1 √1 2

√1 2 + √1 2

]

2

,

− √1 √1 2

2

]

(3.41) .

For .α = π , we have ] [ ] 0 −i cos π2 −i sin π2 = , −i sin π2 cos π2 −i 0 ] [ [ ] sin −π 0 −1 cos −π 2 2 = = . −π − sin −π 1 0 2 cos 2 [

Rπx = .

R−π y

(3.42)

−π

From Eq. (2.51), by applying the first .90◦ pulse, .Rx 2 , on the density matrix in a rotating frame along the x-axis, .ρ(α) is given by (see Fig. 3.6a) [74–76] −π

π

ρ(α) = Rx 2 ρ(0)Rx2 [ ] [1 + h¯ γ B 1 1i =√ × 2 4kB T . 0 2 i 1 ] [ h¯ γ B 1 −i 2k 1 BT , = h¯ γ B 2 i 2k 1 BT where .i =

√

]

0 1 2

−

h¯ γ B 4kB T

[ ] 1 1 −i ×√ 2 −i 1

−1 is the imaginary unit [60, 75].

(3.43)

π

For example, the magnetization vector rotates via a rotation .Ry2 = exp(−i π2 Iy ), by .α = + π2 about the y-axis. The rotation matrix around the y-axis is given by (see Fig. 3.6e) π

−π

ρ(α) = Ry2 ρ(0)Ry 2 [ ] [1 + h¯ γ B 1 1 1 =√ × 2 4kB T . 0 2 −1 1 [ ] hγ ¯ B 1 − 2k 1 BT . = h¯ γ B 2 − 2k T 1 B

0 1 2

−

hγ ¯ B 4kB T

]

[ ] 1 1 −1 ×√ 2 1 1

(3.44)

The second pulse is a .180◦ pulse applied along the x-direction, denoted by .Rπx . The rotation angle is .α = π in Eq. (3.39), which rotates along the .x direction. Therefore, .ρ(α) is given by

Exercises

69 π ρ(α) = R−π x ρ(0)Rx ] [ [ ] [1 ] hγ ¯ B + 4k 0 0 i 0 −i 2 T B = × × hγ 1 ¯ B i 0 −i 0 0 . 2 − 4kB T ] [ 1 − h¯ γ B 0 , = 2 4kB T 1 h¯ γ B + 0 2 4kB T

(3.45)

where the .180◦ pulse leads to the inversion of the populations without any coherence. Therefore, after a .180◦ pulse, the magnetization vector is switched to the .−z direction. For .α = −π along the y-direction, we have π ρ(α) = R−π y ρ(0)Ry ] [ [ ] [1 ] h¯ γ B + 0 0 −1 0 1 2 4k T B = × × h¯ γ B 1 1 0 −1 0 0 . 2 − 4kB T ] [ 1 − h¯ γ B 0 . = 2 4kB T 1 h¯ γ B 0 2 + 4kB T

(3.46)

Chapter 4

Bloch Equation Description with Relaxation Time

4.1 Introduction In the previous chapter, we considered the Bloch equations without taking into account the relaxation process to provide better visualization of the spin system in the rotating frame. In this chapter, we introduce the foundational concept of NMR, which involves the decay and buildup of net magnetization over time, known as the relaxation term. From the perspective of the Bloch equations, we discuss how the spin–lattice and spin–spin relaxation processes appear in the NMR system. Moreover, we explain how the Fourier transformation allows us to understand the connection between the time and frequency domains in the NMR spectrum. We also investigate the dynamics of magnetization behavior from a quantum viewpoint and discuss the density matrix of the NMR system. Also, we introduce the quantum time-energy uncertainty in the NMR system, which leads to minimum detectable resonance line-width in a system.

4.2 General Formalism of the Bloch Equations In the previous chapter, we considered the Bloch equation and the system’s dynamics without including the relaxation. The general form of the Bloch equation, containing the relaxation terms, is defined by

.

M (t)iˆ + My (t)jˆ (Mz (t) − Mz (0)) kˆ d M(t) - − x = M(t) × γ B(t) − , dt T2 T1

(4.1)

where .Mx (t), .My (t), and .Mz (t) are the elements of magnetization in the x, y, and z directions. Also, .Mz (0) is the magnetization at the initial state of the magnetization vector, which is along the z-axis. Moreover, we have two terms related to the © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Khashami, Fundamentals of NMR and MRI, https://doi.org/10.1007/978-3-031-47976-2_4

71

72

4 Bloch Equation Description with Relaxation Time

relaxation time of the magnetization vector. The second term of the Bloch equation is linked to spin–spin relaxation time, .T2 , in the .x − y plane, and the third term is connected to spin-lattice relaxation time, .T1 , along the z-axis. Considering each of the Bloch components in the magnetic field, .B(t) = [Bx (t), By (t), Bz (t)], we attain the rate of magnetization as .

( ) Mx (t) dMx (t) = γ My (t)Bz (t) − Mz (t)By (t) − ,. dt T2 My (t) dMy (t) = γ (Mz (t)Bx (t) − Mx (t)Bz (t)) − ,. dt T2 ( ) Mz (t) − Mz (0) dMz (t) = γ Mx (t)By (t) − My (t)Bx (t) − . T1 dt

(4.2) (4.3) (4.4)

If we consider the static magnetic field along the z-axis, .B-x = 0, B-y = 0, B-z = from the Bloch equation becomes B0 , the rate of change of magnetization .M(t) .

dMx (t) Mx (t) = ω0 My (t) − ,. dt T2

(4.5)

My (t) dMy (t) = −ω0 Mx (t) − ,. dt T2

(4.6)

Mz (0) − Mz (t) dMz (t) = . T1 dt

(4.7)

The first two relations are the Bloch equations along the x-axis and y-axis, denoted by the transverse magnetization rate. The rate of magnetization implies the magnetization changes and spin precession in the transverse plane, which causes a loss of coherence between spins. Equation (4.7) is known as the Bloch equation for z-magnetization, which characterizes the longitudinal component of the magnetization [35, 77, 78]. The rate of change of z-magnetization with time is inversely proportional to .T1 . It is directly proportional to the difference between the equilibrium value of magnetization, .Mz (0), and z-magnetization at time .t, .Mz (t) [24, 79].

4.3 The Longitudinal Magnetization Behaviors in the Spin System When we turn on the RF pulse, the nuclear spins have the same frequency as the RF pulse frequency in the lower energy level, absorb the RF pulse’s energy, and transfer to the higher energy level along the z-axis. After switching off the RF pulse, the net magnetization of the spin-.1/2 sample returns to the lower energy state, known as the longitudinal magnetization, and releases the absorbed energy. This process is called

4.3 The Longitudinal Magnetization Behaviors in the Spin System

73

Fig. 4.1 Scheme of the relaxation time procedure. When the RF pulse is applied to the spin-.1/2 system, some spins resonate at the same Larmor frequency with the RF pulse. The nuclei in the lower energy state (.mI = −1/2) are excited and flipped to the higher energy (.mI = 1/2). Then, the nuclei return to the equilibrium state in the lower energy level and release the absorbed energy, where .T1 value is recorded. .T1 relaxation time determines how quickly the spin of the nucleus becomes parallel to the magnetic field. The red arrow indicates the magnetization vector .Mz of the spin-.1/2 sample, which transfers from the lower to the higher energy level

the longitudinal relaxation time, which is recorded .T1 [57, 80]. Therefore, we can write the longitudinal relaxation time relates to energy exchanges between the two states. We display the longitudinal relaxation time process in Fig. 4.1. Furthermore, the longitudinal component recovers to the original state by emitting energy to the surrounding tissue. We also call it spin-lattice relaxation time, essential in NMR spectroscopy, determining how quickly the nuclear spins become parallel or along the magnetic field. In other words, a constant value of .T1 is recorded due to energy exchange between the spin system and neighboring molecules. We can write the solution of the Bloch equation along the z-axis from Eq. (4.7) as Mz (t) = Mz (0)e

.

− Tt

1

+ M0 (1 − e

− Tt

1

),

(4.8)

where the magnetization returns and reaches its thermal equilibrium at .M0 . To simplify Eq. (4.8), we suppose at time .t = 0 there is no magnetization, where we have .Mz (0) = 0. Thus, the spin magnetization at time t is written as Mz (t) = M0 (1 − e

.

− Tt

1

),

(4.9)

which demonstrates that .Mz (t) grows with time and exponentially approaches the equilibrium value, .M0 [35, 81]. The Eq. (4.9) displays that the magnetization enhances to .(1 − e−1 ) ∼ 63% of its maximum value to register .T1 value. Therefore, the longitudinal magnetization at .t → ∞ reaches its equilibrium value .Mz (∞) = M0 [81, 82]. When relaxation time is very long, we have .t 0. This gives .J < 1/2ω0 . Taking the maximum value .J = 1/2ω0 , we get .1 − 4J 2 ω02 = 0, and, therefore, we can write 2 2 .τ = 1/ω0 . If we set .4J ω 1/ω0 ), which generates a more significant spectral density, resulting in a very short relaxation time .T1 . In Fig. 5.3c, the molecule’s frequency is comparable to the Larmor frequency for a medium molecule size, which is .τ ≈ 1/ω0 . The relation indicates that relaxation time is dependent on the Larmor frequency. Moreover, the power distribution is spread efficiently in space, and a shorter or more efficient .T1 can be recorded. In Fig. 5.3d, we display a small molecule that spins quickly. A small molecule has a shorter correlation time and fast motion. Thus, correlation time is relatively small compared with .1/ω0 (.τ > 1/ω0 , where there is a larger spectral density with a very short relaxation time .T1 . (c) A medium-sized molecule has a relativity moderate motion .τ ≈ 1/ω0 , and an efficient .T1 can be recorded. (d) A small-sized molecule has a fast motion with .τ 1, the second and third terms can be zero, and we only have the first term in the bracket. By considering the spectral density function, the spin–spin relaxation rate is rewritten as .

3μ20 γP2 γP2 h¯ 2 1 [3J (0, τ ) + 5J (ω0 , τ ) + 2J (2ω0 , τ )]. = T2 320π 2 r 6

(5.21)

100

5 Molecular Motion, Correlation, and Relaxation Time

The rate of .T2 depends on .J (0, τ ), .J (ω0 , τ ), and .J (2ω0 , τ ). The term .J (0, τ ) = τ is the highest spectral density at the zero Larmor frequency, which shows that the first term relies on the static magnetic field. Also, Eq. (5.21) suggests that when .τ increases, .T2 decreases. To understand the concepts we have discussed so far, we consider the relaxation time of a pure water sample concerning the spectral density changes and the correlation time in Fig. 5.4a. We investigate the behavior of .T1 & .T2 by modifying .τ at 3 T. We describe the proton–proton interaction in Fig. 5.4b. The correlation time between two proton molecules is shorter compared with .1/ω0 (.τ 1, we have .T1 >> T2 , with a slow tumbling rate related to the large molecule. (b) The diagram shows the proton–proton interaction. The correlation time between two water molecule protons is smaller than .1/ω0

5.8 The Solomon–Bloembergen–Morgan Theory

101

In proton–proton interaction, the relaxation time is dominated by other nearby protons in the sample. Therefore, the number of protons affects .T1 . In Fig. 5.4a, the minimum point on the .T1 curve is given by the inverse of the Larmor frequency, corresponding to the maximum point in the spectral density. The plot illustrates that we have a fast tumbling rate when .ω0 τ > 1, one gets a slow tumbling rate for large molecules in the system. We also consider the effect of .ω0 τ on .T2 . According to the plot, .T2 is a decreasing function of .ω0 τ . For .ω0 τ > 1 we have .T1 >> T2 , where .T2 continuously decreases.

5.8 The Solomon–Bloembergen–Morgan Theory In 1955, Solomon generalized the BPP theory to a theory which is called the Solomon–Bloembergen–Morgan (SBM) theory [19]. Bloembergen and Morgan generalized the SBM theory to include electron spin relaxation in 1961 [103]. The relaxation rate by considering the proton–electron interactions are given by 1 ∝ γP2 γS2 [J (ωP − ωS ) + J (ωP ) + J (ωP + ωS )], T1 .

1 ∝ γP2 γS2 [4J (0) + J (ωP − ωS ) + 3J (ωP ) + 6J (ωS ) + 6J (ωP + ωS )], T2 (5.22)

where .ωP and .ωS are the proton and electron Larmor frequencies, respectively. Since the proton mass is much larger than the electron mass (we have .mP ≈ 2000 mS ), the magnetic moment of the electron is much more than the magnetic moment of the proton (.μS ≈ 2000 μP ). Therefore, an electron has a larger gyromagnetic ratio than a proton, which can be expressed as .γS ≈ 700 × γP from Eqs. (2.6) & (2.7). From Eq. (5.3), we observe that the dipole interaction energy between an electron and proton can enhance the interaction by about 700 times greater than that between two protons in the sample, where the dipole energy reads .E ∝ γP γS ≈ 700 γP γP . Considering the proton–electron interaction by the SBM theory from Eq. (5.22), the relaxation rate .1/T1 is about .490,000 greater than that between two protons, where we obtain .1/T1 ∝ 490,000 γP2 γP2 . We also can attain the rate .1/T2 for the proton– electron interaction as .1/T2 ∝ 490,000 γP2 γP2 . One of the most practical examples of the proton–electron interaction is the gadolinium (Gd) interaction with the proton sample. As we display in Fig. 5.5a, Gd with atomic number 64 has 8 unpaired electrons in .4f7 5d1 electron shells. After donating three electrons, we have .Gd+3 with seven unpaired electrons in .4f7 electron shell, as illustrated in Fig. 5.5a.

102

5 Molecular Motion, Correlation, and Relaxation Time

Fig. 5.5 (a) Scheme of the gadolinium (Gd) electron structure with eight unpaired electrons in 7 and .5d1 electron shells. After donating three electrons, .Gd+3 has 7 unpaired electrons in .4f7 electron shell. (b) Interaction between a proton in the water sample and Gd.+3 can deform the magnetic field over a larger space

.4f

Thus, Gd.+3 with seven unpaired electrons in orbital .f (.[Xe]4f7 ) can be an ideal element to add into the water sample to increase the magnetic moments of the sample significantly. Due to its size, .Gd+3 can deform the magnetic field in a larger space and raise the interaction rate. This shows the correlation time can reach +3 interactions, which results in a shorter .T and a .τ ≈ 1/ω0 in the proton-Gd. 1 brighter image. Therefore, Gd.+3 is known as .T1 contrast or enhancement agent utilized in clinical studies. We illustrate a scheme of the correlation of the protonGd.+3 interactions in Fig. 5.5b. +3 .Gd also is known as the most common contrast agent for recording a shorter .T2 relaxation time, resulting in a darker image. The spin signal decays rapidly in the proton–electron sample, meaning the precession frequency rate changes and forms the transverse relaxation time, where we record a shorter .T2 . It can be applied to many biological tissues, where .T2 is always shorter than .T1 . This means dephasing the magnetization is faster than re-phasing the magnetization in the biological sample. For instance, in a pure water sample, the spins tumble rapidly without dephasing, which means that .T2 ≈ T1 . In the solids sample, there is no significant

Exercises

103

Fig. 5.6 The chemical structure of gadobenate dimeglumine (.Gd+3 -BOPTA) and gadoterate meglumine (.Gd+3 -DOTA)

tumbling rate. Thus, .T2 is very small. Therefore, a solid sample cannot be imaged with MRI and NMR scanner. There are several gadolinium-based studies, such as gadobenate dimeglumine (.Gd+3 -BOPTA) and gadoterate meglumine (.Gd+3 -DOTA), which are used in clinical applications. In Fig. 5.6, we display the chemical structure of these wellknown gadolinium agents used in experiments.

Exercises Problem 1 For proton-nitrogen interactions (P-N), if the proton is .1.01 Å away from a nucleus.

15 N

.

(a) Plot .T1 versus .B-0 (from .0.06 T to 15 T), by considering .τ (.10−12 , .10−10 , .10−8 , and .10−7 s). (b) Plot .T2 versus .B-0 (from .0.06 T to 15 T), by considering .τ (.10−12 , .10−10 , .10−8 , and .10−7 s). (c) Plot .T1 and .T2 versus .B-0 (from .0.06 T to 15 T), by considering .τ (.10−8 s). Gyromagnetic ratio of nitrogen is .γ 15 N = −4.316 MHz .Tesla−1 .

104

5 Molecular Motion, Correlation, and Relaxation Time

Solution 1 (a) We plot T1 versus B-0 in Fig. 5.7a. For smaller correlation time, τ = 10−12 s (red) and τ = 10−10 s (blue), we observe T1 plot is constant. For larger correlation time, τ = 10−8 s (green) and τ = 10−7 s (black), T1 changes from the smaller magnetic field to the higher magnetic field. (b) We plot T2 versus B-0 in Fig. 5.7b. For τ = 10−12 s (red) and τ = 10−10 s (blue), there is no dependence on the magnetic field. For τ = 10−8 s (green) and τ = 10−7 s (black), we have a slight dependence of T2 on B-0 . (c) We plot both T1 & T2 versus B-0 at τ = 10−8 s (green) in Fig. 5.7c. As we can observe, at a low magnetic field, there is an overlap between the T1 and T2 curves. On the other hand, as the magnetic field increases, a separation occurs between the T1 and T2 curves, indicating that T1 is greater than T2 .

(a)

τ =10-12 s τ =10-10 s τ =10-8 s τ =10-7 s

(b)

102

T2 (s)

T1 (s)

102

100

100

τ =10-12 s τ =10-10 s τ =10-8 s τ =10-7 s

10-2 0

5

10

→

10-2 15

0

5

B0 (T) 100

(c) t =10-8 s T1, T2(s)

→

B0 (T)

10-1

T1 T2

10-2

10-1

100 →

B0 (T) Fig. 5.7 Problem 1 (a)–(b) and (c)

101

10

15

Exercises

105

Fig. 5.8 Problem 2

Problem 2 From the previous problem, consider the behavior .T1 and .T2 versus .ω0 τ for protonnitrogen interactions. In this part, .τ is from .10−12 to .10−5 s, and .B-0 is at 7 T, 3 T, and 1 T.

Solution 2 In Fig. 5.8, we plot .T1 (solid line) and .T2 (dash line) versus .ω0 τ for .B-0 being 7 T, 3 T, and 1 T. In the .T1 curve, the curve starts from a shorter correlation time, reaches the minimum point, and then increases. The .T1 curve has a minimum point at the position .ω0 τ ≈ 1. We observe a larger .T1 for 7 T. On the other hand, there is no minimum point for the .T2 curve, and the curve decreases continuously.

Problem 3 By considering the radius of the hydrogen and oxygen atom .r = rH = rO ≈ 0.6 Å (this length is in Angstrom, .Å) estimate the radius of a water molecule, which is named R in Fig. 5.9.

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5 Molecular Motion, Correlation, and Relaxation Time

Fig. 5.9 Scheme of the inside of water molecule, which contains two hydrogen atoms and one oxygen atom

Solution 3 The water molecules contain two hydrogen atoms and one oxygen atom, as shown in Fig. 5.9. To calculate the radius of the water molecule, we model the water molecule as a sphere, where its radius, R, from Fig. 5.9 can be written as R = r + x.

.

(5.23)

Also, .r is the radius of a hydrogen or oxygen atom, and x is the distance from the center of the water molecule to the center of the hydrogen or oxygen atoms, which is given by .x = √2 r. Therefore, R is written as 3

( R=

.

) ( ) 2 2 √ + 1 · r = √ + 1 · 0.6 Å ≈ 1.3, Å. 3 3

(5.24)

Chapter 6

Chemical Shift, NMR Spectroscopy, and Beyond

6.1 Introduction The essential of NMR technology comes from the fundamental concept of quantum spin. The NMR technique can identify an unknown sample using the nuclear spin properties. In this chapter, we introduce NMR spectroscopy for investigating various atomic isotopes with a net nuclear spin. This technique can help to understand the nuclear precession and any record of the electron densities in the complex molecules. We introduce the chemical shift and NMR sensitivity of different atomic isotopes in this chapter. We also discuss spectroscopy techniques for NMR and MRI systems. We consider various spectroscopy methods and their specific applications in medical studies. In particular, we discuss .1 H, .13 C, .31 P, .15 N, .19 F, and .129 Xe isotopes.

6.2 Electron Density Shielding The magnetic field experienced by a nucleus in a molecule is affected by the distribution of electrons around it. Electrons generate their own magnetic fields and interact with the magnetic field of the nucleus, resulting in a shift in the resonance frequency of the nucleus. This shift is proportional to the strength of the external magnetic field provided by the electrons and is known as the chemical shift [104, 105]. The chemical shift is denoted by .δ and is usually expressed in parts per million (ppm). The chemical shift can determine molecules’ structure and functional groups in the NMR spectrum signal. As we present in Fig. 6.1, all protons are surrounded by electrons that shield them from the magnetic field. Therefore, a higher electron density around the nucleus can induce a more significant shielding effect. In other words, the amount of shielding is directly associated with the electron density © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Khashami, Fundamentals of NMR and MRI, https://doi.org/10.1007/978-3-031-47976-2_6

107

108

6 Chemical Shift, NMR Spectroscopy, and Beyond

Fig. 6.1 (a) Scheme of the proton in the hydrogen atom subjected to an external magnetic field, B-0 . (b) The diagram represents the electron cloud surrounding the hydrogen atom, and the nucleus feels an induced current by applying B-0 . The induced magnetic field, B-ind , is opposite in the direction to B-0 . The yellow sphere shows the proton in the hydrogen atom, and the blue cloud shows the electron density around the proton in the hydrogen atom

around .1 H nucleus. Thus, the higher shielded electrons need a higher external magnetic field to perform resonance [55, 83]. According to Lenz’s law, changing the magnetic field can induce a current in the magnet coil that surrounds the sample, which is directed to oppose the induced current. The local field around the nucleus is called the effective magnetic field, .B-eff . -eff experienced by a nucleus is the sum of .B-0 and the induced magnetic field, .B-ind . .B Therefore, .B-ind is in the opposite direction to .B-0 , where the electron density of .1 H atom influences .B-0 . In this description .B-ind is given by B-ind = −σ B-0 ,

.

(6.1)

where .σ is the shielding constant. .B-eff is generally less than .B-0 and can be expressed as B-eff = B-0 + B-ind .

.

(6.2)

6.2 Electron Density Shielding

109

Fig. 6.2 Scheme of the shielding and de-shielding in an external magnetic field. From the Zeeman effect, the energy difference between two states is .AE = h¯ γ B0 . When .B-eff is less than .B-0 , the nucleus becomes shielded, and when .B-eff is more than .B-0 , it becomes de-shielded

By replacing .B-ind , we have B-eff = B-0 − σ B-0 .

.

(6.3)

Thus, the dimensionless shielding constant, .σ , is determined by σ =

.

B-0 − B-eff . B-0

(6.4)

As we illustrate in Fig. 6.2, when .B-eff is less than .B-0 , the nucleus is shielded, a process known as chemical shielding. Alternatively, when .B-eff is much larger than .B-0 , the nucleus is considered de-shielded. In the shielding part, the electron density circulates about the direction of .B-0 and opposes the external magnetic field, reducing .B-eff . As we noted earlier, the nucleus in different local environments experiences a slightly different Larmor frequency from .ω0 , resulting in the effective Larmor frequency, .ωeff , defined by ωeff = γ B-eff = γ B-0 (1 − σ ).

.

(6.5)

Considering the chemical shift of the sample, .δs , with the resonance Larmor frequency, .ωs , is written as δs ≈

.

ωs − ωref , ωref

(6.6)

where .ωref is the Larmor frequency of an internal reference. Therefore, the frequency difference between the sample and reference can define the chemical shift

110

6 Chemical Shift, NMR Spectroscopy, and Beyond

of the sample in ppm [39, 52]. The chemical shift in ppm for the sample can be converted to Hz according to δs (ppm) = 106 ×

.

ωs − ωref . ωref

(6.7)

If we consider two different samples in an external magnetic field, they experience different Larmor frequencies .ω1 and .ω2 of samples 1 and 2, respectively. We can attain the chemical shift difference between the two samples as Aδsample = δ2 − δ1 ω2 − ωref ω1 − ωref − 106 × ωref ωref ω2 − ω1 = 106 × ωref

= 106 × .

= 106 ×

(6.8)

Aωsample , ωref

where .δ1 and .δ2 are the chemical shifts of samples 1 and 2, respectively. Now, multiplying both sides of Eq. (6.8) by .ωref , we have Aωsample = ω2 − ω1 = 10−6 × ωref × (δ2 − δ1 ).

.

(6.9)

Generally, from Eq. (2.15), we can write AE ∝ Aω ∝ Aδ.

.

(6.10)

The difference between the energies of the two states is in terms of differences in the Larmor frequencies. Also, the difference between the two frequencies is in terms of the difference between the chemical shifts [1, 21, 22].

6.3 The Chemical Shifts of Fat and Water Molecules An appropriate example in an NMR experiment is to consider the different environments around fat and water molecules, which come from their diverse structures. As we present in Fig. 6.3a, the oxygen atom in the water molecule reduces the electron shielding around the hydrogen nucleus. This is due to the high electronegativity of the oxygen atom. This fact highlights the significant impact of the magnetic field on the hydrogen atom [106–109]. On the other hand, as shown in Fig. 6.3b, hydrogen atoms are settled within a long carbon chain in a fat molecule. The carbon atom is less electronegative than the oxygen atom. Therefore, the carbon atom has no significant effect on the hydrogen

6.3 The Chemical Shifts of Fat and Water Molecules

111

Fig. 6.3 Scheme of the structures of water and fat molecules. (a) The water molecule contains two hydrogens and one oxygen atom. The oxygen atom can attract electron density around the hydrogen atom, leaving the hydrogen nucleus less shielded. (b) The fat molecule contains hydrogen, oxygen, and a long carbon chain. The carbon atom is less electronegative than the oxygen atom. Thus, the hydrogen atom in the fat is more shielded than the hydrogen atom in the water molecule. The white sphere represents the oxygen atom. The yellow sphere is the hydrogen atom. The small pink sphere is the electron atom. The red sphere is the carbon atom. The dashed line shows the electron shielding. (c) Due to their different structures, the water and fat are detected at different ppm in the NMR spectrum

atom’s electron density. Hence, the nucleus remains shielded from the impact of the magnetic field. Thus, the Larmor frequency of hydrogen atoms in water is higher than hydrogen in fat molecules. Therefore, fat and water molecules have different NMR spectrums shown at separated ppm values. To illustrate, we consider the frequency difference between the hydrogen atoms of fat and water molecules at 3 T and .1.5 T magnetic fields with the Larmor frequencies of 128 and 64 MHz, respectively. The difference in chemical shift between fat and water molecules from the experiment is .3.5 ppm [110]. Therefore, the frequency differences from Eq. (6.9) at 3 T are written as Aω = (128 × 106 ) × (3.5 × 10−6 ) = 448 Hz.

.

(6.11)

Moreover, as depicted in Fig. 6.3c, the fat and water frequency differences are 224 Hz at .1.5 T. The relation indicates that the difference between the two peaks at 3 T is more significant than that at .1.5 T. Therefore, by increasing the magnetic

112

6 Chemical Shift, NMR Spectroscopy, and Beyond

field, the Larmor frequency enhances. Thus, the distinctions in frequency between carbon and oxygen in hydrogen atoms increase.

6.4 Electron Shielding for Organic Chemicals To further explore the chemical shift concept, we discuss the examples of electron shielding for two important organic molecules. We introduce two types of organic compounds that are placed in an external magnetic field. We represent the electron shielding of Alkenes in Fig. 6.4 and Alkyne molecules in Fig. 6.5 [42]. We briefly consider the effects of the magnetic field on the electron clouds around the hydrogen and carbon atoms of Alkene and Alkyne molecules and compare their electron shielding [47, 111]. An Alkene molecule is an unsaturated structure of hydrocarbons that includes double bonds between carbon atoms, which is called the Pi bond, as shown in Fig. 6.4. Also, the Alkene molecule has two hydrogen atoms on each side, called olefinic hydrogen atoms. When an Alkene molecule is exposed to an external magnetic field, the electron density tends to induce the magnetic field that opposes the external magnetic field at the Pi bond portion, where we arrive at .B-eff = B-0 − B-ind . Thus, the effective magnetic field decreases around the Pi bond and becomes less than .B-0 .

Fig. 6.4 Scheme of the electron shielding of Alkene molecule. An Alkene molecule has a double bond between carbon atoms in the structure. The red sphere shows the carbon atom, and the hydrogen atom is in the yellow sphere. The blue cloud shows the electron cloud around the atom or sigma bond. The green color shows the Pi bond, which has two electron density clouds above and below the molecule. Also, the Alkene molecule is called the Pi bond effect

6.5 NMR Sensitivity and Receptivity

113

Fig. 6.5 Scheme of the electron shielding in the Alkyne molecule. Alkyne molecules have at least one carbon–carbon triple bond in the chemical structure. The red sphere shows the carbon atom, and the hydrogen atom shows the yellow sphere. The blue cloud shows the electron cloud around the atom or sigma bond

On the other hand, the olefinic hydrogen atoms are present on each side of an Alkene molecule. The induced magnetic field is created in the same direction as -0 at that area. Thus, .B-eff for olefinic hydrogen atoms is the sum of the external .B magnetic field and induced magnetic field, where we have .B-eff = B-0 + B-ind . The effective magnetic field increases at the position of the olefinic hydrogen atoms, which is greater than .B-0 [112]. An Alkyne molecule is another unsaturated hydrocarbon containing at least one carbon–carbon triple bond that we show in Fig. 6.5. The situation for hydrogen atoms attached to carbon–carbon triple bonds differs from an Alkene molecule. Unlike Alkene molecules, hydrogen atoms are in the same direction as an external magnetic field in this case. In other words, hydrogen atoms are in the same direction as each carbon atom. Thus, when an external magnetic field is applied, hydrogen atoms induce their magnetic field, opposing .B-0 . Therefore, the induced magnetic field decreases the effective magnetic field at the position of the hydrogen atoms. As a result, .B-eff is smaller than .B-0 in this molecule [48].

6.5 NMR Sensitivity and Receptivity NMR sensitivity is an essential concept that determines the performance of the NMR system. NMR signal is more sensitive when there is a higher gyromagnetic ratio, stronger magnetic field, and lower temperatures. The sensitivity of NMR relies on the spin population in the energy levels. From the Boltzmann equation, the more

114

6 Chemical Shift, NMR Spectroscopy, and Beyond

significant the energy difference, the more separation in spin populations and, thus, the higher NMR sensitivity [106–109]. Intrinsic NMR sensitivity, known as receptivity, is another important concept that needs to be considered. The receptivity is related to the number of protons in the sample and is proportional to the gyromagnetic ratio. Therefore, we have Receptivity ∝ γ 3 I (I + 1),

(6.12)

.

where .γ 3 is linked to the sensitivity of the spin system. Considering the natural abundance and sensitivity, the receptivity of a nucleus is calculated by Receptivity = Sensitivity × Natural abundance.

.

(6.13)

In Table 6.1, we compare the gyromagnetic ratio of .1 H atom (.γ 1 H) with the gyromagnetic ratio of .13 C atom (.γ 13 C). We also consider the gyromagnetic ratio’s effect on the nucleus sensitivity. To have an equal number of nuclei, we calculate the sensitivity and receptivity related to .13 C nucleus [1, 32]. From the above table, we calculate the receptivity of .1 H atom over .13 C atom as

.

Receptivity 1 H = Receptivity 13 C

(

γ 1H γ 13 C

)3

( ×

) Nat.Abd(%) 1 H , Nat.Abd(%) 13 C

(6.14)

where from the gyromagnetic ratio of .1 H atom and .13 C atom we have .

42.58 γ 1H = ≈4 10.7084 γ 13 C

(6.15)

The ratio indicates that .1 H atom is about 64 as sensitive as .13 C nucleus, as the effect of the gyromagnetic ratio. Also, because the natural abundance of .1 H atom is about .90.9 times larger than .13 C atom, the receptivity of .1 H atom is .≈.5.82 .×103 times larger than that of .13 C atom. Table 6.1 Properties of .1 H and .13 C nucleus with the spin quantum number .1/2 at a constant magnetic field. The sensitivity and receptivity numbers are calculated over .13 C nucleus Nucleus 1 . H 13 . C

(MHz Tesla−1 ) .42.58 .10.7084 .γ

Nat. Abd (%)

Sensitivity

Receptivity

.100.0

.6.4 .× .10

.5.82 .× .10

.1.11

1

1

1

3

at 3 T (MHz) 128 . 32 .ω0

6.6 Proton-NMR Spectroscopy

115

6.6 Proton-NMR Spectroscopy In this section, we discuss the properties of proton-NMR spectroscopy (.1 H-NMRS) for medical studies. .1 H-NMRS is an important technique to determine different types of hydrogen atoms that are presented in the sample. .1 H-NMRS is a practical method for knowing metabolic pathways in living cells. Moreover, it can be used to analyze various kinds of treatments. For instance, .1 H-NMRS is a beneficial procedure for investigating new drug therapies [113]. .1 H-NMRS is a non-invasive method to detect metabolites and monitor dynamic metabolism in the brain and nervous system [113–115]. Also, it can be used as a diagnostic tool for children and infants [115–117]. Due to .1 H atom being an abundant isotope in human bodies, .1 H-NMRS is the primary part of medical imaging in human studies [61, 118]. An NMR spectrum shows the intensity of a signal as a function of chemical shift, measured in ppm. .1 H-NMR spectrum presents many peaks from 0 to 14 ppm. In this technique, the biological sample is dissolved in a solvent as the .1 H-NMR reference. As an example, we represent deuterium oxide (.D2 O) in Fig. 6.6a.

Fig. 6.6 Scheme of the standard internal reference for different types of NMR spectroscopy. (a) Deuterium oxide (.D2 O) for .1 H-NMR, (b) TMS (.Si(CH3 )4 ) for .13 C-NMR, (c) phosphoric acid (.H3 PO4 ) for .31 P-NMR, (d) nitromethane (.CH3 NO2 ) for .15 N-NMR, (e) trichlorofluoromethane (.CFCl3 ) for .19 F-NMR, and (f) xenon oxytetrafluoride (.XeOF4 ) for .129 Xe-NMR

116

6 Chemical Shift, NMR Spectroscopy, and Beyond

As we mentioned, analyzing the NMR spectrum provides information about the number of signals and their positions in the sample. The data attained from the NMR spectrum shows the number of hydrogen atoms that yield the peaks and their signal intensity. The NMR signal can show the number of nuclei interacting with neighboring hydrogen atoms, known as spin splitting. Usually, NMR signals present as groups of peaks. These peaks show multiple signals because of the effects of neighboring atoms. In an NMR spectrum, the distance between two peaks is determined by .J coupling. The .J -coupling delivers details in a splitting pattern related to chemical bonds and the amount of interaction between two hydrogens in the sample, according to the .n + 1 rule, where n is the number of protons in the nearby nuclei. In Fig. 6.7, we introduce various splitting patterns by the .n + 1 rule. The origin of the .n + 1 rule comes from the spin properties and their interaction with the surrounding spins. Considering the effect of an external magnetic field on a proton and its surrounding spins, we can investigate the proton energy splitting in the system. Each neighboring proton, denoted as n, involves energy splitting, resulting in .n + 1 distinct peaks in the NMR spectrum. For example, when no hydrogen atom is around the nuclei, the splitting signal from the .n + 1 rule is a singlet. A singlet signal means the nuclei are not coupled to

Fig. 6.7 Scheme of the splitting patterns in NMR spectroscopy. The diversity of the signal is calculated by the .n + 1 rule. Singlet signals are not coupled to any protons. Doublet signals are coupled to one proton. Triplet signals are associated with two protons. Quartet signals are next to three protons. The coupling constant, .J , is the distance between the peaks in a splitting pattern. Note that .Ha and .Hb are hydrogen nuclei

6.8 Phosphorus-NMR Spectroscopy

117

neighboring protons. Similarly, in doublet signals, the nuclei are connected to one proton. Moreover, in triplet signals, the nuclei are related to two protons, and so on. Therefore, if we have a complex or a simple molecule in the sample, the number of protons around the molecule can explain the signal output. Each group of proton connections can absorb different frequencies and produce a specific chemical shift along the x-axis in ppm. Moreover, each group of protons in a molecule communicates with adjacent nuclei or different types of protons by spin–spin coupling, which causes the splitting pattern of the signal.

6.7 Carbon-NMR Spectroscopy NMR is also a practical method for analyzing carbon-based chemical samples, such as living systems and organic compounds. Only about .1.11% of natural carbon atoms have .1/2 quantum spin number with .γ = 10.7084 MHz .Tesla−1 , as we present in Table 6.1. Due to the lower gyromagnetic ratio, .13 C nucleus is less sensitive than .1 H-NMR. The .13 C-NMR sample should be mixed with a solvent as a reference compound containing carbon such as TMS (.Si(CH3 )4 ) at .0.5% concentration that makes protons shielded, as we present in Fig. 6.6b. Compared with .1 H-NMR spectrum, .13 C-NMRS peaks are spread from 0 to 200 ppm, which assists in the detection of distinct peaks with a low possibility of signal overlapping in NMR spectroscopy. 13 . C-NMRS tells us about the number of carbon and the kind of carbon we have at a specific chemical shift in the sample. Moreover, by considering .13 C-NMRS, we can find the carbon–carbon and the carbon–proton couplings in the sample, and we can distinguish different splitting patterns in the molecule. 13 13 13 . C-NMRS with . C-labeled substrate is a basic technique utilized to trace . C metabolism in the human body. The main substrates used for in vivo or clinical studies are .13 C glucose [119–122], .13 C glutamine [123–125], and .13 C alanine [122, 126, 127], which are the primary sources of energy in the brain, heart, and various types of cancer diseases.

6.8 Phosphorus-NMR Spectroscopy The .31 P nucleus has .100% natural abundance with .γ = 17.235 MHz .Tesla−1 , which behaves like .1 H-NMRS, as we present in Table 6.2. However, .31 P chemical shifts cover a broader spectrum compared with .1 H-NMRS. To attain .31 P-NMRS signal, the NMR sample should be combined with a solvent that contains phosphorus, such as phosphoric acid (.H3 PO4 ), as is shown in Fig. 6.6c. Moreover, .31 P-NMRS has less sensitivity compared with .1 H-NMRS, and its peak is sharper than the peak of 13 . C-NMRS.

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6 Chemical Shift, NMR Spectroscopy, and Beyond

Table 6.2 Summary of various nuclei active in NMR spectroscopy (NMRS) methods and their physical properties. Note that the sensitivity and receptivity are calculated over .13 C nucleus Nucleus

.γ

(MHz Tesla−1 )

13 . C

.10.7084

31 P

.17.235

15 N

.−4.316

. .

19 . F 129 Xe

.

.40.078 .−11.777

Nat. Abd (%) .1.11 .100.0 .0.37 .100.0 .0.26

Sensitivity

Receptivity

1

1

.4.2

.3.8 .× .10

at 3 T (MHz)

32 2

−2

.ω0

.51.7

−2

.−12.9

−1

.−35.3

.−6.5 .× .10

.−2.2 .× .10

1 .5.2 .× .10

3 .4.8× .10

.−1.3

.−3.1 .× .10

.120.2

The high natural abundance and large .γ make .31 P-NMRS ideal for studying different systems, including phospholipid liposomes [128], investigating different food characteristics in food science like milk, green tea, and other foods [129, 130]. It is also helpful in detecting cancer tumors in in vivo models, such as breast cancer [131] and lung cancer [132]. Recently people have been interested in learning more about energy metabolism in the human brain at high magnetic field .9.4 T [133, 134] and investigated the difference between gray and white matter metabolite concentrations using phosphorous-MRS.

6.9 Nitrogen-NMR Spectroscopy 15 N-NMRS is an effective instrument for providing valuable information about sam-

ples that include nitrogen atoms like RNA/DNA nucleobases [135], the molecular structure of organic compounds [136], and soil organic matter [137]. 15 N-NMRS is much less sensitive compared with 1 H-NMRS and 13 C-NMRS, due to the low natural abundance (37%) of 15 N nucleus and its small γ value (γ = −4.316 MHz Tesla−1 ), as we present in Table 6.2. Moreover, 15 N-NMRS chemical shift is significantly wide compared with 13 C and 1 H-NMRS, ranging from 0 to 900 ppm. In NMR studies with 15 N atom, nitromethane (CH3 NO2 ) is utilized as a standard external reference, as demonstrated in Fig. 6.6d.

6.10 Fluorine-NMR Spectroscopy 19 F-NMRS is a useful instrument for investigating several amino acids, nucleotides,

and sugars which contains fluorine. The 19 F nucleus has about 100% natural abundance with large γ = 40.078 MHz Tesla−1 , which has high sensitivity and broader chemical shift dispersion (varying from −300 to 400 ppm) compared with 1 H-NMRS, as we present in Table 6.2. Therefore, 19 F-NMRS is a reasonable method to examine important compounds of fluorine-containing pharmaceuticals

Exercises

119

[138, 139]. In 19 F-NMRS, Trichlorofluoromethane (CFCl3 ) is usually used as a typical reference. The structure of CFCl3 is displayed in Fig. 6.6e.

6.11 Xenon-NMR Spectroscopy The .129 Xe atom has .1/2 quantum spin with a natural abundance of .26% with −1 .γ = −11.777 MHz .Tesla , as we present in Table 6.2. The .129 Xe atom has medium sensitivity nuclei with a vast chemical shift, ranging from .−400 to 5400 ppm related to gas, which aids in identifying molecular samples and distinguishes different tissues signal. .XeOF4 or .Xe gas at low pressure is used as an NMR reference. The structure of .XeOF4 is shown in Fig. 6.6f.

Exercises Problem 1 The mass of .14 N is about .14.003 AMU (first isotope), .15 N is about .15.000 AMU (second isotope), and the average atomic mass of nitrogen is about .14.007 AMU. What is the percentage of .15 N natural abundance?

Solution 1 The sum of these two isotopes equals 100% of all the elements found in nature. Therefore, we can write Mass1 (x) + Mass2 (1 − x) = Massave .

.

(6.16)

Mass1 is about .14.003 AMU, .Mass2 is about .15.000 AMU, .Massave is about .14.007 AMU, and x is unknown relative abundance. Therefore, we put the numbers in Eq. (6.16), which is given by .14.003(x)+15(1−x) = 14.007. Then, x is calculated as .x ≈ 0.996 AMU. The percent abundance for .14 N is .x ≈ 99.6% and .15 N is .(1 − x) ≈ 0.004 ≈ 0.4%. .

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6 Chemical Shift, NMR Spectroscopy, and Beyond

Fig. 6.8 Problem 2

Problem 2 Represent the receptivity of .1 H, .15 N, .129 Xe, .3 He, .19 F, and .31 P compared with .13 C as a bar plot. Also, calculate the ratio of the receptivity of xenon and helium over carbon. Note: Gyromagnetic ratio of .3 He is .−32.43 MHz .Tesla−1 , and the natural abundance is .0.000137%.

Solution 2 In Fig. 6.8a, the bar plot of .1 H, .15 N, .129 Xe, .3 He, .19 F, and .31 P relative to .13 C is illustrated. Note that we consider the receptivity compared with .13 C atom at the constant magnetic field and the room temperature. We plotted the receptivity of .15 N, .129 Xe, and .3 He atoms compared with .13 C’s receptivity in Fig. 6.8b. If we compare xenon and carbon receptivity, we can calculate the ratio of xenon over carbon as .

)3 ( ) Nat.Abd(%)129 Xe = −1.3×0.24 = −0.31. × Nat.Abd(%)13 C (6.17) Furthermore, the ratio of helium over carbon is written as

Receptivity129 Xe = Receptivity13 C

(

γ 129 Xe γ 13 C

Receptivity3 He = . Receptivity13 C .

(

γ 3 He γ 13 C

)3

( ×

Nat.Abd(%)3 He Nat.Abd(%)13 C

= −28 × 1.2 × 10−4 = −3.5 × 10−3 .

)

(6.18)

Chapter 7

Spin-Echo and Spin-Lock Pulse Sequences in MRI System

7.1 Introduction So far, we have investigated the physics behind the NMR technology and its application and properties in biological sciences. Now, we shift our focus to understanding magnetic resonance imaging (MRI) as an efficient medical imaging technique established based on NMR technology. Using powerful magnetic fields and radio waves, MRI makes it possible to visualize the tissues in the human body with a high signal-to-noise ratio, which is very useful for medical applications. This chapter explores some essential elements of the MRI technique. We mainly focus on the spin-echo and spin-lock pulse sequences. We will utilize our knowledge of quantum mechanics, materialized so far, to derive the density matrix needed for developing a spin-echo pulse sequence which helps to improve our understanding of this essential technique. Also, we consider the behavior of spin magnetization from a quantum mechanics perspective for different biological samples.

7.2 MRI Scanner The human body comprises .70% water, containing hydrogen and oxygen atoms. The hydrogen nuclei are the most abundant atoms in the human body. These nuclei are used in magnetic resonance imaging (MRI), a non-invasive imaging technique, to visualize nuclei of atoms inside the body and generate powerful signals [36, 90]. MRI can record high-resolution images of soft tissues like the brain, heart, and cancer tumors as a diagnostic method. The most significant advantage of MRI is its high sensitivity due to the strong magnetic field used in this technology. MR imaging relies on the main magnetic field, which is typically a constant field as strong as .1.5–.9.4 T for medical usage. However, it is worth mentioning that higher field strengths, such as 7 and .9.4 T, are © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Khashami, Fundamentals of NMR and MRI, https://doi.org/10.1007/978-3-031-47976-2_7

121

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7 Spin-Echo and Spin-Lock Pulse Sequences in MRI System

Fig. 7.1 Scheme of the magnetic resonance imaging (MRI). The direction of the main magnetic field is along the z-axis. Three gradient coils in the .x-, .y-, and z-axis are known as the X, Y, and Z gradients. Moreover, the RF receiver coil is placed closest to the human body or the subject

often used for research purposes, while clinical MRI scanners commonly operate at 1.5 T or 3 T. In Fig. 7.1, we depict various parts of the MRI scanner. A crucial part of the MRI scanner is the radio frequency (RF) coil, which is closest to the body and transfers the RF waves into the human body or the subject for spin excitation and receives signals from the body to yield an image on the screen. The RF coil is designed to enhance the signal-to-noise ratio, which in turn allows better diagnostic MR images to be generated. To comprehend the RF wave frequency, we demonstrate the range of the electromagnetic (EM) spectrum from low energy levels, radio waves, to high energy levels, known as X-ray and gamma waves, in Fig. 7.2. As shown in Fig. 7.2, the ultraviolet (UV), X-rays, and gamma rays are in the range of .1016 –.1024 Hz as a radiated energy in the EM spectrum. These rays have enough energy to ionize atoms. The ionizing radiation may damage systems like DNA and can cause specific cancer and other health issues due to the interaction with living cells. On the other hand, the RF waves are in the range of .106 –.108 Hz, and they are known as non-ionizing radiation. Therefore, due to the small frequency of the RF waves, they do not have severe effects on tissues [140, 141]. However, this does not mean the RF wave is entirely harmless to the human body. In MRI, an RF pulse at the specific frequency of the hydrogen nucleus must be applied to excite the hydrogen atom. The rest of the MRI active nuclei do not get

.

7.2 MRI Scanner

123

Fig. 7.2 Scheme of the spectrum of electromagnetic (EM) radiation. The regions include radio waves, microwaves, infrared, light, ultraviolet (UV), X-rays, and gamma rays. The UV, X-rays, and gamma rays are in the range of .1016 –.1024 Hz as ionizing radiations. The RF waves are in the range of .106 –.108 Hz as non-ionizing radiation, which is used in an MRI scanner

excited with the hydrogen frequency. This is due to the fact that their gyromagnetic ratios differ from the hydrogen atom’s frequency. As a reminder for the reader, in Fig. 7.3a–d, we provide a summary of the process steps once the RF pulse is applied in the spin system. When a .90◦ RF pulse is applied along the .x ' -axis in the rotating frame, the magnetization .Mz from the equilibrium point rotates to the .y ' -axis. The magnetization vector .Mxy precesses in the transverse plane, and the signal decays exponentially. The magnitude of .Mxy determines the strength of the detected signal, which records the FID signal. Therefore, the RF coil receives generated signals in the MRI machine to create images by the FT method on the detector computer. Another part of the MRI scanner is related to the three gradient magnetic fields along the .x-, .y-, and z-axis. These gradient fields are weaker than the main magnetic field, .B-0 . The gradient magnets develop a magnetic field over .B-0 in the .x-, .y-, and z-axis, as shown in Fig. 7.1. The gradient field is expressed as .B(x, y, z), which produces the precession frequency .ω(x, y, z), which is a function of data location. The gradient magnetic field is added to the main magnetic field at the data location, which can help to excite the spins at the specific location and generate the MR image. The three gradient coils are related to a specific MRI slice, called axial, coronal, and sagittal slices. The data from the Z gradient that runs along the long axis can create the axial slices from the head to the foot of the subject. The data from the Y gradient can produce coronal slices from the front to the back of the subject. Finally, the sagittal slice from the X gradient can generate an image from the left to the right side of the subject [142, 143].

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7 Spin-Echo and Spin-Lock Pulse Sequences in MRI System

Fig. 7.3 A summary of the process after applying an RF pulse in an MRI system. (a) An RF pulse with flip angle .90◦ along the .x ' -axis is applied to .Mz at the equilibrium point. (b) .Mz rotates to the .y ' -axis and starts to lose the coherence. (c) The scheme of the RF pulse in the time domain. The excitation pulse is not a perfect rectangle shape in practice. It can be performed by a pulse envelope shape. (d) The exponential decay of the FID signal from (b)

7.3 Spin-Echo Pulse Sequence in MRI System One of the essential elements in understanding MRI signals is the pulse sequence setup. Pulse sequence enables us to control how the system applies the RF pulses. This way, the quality of a series of images can be improved. The spin-echo pulse sequence can be critical in constructing a high-quality MR image. The MRI scanner can produce an image by repeating the RF pulse several times. In this section, we consider the dynamics of a spin-.1/2 system, where the magnetization vector is subjected to one or more RF pulses. The spin-echo pulse sequence refers to a .90◦ RF pulse followed by one or more ◦ .180 RF pulses to the spin system. This process is shown in Fig. 7.4(1)–(4). In Fig. 7.4(1), we display two essential parameters in the spin-echo pulse sequence, which are the repetition time, denoted as .TR , and the time to echo, denoted as .TE .

7.3 Spin-Echo Pulse Sequence in MRI System

125

Fig. 7.4 Scheme of the spin-echo pulse sequence. (1) The diagram represents a 90◦ RF pulse at t = 0 and t = TR and one or more 180◦ RF pulses from t = TE /2. (2) The Mz vector reduces from the start point, M0 , and grows along the z-axis until the next 90◦ RF pulse at t = TR . (3) The longitudinal magnetization flips back into the x − y plane and starts to dephase and record T2∗ decays and produce the FID signal. (4) The spin-echo signal measures at t = TE point

TR is a parameter to calculate the scan time, defined as the time between two .90◦ RF pulses given in milliseconds (ms). On the other hand, .TR also determines the .T1 relaxation time, which is the time that magnetization takes to reach its maximum point [138, 144]. If a .90◦ RF pulse is applied at each .TR , the magnitude of .Mz according to Eq. (4.9) at .t = TR is given by .

( ) T − R Mz (TR ) = M0 1 − e T1 .

.

(7.1)

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7 Spin-Echo and Spin-Lock Pulse Sequences in MRI System

As shown in Fig. 7.4(2) and (3), a .90◦ RF pulse is applied along the x-axis to the magnetization .Mz = M0 at .t = 0, which is turned the magnetization point .90◦ into the .x − y plane, from .Mxy = 0 at start point .t = 0. Then, .Mxy vector precesses at the .x − y plane and produces the FID signal, where the initial intensity of the signal is proportional to the magnitude of .Mxy . The FID signal decays exponentially, and the amplitude decreases over time. Another critical parameter in the spin-echo pulse sequence is .TE , which is the time between a .90◦ RF pulse and an echo-formed signal in the MRI sequence, where an echo signal occurs. .TE is related to magnetization decay during the signal formation, which is also recorded in milliseconds (ms). MRI results representing different tissues have different .T1 and .T2 relaxation times. Thus, the spin system experiences different .TR and .TE . By controlling the parameters in the spin-echo pulse sequence, we can determine .T1 and .T2 values for the specific anatomy [138, 144]. The FID signal reverses and recovers the magnetization vector by applying a ◦ .180 RF pulse at .t = TE /2. The magnetization relaxes along the .−z-axis and then returns to the .+z-axis until it reaches its original value .M0 . By applying another ◦ .180 RF pulse, the magnetization vector refocuses and forms a second spin-echo. Therefore, the echo signal is created at .t = TE , called the echo signal with .T2 decay rate. Repeating this process can build multiple spin echoes and accurately measure .T2 decay rate. From Eq. (4.20), .Mxy at .t = TE is written as Mxy (TE ) = Mxy (0)e

.

T

− TE 2

.

(7.2)

In a homogeneous field, the FID signal decays and records .T2 relaxation because of the interaction between the spins. On the other hand, due to a macroscopic magnetic field non-uniformity in the system, the magnetization vector decays in a time with .T2∗ rate [145, 146]. As we present in Fig. 7.4(4), in a non-uniform system, the spin’s magnetization experiences slightly different magnetic fields and frequencies, which causes the spins to dephase or become out of phase quickly. Therefore, the system decays faster than .T2 , which is determined by .T2∗ rate. We can say .T2∗ > T1 ), the spin magnetization has enough time to recover or reach the equilibrium point at a longer time. Thus, the magnetization fully recovers and reaches its initial value, .M0 , which does not provide .T1 image contrast for the tissue. This means we cannot distinguish two tissues at very long .TR . In a different situation, if .TR is close to .T1 (.TR ≈ T1 ), the spin magnetization does not have enough time to recover and reach its maximum value. From the diagram, we have the gray and white matters that can be distinguished on the image, resulting in image contrast or .T1 -weighted image. In Fig. 7.10, we consider the transverse magnetization changes for brain tissues in a low and high magnetic fields. The transverse magnetization curves decay exponentially with a time constant .T2 . The magnetization starts at .Mxy (0), its maximum value, and decays from its initial value in the .x − y plane according to

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7 Spin-Echo and Spin-Lock Pulse Sequences in MRI System

Fig. 7.9 The fraction of longitudinal magnetization for human brain tissues. The magnetization ratio .Mz recovers with .T1 relaxation time. We represent the effect of short and long .TR on magnetization behavior

Eq. (7.2). The plot also represents that a higher magnetic field decays faster than a lower magnetic field in brain tissues. We also show the effect of .TE parameter on magnetization changes in the figure, which provides insights into the influence of the echo time on the behavior of magnetization. If .TE is close to .T2 (.TE ≈ T2 ), the spin magnetization has enough time to decay and get to its .37% of the initial value. Therefore, as we display, the signal reaches its maximum decay for both magnetic fields and records a different signal contrast between the gray and white matters, where we can distinguish different brain parts. This is named .T2 -weighted image contrast. On the other hand, if .TE is too short (.TE > T2∗ . Therefore, we can say .T1ρ is sensitive to the system’s physical processes and molecular motion.

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7 Spin-Echo and Spin-Lock Pulse Sequences in MRI System

Fig. 7.13 Scheme of .T1ρ relaxation time compared with .T2∗ relaxation time. .T1ρ decays in a longer time than .T2∗ relaxation time

From Eqs. (5.18) and (5.21), we have the density function for .T1ρ as J (ω1 ) =

.

τ , 1 + ω12 τ 2

(7.12)

where .J (ω1 ) = J (γ B1 ) is associated with the Larmor frequency of the spin-locking pulse. Thus, .T1ρ from the Solomon equation is given by [150, 151] .

1 ∝ J (ω1 ) + J (ω0 ) , T1ρ

(7.13)

where .ω0 represents the Larmor frequency of the main magnetic field. If we suppose ω1 to be very small, we get

.

.

1 1 ∝ J (0) + J (ω0 ) ∝ . ω1 →0 T1ρ T2 lim

(7.14)

Exercises

137

According to the above relation, .T1ρ is equal to .T2 for very small spin-lock Larmor frequency .ω1 . Typically, .T1ρ is employed in the macromolecule dynamics system. This relaxation parameter delivers practical information about the interactions and motion of macromolecules, such as proteins and nucleic acids, by examining their relaxation behavior in the transverse plane. Also, the .T1ρ method demonstrates sensitivity to water–protein interactions, making it a valuable tool for analyzing articular cartilage [152]. This sensitivity arises from the impact of water–protein interactions on the relaxation dynamics of articular cartilage, allowing for the description of its structural and compositional properties. Recently, people have been interested in learning about the molecular motion properties of myocardial tissue [153, 154] by applying a spin-lock pulse sequence. This method shows the contrast between infarct and healthy myocardium significantly. Furthermore, .T1ρ is a practical method for various clinical applications like liver disease, Alzheimer’s and Parkinson’s disease, and musculoskeletal systems [155–157].

Exercises Problem 1 Suppose a .90◦ RF pulse with a duration time of 20 .μs is applied. The delay time between .90◦ and .180◦ is 100 .ms, and a duration time for .180◦ RF pulse is 20 .μs. Moreover, the delay time between the .180◦ RF pulse and the signal echo is 100 .ms. Calculate the total time it takes to create a spin-echo pulse sequence.

Solution 1 The sum of all steps in Fig. 7.4 gives us the time to complete one spin-echo pulse sequence. Total time = 20 μs + 100 ms + 20 μs + 100 ms = 200.004 ms.

.

(7.15)

Problem 2 −

t

According to Eq. 7.11, we have .MSL = M0 e T1ρ . What is the rate of change of .MSL at .t = 0 and .t = T1ρ ? How does the rate change between these two time moments?

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7 Spin-Echo and Spin-Lock Pulse Sequences in MRI System

Solution 2 The rate of change of .MSL is determined by the time evolution of .MSL which is given by .

t M0 − T1ρ dMSL =− . e dt T1ρ

(7.16)

At .t = 0, the rate is determined by .

( ) t M0 − T1ρ M0 dMSL =− = lim − e . t→0 dt t→0 T1ρ T1ρ lim

(7.17)

On the other hand, at .t = T1ρ , we have .

M0 −1 dMSL =− e . dt T1ρ

(7.18)

The rate has a negative value, and it starts from .−M0 /T1ρ and approaches zero at infinity with the exponential rate.

Chapter 8

Gradient-Echo Pulse Sequence in MRI System

8.1 Introduction So far, we have discussed some important elements of MRI technology. In particular, we have analyzed the spin-echo and spin-lock pulse sequences. In the present chapter, we discuss the gradient-echo pulse sequence. The gradient-echo pulse sequence is crucial for creating a high-quality MR image. Here, we investigate different parts of the gradient-echo pulse sequence properties and consider how this makes the generation of MR images possible. Also, we investigate gradient coils and their properties and introduce some practical coils which are used in experiments. Moreover, we briefly introduce the method to sample “.K-Space” of the MRI sequence to generate an MR image. Also, we introduce the signal-to-noise ratio of MR images and the formalism of the magnetization behavior in the MRI sequence.

8.2 Gradient Coils’ Properties Before introducing the gradient-echo pulse sequence, we need to discuss the gradient coils in the magnet. As we mentioned before, the MRI scanner contains three gradient coils to build a gradient field in three orthogonal directions, x-, y-, and z-axis inside the magnet bore [59, 158, 159]. The magnitude of the gradient fields is characterized as .Gx , .Gy , and .Gz , respectively. The unit of gradient field is −1 (Tesla per meter). .Tm

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Khashami, Fundamentals of NMR and MRI, https://doi.org/10.1007/978-3-031-47976-2_8

139

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8 Gradient-Echo Pulse Sequence in MRI System

Accordingly, the gradient field along the x-, y-, and z-axis from .∇B is given by dBz , dx dBz , . Gy = dy Gx =

Gz =

(8.1)

dBz . dz

ˆ Thus, the magnitude of Note that the magnetic field is .B- = Bx iˆ + By jˆ + Bz k. the total magnetic field is written as / Bx2 + By2 + Bz2 . (8.2) .B = The gradient field is expressed as the change of the magnetic field with respect to the change in distance, which is written as B-z = B-0 + Gx (t)x + Gy (t)y + Gz (t)z,

.

(8.3)

where .Gx (t) and .Gy (t) are the time-dependent transverse gradients along the xand y-axis, and .Gz (t) is the time-dependent longitudinal gradient along the z-axis. Also, .B-0 describes the main magnetic field strength along the z-axis, which is the static and uniform magnetic field yielded by the MRI scanner. For example, when we turn on the gradient field along the z-axis, the spin feels a new magnetic field in that direction, which is given, from Eq. (8.1), by B-z = B-0 + Gz (t)z,

.

(8.4)

where z is the slice thickness, as we present in Fig. 8.1. The precession frequency on spins at the specific slice thickness z is written as

Fig. 8.1 Scheme of the gradient coil along the z-axis. The .Gz is the magnitude of the gradient field along the z-axis. The frequency of the main magnetic field at slice thickness .z = 0 is .ω0 . The precession frequency of spins at slice thickness z is .ω(z)

8.2 Gradient Coils’ Properties

141

ω(z) = γ Bz = ω0 + γ Gz (t)z,

.

(8.5)

where .ω0 = γ B0 is the frequency of the main magnetic field at slice thickness z = 0. The slice thickness from Eq. (8.5) is expressed as

.

z=

.

Aω ω(z) − ω0 , = γ Gz (t) γ Gz (t)

(8.6)

where z is directly proportional to .Aω at a constant gyromagnetic ratio, measured in millimeters (mm). Similar equations can be written for the gradient’s x- and y-axis, .Gx (t) and .Gy (t), respectively. Note that the gradient fields can be adjusted in the MRI sequence by turning them on and off. The fast-changing gradient coils inside the magnet usually produce load sound in the MRI scanner experiment [141, 144]. Maxwell and Helmholtz’s coils are typically utilized in MRI systems to enhance and modify magnetic field characteristics. The Maxwell coil, named after the Scottish physicist James Clerk Maxwell [160], is used to improve magnetic field along the z-axis. It contains two loops with opposite current directions. If we add the gradient field along the z direction to the main magnetic field, as explained in Eq. (8.4), the total magnetic field increases linearly along the z-axis. Typically, Biot–Savart’s law can be written for the magnetic field of any wire magnet coil design at an arbitrary point P as - = μ0 i .B 4π

f

dS × rˆ , r2

(8.7)

where .r is the distance from the small element on the wire to the arbitrary point P . Also, .dS is the distance from the small element to the arbitrary point P , and .rˆ is the unit displacement vector from the small element on the wire to the arbitrary point. −7 VsA−1 m−1 . .μ0 is the permeability of free space, which is equal to .4π × 10 Accordingly, the Maxwell coil containing two loops of radius a is placed in distance d along the z-axis, where the current i is passed through each loop in the opposite direction, as we have shown in Fig. 8.2. The total magnetic field is given by 2 - = B-1 − B-2 = μ0 ia .B 2

[

] 1 1 − 2 , (z12 + a 2 )3/2 (z2 + a 2 )3/2

(8.8)

where .z1 = z − d/2 and .z2 = z + d/2. Therefore, the total magnetic field will be zero at the center of the Maxwell coil, where .z = 0, as shown in Fig. 8.2. In the plot, we present the magnetic field of each coil and get the maximum value at the center of each coil. On the other hand, Helmholtz coils, named after German physicist Hermann von Helmholtz [161], which consist of two loops with the same current direction, are widely used to produce a highly homogeneous main magnetic field. This type of

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8 Gradient-Echo Pulse Sequence in MRI System

Fig. 8.2 The Maxwell coil along the z-axis. Coils 1 and 2 with the opposite direction of current i are placed in distance d along the z-axis. All distances are calculated from point P as an arbitrary point along z

coil is necessary for proper MR imaging and spectroscopy signals, as it provides uniformity of the precession frequency of nuclear spins. The total magnetic field is the sum of the magnetic field for coil 1 and the magnetic field for coil 2. The distance between the two coils is d. Therefore, the total magnetic field is written as .B- = B-1 + B-2 . We have the total magnetic field as 2 - = B-1 + B-2 = μ0 ia .B 2

[

] 1 1 + 2 , (z12 + a 2 )3/2 (z2 + a 2 )3/2

(8.9)

8.2 Gradient Coils’ Properties

143

Fig. 8.3 The two-loop coil with the same current direction. (a) The magnetic field vs. distance for two loops with the same current direction is displayed. The radius of loops is .a = 0.1 m. The current in loops is .i = 1 A. The loops are separated by a distance d. (b) We display the total magnetic field as dependent on the distance between the two loops. For example, the black curve is for .d = a, the red curve is for .d < a, and the blue curve is for .d > a

where .z1 = z − d/2 and .z2 = z + d/2. In Fig. 8.3a, we display the magnetic field that can be maximized at the center of each coil. In Fig. 8.3b, we illustrate the total magnetic field that can be maximum in the middle of the two loops when the distance between the two coils is .d = a. At this point, we can record the maximum of the total magnetic field, where the field can spread in a broad range uniformly. In this case, we can use the Helmholtz coil sufficiently. In contrast, we show that when .d < a, the total magnetic field is still maximum, but its distribution spreads in a narrow range. On the other hand, for .d > a, the total magnetic field is not uniform. This configuration does not provide a homogeneous magnetic field, as the field power changes significantly across the area of interest.

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8 Gradient-Echo Pulse Sequence in MRI System

8.3 Gradient-Echo Pulse Sequence A gradient-echo pulse sequence is a collection of timing and stopping points, which can process the spin magnetization interactions with the gradient field. From the Zeeman effect, the gradients affect the hydrogen atom in the sample to feel various magnetic fields along the specific direction, which changes the Larmor frequency of the nuclei in the selected area. The MR image is a combination of magnetic resonance and the specific data location we select. We must combine the Larmor frequency difference and the required sample with the gradient field to create an MR image on the screen. Thus, the gradient field is applied to a specific region along the z-axis, which aligns the nuclei in that region in a particular direction. This process is known as slice selection (SS). Furthermore, the excited nuclei are encoded in the two-dimensional plane along the x- and y-axis, called frequency encoding (FE) and phase encoding (PE) [162, 163]. As depicted in Fig. 8.4, the slice selection obtains data from a specific slice and desired slice thickness by applying a magnetic gradient field during the RF pulse. During this process, only those spins whose precession frequencies correspond to the band of the RF’s frequency, called the RF pulse’s bandwidth, are excited and tipped into the transverse plane. Thus, the RF pulse energy transfers at a specific location called the transmit bandwidth of the RF pulse (tBW) [80, 141, 164].

Fig. 8.4 Scheme of the slice selection along the z-axis at the constant magnetic field. The RF pulse is applied to the spins to produce resonance in the sample. Only proton’s nuclei with the particular frequency precession are processed. Therefore, the excitation pulse happens to a specific region around the spectral width. This area can be selected for MRI sequence. The difference in Larmor frequencies for the selected area is .Aω. The difference in space along the z-axis is .Az

8.4 Converting Time Domain FID Signal into a Frequency Domain Signal

145

According to Eq. (8.6), the slice thickness is directly proportional to the difference in Larmor frequencies (.Aω) and inversely proportional to the gradient strength along the z-axis (.Gz ). We can build a thicker or a thinner slice by changing a gradient strength along the z-axis with the magnitude .Gz or changing the range of transmitted RF pulse into the subject, related to the Larmor frequency difference. Another aspect is considering the carrier frequency or the frequency of the RF pulse at the region of interest for a uniform excitation pulse. In this way, the RF pulse is subjected at the center of the area of interest, which can excite the samples uniformly, meaning the excitation pulse properly covers the slice selection area. Therefore, the spin magnetization of the sample is uniformly flipped into the .x − y plane, resulting in the output signal having similar intensity, which can enhance the signal-to-noise ratio value. How can we excite the spins if the Larmor frequency is not uniform in space? As we have learned, we have all excited spins out of phase in the nonuniform frequency, resulting in magnetization canceling out on average. Thus, we need to make the excited spin direction in phase. To do so, we can refocus all excited spins in one direction by addressing an opposite gradient field along the z-axis to all spins. This technique is called slice refocusing gradient. After the slice selection along the z-axis, we can encode or localize the spins along the second direction by applying a PE gradient. We illustrate these steps in Fig. 8.5(1)–(3). Instead of applying a gradient during the excitation or acquisition (.AQ) time, we can use a gradient field for a short time between these two times, which localizes the spins at .TE /2 interval time. In the following section, we explain how to sample a PE gradient. If we do not apply a PE gradient, all magnetization vector points are initially along the same direction after the RF excitation pulse. However, the magnetization vectors point along the y-axis using a PE gradient. Following this step, we represent a method to localize the spins along the x-axis in Fig. 8.5(4). Through this step, the subject senses different magnetic fields, and the spins precess at different frequencies. This process happens with the FE gradient along the x-axis. The signal echo is centered in the middle of the FE gradient at .TE interval or readout time. The amplitude of the FE gradient defines the field-of-view (FOV) size, which relies on the area size or the matrix size coating the FE gradient during the scan and is measured in millimeters (mm).

8.4 Converting Time Domain FID Signal into a Frequency Domain Signal In this part, we consider the effect of the FT method on the FID signal and investigate the process of generating data space after applying the gradient field. Initially, upon slice selection along the z-axis, the output signal or the FID signal is two-dimensional for the variables x and y, which are related to the selected area, and can be written as

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8 Gradient-Echo Pulse Sequence in MRI System

Fig. 8.5 Scheme of a gradient-echo pulse sequence. (1) The figure presents the RF pulse at .t = 0 and the acquisition time (.AQ). (2) The slice selection (SS) along the z-axis during the RF pulse. (3) The phase encoding (PE) along the y-axis at .TE /2 time point before producing the signal echo. (4) The frequency encoding (FE) along the x-axis at .TE time point during the signal echo and the .AQ time. The blue line shows the “zero-gradient” point, and the yellow arrow shows the PE gradient changes from the highest gradient to zero point (blue line) in a positive direction (it leads to rephasing) and from zero point to the lowest gradient in the negative direction (it shows dephasing)

f

h(x, y)e−iω0 t dxdy,

S(t) =

.

(8.10)

all sample

where .ω0 is the uniform Larmor frequency applied to the spins. Also, .h(x, y) denotes the final data distribution of all spins involved in the spatial direction, which generates the final image. The function .S(t) represents the information carried by the FID signal. Note that the integral over the “all sample” is determined by the maximum and minimum values along the x and y directions. Considering the FT method, the transformed signal is expressed as f

h(x, y)e−iωu x e−iωv y dxdy,

h(u, v) =

.

all sample

(8.11)

8.4 Converting Time Domain FID Signal into a Frequency Domain Signal

147

where .ωu = 2π u and .ωv = 2π v represent the frequencies along the x- and the y-axes. From the above relations, the time domain FID signal is related to −iω0 t term and can be converted into the frequency domain signal rep.h(x, y) e resenting .h(u, v)e−iωu x e−iωv y . Thus, we generate the two-dimensional frequency domain of the FID signal, which is named the MRI “.K-Space.” We simulate the above equation once we have the gradient field. After turning on the gradient field along the x- and y-axes, the spin system feels a new frequency dependent on spatial location as .ω(x, y), where we replace .ω0 with .ω(x, y) = γ (B0 + Gx x + Gy y) from Eq. (8.3). Therefore, the signal can be given as f

h(x, y)e−iγ (B0 +Gx x+Gy y)t dxdy,

S(Gy , ty , Gx , tx ) =

.

(8.12)

all sample

where .S(Gy , ty , Gx , tx ) represents the output signal with the applied gradient fields along the x and y directions. Since phase term, .exp (−iγ B0 t), is not dependent on x and y directions, we can take it out of the integral, resulting in S(Gy , ty , Gx , tx ) = e−iγ B0 t

f

h(x, y)e−iγ Gx xtx e−iγ Gy yty dxdy.

.

(8.13)

all sample

After the demodulation step, .exp (−iγ B0 t) phase term can be removed. By comparing with Eq. (8.11), we can establish a connection between u and the gradient field along x-axis .Gx and between v and the gradient field along y-axis .Gy . Thus, we can express .u = kx = γ Gx tx as the “.K-Space” data along the x-axis with time duration .tx and .v = ky = γ Gy ty as the “.K-Space” data along the y-axis with time duration .ty . Consequently, we can determine the signal as f S(Gy , ty , Gx , tx ) =

.

h(x, y)e−ixkx e−iyky dxdy.

(8.14)

all sample

The dimension of “.K-Space” is determined by .kx , .ky , and .kz with the unit of . cm−1 . The number of data points in “.K-Space” is named .Nkx , Nky , and .Nkz , corresponding to .Nx , Ny , and .Nz in the image space, respectively. We display these parameters in Fig. 8.6 and demonstrate the FT method. The dimension of each data point along the .kx - and .ky -axis is depicted as .Ax and .Ay, respectively. The desired image size is determined by Ax =

1 1 1 = = , (+kx max ) − (−kx max ) 2kx max kF OVx

Ay =

1 1 1 = = , 2ky max kF OVy (+ky max ) − (−ky max )

.

(8.15)

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8 Gradient-Echo Pulse Sequence in MRI System

Fig. 8.6 Scheme of the two-dimension “.K-Space.” The number of data points in “.K-Space” is named .Nkx and .Nky , corresponding to the image space as .Nx and .Ny , respectively. The dimension of each data point along .kx and .ky is denoted as .Ax and .Ay in the image space. .+kymax and .−kymax are the highest positive and negative values along the y-axis in “.K-Space.” The difference between these two points is .kF OVy . Furthermore, we have .FOVy along the y-axis. Similarly, we have these points for the x-axis

where .+kxmax and .−kxmax are the highest positive and negative values along the x-axis in “.K-Space,” respectively. Therefore, the difference between these two points is described as .kF OVx , describing FOV along the .kx -axis. Similarly, these parameters are applicable to the y-axis. The connection between the desired FOV in “.K-Space” and the image resolution dimension is determined by Akx =

1 , xmax − xmin

Aky =

1 , ymax − ymin

.

(8.16)

where .Ak is the width of two data points in “.K-Space.”

8.5 Sampling Data Space in Gradient-Echo Pulse Sequence In this section, we consider an example of filling “.K-Space” as a .6 × 6 matrix for the illustration, as presented in Fig. 8.7. We display a “.K-Space” that has a direction along the x-axis (rows) related to the FE gradient and a direction along the y-axis (columns) corresponding to the PE gradient. The purpose is to fill the entire “.KSpace” with data points from top to bottom of the matrix.

8.5 Sampling Data Space in Gradient-Echo Pulse Sequence

149

Fig. 8.7 Scheme of the filling “.K-Space” with data points. (1) The diagram indicates that by repeating .TR six times, how the entire “.K-Space” is completed. (2) displays the signal amplitude on “.K-Space.” (3) introduces the voxel volume characteristics. (4) points the data from “.K-Space” converted to a data space by applying the FT method. The blue line shows “zero-gradient,” and the purple cubic shows “.K-Space”

As we present in Fig. 8.7(1), once we select the slices along the z-axis, we apply six steps of the FE in the x-axis and PE gradient in the y-axis to form a .6 × 6 (frequency .× phase) matrix. For each line of the matrix, one .TR is used from the spin echo pulse sequence, which fills the first row of the matrix for a specific number of slices. The first row of data is filled by turning on the FE gradient and applying the highest PE gradient. The rest of the row data is recorded using the same FE and data sampling by repeating the process. The only change in this process is to alter the PE gradient every .TR . Therefore, the PE gradient decreases to fill the second row of the matrix until it reaches the center of “.K-Space,” which is called “zero-gradient.” In Fig. 8.7(2), we express the center of “.K-Space” with a high signal amplitude. The PE gradient increases in the negative direction from the center of “.K-Space” until it satisfies the whole space. By involving the FT method, “.K-Space” data converts to the actual space and vice versa. By adding a new .TR for each row, the entire “.K-Space” is completed, which means that for a matrix size of 6 .× 6, we need to apply 6 .×TR to finish the scan. Hence, the scan time depends on .TR , the PE gradient, and the number of excitation (NEX) or the number of signal averages (NSAs).

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8 Gradient-Echo Pulse Sequence in MRI System

To be more precise, the scan time is given by Scan Time = TR × Number of PE × Number of Excitation (NEX),

.

(8.17)

where NEX is the number of times a matrix line is filled with data points. Besides the NEX, the voxel volume is determined by the slice thickness and pixel area size in the reconstructed image. These factors can affect the image quality and scan time, influencing the resulting signal-to-noise ratio. A pixel area is a unit of a digital image in two-dimensional size, as we display in Fig. 8.7(3). Therefore, a larger voxel volume includes more spins or sample tissues. Consequently, more signals are detected on the screen. By adjusting some parameters in MRI planning, we can improve the signal-tonoise ratio significantly. The signal-to-noise ratio depends on FOV and “.K-Space” parameters and can be determined as Signal-to-noise ratio = ρ ×

.

/ .

×z×

NSA ×

FOV(Frq) FOV(Ph) × matrix(Ph) matrix(Frq)

1 × matrix(Frq), BW

(8.18)

where .FOV(Ph) and .FOV(Frq) are along the phase and frequency encoding directions, measured in millimeters (mm). Also, .matrix(Ph) and .matrix(Frq) are the number of matrices in the phase and frequency encoding direction. Moreover, BW refers to the bandwidth of the excitation pulse, and .ρ represents the spin density. From the above relation, if we want to increase the signal-to-noise ratio of the signal, we can increase the voxel volume, increase FOV, decrease the number of matrices along the phase and frequency direction, and increase the slice thickness. For instance, by receiving √ a smaller BW by a factor of f , we can reduce the signal’s noise by a factor of .1/ f . The final image is obtained on the screen after recording “.K-Space” data, followed by the FT method, as we present in Fig. 8.7(4). Note that “.K-Space” data are not the final image. We must apply the FT method to create an image. We also present all three gradients in Fig. 8.8. The spatial localization indicates that the phase encoding is perpendicular to the readout period and the slice selected. An essential property of “.K-Space” is the complex conjugate symmetry data, as shown in Fig. 8.9. A complex signal has real and imaginary components related to the magnitude and phase of the data points in space. Therefore, if we record only half of the data on “.K-Space,” we can fill the rest of the space by flipping the conjugate of data points. This technique is known as the half-scan method in MRI sampling, which reduces the scan time or .AQ time. Note that the conjugate symmetry of “.KSpace” is located diagonally, where the complex conjugate of data points .a + ib is .a − ib.

8.6 Maximizing the Signal Quality for Gradient-Echo Imaging

151

Fig. 8.8 Scheme of the spatial localization in three-dimensional. Phase encoding is perpendicular to the readout period and the slice selected

Fig. 8.9 Scheme of the half-scan method in “.K-Space” sampling. The conjugate symmetry data of “.K-Space” are placed diagonally. The complex data point is .a + ib, and the complex conjugate of its data point is .a − ib

8.6 Maximizing the Signal Quality for Gradient-Echo Imaging Optimizing the flip angle by introducing the Ernst angle is a practical approach to improve image quality, signal-to-noise ratio, and image contrast. In this method, after applying the first .TR with a flip angle smaller than .90◦ , .α < 90◦ , the longitudinal magnetization’s spin changes along the z-axis, from Eq. (4.8), as

152

8 Gradient-Echo Pulse Sequence in MRI System

Mz (t) = M0 cos(α)e

.

− Tt

1

( ) − t + M0 1 − e T 1 .

(8.19)

We assume that the transverse magnetization is zero when we apply the second RF pulse. Therefore, at .t = TR , we have the longitudinal magnetization as Mz (TR ) = M0 cos(α)e

.

T

− TR 1

( ) T − R + M0 1 − e T 1 .

(8.20)

Thus, after applying n-th RF pulse with flip angle .α, the longitudinal magnetization at .t = nTR is expressed as Mz (nTR ) = Mz ((n − 1)TR ) cos(α)e

.

T

− TR 1

( ) T − TR 1 . + M0 1 − e

(8.21)

Note that .Mz (nTR ) is the magnetization at .t = nTR based on the magnetization at .t = (n−1)TR . If we consider a steady state, we have .Mz (nTR ) = Mz ((n−1)TR ). Therefore, the longitudinal magnetization after excitation pulse with flip angle .α is written as [165] 1−e

'

M = M0

.

1−e

T

− TR 1

T − TR 1

(8.22)

,

cos α

where .M ' = Mz (∞) is at a steady state along the z-axis. The transverse magnetization is characterized by .sin(α) and decay with .T2∗ rate, which is given by 1−e

S = S0

.

1−e

T

− TR

T

1

sin(α) e

T

− TR 1

− TE∗ 2

,

(8.23)

cos α

where .S0 is the longitudinal magnetization after applying the RF pulse with flip angle .α in the steady state. Also, S is the final signal intensity of gradientecho imaging. Moreover, .sin(α) indicates the signal intensity flipped toward the transverse plane. The exponential part is related to signal decay at .T2∗ rate. Therefore, the above equation is maximum at .

cos αopt = e

T

− TR 1

,

(8.24)

where .αopt is the optimal flip angle known as the Ernst angle, which is given as αopt = cos

.

−1

( T ) − R e T1 .

(8.25)

8.7 Echo-Planar and Spiral Imaging Sequence

153

Fig. 8.10 (a) The optimal flip angle versus .TR /T1 is presented. (b) We assume .α1 < α2 , so we can write .M cos(α1 ) > M cos(α2 ). This shows that .M cos(α1 ) returns to its original state faster than .M cos(α2 )

From the above relation, the maximum value of the flip angle depends on .TR . For instance, if .T1 = 1400 and .TR = 2000 ms, the Ernst angle is equal to ◦ .≈1.3288[rad] ≈ 76.1 . Figure 8.10a illustrates the dependence of the optimal flip angle on .TR /T1 . As is presented in the plot, when the ratio of .TR /T1 is very small, the optimal flip angle is less than .70◦ . In contrast, when the ratio of .TR /T1 is very large, the optimal flip angle is around .90◦ . We also observe that the gradient-echo is not only a function of .TR but is also affected by the flip angle. Thus, choosing the right flip angle for excitation can decrease the scan time and improve the signal quality. In Fig. 8.10b, we compare two optimal flip angles, .α1 and .α2 , to visualize the magnetization behavior after applying the optimal RF pulse. We assume .α1 < α2 , and therefore, for the longitudinal magnetization we have .M cos(α1 ) > M cos(α2 ), which means that .M cos(α1 ) spends less time to return to its original state (.Mz ) than .M cos(α2 ).

8.7 Echo-Planar and Spiral Imaging Sequence We briefly introduce two practical sampling methods of “.K-Space” in the MRI sequence, the echo-planar, and spiral imaging. The echo-planar imaging (EPI) sequence is the most efficient and quickest method developed to generate MR imaging, which was introduced by Peter Mansfield in 1977 [22]. The EPI sequence acquires a single RF excitation pulse or .TR to complete the entire space. Therefore, the EPI sequence needs a range of 20–45 ms to fill the entire slice or volume [162, 166, 167]. In contrast, gradient-echo sampling takes several minutes to fill a whole volume or collect a slice. As shown in Fig. 8.11a, the EPI operates the gradient along the x-axis, .Gx , to dephase the spins or apply the reverse of the .Gx to rephase the spins, instead of

154

8 Gradient-Echo Pulse Sequence in MRI System

Fig. 8.11 (a) Scheme of the echo-planar imaging (EPI) sequence. EPI encodes spatial information in a single RF excitation pulse using FE and PE gradients. Different phase shifts are generated by applying gradient blips in the positive y-axis for each readout. (b) Scheme of the spiral imaging sequence. Using a spiral trajectory, “K-Space” starts filling from the center to the outward of space. The gradients are applied continuously in both directions of the “K-Space”

using a .180◦ pulse. We apply negative x- and y-axis gradients once the spins are excited at the center of “.K-Space.” Then, the gradients move down the sampling step in the negative x- and y-axis in the lower left corner “.K-Space” and start to acquire the data along the positive x-axis followed by a tiny blip in the positive y-axis for each readout, until to fill the entire space. There is also another way to fill “.K-Space,” which used a spiral sequence as is shown in Fig. 8.11b. In this method, we apply the RF excitation pulse at a certain slice, and the frequency and phase gradient oscillate and enhance the amplitude along the x- and y-axis to yield a spiral pattern from the center of “.K-Space.” The gradient field along x- and y-axis is given by Gx = G0 t sin(ωt), .

Gy = G0 t cos(ωt).

(8.26)

Exercises

155

Exercises Problem 1 Calculate (a) the scan time and (b) the signal-to-noise ratio for a spin echo sequence with the following parameters: .TR = 500 ms, .TE = 24 ms, .40 × 48 matrix, 5 number of scan average, 10 mm slice thickness, FOV 20 cm, and the BW is 20 Hz/pixel.

Solution 1 (a) From Eq. (8.17), the scan time is given by Scan Time = TR × Number of PE × NSA = 500 × 40 × 5 = 100,000 ms. (8.27) (b) To calculate the signal-to-noise ratio, we have .

Signal-to-noise ratio ∝

.

/ .

×z×

NSA ×

FOV (Ph) FOV (Frq) × matrix (Ph) matrix (Frq) 1 × matrix (Frq). BW

We can calculate phase and frequency dimensions as .

FOV (Ph) 200 = = 5 mm, matrix (Ph) 40

FOV (Frq) 200 = = 4.17 mm. matrix (Frq) 48

(8.28)

Therefore, we get for a 40 × 48 image resolution, the pixel size is 5 × 4.17 mm. By substituting in the signal-to-noise ratio relation, we have / Signal-to-noise ratio ∝ 5 × 4.17 × 10 ×

.

5×

1 × 48 ≈ 722. 20

(8.29)

Problem 2 (a) The magnetization at the z direction is given by Eq. (8.19). Find the rate of magnetization changes at t = 0 and t = TR . (b) Find the optimal flip angle from Eq. (8.23).

156

8 Gradient-Echo Pulse Sequence in MRI System

Solution 2 (a) The magnetization is given by Mz (t) = M0 cos(α)e

.

− Tt

1

( ) − t + M0 1 − e T 1 .

The rate of the magnetization change is given by the derivative of Mz (t) with respect to time. So, we have .

M0 dMz (t) M0 − Tt − t =− cos(α)e T1 + e 1 dt T1 T1 =

M0 − t (1 − cos(α))e T1 . T1

Note that dMz (t)/dt is always a positive quantity for non-zero α, which indicates the fact that Mz (t) increases as time passes. Now, at t = 0, we attain M0 0 dMz (t)/dt = M T1 (1 − cos(α)), and at t = TR , we attain dMz (t)/dt = T1 (1 − −

TR

cos(α))e T1 . (b) To calculate the optimal flip angle, we need to maximize the output signal of S with respect to α. Thus, we consider S as sin(α)

S=A

.

1−e

(8.30)

,

T

− TR

cos α

1

where A is the constant part S0 exp(− TTE∗ )(1 − exp(− TTR1 )), which is not 2 dependent on the flip angle α. The derivative of S with respect to α is dS = . dα

cos(α) − e (1 − e

T − TR 1

T

− TR 1

.

(8.31)

cos(α))2

We have the derivative equal to zero to find the optimal point, as dS/dα = 0. Solving for α, we have the optimal flip angle as .

cos αopt = e

T

− TR 1

.

(8.32)

Chapter 9

Spin–Spin Coupling

9.1 Introduction In this chapter, we consider the system of two coupled spins. The two spins are connected through the magnetic dipole moment interactions, which results in the spin–spin coupling between the two subsystems in the system. We establish Hamiltonian’s eigenstates and discuss the system’s energy transition state structure. We show how such a formalism can explain the spectrum of two interacting spins that are observed in the experiment. We consider various settings in the system, such as the weak and the strong coupling scenarios. We also discuss the relation of such interaction with the homonuclear and heteronuclear systems and outline how homonuclear and heteronuclear systems provide different signals, which can be explained by the spin–spin coupling of the systems. This formalism provides the quantum mechanical description of a vast set of NMR and MRI principles and modern techniques. One of the interesting techniques that can be rooted in the spin–spin coupling principles of quantum mechanics is the hyperpolarization technology that we will discuss later.

9.2 Density Matrix and Hamiltonian for the Two-Spin System Here, we consider the system of two coupled spins. In this setting, the two spins are connected through the magnetic dipole moment, which is related to the spin–spin coupling in the system. We denote the operator of the first spin as .I1 and the operator of the second spin as .I2 . The interaction between the two spins is determined by the coupling constant .J , which is expressed in .s−1 or Hz [168].

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Khashami, Fundamentals of NMR and MRI, https://doi.org/10.1007/978-3-031-47976-2_9

157

158

9 Spin–Spin Coupling

The Hamiltonian of the two coupled spins can determine the energy transition between two spin systems, which is given by H = Hz + HJ ,

(9.1)

.

where .Hz represents the Zeeman interaction along the z-axis, and .HJ represents the J -coupling interaction between the two spins. Under the high field approximation, the contribution from .HJ will be very small compared with that from .Hz , where we can effectively assume .J = 0 in all equations. The general form of the Hamiltonian of the system can be written as (by setting .h ¯ = 1) [169–171] .

H=

.

1 1 ωE (I1z + I2z ) + ωδ (I1z − I2z ) + J I1 · I2 . 2 2

(9.2)

We can also express the spin Hamiltonian as H=

.

1 1 (ωE + ωδ )I1z + (ωE − ωδ )I2z + J I1 · I2 , 2 2

(9.3)

where we have ωE = ω1 + ω2 ,

.

ωδ = ω1 − ω2 .

(9.4)

The .ωE and .ωδ are the sum and difference of the Larmor frequencies of spins 1 and 2 in Hz units. Moreover, .I1z and .I2z are the z-components of the spin operators of spins 1 and 2. Furthermore, we can write ωi = ω0 (1 − σi ) − ωRF ,

.

i = {1, 2},

(9.5)

where .σi is the chemical shift constant elements for spin .Ii , which is presented in rad .s−1 , and .ω0 = γi B0 is the Larmor frequency for spin .Ii in the static magnetic field. Thus, the Larmor frequencies for spins .I1 and .I2 are written as

.

ω1 = ω0 (1 − σ1 ) − ωRF , .

ω2 = ω0 (1 − σ2 ) − ωRF ,

(9.6)

where .ωRF is the RF pulse carrier frequency. The last part of the spin Hamiltonian is related to the interaction of the two spins in the system with the .J coupling constant. The dot product of spin angular momenta 1 and 2 can determine the energy levels of the two spins in the system, which can help us to understand the signal magnitude. Considering the matrix representation of the Hamiltonian, we can write the Hamiltonian of the two-spin system as (see Appendix B for the details)

9.3 The Energy Levels and Transition Energy in the Two-Spin System

159

⎛

⎞ +ωE + 12 J 0 0 0 ⎟ 1⎜ J 0 0 +ωδ − 21 J ⎜ ⎟. .H = 1 ⎝ ⎠ 0 0 J −ωδ − 2 J 2 1 0 0 0 −ωE + 2 J

(9.7)

Accordingly, if we consider a weak coupling interaction, where .J is very small, the two off-diagonal elements can be neglected in the above Hamiltonian. However, this term plays a significant role in the strong coupling interaction.

9.3 The Energy Levels and Transition Energy in the Two-Spin System Considering a two-spin system, the number of the degeneracy of the system is calculated as I = (2I1 + 1)(2I2 + 1),

.

(9.8)

where .I1 and .I2 are spin quantum numbers of spins 1 and 2. Therefore, for two spin-.1/2, we have four degenerate energy states, where we can detect four distinct peaks in the NMR spectrum. To analyze the physics of the system, we first determine the eigenvalue and eigenstates of the Hamiltonian. The details of the calculations are given in Appendix C. Thus, from the Hamiltonian, the eigenstates of two interacted spins are calculated as |φ1 > = |ββ>, |φ2 > = cos θ |βα> + sin θ |αβ>, .

|φ3 > = − sin θ |βα> + cos θ |αβ>,

(9.9)

|φ4 > = |αα>, where the angle .θ is related to the coupling constant, .J , which is determined by J , D . ωδ cos 2θ = , D sin 2θ =

(9.10)

where D is a positive quantity, which we represent in Fig. 9.1a, given by D=

.

/ ωδ2 + J 2 .

(9.11)

160

9 Spin–Spin Coupling

Fig. 9.1 (a) The geometric diagram of the coupling angle. (b) The Zeeman energy levels for the two-spin are .{|φ1 >, |φ2 >, |φ3 >, |φ4 >}. The energy differences between the two coupled spins are named .E1 , .E2 , .E3 , and .E4 . The spin transition between .3 ↔ 1, .2 ↔ 4, .4 ↔ 3, and .2 ↔ 1 is named .W1 . The spin transition between .1 ↔ 4 is named .W2 and between .2 ↔ 3 is called .W0

From Eq. (9.10), we can also find .

tan 2θ =

J . ωδ

(9.12)

The eigenvalues related to these eigenstates are given by .E1 , .E2 , .E3 , and .E4 . Therefore, the related eigenvalues are expressed as

9.3 The Energy Levels and Transition Energy in the Two-Spin System

E1 E2 .

E3 E4

) ( 1 1 = ωE + J , for |φ1 > 2 2 ( ) 1 1 D − J , for |φ2 > = 2 2 ) ( 1 1 D + J , for |φ3 > =− 2 2 ( ) 1 1 −ωE + J , for |φ4 >. = 2 2

161

(9.13)

We represent the diagram of the two interacting spins, .I1 and .I2 , in Fig. 9.1b. We show the Zeeman energy levels of the system and the spin coupling in the two-spin setting in Fig. 9.1b. The general form of the transition energy of the coupled spin system is calculated from Eq. (9.13), which gives 1 (−ωE 2 1 = E2 − E1 = (−ωE 2 1 = E2 − E4 = (−ωE 2 1 = E3 − E1 = (−ωE 2

E4↔3 = E4 − E3 = E2↔1 .

E2↔4 E3↔1

1 + J ) + D, 2 1 − J ) + D, 2 1 + J ) − D, 2 1 − J ) − D. 2

(9.14)

For instance, .E4↔3 is related to the energy transition between energy levels 4 and 3 in the system, which is shown in Fig. 9.1b. The frequency transition is determined by .Aω = AE/γ . Therefore, we have E4↔3 ≥ E2↔1 ≥ E2↔4 ≥ E3↔1 .

.

(9.15)

The general form of energy difference between .E2↔4 and .E2↔1 is written as |AE| = |E2↔4 − E2↔1 | = D − J .

.

(9.16)

The energy difference between .E2↔4 and .E3↔1 is given by |AE| = |E2↔4 − E3↔1 | = J .

.

(9.17)

162

9 Spin–Spin Coupling

9.4 The Free Induction Decay for the Two-Spin System In this section, we calculate the FID signal for a two-spin system, which is excited by a .90◦ pulse along the x-axis. From our definition of the FID signal, we express a complex value of the two magnetization elements along the x- and y-axis. Therefore, we have the observation of the spin magnetization as = + i,

.

(9.18)

where . = Tr(ρMx ) and . = Tr(ρMy ) are the expected values of magnetization vector along the x- and y-axis, respectively. To attain the above relation, we have to return to our definition of the density matrix in Eq. (2.53), which for a single spin we presented as .ρ(0) = (I + Iz )/2, where .I is the identity matrix and .Iz is the z-component of the spin. Thus, for a two-spin system, we have ρ(0) =

.

1 1 (I1 + I2 + Iz ) = (I + I1z + I2z ), 2 2

(9.19)

where .I = I1 + I2 and we can define .I1z + I2z = Fz along the z-axis. Thus, we can write ρ(0) =

.

1 (I + Fz ). 2

(9.20)

The complex FID components along the x- and y- axes are written as = + i,

.

(9.21)

where we can define .I1x + I2x = Fx along the x-axis and .I1y + I2y = Fy along the y-axis. Therefore, we can write ∝ .

(9.22)

.

After rotating .ρ(0) for .90◦ along the x-axis, from Eq. (2.57), and letting the system evolve with the Hamiltonian, we have ρ(t) = e−i Ht e− 2 π Fx ρ(0)e+ 2 π Fx e+i Ht . i

i

.

(9.23)

Thus, from our earlier analyses, we can write the time evolution operator with the Hamiltonian as E −i Ht .U = e = e−iEi t |φi > 0, ωδ > 0, then inner peaks for .E2↔4 and .E2↔1 have high intensity determined by .1 + sin 2θ , while outer peaks for .E3↔1 and .E4↔3 have low intensity determined by .1 − sin 2θ , which can also be seen from the equation above. This factor is known as the “roofing effect” in the NMR spectrum.

9.5 The Strong Spin–Spin Coupling in the System Now, we consider the strong coupling limit in our general description of a two-spin system. For a strong coupling regime where .J >> ωδ , we can approximate Eq. (9.11) as

164

9 Spin–Spin Coupling

Fig. 9.2 Scheme of the spectrum of two interacting spins. The diagram shows that the frequency increases from the right to the left. We can write .E4↔3 ≥ E2↔1 ≥ E2↔4 ≥ E3↔1 . The frequency difference from .E2↔4 and .E2↔1 is .D − J . The frequency difference from .E2↔4 and .E3↔1 and similarly .E4↔3 and .E2↔1 is .J . Moreover, the distance between .E2↔4 and .E4↔3 is D. Furthermore, the signal intensity is proportional to .1 ± sin 2θ

D=

.

/

(

ωδ2

+ J2

1 =J 1+ 2

(

ωδ J

)2 ) =J +

1 ωδ2 . 2 J

(9.28)

Therefore, we have D−J =

.

1 ωδ2 . 2 J

(9.29)

By substituting the above relation in Eq. (9.16), we get the energy difference between .E4↔2 and .E2↔1 proportional to .ωδ2 . Since .ωδ2 is a small value and we can ignore it, thus, we can write .D = J for a strong coupling system. Therefore, under the above condition, the eigenvalues of the system are expressed as E1 =

.

) ( 1 1 ωE + J , . 2 2

1 J ,. 4 3 E3 = − J , . 4 E2 =

(9.30) (9.31) (9.32)

9.5 The Strong Spin–Spin Coupling in the System

) ( 1 1 E4 = −ωE + J . 2 2

165

(9.33)

The energy differences from Eq. (9.14) for strong coupling condition are given by 1 (−ωE + 2J ) , 2 1 = − ωE , 2 1 = − ωE , 2 1 = − (ωE + 2J ) . 2

E4↔3 = E2↔1 .

E4↔2 E3↔1

(9.34)

In Fig. 9.3, we represent the diagram of a strongly coupled system, where the distance between the two middle peaks is very small due to the small value of .ωδ . However, the outer peaks are relatively far since the coupling constant is large.

Fig. 9.3 Scheme of the spectrum of two strongly interacted spins. The frequency difference from and .E2↔1 is .D − J = ωδ2 /2J . The frequency difference from .E2↔4 and .E3↔1 and similarly .E4↔3 and .E2↔1 is .J for strong coupling. Moreover, the distance between .E3↔1 and .E4↔3 is 2 .2J + ωδ /2J .E2↔4

166

9 Spin–Spin Coupling

In a strong coupling regime, .tan 2θ in Eq. (9.12) is very large, where we can approximately set .θ = π/4, and thus, the eigenstates of the system are written as |φ1 > = |ββ>, 1 |φ2 > = √ (|βα> + |αβ>), 2 . 1 |φ3 > = √ (|αβ> − |βα>), 2

(9.35)

|φ4 > = |αα>, where the eigenstates of .|φ1 >, .|φ2 >, and .|φ4 > are considered as the triplet states. Moreover, the eigenstate of .|φ3 > is denoted as the singlet state. The eigenstates in Eq. (9.35) can be arranged according to their total angular momentum quantum number .I = 1 for angular number .mI . Therefore, we have the triplet states for .mI = 0, ±1 as |φ1 > = |ββ>, .

mI = +1

1 |φ2 > = √ (|βα> + |αβ>), 2 |φ4 > = |αα>,

mI = 0

(9.36)

mI = −1.

For .I = 0, known as the singlet state with the angular number .mI = 0, we have .

1 |φ3 > = √ (|αβ> − |βα>). 2

(9.37)

The question we can ask here is why the intensity of .E3↔1 and .E4↔3 is smaller than that of .E4↔2 and .E2↔1 ? From the quantum rules, the transition from angular number .I = 1 (the triplet states) to .I = 0 (the singlet state) is forbidden due to the changing spin .I beside energy transition. Therefore, the output signals are weaker for these two transitions. In other words, the transition from triplet to singlet state is not allowed.

9.6 The Weak Spin–Spin Coupling in the System We can also consider the weak coupling condition in the spin system, where the two spins cannot strongly interact with each other in the sample. If the two spins are weakly coupled, we can say .J , .

|φ3 > = |αβ>,

(9.42)

|φ4 > = |αα>.

9.7 Quantum Transition in the Two-Spin System The spin Hamiltonian represents the Zeeman energy levels of the two spins coupled weakly in the system with the basis given by .{|ββ>, |βα>, |αβ>, |αα>}. In Fig. 9.1b, we represent the quantum transition of spin between the two states. There are three types of transitions in the spin system. For example, the transition between .|αβ> and .|ββ> is a single quantum transition, with only one flip of the first spin, from .α to .β state. This transition is named .W1 , which shows the probability of the Zeeman spin transition between two states. In .W1 transition, the difference of magnetic quantum numbers is .AmI = 1. Moreover, the spin that flips its state is known as active spin, and the nuclear spin that remains in the same state is called passive spin. Also, in the transition from .|αα> to .|ββ>, two aligned spins simultaneously flip, which is called double quantum transition and is denoted as .W2 . In .W2 spin transition, the first spin changes from .α to .β state, and also, the second spin flips from .α to .β state. In the .W2 transition, the difference of magnetic quantum numbers is .AmI = 2. Furthermore, the transition between .|αβ> and .|βα> is called zero quantum transition, where both anti-parallel spins simultaneously flip, which is called .W0 . In the .W0 transition, the difference of magnetic quantum numbers is .AmI = 0. To illustrate, we suppose the proton nuclei resonating in the system at 600 .MHz. We only need 600 .MHz power to flip a spin between two states for a single quantum transition. For a double quantum state, we need .2 × 600 = 1200 .MHz for having a flip-flop spin transition. Also, for zero quantum transition, we need a range of energy between 100 and 1200 .Hz, which is the resonance frequency difference between two protons. Therefore, for the transition energies of the system, we can write W2 > W1 > W0 .

.

(9.43)

The above relation shows that a double or a single transition needs more energy than a zero quantum transition. Another important concept in the two-spin interaction is related to the spin transition between .W2 and .W0 , known as the cross-relaxation rate. The difference between the two spin transitions is defined as cross-relaxation rate, which is given by

9.8 The Homonuclear and Heteronuclear Systems

σ = W2 − W0 .

.

169

(9.44)

We can express a connection between the cross-relaxation rate and the spectral density function as σ ∝

.

1 [6J (2ω0 , τ ) − J (0, τ )], r6

(9.45)

where r is the inter-nuclear distance between the two spins, and .σ is the crossrelaxation rate between spin 1 and spin 2. Therefore, .σ is inversely proportional to .r 6 . Moreover, the first term of the cross-relaxation rate is a double quantum transition, and the second is a zero quantum transition.

9.8 The Homonuclear and Heteronuclear Systems In the NMR spin interaction systems, there are two important types of nuclei interactions, known as homonuclear and heteronuclear. The homonuclear is between two identical types of nuclei, where the two gyromagnetic ratios are equal (e.g., 1 1 . H − H), whereas the heteronuclear is between two different types of nuclei with two different gyromagnetic ratios (e.g., .1 H − 13 C). In this section, we consider the energy level and different energy transitions for 1 1 1 13 C interactions in Fig. 9.5a and b. We calculated the energy . H − H and . H − levels for the homonuclear and heteronuclear systems. For a .1 H − 1 H system, where .γ1 = γ2 = γ , we have the energy diagram as represented in Fig. 9.5a. The Larmor frequencies of protons are of the order of hundreds of MHz, while .J -couplings are in Hz. Thus, the .J -coupling magnitude is smaller than the Larmor frequency of the proton. For .1 H − 1 H interaction, we have .ω1 = ω2 = ωH , and therefore .ωδ = 0 and .ωE = 2ωH in the energy level equations. We also have .D = J from Eq. (9.11) for 1 1 . H − H interaction. Thus, we can calculate the energy levels from Eq. (9.13) as 1 E1 = ωH + J , . 4 1 E2 = J , . 4 3 E3 = − J , . 4 1 E4 = −ωH + J . 4

.

(9.46) (9.47) (9.48) (9.49)

This shows that .E2 is slightly more than .E3 , which depends on the .J -coupling constant in the order of Hz. However, their energy is significantly different from

170

9 Spin–Spin Coupling

Fig. 9.5 (a) Energy diagram of a homonuclear system for two protons interaction. In this case, we have a strong interaction since the condition .J >> ωδ is satisfied. (b) Energy diagram of a heteronuclear system for a proton–carbon interaction. In .W2 , .E1 − E4 ≈ 5ωC . For .W0 , .E3 − E2 ≈ 3ωC

the other two states. We illustrate the transition energy between the two states in Fig. 9.5a. For instance, .|E2↔1 | = |E2↔4 | = ωH and .|E3↔1 | = ωH + J , and .|E3↔4 | = ωH − J . An important observation is that since .ωδ = 0, we attain .θ = π/4, and thus the eigenstates are given by Eq. (9.35). The physical structure of the heteronuclear systems is usually different. In these systems, we normally have weak couplings, and the correlation between the two spins is not very significant. For example, for the heteronuclear interaction in .1 H − 13 C system, one has .γ ≈ 4γ . The Larmor frequencies differ significantly; thus, H C they record different energy levels, as shown in Fig. 9.5b. Therefore, we have .ωδ ≈

Exercises

171

3ωC and .ωE ≈ 5ωC in the energy level equations. In this case, we have a small J -coupling constant compared with .ωδ , since .D ≈ ωδ ≈ 3ωC . By placing the above conditions in Eq. (9.13), we have the energy levels as

.

5ωC 2 3ωC E2 ≈ + 2 3ωC E3 ≈ − 2 5ωC E4 ≈ − 2 E1 ≈ +

.

1 + J ,. 4 1 − J ,. 4 1 − J ,. 4 1 + J. 4

(9.50) (9.51) (9.52) (9.53)

Therefore, we have the energy difference between states, such as .E2↔4 ≈ 4ωC − J /2 and .E1↔3 ≈ 4ωC + J /2. Moreover, the energy difference between .|αβ> and 1 13 C due to the .|βα> is .E2↔3 ≈ 3ωC , which is very large in the interaction of . H − Larmor frequency differences. In the .W2 transition, we have the spin transition between .|ββ> and .|αα>, where we can write .E1 −E4 ≈ 5ωC . On the other hand, for the .W0 transition, the spin transition happens between .|βα> and .|αβ>, where the energy difference gives .E3 − E2 ≈ 3ωC . Alternatively, we have the spin transition between .|αα> and .|βα> as the .W1 transition, with energy difference .E3 − E4 ≈ ωC − J /2. Comparing with the .W2 transition, we see that the energy difference between .|ββ> and .|αα> is approximately four times larger than the energy difference between .|βα> and .|αα> for the carbon state.

Exercises Problem 1 Considering the orthonormal basis ( .

p+ q+

)

find .tan θ that satisfies .H'

( =

(

p+ q+

cos θ sin θ

) ( ) ) ( p− − sin θ , , = + cos θ q−

)

( = λ+ H' =

.

(

(9.54)

) p+ , where .H' from Appendix C is q+ αv v β

) .

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9 Spin–Spin Coupling

Also, .λ± are .

λ± =

1 [ωE ± D] . 2

Solution 1 From the above equation, we have { .

αp+ + vq+ = λ+ p+ , vp+ + βq+ = λ+ q+ .

(9.55)

Here, we solve for .λ+ . By replacing .p+ and .q+ in the above relation, we have α cos θ +

.

J sin θ = λ+ cos θ, 2

(9.56)

where we suppose .v = J /2 in the equation. Then, we can write as .

J sin θ = (λ+ − α) cos θ 2

⇒

J tan θ = λ+ − α. 2

(9.57)

Then, by replacing .λ+ , we have .

tan θ =

D − wδ . J

(9.58)

On the other hand, from the second part of Eq. (9.55), we have .

J cos θ + β sin θ = λ+ sin θ. 2

(9.59)

Then, we have similarly ([ .

J = 2 (λ+ − β) tan θ

⇒

J =2

] ) 1 (α + β) + D − β tan θ. 2

(9.60)

Then, we have .

tan θ =

J . wδ + D

By multiplying and dividing to .D − wδ , we have

(9.61)

Exercises

173

.

tan θ =

D − wδ J D − wδ = . × D − wδ wδ + D J

(9.62)

Thus, both relations are satisfied by the eigensystem equation.

Problem 2 Calculate the energy difference between a proton–nitrogen interaction. For .15 N, −1 .γ15 N is .−4.316 MHz Tesla . The gyromagnetic ratio of proton to nitrogen is approximately 10.

Solution 2 Hint: Follow our calculation steps for proton–carbon interaction in Sect. 9.8.

Problem 3 Show that states .|φ2 >, .|φ3 >, .|φ1 >, and .|φ4 > with |φ1 > = |ββ>, |φ2 > = cos θ |βα> + sin θ |αβ>, .

|φ3 > = − sin θ |βα> + cos θ |αβ>, |φ4 > = |αα>

are orthogonal to one another.

Solution 3 Hint: States .|φ> and .|ψ> are orthogonal if and only if . = 0.

(9.63)

Chapter 10

Hyperpolarized MRI Technique and Its Application in Medical Science

10.1 Introduction In this chapter, we introduce the hyperpolarized (HP) MRI and consider different applications of this interesting technique in MRI. In particular, we investigate the quantum view behind the HP MRI and discuss the practical polarization methods, like the nuclear Overhauser effect (NOE), dynamic nuclear polarization (DNP), and gas polarization techniques. These methods make it possible to prepare the system in a particular statistical distribution that enables signal enhancement in MRI. Consequently, one can extract useful information about the relaxation rate of two nuclear spins within a system that can help to determine the molecular structure in biology and enable the measurement of the correlation distance between the two spins. Here, we discuss the utilization of hyperpolarization technology to polarize electrons and .13 C nucleus in medical science. We also discuss the utility of polarizing gases, such as .129 Xe and .3 He, at both low and high magnetic fields for the HP MRI techniques.

10.2 The Spin Population in Hyperpolarized MRI Technique The main weakness of NMR or MRI is its low sensitivity, which is described by the Boltzmann distribution at thermal equilibrium. Considering the relatively weak signals obtained from NMR or MRI spectroscopy, hyperpolarized (HP) MRI has recently been developed to overcome the challenge. This development makes it possible to get information about sensitive tissues like the brain and human cardiovascular and lungs to attain high-intensity MRI signals. Furthermore, HP MRI is a valuable technique for improving the signal-to-noise ratio of MR images by enhancing the spin polarization method. Increasing the © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Khashami, Fundamentals of NMR and MRI, https://doi.org/10.1007/978-3-031-47976-2_10

175

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10 Hyperpolarized MRI Technique and Its Application in Medical Science

spin polarization and magnetization can accumulate a sharper signal from the spin population distribution compared with its thermal condition. Therefore, enhanced polarization techniques are essential, especially at low-field MRI. Moreover, if we polarize the sample at a more powerful magnetic field and low temperature, it can improve the signal-to-noise ratio by a factor of the applied magnetic field. In Fig. 10.1, we schematically present the Zeeman energy level differences between thermal and hyperpolarization methods. As can be seen from the figure, there are slightly more spins at the lower energy level in the thermal equilibrium. This difference is quite large for HP distribution. Therefore, HP MRI significantly increases the spin population at the lower energy level, which enhances the longitudinal relaxation rate. Imaging techniques based on proton-MRI are not always the most favorable method, especially for soft tissues like lungs and heart [43, 173, 174]. To overcome this, recently HP .13 C, .129 Xe, and 3 . He-MRI, for clinical applications, have attracted much attention [175–177]. According to the Boltzmann distribution, .129 Xe or .13 C gives small spin polarization of about .≈10−6 , at thermal equilibrium, as we presented in Fig. 2.5. Thus, a thermalized spin sample provides a weak signal with low resolution for MR imaging, as shown in Fig. 10.1 [53, 176]. However, highly polarized spin samples improve the signal about .≈104 times as is achievable with HP techniques [173, 176].

Fig. 10.1 Scheme of the difference between spin population for hyperpolarization and thermal polarization methods. The intensity of the MRI signal is proportional to the population difference between the two states. In unpolarized spins, the population difference is very small. Therefore, the final signal has a very low signal-to-noise ratio and small intensity. On the other hand, when we polarize the spins, the population difference changes significantly. Thus, we get a sharp MRI signal at the end

10.2 The Spin Population in Hyperpolarized MRI Technique

177

Fig. 10.2 Scheme of the process of the longitudinal magnetization relaxation to record .T1 rate in the HP mechanism

In Fig. 10.2, we depict the process of the relaxation of the spin system from a polarized state to an unpolarized state, which records the .T1 rate. Therefore, the spin polarization decays to its thermal equilibrium state by rate .T1 . Thus, the longitudinal magnetization recovers to the thermal equilibrium state after each excitation pulse. In Chap. 2 we calculated the spin polarization of the sample from Eq. (2.23), which results in polarization value depending on the external magnetic field, temperature, and the Larmor frequency. In other words, we have p ∝ ω0 = γ hB ¯ 0,

.

p∝

1 . T

(10.1)

For instance, the relation between the gyromagnetic ratios of proton and carbon is .γP ≈ 4 × γC , and the gyromagnetic ratio of electron compared with proton is .γS ≈ 700 × γP . By considering the above relation, the polarization of electron compared with proton is given by .

arctanh(pS ) γS = 700, = arctanh(pP ) γP

(10.2)

where .pS and .pP are the polarization of electrons and the proton’s sample. The above equation represents that the polarization of a sample with the electron is larger than the polarization of a proton sample in the same temperature and magnetic field. In other words, .

arctanh(pS ) >> arctanh(pP ).

(10.3)

Therefore, we can calculate the polarization .pS compared with .pP as [( pS = tanh

.

γS γP

)

] · arctanh(pP ) = tanh[700 · arctanh(pP )].

(10.4)

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10 Hyperpolarized MRI Technique and Its Application in Medical Science

Also, the carbon .pC polarization compared with .pP gives [( pC = tanh

.

γC γP

)

] [ ] 1 · arctanh(pP ) = tanh · arctanh(pP ) . 4

(10.5)

Therefore, the ratio of the electron’s polarization to carbon’s polarization is .

pS = 4 × 700, pC

(10.6)

and the ratio of the proton’s polarization to carbon’s polarization is .

pP = 4. pC

(10.7)

In Fig. 10.3, we analyze the spin population difference between electron, carbon, and proton at room temperature from Eq. (2.24). As can be seen, due to the larger

Fig. 10.3 The effect of magnetic field (.0.06 T, .1.5 T, 3 T and 7 T) on the spin population between two states for .1 H (blue line), .13 C (red line), and electron (purple line) at the room temperature 300 K

10.3 Different Hyperpolarization Techniques

179

gyromagnetic ratio of electrons, we have a higher spin population at the lower and higher states compared with proton and carbon. We are mainly interested in using HP MRI for medical applications to enhance the signal-to-noise ratio of low-sensitivity nuclei like carbon. In the following section, we introduce the process of polarizing carbon, xenon, and helium and their applications in clinical settings.

10.3 Different Hyperpolarization Techniques Our earlier discussion shows that enhancing the population difference between the two states can significantly affect the signal intensity and signal-to-noise ratio. There are several methods, like the nuclear Overhauser effect (NOE), dynamic nuclear polarization (DNP), and gas polarization techniques, to achieve a polarized sample in studies. Albert Overhauser discovered the polarizing nuclei in metals in 1953 [178], called the nuclear Overhauser effect (NOE) technique. The NOE explains how one can attain a significantly high polarization in a spin system. The importance of the NOE experiment is to find the molecule’s structure in biology and measure the correlation distance between the two spins in the system, where we can find the interaction of the two spins in small molecules, DNA, and protein. NOE depends on the spin–lattice relaxation as well as the dipole–dipole interactions between the two spins, as we explained in Chap. 9. In NOE, the two spins, S or the source spin and I or the investigating spin, are correlated and connected. When the RF pulse saturates the source spin at a specific frequency, it can change the investigating spin signal intensity in other space, where we can detect a sharper signal from spin I . In Fig. 10.4a, we present the process of the NOE system according to the spin population. When the RF pulse applies to the source spin S, which initially has a non-equal spin population, it can change the spin population and transfer the system from the equilibrium state to the saturation point, where the population difference (.δ) is zero. Once the RF pulse is switched, the spin system S returns to the equilibrium state, where we can record the relaxation rate. As we mentioned, the system has three types of spin relaxation rates. The NOE leakage occurs when the spins relax through the single quantum transition, .W1 , where the relaxing of source spin cannot affect the signal intensity of spin I . On the other hand, in a two-spin system, we also have the cross-relaxation terms, where the spins can relax via .W2 and .W0 . In the .W2 relaxation rate, spins S and I flip from .β to .α state. When saturated spin, S, transfers to the ground state, the coupled spin, I , moves to the lower energy level, which increases the population difference for spin I . This relaxation process results in detecting a sharper signal for spin I , as we present in Fig. 10.4b and c.

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10 Hyperpolarized MRI Technique and Its Application in Medical Science

Fig. 10.4 Scheme of the NOE system according to the spin population. (a) When the RF pulse is applied to the source spin, S, it can change the spin population and transfer the system from the equilibrium to the saturation point, where the population difference is zero (.δ = 0). Once the RF pulse is switched, the spin system returns to its equilibrium, where we can record the relaxation time. (b) Spins can relax via single, .W1 , double, .W2 , and zero, .W0 , quantum transitions. (c) If .W2 >> W0 , the signal intensity for spin I will be larger than the original signal. If spin transition I .W0 is larger than .W2 (.W0 >> W2 ), we can detect a smaller signal intensity for spin I . .W1 is the single quantum transition for spin I

If .W2 >> W0 , the NOE enhancement can be a positive number from Eq. (9.44), where the signal intensity is higher than the original signal. The positive NOE is typically related to small molecules with short correlation rates. In contrast, if the source spin relaxes through the zero quantum transition, .W0 , from .β to .α state, the coupled spin, I , flips the opposite way from .α to .β. In this relaxation process, the population difference for spin I decreases, detecting a smaller signal than the original signal. In other words, if .W0 is larger than .W2 (.W0 >> W2 ), the NOE enhancement has a negative value, which is normally related to a long correlation rate and large molecules such as proteins. As mentioned earlier, the cross-relaxation rate is the difference between double and zero quantum transitions, .σ = W2 − W0 . We can also define the total relaxation

10.4 Hyperpolarized Carbon MRI and Its Applications in Medical Imaging

181

rate for spin I as .σI = W0 + 2W1I + W2 , where .W1I is the single relaxation rate for spin I . Thus, the NOE enhancement, denoted as .ηIS , from the Solomon equation can be written as [19] ηIS =

.

W2 − W0 γS γS σ = . γI W0 + 2W1I + W2 γI σI

(10.8)

Another practical method to polarize a sample is the dynamic nuclear polarization (DNP) technique. The theory behind the DNP is similar to NOE and is related to the electron–nucleus cross-relaxation rate [178]. This method is mostly used to polarize .13 C nucleus. In the DNP method, the unpaired electrons or free radicals are highly polarized at low temperatures and high magnetic fields. The polarized electron state transfers into .13 C state by microwave (MW) pulse at frequency .ωMW . The MW frequency, .ωMW , is close to the electron frequency, .ωe , in .W1 transition, in which electron transfers from .α to .β state and losses its spin excitation. On the other hand, the electron can get excited through .W0 and .W2 transitions at the MW frequency, which is determined by .ωMW = ωe ± ωn , where .ωn is related to the nucleus frequency. Thus, the .ωMW can enhance carbon polarization. After polarizing the sample, the polarized sample gets injected at approximately ◦ .37 C into the subject. Due to the low gyromagnetic ratio and low natural abundance of .13 C, DNP enhances the polarization level up to 20–40% [179, 180]. We discuss the scheme of the DNP process and consider some applications of it in medical science later in this chapter. The spin-exchange optical pumping (SEOP) is a powerful method for gas polarization. The SEOP method is usually used for .129 Xe and .3 He gases [174, 181, 182]. This method enables the hyperpolarization of gases in abnormal lung ventilation. We discuss the SEOP method for .129 Xe nucleus later.

10.4 Hyperpolarized Carbon MRI and Its Applications in Medical Imaging One of the promising medical imaging techniques, which is used in clinical applications, is HP .13 C MRI. The HP .13 C MRI provides a universal technique for analyzing cancer metabolism based on .13 C analysis that is one of the most important tools for biological and, more specifically, cancer study [183–185]. Several carbon substrates are used to visualize real-time .13 C metabolism in different types of diseases like the brain, liver, and breast tumor [184, 186, 187]. The most typical type of .13 C substrates is utilized in medical imaging that are .[1 − 13 C] pyruvate, .13 C glucose, .13 C urea, and .13 C lactate. As was mentioned, the DNP technique is the most practical method for polarized carbon nuclei. We present a summary of .[1 − 13 C] pyruvate polarization steps in

182

10 Hyperpolarized MRI Technique and Its Application in Medical Science

Fig. 10.5 Scheme of [1 − 13 C] pyruvate polarization process by DNP method. The unpaired electrons are highly polarized at low temperatures and high magnetic fields. The polarized electron state transfers into 13 C state by microwave pulse. After dissolution steps and temperature checking, the polarized sample is ready for injection into the subject for MR imaging. The intensity of the HP signal is greater than the thermal signal, enhancing the signal-to-noise ratio of the metabolic signal in MRI and spectroscopy results

Fig. 10.5. For instance, HP .[1 − 13 C] pyruvate is used as a non-invasive tracking technique to detect damage and any metabolic changes in the human cells [121, 188]. .[1 − 13 C] pyruvate lends a valuable substrate to track pyruvate dehydrogenase (PDH) and lactate dehydrogenase (LDH) activities and also analyze .13 C bicarbonate production [183, 187, 189, 190], as we demonstrate in Fig. 10.6. Furthermore, .13 C glucose substrates are used to investigate glycolysis activity and pentose phosphate pathway (PPP) in cancer disease and traumatic brain injury (TBI), respectively [191, 192]. Recently, altered .13 C glucose metabolism was also investigated to analyze metabolic changes in TBI patients, which can detect .13 C bicarbonate signal more quickly and measure the rate of oxidative phosphorylation at the location of the injury, besides the PPP pathway activity analysis [184, 193]. We present a summary of the metabolic pathway of glucose in Fig. 10.6. The diagram shows how glucose converts to pyruvate and lactate in the cell membrane.

10.5 Hyperpolarized Helium and Xenon MRI and Their Applications in. . .

183

Fig. 10.6 Scheme of the 13 C glucose metabolic pathway. Most organisms rely on the breakdown of glucose to provide energy to their cells. The complex molecules transfer through specific transporters into the cell membrane and break down into simpler molecules via an enzyme to release energy, known as adenosine triphosphate (ATP). In the presence of O2 , normal cells consume glucose and convert it to pyruvate and carry the final product via the Krebs cycle (TCA or citric acid cycle) and oxidative phosphorylation in the mitochondria, producing 36 moles of ATP, carbon dioxide, and water. The mitochondria diagram describes that the resulting pyruvate from the glycolysis pathway carries the metabolic system through the TCA cycle to undergo oxidative phosphorylation

10.5 Hyperpolarized Helium and Xenon MRI and Their Applications in Medical Imaging The hyperpolarization technology is not limited to carbon-based protocols. Gases like .129 Xe and .3 He are also used for such a task at low and high magnetic fields [184, 194, 195]. More specifically, .3 He-MRI as a noble gas isotope has been used in human studies since 1995 [173]. In the particular case of lung study, it is challenging to visualize it with proton-MRI due to the low lung proton density. Therefore, HP .129 Xe and .3 He MRI are a valuable method for mapping lung ventilation and overcoming this problem. Gas MRI techniques are more sensitive than proton-MRI, with fewer side effects than the CT scan. More specifically, inhaling any of these gases as a contrast agent to visualize MRI signals from the lungs is safer than a gadolinium contrast agent for proton-MRI to improve .T1 relaxation time for tissues or blood [176, 176, 198]. Thus, these noble gases provide a powerful method for lung, brain, and kidney medical

184

10 Hyperpolarized MRI Technique and Its Application in Medical Science

Fig. 10.7 Scheme of the spin-exchange optical pumping (SEOP) method for .129 Xe gas. The SEOP method has two main steps. First, rubidium metal (. Rb) is polarized by infrared laser light with a wavelength of .λ = 794.7 nm. Then, spin exchange occurs between the polarized . Rb state and the 129 Xe state through hyperfine interaction. The collisions between unpaired electrons in the . Rb and . 129 Xe gas can polarize the gas .

imaging and have the ability to dissolve in tissues and red blood cells shortly after inhaling [199, 199, 200]. The HP MRI of noble gases provides a signal with a high signal-to-noise ratio compared with other standard HP MRI methods [147, 195, 201, 202]. The main advantage of gas HP MRI in lung imaging is their sensitivity at low-field MRI. Several studies have investigated low-field MRI to visualize lung anatomy and function in the literature. As we noted earlier, the spin-exchange optical pumping (SEOP) is a powerful method for polarizing gases like .129 Xe and .3 He. We display a summary of hyperpolarization mechanism for .129 Xe experiment in Fig. 10.7. The diagram displays the optical pumping and spin exchange process on .129 Xe nucleus before HP MRI injection. According to Fig. 10.7, there are two main steps to polarize .129 Xe by the SEOP method. In this method, the required wavelength to polarize an alkali metal rubidium (. Rb) is the infrared laser light with .λ = 794.7 nm, followed by a spin exchange between . Rb state and .129 Xe state via the hyperfine interaction [196, 197].

10.5 Hyperpolarized Helium and Xenon MRI and Their Applications in. . .

185

In the spin exchange step, the collision between an unpaired electron of . Rb and 129 Xe gas can polarize the sample in a few hours [203]. 129 Xe, besides having all advantages of .3 He gas, has several other benefits for . human study applications, such as its lower operating cost, significant chemical shift, and higher polarizability properties [54, 204, 205]. Moreover, .129 Xe gas has a fast spin-exchange speed and requires a low concentration HP sample, which makes it an excellent HP candidate to produce a strong signal in the lung imaging [206, 207]. The magnetic resonance chemical shifts of .129 Xe appear near 0 .ppm (air space), 197 .ppm for blood plasma or barrier, and 218 .ppm for red blood cells [208–210]. Furthermore, .T1 relaxation time of HP .129 Xe is sufficiently long in the blood and is recorded in a range of 3.4–7.8 s in .1.5 T at the room temperature [176, 198, 211]. Another method used to polarize gas is the metastability exchange optical pumping (MEOP) method, which is only practical for polarizing .3 He nucleus [177, 212]. The MEOP method does not utilize . Rb metal and only applies laser light with .λ = 1083 nm to polarize the electron state. Therefore, the polarized electron state transfers into .3 He state, which happens faster than the SEOP method. In this part, we introduce some applications of HP MRI gases at low fields. Tsai et al. investigated open-access pulmonary imaging based on HP .3 He at low-field .6.5 mT with a resonance frequency of 210 KHz [213]. They considered the effect of different types of posture on the final result. Also, 2D and 3D MRI images with high signal-to-noise ratios were performed in their study. Their results demonstrated that the open-access lung imager provides posture-dependent pulmonary functional imaging with a resolution and sensitivity sufficient for diagnosis [214]. Furthermore, similar studies were also performed in [175, 215, 216]. Moreover, Salerno et al. introduced new technologies to consider the role of magnetic-susceptibility-induced field inhomogeneities on interleaved-spiral and interleaved-echo-planar lung images at .0.54 T and .1.5 T. They measured .T2∗ with HP .3 He gas at .0.54 T and compared their results with the one corresponding to .1.5 T [217, 218]. Their results revealed that by decreasing magnetic susceptibility at a lower field, the average of .T2∗ relaxation time for .3 He gas in the healthy human lung is .26.8 ± 1.5 ms (for .1.5 T) and .67.9 ± 1.3 ms (for .0.54 T), demonstrating approximately three times increase of the relaxation time at the low-field case. As we mentioned earlier, a big challenge in HP MRI noble gases at low-field is their sensitivity to electrical noise, such as RF coils [195, 219]. To reduce the noise and improve the signal-to-noise ratio, Dominguez et al. used Litz wire, in their setup, compared with the copper wire in phantoms and in vivo in rat lungs [220]. Their results showed a signal-to-noise ratio improvement of up to .131% for 129 Xe and .42% for .3 He, compared with the conventional copper wire at .73.5 mT. . .

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10 Hyperpolarized MRI Technique and Its Application in Medical Science

Exercises Problem 1 Suppose a carbon nucleus signal enhances up to 5000 after the hyperpolarization method. If the original signal from the carbon nucleus was 10 units, what is the new signal strength after the hyperpolarization method?

Solution 1 To calculate the signal strength, we have Signal strength after HP method =

.

Original signal strength × HP enhancement number. Therefore, we can calculate the signal strength after the hyperpolarization method, such that the new signal will be .5000 × 10 = 50,000 units.

Problem 2 For the SEOP method, it was discussed that the infrared laser light with a wavelength of .λ = 794.7 nm is used. Find the energy carried by a photon that has this wavelength.

Solution 2 The energy of a photon with the frequency .ν is given by E = hν.

.

On the other hand, .ν = c/λ, and thus we have c E=h . λ

.

(10.9)

We have .h = 6.626 × 10−34 J.S and .c = 3 × 108 m/s. Therefore, we find the energy of a photon as E = 6.626 × 10−34

.

3 × 108 = 2.5 × 10−19 J. 794.7 × 10−9

(10.10)

Chapter 11

Some Specific NMR and MRI Techniques

11.1 Introduction In this chapter, we aim to highlight some practical and powerful NMR and MRI techniques, such as superconducting quantum interference devices (SQUIDs), earth’s magnetic field NMR (EFNMR), and NMR cryoporometry (NMRC). These techniques have widespread applications in material science, biology, and medicine. At the end of this chapter, we summarize how we process NMR or MRI experiments in in vitro and in vivo setups.

11.2 Superconducting Quantum Interference Devices One of the principal advantages of low-field MRI is that it provides a low-cost alternative to high-field instruments for MR imaging. The frequency of operation is much lower for low-field systems. For instance, a low-field MRI with .0.06 T operates at approximately .2.46 MHz frequency, while a 3 T MRI operates closer to 128 MHz. Moreover, the magnetic field provided at the lower field is significantly homogeneous, which makes it an excellent clinical research resource [29, 195]. Superconducting quantum interference devices (SQUIDs) are a common method used for various research fields like geophysics, neuroscience, material science, NMR research, and many other research areas that need a very low-field power. SQUIDs are a physical phenomenon based on the Josephson effect, named after the physicist Brian David Josephson that discovered the Josephson effect 1962 [221]. SQUIDs contain a superconducting loop with one or more Josephson junctions to measure the slightest changes in a magnetic field. The RF SQUID contains only one Josephson junction, and DC SQUID contains two or more Josephson junctions. We represent a simple RF SQUID and a DC SQUID scheme in Fig. 11.1a,b, respectively. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Khashami, Fundamentals of NMR and MRI, https://doi.org/10.1007/978-3-031-47976-2_11

187

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11 Some Specific NMR and MRI Techniques

Fig. 11.1 Scheme of the superconducting quantum interference devices (SQUIDs). (a) A single Josephson junction, known as an RF SQUID. The magnetic flux of the loop is φ. (b) SQUID with a direct current is a standard setup utilized in medical research, known as DC SQUID. It includes two parallel Josephson junctions inserted in a superconducting loop. The voltage is a function of the magnetic flux (φ), which is measured and used to calculate the flux quantum

From Maxwell’s and Ampere’s equations, magnetic flux .φ passing through the area .S with magnetic field B is equal to the integral of the vector potential A with the loop circumference .C from point 2 to 1 in the figure, satisfying the Stoke’s theorem, which is written as f φ=

f B · dS =

.

S

f A · dS =

C

1

A · dS.

(11.1)

2

The electric current density of the system, .j (r), in terms of the macroscopic wave function .w = |w0 |eiθ is given by [222, 223] j (r) =

.

−e |w|2 (h∇θ + 2eA), ¯ me

(11.2)

where e and m are the charge and mass of the particle in the presence of a magnetic field, respectively. The first term of the above equation, .h∇θ ¯ , comes from the momentum term [224], and the second term, .2eA, is the contribution of the vector potential A and charges e. By assuming the electric current density .j = 0 inside the superconductor, and .|w|2 = w02 to be a nonzero constant term, we get h∇θ + 2eA = 0. ¯

.

(11.3)

11.2 Superconducting Quantum Interference Devices

189

We integrate both sides of the above equation around the loop .C and apply Stoke’s theorem as f h¯

2

.

1

∇θ · dl = hAθ = h(θ ¯ ¯ 2 − θ1 ).

(11.4)

Thus, f hAθ = −2e ¯

f A · dl = −2e

.

C

∇ × A · dS.

(11.5)

S

By considering .∇ × A = B and Eq. (11.1), we have f hAθ = −2e ¯

B · dS = −2eφ.

.

(11.6)

S

Therefore, the magnetic flux .φ is determined as φ=

.

−h¯ Aθ, 2e

(11.7)

where .Aθ is the phase change of the charge, which with a complete trip around .C is equal to .±2π n, n is an integer number. Therefore, by substituting in Eq. (11.6), we have φ=

.

nh −h¯ (±2π n) = ± , 2e 2e

(11.8)

where .h¯ = h/2π . Thus, the smallest unit of the flux quantum that was predicted by Fritz London [225] is φ0 =

.

6.626 × 10−34 h = = 2.0678 × 10−15 Wb = 2.0678 × 10−15 Tm2 , 2e 2 × 1.602 × 10−19 (11.9)

where the charge is .e = 1.602 × 10−19 Coulomb. The flux quantum from the above relation indicates that the magnetic flux is a quantized parameter like charge [226, 227]. SQUID NMR technique is sensitive to the minor magnetic field signals from the heart and brain [228, 229]. The SQUID NMR method delivers outstanding information for spectroscopy and imaging. Therefore, this technology is mainly used for scientific research and medical applications. Depending on their applications in science, SQUIDs have some advantages and disadvantages [230, 231]. In Table 11.1, we provide a list of some benefits and drawbacks of SQUIDs. Typically, low-field or ultra-low-field magnetic fields have numerous benefits, but their sensitivity to environmental noise can pose significant challenges for

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11 Some Specific NMR and MRI Techniques

Table 11.1 A list of some drawbacks and benefits of SQUIDs The benefits of SQUIDs

The drawbacks of SQUIDs

Sensitive to magnetic field of about −15 T [232] .10 Detect frequency range from DC to GHz The noise level is about 2 .fT .Hz−1/2 [233] Suitable for using in limited space Stable device for a long time Energy-efficient

Need careful structure setup Requires a very low temperature .≈ 4.2 K [229] Sensitive to the environment’s magnetic field Detect a very narrow bandwidth Need protection from environment noise Expensive to purchase

researchers. Nevertheless, many studies have attempted to handle this issue, and here, we review some of the most promising approaches. One shortcoming of a low-filed NMR and MRI technology is that it is sensitive to noise originating from the environment, as we mention in Table 11.1 for SQUIDs, known as external electromagnetic interference (EMI) signals. There are several developments to prevent and remove EMI during the scan time. For instance, Huang et al. showed that the SQUID magnetometer technology could detect environmental noise and clear any signal around the MRI coil after applying the adaptive suppression method. Using this method, they significantly improved the imaging bandwidth [195]. O’Reilly et al. adjusted a conductive cloth over the subject to reduce additional noise in the system [201]. Moreover, Liu et al. developed a commercial method for a Hyperfine scanner to eliminate the unwanted magnetic field and RF shielding for the brain at a .0.055 T MRI scanner. They also successfully applied .T1 and .T2 -weighted imaging on the system [147]. Recently, Srinivas et al. introduced a dynamic EMI remover, external dynamic inTerference estimation and removal (EDITER), to remove image artifacts. They investigated for phantom in a shielded environment with an 80 mT scanner and in vivo brain experiment in an unshielded environment at .47.5 mT. The results indicated the image quality improved, and the signal-to-noise ratio enhanced up to a factor of 9 for both EMI environments. The advantage of this model is that it does not need to calibrate data or adjust the imaging sequence at the end of each acquisition time [202]. To overcome the sensitivity of low-field or ultra-low-field magnetic fields, the HP method, as described in Chap. 10, can enhance the signal intensity and improve the signal-to-noise ratio of the signal significantly [234–238].

11.3 Earth’s Magnetic Field NMR Low-field NMR has many applications, including Earth’s magnetic field NMR (EFNMR). This technique takes advantage of the unique characteristics of Earth’s

11.4 NMR Cryoporometry for Porous Materials

191

magnetic field, which is simultaneously weak and uniform, making it ideal for highresolution NMR spectroscopy [239, 240]. Despite the weak magnetic field, which can affect NMR signal intensity due to the spin polarization level being directly proportional to the applied magnetic field from the Boltzmann distribution, the uniformity of Earth’s magnetic field can make the EFNMR an ideal instrument for practical applications. This uniformity, combined with the accessibility and global availability of the Earth’s magnetic field, makes it a valuable resource for low-field NMR studies [241–243]. One powerful benefit of EFNMR is its low cost compared with high-field NMR instruments. Instead of using large magnets with a magnetic field strength of .5,000– .30,000 Gauss, EFNMR relies on Earth’s .0.5 Gauss magnetic field, making it a cost-effective alternative for NMR experiments. EFNMR is user-friendly, making it an applicable device for spectroscopy research and imaging in unconventional locations. Studying spin–lattice and spin–spin relaxation time delivers valuable insights into the system’s dynamics in the presence of a magnetic field. As mentioned, .T1 and .T2 depend strongly on the magnetic field and sample temperature. Therefore, EFNMR is an exciting research opportunity to study proton relaxation times in Earth’s magnetic field. For example, the earth’s magnetic field near the equator is about 30 .μT , and near the poles is about 60 .μT . Therefore, the Larmor frequency of the proton is about .1.3 .KHz near the equator and .2.5 KHz near the poles [244]. Moreover, recently, people have used the hyperpolarization method at Earth’s magnetic field to improve the output signal from biological samples [236, 245].

11.4 NMR Cryoporometry for Porous Materials NMR Cryoporometry (NMRC) is a non-destructive and non-invasive method that utilizes NMR spectroscopy to consider the pore-size distribution of materials. NMRC depends on the principles of NMR and relaxation time to investigate the melting-point depression of a material, which relies on the pore’s size, shape, and surface properties. Moreover, NMRC has numerous applications in various fields, such as gas separation and rock analysis. J.W. Gibbs, J. Thomson, W. Thomson, and J.J. Thomson developed the basis for the theory of the NMRC in porous materials [246]. They introduced the meltingpoint depression, .ATm , as ATm = Tm − Tm (x) =

.

4σ Tm , xAHf ρ

(11.10)

where .Tm is the melting point for the liquid or bulk sample, and .σ is the surface energy at the liquid-solid interface. Also, .Tm (x) is the crystal’s or solid’s melting temperature with a linear diameter x, and .AHf is the material enthalpy of fusion.

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11 Some Specific NMR and MRI Techniques

Moreover, .ρ is the density of the solid or crystal. The Gibbs–Thomson equation can be written as ATm =

.

k , x

(11.11)

where k is a calibration constant that can determine the influence of pore geometry [247, 248]. The quantitative method for NMRC is established on the Gibbs–Thomson equation. The melting curve converts to the pore-size distribution curve, which is linked to the pore volume .v(x), and is named the pore-size distribution. The differentiation of the pore volume with respect to the pore diameter x is determined as .

dv dv dTm (x) = , dx dTm (x) dx

(11.12)

where .dv/dTm (x) determines the volume variation of all pores with a melting-point depression of .Tm (x) [249, 250]. The proton NMR is a powerful method in NMRC since the liquid typically includes water. The signal intensity of materials is measured by creating the FID signal and using the spin-echo pulse sequence. We need to mention that the effect of the inhomogeneous magnetic field can change NMR spectroscopy results. The spin-echo pulse sequence is a practical method to eliminate this drawback. As we discuss in Chap. 5, the type of material affects the molecule’s correlation time and tumbling rate. From the correlation time of a molecule, we can consider the relaxation time of the sample, which is related to the geometry of the pore’s material, the amount of sample volume, and the volume of liquid is involved in the experiment. .T2 relaxation time depends on the surface chemistry of the sample, which considers the loss of coherence of the magnetization. It is related to a spherical pore’s surface-to-volume ratio, .S/V . Generally, the rate of .T2 by considering the surface parameters is written as .

1 1 1 εS = + · , T2,obs T2,bulk T2,surf V

(11.13)

where .T2,obs , .T2,bulk , and .T2,surf are the observed, the bulk liquid, and the surface of the pore relaxation time, respectively. The bulk relaxation is usually larger than the surface relaxation time. Also, .ε is the thickness of the adsorbed surface of the water molecules. Moreover, the surface relaxivity, .ρ, can determine by ρ=

.

which is in .1/ms unit.

ε T2,surf

,

(11.14)

11.4 NMR Cryoporometry for Porous Materials

193

Fig. 11.2 (a) Scheme of a spherical pore material filled with liquid. (b) .T2 relaxation time, and (c) relaxation rate of the sample when the surface area to volume ratio changes. Typically, the bulk relaxation is larger than the surface relaxation time. In this plot, we consider .T2,bulk = 10 ms and .T2,obs = 5 ms .T2

When the surface area to volume ratio increases, the second term of Eq. (11.13) enhances. Therefore, the observed relaxation time decreases. The relaxation rate is the inverse of the relaxation time (where .R2 = 1/T2 ), which is a linear function of the .S/V ratio. Also, if .T2,bulk >> T2,surf , we can ignore the first term in the above equation, and therefore, we obtain the observed relaxation rate is directly proportional to .S/V ratio. In Fig. 11.2a, we display surface and bulk regions for relaxation time, showing that transverse relaxation can happen due to interactions within the bulk sample and with the surface. In Fig. 11.2b,c, we consider .T2 relaxation time and .T2 relaxation rate of the sample once .S/V ratio changes in the units of .ε. Moreover, we can consider the above equations for analyzing .T1 relaxation time in the pore system. It is worth noting that EFNMR can be a valuable tool for measuring .T1 and .T2 relaxation times of porous materials at low fields [251].

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11 Some Specific NMR and MRI Techniques

11.5 A Summary of Processing NMR and MRI Experiment The NMR or MRI sample is collected from either cell culture (in vitro) or animal and human model (in vivo) experiments, as we present in Fig. 11.3a. The isotope molecule such as .1 H, .13 C, .31 P, .15 N, and .19 F is dissolved into special cell culture media and added to cell culture dishes to track their metabolism in the cell. As we mentioned earlier, all isotope molecules have specific standard external references, which are utilized to normalize the output signal from the scanner [252, 253]. After doping the powder in the cell culture dishes, the extracted cell sample is transferred into the NMR tube and placed in the magnet. Therefore, the final NMR spectrum is processed and displayed on a monitor for further metabolic analysis, as shown in Fig. 11.3b. Moreover, as we mentioned earlier, the polarized or nonpolarized sample is prepared for an MRI scan and injected into the body’s soft tissue for in vivo study [253–256]. The NMR technique aims to precisely detect and quantify the metabolic changes in the cells or organs. For MR imaging, depending on the type of study performed, the sample is prepared, and an MRI scanner images the subject. Then, the MRI files are saved in a digital imaging and communications in medicine (DICOM) format and considered for further analysis.

Fig. 11.3 Scheme of the in vitro and in vivo NMR and MRI experiment (a) Sample collection. (b) Data collection

Exercises

195

Fig. 11.4 (a) MRI Image processing. (b) Data analysis

Several computing methods, like image processing in MATLAB, ImageJ software, and FT NMR method, are used to process an MR image on the computer. For instance, the area of interest is detected in the MATLAB segmentation code manually or automatically, and after some analysis, the data value is exported. Then, the final output of data, known as “Voxel” and “Pixel” information, as we explained before, is plotted. We present the scheme of the summary of analysis steps in Fig. 11.4.

Exercises Problem 1 The critical magnetic field denoted as .Bc (T ) represents the maximum magnitude of the applied magnetic field at temperature T , above which the material loses its superconductivity [257]. The temperature dependency of the critical magnetic field can be characterized by

196

11 Some Specific NMR and MRI Techniques

[

(

Bc (T ) = Bc (0) 1 −

.

T Tc

)2 ] .

(11.15)

Here .Bc (0) is the critical magnetic field at zero temperature. A superconducting wire has a critical temperature, .Tc , of 10 K and a critical magnetic field of .Bc (0), 2 T. If the wire is cooled to 5 K, what will be the highest applied magnetic field for which the wire will still be a superconductor?

Solution 1 For .Tc = 10 K, .Bc (0) = 2 T, and .T = 5 K. [

(

Bc (T ) = 2 1 −

.

5 10

)2 ] = 1.5 T .

(11.16)

Problem 2 Using the equation introduced in the previous problem, find an explicit expression for the temperature T , and find the maximum temperature that a metal remains superconductor, if it has .Bc (0) = 4 T and .Tc = 6 K and is placed in a magnetic field of 3 T.

Solution 2 ( .

T Tc

)2 =1−

B , Bc

(11.17)

where we have / T . = Tc

1−

B . Bc

(11.18)

B . Bc

(11.19)

Therefore, we have / T = Tc 1 −

.

Once, B/Bc = 3/4, therefore, we obtain T = Tc /2 from the above relation. For Tc = 6 K, we obtain T = 3 K.

Appendix A

Direct Product

A general quantum state of a single spin on this basis was given as |ψ> = α| ↑> + β| ↓>.

.

(A.1)

This can be expressed, in a vector form, as |ψ> = α

.

( ) ( ) ( ) 1 0 α +β = . 0 1 β

(A.2)

Now, if .|ψ> = α| ↑> + β| ↓> and .|φ> = a| ↑> + b| ↓>, the product state .|φ> ⊗ |ψ> can be written in the vector form as ⎛ ( )⎞ ⎛ ⎞ α aα ( ) ( ) ⎜ a β ⎟ ⎜ aβ ⎟ a α ⎟ ⎜ ⎟ . |φ> ⊗ |ψ> = ⊗ =⎜ ⎝ ( α ) ⎠ = ⎝ bα ⎠ . b β b β bβ For a two-spin system with .2 × 2 matrices, let us define two matrices as ( .

A=

ab c d

)

and ( .

B=

αβ γ δ

) .

The direct product of these two matrices is determined as

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Khashami, Fundamentals of NMR and MRI, https://doi.org/10.1007/978-3-031-47976-2

197

198

A Direct Product

) ( ⎛ ( αβ α ( ) ( ) a b ⎜ ab αβ γ δ γ .A ⊗ B = ⊗ =⎜ ⎝ (α β ) (α c d γ δ c d γ δ γ ⎛

.

aα ⎜ aγ =⎜ ⎝ cα cγ

aβ aδ cβ cδ

bα bγ dα dγ

⎞ bβ bδ ⎟ ⎟. dβ ⎠ dδ

)⎞ β δ ⎟ )⎟ β ⎠ δ

(A.3)

Appendix B

The Density Matrix and Hamiltonian Elements for the Two-Spin System

In Sect. 9.2, we have a matrix form of the Hamiltonian. Here, we provide the details to arrive at the matrix form of the Hamiltonian. The spin matrices for the .I = 1/2 (setting .h¯ = 1) are given by Iix =

.

] [ 1 01 2 10 i

] [ 1 0 −i 2 i 0 i

Iiy =

Iiz =

] [ 1 1 0 , 2 0 −1 i

(B.1)

which using the Pauli matrix representations we can write Iiz =

.

1 σiz , 2

S=

1 σ. 2

(B.2)

The total Hamiltonian of the system was given as H=

.

1 1 (ωΣ + ωδ )I1z + (ωΣ − ωδ )I2z + J I1 · I2 . 2 2

(B.3)

We can write this Hamiltonian as .H = Hz + HJ , with Hz =

.

1 1 (ωΣ + ωδ )I1z + (ωΣ − ωδ )I2z , 2 2

(B.4)

and HJ = J I1 · I2 .

.

(B.5)

From the properties of spin operators, we know that .Iz acts on the individual spin states .|α> and .|β> as Iz |α> =

.

1 |α>, 2

1 Iz |β> = − |β>. 2

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Khashami, Fundamentals of NMR and MRI, https://doi.org/10.1007/978-3-031-47976-2

(B.6) 199

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B The Density Matrix and Hamiltonian Elements for the Two-Spin System

We also know that the spin states are orthonormal. In other words, = = 1,

= = 0.

.

(B.7)

Also, we have I1z ± I2z = I1z ⊗ I2 ± I1 ⊗ I2z ,

(B.8)

.

in which .I1 and .I2 are the identity matrices of the first and the second spins, respectively. Similarly, we have I1x = I1x ⊗ I2 , .

(B.9)

I2x = I1 ⊗ I2x .

(B.10)

.

Therefore, we calculate the matrix elements of the free part of the Hamiltonian as .

=

ωΣ , 2

=

ωδ , 2

(B.11)

and, .

=

−ωδ , 2

=

−ωΣ , 2

(B.12)

and the rest are zero. The resulting matrix is ⎛ ωΣ 2

0

⎜ 0 ωδ 2 Hz = ⎜ ⎝ 0 0 0 0

.

0 0

−ωδ 2

0

⎞ 0 0 ⎟ ⎟. 0 ⎠

(B.13)

−ωΣ 2

To determine the interaction part, we note that the inner product of the operators for the .J coupling part is given as I1 · I2 = I1x ⊗ I2x + I1y ⊗ I2y + I1z ⊗ I2z .

.

(B.14)

This can be determined by calculating each single term. For example, we have

I1x ⊗ I2x =

.

1 2

(

01 10

) ⊗

1 2

(

01 10

)

⎛

0 1⎜ 0 = ⎜ 2 ⎝0 1

0 0 1 0

0 1 0 0

⎞ 1 0⎟ ⎟. 0⎠ 0

Therefore, we can calculate each term and insert in Eq. (B.14), to get

(B.15)

B The Density Matrix and Hamiltonian Elements for the Two-Spin System

⎛

1 0 1⎜ ⎜ 0 −1 .I1 · I2 = 2 ⎝0 2 0 0

0 2 −1 0

⎞ 0 0⎟ ⎟. 0⎠ 1

We insert these elements into the Hamiltonian to arrive at its matrix form.

201

(B.16)

Appendix C

The Eigenstate and Eigenvalue of the Two-Spin System

The Hamiltonian is .4 × 4, which we represent as ⎛

h11 ⎜ 0 .H = ⎜ ⎝ 0 0

0 h22 h32 0

0 h23 h33 0

⎞ 0 0 ⎟ ⎟. 0 ⎠ h44

(C.1)

To find the eigenstates of the Hamiltonian, we note that .h11 and .h44 are two eigenvalues since all other elements in their corresponding column and rows are zero. In other words, .λ1 = h11 with the eigenstate .≡ |ββ>, and also the eigenvalue .λ4 = h44 has the eigenstate .≡ |αα>. To determine the eigenvalue .λ2 and .λ3 , we simply need to find the eigensystem problem for H' =

(

.

h22 h23 h32 h33

)

( =

αv v β

) .

(C.2)

Thus (

α−λ v . det v β −λ

) = 0,

(C.3)

which gives (α − λ)(β − λ) − v 2 = 0 .

λ2 − (α + β)λ + αβ − v 2 = 0.

(C.4)

Then, for .aX2 + bX + C = 0, we have eigenvalue as

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 F. Khashami, Fundamentals of NMR and MRI, https://doi.org/10.1007/978-3-031-47976-2

203

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C The Eigenstate and Eigenvalue of the Two-Spin System

√ √ −b ± b2 − 4ac A −b ± . λ± = = . 2a 2a

(C.5)

Thus, we have

.

λ± =

(α + β) ±

/

) ( (α + β)2 − 4 αβ − v 2 2

,

where .A = (α + β)2 − 4αβ + 4v 2 = (α − β)2 + 4v 2 = ωδ2 + 4v 2 . So, we find [ ] / 1 1 2 2 ωΣ ± ωδ + 4v = [ωΣ ± D] , . λ± = 2 2

(C.6)

(C.7)

where .ωΣ = ωα + ωβ and .ωδ = ωα − ωβ . Also, we have from Eq. (9.11) D=

/

.

wδ2 + 4v 2 =

/

ωδ2 + J 2 ,

(C.8)

where we suppose .2v = J in the equation. Now, we find the eigenstates of the system. We define the state for Hamiltonian as ( ) ( ) p± ' p± = λ± , .H (C.9) q± q± where .|q± |2 + |p± |2 = 1, and .p± and q± are real values. We consider the orthonormal basis as ( ) ( ) ( ) ( ) p+ cos θ − sin θ p− = = . (C.10) , . q+ q− sin θ + cos θ Therefore, by replacing the above equation, we have ( .

αv v β

)(

p+ q+

)

( = λ+

p+ q+

) .

(C.11)

Thus, we have { .

αp+ + vq+ = λ+ p+ , vp+ + βq+ = λ+ q+ .

(C.12)

If we multiply both sides of the above equations to .q+ and .p+ , respectively. We have

C The Eigenstate and Eigenvalue of the Two-Spin System

{ .

q+ × (αp+ + vq+ ) = λ+ q+ p+ , p+ × (vp+ + βq+ ) = λ+ q+ p+ .

205

(C.13)

) ( 2 2 + (α − β)p q = 0. Therefore, we Then, we subtract them and attain .v q+ − p+ + + obtain .

v p+ q+ v = . = 2 − q2 α − β c p+ +

(C.14)

From Eq. (C.10), we have 2 sin θ cos θ .

cos2 θ − sin2 θ

=

2v 2v ⇒ tan 2θ = . ωδ ωδ

(C.15)

Thus, we prove .

tan 2θ =

J , ωδ

(C.16)

which is Eq. (9.12). Therefore, we can find

.

2v sin 2θ = √ , A ωδ cos 2θ = √ . A

(C.17)

J , D ωδ cos 2θ = . D

(C.18)

Thus, we have sin 2θ = .

So, we proved Eq. (9.10).

Appendix D

The Free Induction Decay Signal Elements for the Two-Spin System

To calculate the unitary time evolution operator of the system, we define the Hamiltonian of the system in its diagonal form as H = E1 |φ1 > + sin θ |αβ>)(cos θ )(− sin θ