X-Nuclei Magnetic Resonance Imaging

Guillaume Madelin

• Éditeur: Jenny Stanford Publishing
• Année: 2022
• ISBN:
  9781000047660
• Édition: 1
• Sujet: Medicine
• Sous-sujet: Biochemistry in Medicine

Standard magnetic resonance imaging (MRI) is a prominent clinical imaging modality used to diagnose and study diseases in vivo. It is principally based on the detection of the nuclei of hydrogen atoms (the proton; symbol 1H) in water molecules in tissues. X-nuclei MRI (also called non-proton MRI) is based on the detection of the nuclei of other atoms (X-nuclei) in the body, such as sodium (23Na), phosphorus (31P), chlorine (35Cl), potassium (39K), deuterium (2H), oxygen (17O), lithium (7Li), and fluorine (19F) using modified software and hardware. X-nuclei MRI can provide fundamental, new metabolic information related to cellular energetic metabolism and ion homeostasis in tissues that cannot be assessed using standard hydrogen MRI.

This book is an introduction to the techniques and biomedical applications of X-nuclei MRI. It describes the theoretical and experimental basis of X-nuclei MRI, the limitations of this technique, and its potential biomedical applications for the diagnosis and prognosis of many disorders or for quantitative monitoring of therapies in a wide range of diseases. The book is divided into four parts. Part I includes a general description of X-nuclei nuclear magnetic resonance physics and imaging. Part II deals with the MRI of endogenous nuclei such as 23Na, 31P, 35Cl, and 39K; Part III, the MRI of endogenous/exogenous nuclei such as 2H and 17O; and Part IV, the MRI of exogenous nuclei such as 7Li and 19F. The book is illustrated throughout with many representative figures and includes references and reading suggestions in each section. It is the first book to introduce X-nuclei MRI to researchers, clinicians, students, and general readers who are interested in the development of imaging methods for assessing new metabolic information in tissues in vivo in order to diagnose diseases, improve prognosis, or measure the efficiency of therapies in a timely and quantitative manner. It is an ideal starting point for a clinical or scientific research project in non-proton MRI techniques.

Part I MR Physics and Imaging of X-Nuclei

Chapter 1 Spin Dynamics in NMR

1.1 Introduction

In this chapter, I will give an broad overview on the basics in quantum physics and spin dynamics used to interpret and simulate nuclear magnetic resonance (NMR) experiments. More details can be found in references [1–13].

1.2 Quantum Spin States and Density Operator

A quantum system (such as an individual particle, or an ensemble of individual particles) in a pure quantum state can be represented by a vector:

$|\psi \rangle ={\displaystyle \sum _m{a}_m|m\rangle },$ (1.1)

which is linear supersposition of all the elements of the vector basis states |m〉, where the coefficients a_m are complex and represent an “amplitude of probalility.” In the case of a spin with angular momentum quantum number I, this basis states |m〉 are labelled by the azimuthal quantum number m that can take values [–I, –I + 1,..., I – 1, I], which are eigenvalues of the angular momentum component I_z along the z-axis such that I_z |m〉 = m |m〉. In that case, the expectation value of an observable A is defined by

$\langle \psi \left|A\right|\psi \rangle ={\displaystyle \sum }_{m,m^{\prime }}{a}_m^{*}{a}_{m^{\prime }}\langle m^{\prime }|A|m\rangle .$ (1.2)

For example, if $A=I_z,\langle \psi \left|A\right|\psi \rangle ={{{\displaystyle \sum }}^{\text{}}}_m\left|a_m^2\right|m,where\left|a_m^2\right|$ can be interpreted as the probability to measure the system in quantum state m with the operator I_z.

However in NMR, we are dealing with macroscopic samples of large sub-ensembles of spins in different pure states, which can be interacting with each other and with external magnetic fields. This large mixed quantum system cannot be described with a simple “pure state” vector |ψ〉, but as a statistical average over an ensemble of pure independent systems, called the density operator:

$\rho =\overline{|\psi \rangle \langle \psi |}={\displaystyle \sum }_{\psi }{p}_{\psi }|\psi \rangle \langle \psi |,$ (1.3)

where the overbar symbol indicates the average of all members of the macroscopic ensemble, and p_ψ the probability of the system to be in pure state |ψ〉, and Σ_ψ p_ψ = 1. We therefore can measure the average (macroscopic) expected value of an operator A as:

$\langle A\rangle =\overline{\langle \psi \left|A\right|\psi \rangle }=\text{Tr(}\rho A)=\text{Tr(}{A}_{\rho }).$ (1.4)

The density operator ρ can be developed as a function of the elements of vector basis state |m〉:

$\rho ={\displaystyle \sum _{m,m^{\prime }=\mathrm{I}}^{-\mathrm{I}}{\rho }_{m,m^{\prime }}\langle m^{\prime }\left|A\right|m\rangle \equiv {\displaystyle \sum _{i,j=1}^N{{\rho }^{\prime }}_{i,j}\langle i\left|A\right|j\rangle }}$ (1.5)

with ${\rho }_{m,m\text{'}}=\overline{a_m^{*}a{}_{m\text{'}}}$ , and represented by the density matrix:

$\begin{array}{l}\rho =\left(\begin{array}{ccccc}{\rho }_{I,I}& {\rho }_{I,I-1}& {\rho }_{I,I-2}& \cdots & {\rho }_{I,-I}\\ {\rho }_{I-1,I}& {\rho }_{I-1,I-1}& {\rho }_{I-1,I-2}& \cdots & {\rho }_{I-1,-I}\\ {\rho }_{I-2,I}& {\rho }_{I-2,I-1}& {\rho }_{I-2,I-2}& \cdots & {\rho }_{I-2,-I}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ {\rho }_{-I,I}& {\rho }_{-I,I-1}& {\rho }_{-I,I-2}& & {\rho }_{-I,-I}\end{array}\right)\\ \equiv \left(\begin{array}{ccccc}{{\rho }^{\prime }}_{1,1}& {{\rho }^{\prime }}_{1,2}& {{\rho }^{\prime }}_{1,3}& \cdots & {{\rho }^{\prime }}_{1,N}\\ {{\rho }^{\prime }}_{2,1}& {{\rho }^{\prime }}_{2,2}& {{\rho }^{\prime }}_{2,3}& \cdots & {{\rho }^{\prime }}_{2,N}\\ {{\rho }^{\prime }}_{3,1}& {{\rho }^{\prime }}_{3,2}& {{\rho }^{\prime }}_{3,3,}& \cdots & {{\rho }^{\prime }}_{3,N}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ {{\rho }^{\prime }}_{N,1}& {{\rho }^{\prime }}_{N,2}& {{\rho }^{\prime }}_{N,3}& \cdots & {{\rho }^{\prime }}_{N,N}\end{array}\right)\end{array}$ (1.6)

where the matrix ρ′ is just another representation that is commonly used for the simplicity of its indices for the basis states, which are defi...