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Magnetic Resonance Technology
Hardware and System Component Design
By Andrew G Webb The Royal Society of Chemistry
Copyright © 2016 The Royal Society of Chemistry
All rights reserved.
ISBN: 978-1-78262-359-5
CHAPTER 1
The Principles of Magnetic Resonance, and Associated Hardware
ANDREW WEBB
C.J. Gorter Center for High Field MRI, Department of Radiology, Leiden University Medical Center, Leiden, The Netherlands
1.1 Introduction
The diversity of magnetic resonance (MR) experiments is enormous, ranging from simple one-dimensional proton nuclear magnetic resonance (NMR) spectroscopy through multi-dimensional multi-nuclear spectra to full three-dimensional magnetic resonance imaging (MRI) of morphology and function in animals and humans. Some examples of the types of data produced from different MR experiments are shown in Figure 1.1.
Despite the widely different information content of these data, the fundamental hardware systems for NMR spectroscopy (in both the liquid and solid states) and MRI (human and animal) are very similar. The basic components include the following:
(i) The magnet, which polarizes the nuclei and produces a net magnetization within the sample.
(ii) The radiofrequency coil(s), which transmit pulses of electromagnetic (EM) energy into the sample and detect the precessing magnetization, which constitutes the MR signal.
(iii) Magnetic field gradient coils, which induce a spatial dependence of the nuclear precession frequency and can be used for coherence order selection, measurements of diffusion, and MRI.
(iv) Shim coils, which are used to produce as homogeneous a magnetic field as possible throughout the sample, (v) The receiver electronics and circuitry, which amplify, filter and digitize the MR signal for data storage and post-processing.
The physical arrangement of components (i) to (iv) is shown in Figure 1.2 for a vertical bore magnet.
In addition, there are a series of electronic components used to switch the gradients on and off, to produce high power RF pulses, and to amplify and digitize the signal. A simplified block diagram of a generalized MR system is shown in Figure 1.3.
Table 1.1 gives an idea of the characteristics and performance of components in typical commercial NMR and MRI systems. The system performances of each of the components in the table are explained in greater detail in the relevant sections throughout the book.
In the following sections in this chapter the basic phenomena involved in magnetic resonance are explained briefly, with links to the relevant system hardware. There are a large number of MR books dealing with the basic theory of high-resolution liquid-state NMR, solid-state NMR and MRI, and readers are advised to consult these tomes for much more in-depth analyses of different aspects of basic MR theory.
1.2 The Superconducting Magnet and Nuclear Polarization
The role of the magnet is to polarize the nuclei to produce a net magnetization within the sample. For NMR spectroscopy and MRI, almost all magnets are superconducting. The magnetic field should be temporally stable and homogeneous to within parts-per-billion (ppb) throughout the sample. Most magnets are actively shielded, i.e. the fringe field does not extend significantly outside the physical dimensions of the magnet itself. Magnet design is considered in detail in Chapter 2 of this book, as well as Appendix 1A at the end of this chapter.
All nuclei with an odd atomic weight and/or an odd atomic number possess a fundamental quantum mechanical property termed "spin" and are termed "spin-active" or "NMR-active". The most important spin-active nuclei include 1H, 13C, 15N, 23Na, 17O, 31P and 2H. Notably spin-inactive are nuclei such as 16O and 12C. Considering the proton as the simplest example, the property of spin can be viewed as the proton spinning around an internal axis of rotation giving it a certain value of angular momentum (P). Since the proton is a charged particle, this rotation results in a magnetic moment (μ). This magnetic moment produces an associated magnetic field, which has a configuration similar to that of a bar magnet. The magnitude of P is quantized in terms of the nuclear spin quantum number (7):
[MATHEMATICAL EXPRESSION OMITTED] (1.1)
where h is Planck's constant (6.63 x 10-34 Js). In the following analysis a spin 1/2 nucleus (I = 1/2) is considered, corresponding to 1H, 13C, 15N, and 31P in the previous list. In this case:
[MATHEMATICAL EXPRESSION OMITTED] (1.2)
The magnitudes of the magnetic moment and the angular momentum of the proton are related by:
[MATHEMATICAL EXPRESSION OMITTED] (1.3)
where γ is the nuclear gyromagnetic ratio, and has a specific value for different nuclei, with protons having the highest γ (with the exception of tritium). For protons therefore:
[MATHEMATICAL EXPRESSION OMITTED] (1.4)
μ contains three components (μx, μy and μz), each of which can have any value within the conditions governed by eqn (1.4): this situation is shown in Figure 1.4(a). However, in the presence of a strong magnetic field, B0, μz is quantized with values given by:
μz = γh/2π mI (1.5)
where mI is the nuclear magnetic quantum number, and can take values I, I – 1 ... –I. In the case of a proton, m1 = [+ or -]1/2 and so:
μz = [+ or -]γh/4π (1.6)
The orientation of μ with respect to B0 is shown in Figure 1.4(b). The interaction of the static magnetic (B0) field with μz results in Zeeman splitting, producing two energy levels: one in which μz aligns parallel to B0 (the lower energy state) and the other anti-parallel (the higher energy state).
The net magnetization, M0, of a sample containing Ns protons is proportional to the difference in populations between the two energy levels, which is dictated by Boltzmann's equation:
[MATHEMATICAL EXPRESSION OMITTED] (1.7)
Eqn (1.7) shows that the net polarization of a sample is proportional to the strength of the main magnetic field. However, since the energy difference between the two levels is very small, so is the population difference. For example, at an operating magnetic field of 11.7 tesla for every one million protons, there is a population difference of only ~40 protons between the parallel and anti-parallel orientations.
1.3 The Transmitter Coil to Generate Radiofrequency Pulses
In order to detect an MR signal, energy must be applied to the nuclear spin system to stimulate transitions between the two energy levels. A pulse of EM energy is applied at the specific resonance frequency (f0), which corresponds to the energy difference between the two levels via:
hf0 = ΔE = γhB0/2π (1.8)
The resonance frequency in Hz, or resonance angular frequency (ω0) in radians per second, is therefore given by:
f0 = γB0/2π, ω0 = γB0 (1.9)
The pulsed EM wave consists of both magnetic and electric field components, and it is the magnetic component that interacts with the nuclear magnetization. The EM energy, termed a radiofrequency (RF) pulse, is transmitted via an RF coil. As shown in Figure 1.5, the magnetic field component, B+1, of the EM wave from the RF coil must be created in a direction perpendicular to B0 in order to interact with the magnetization.
Using classical mechanics, the action of the RF pulse applied along one axis produces a torque perpendicular to that axis. As shown in Figure 1.5(b), the angle a by which the magnetization is rotated is proportional to the product of the strength of the applied RF field and the time, [MATHEMATICAL EXPRESSION OMITTED], for which it is applied.
[MATHEMATICAL EXPRESSION OMITTED] (1.10)
After application of an RF pulse with tip angle a about the x-axis, the net magnetization can be expressed as three different vectors.
Mz = M0 cos α, Mx = 0, My = M0 sin a (1.11)
The geometric design of the RF coil is determined by the requirement that the B+1 field produced by the coil must be perpendicular to B0. There are a large number of different geometries, which are discussed in detail in Chapter 3. One example, widely used in high-resolution liquid-state NMR, is the "saddle" coil, shown in Figure 1.6. The long axis of the coil is coincident with B0, and current flowing through the copper conductor produces a B+1 field with the required orientation. The RF coil is tuned and impedance matched to 50 Ω at the Larmor frequency for high efficiency. The RF coil may be tuned to more than one frequency, as also discussed in Chapter 3.
Power to the RF coil is supplied by an RF amplifier, the general specifications of which include:
(i) Amplifiers for liquid-state and solid-state NMR experiments must be able to produce pulses as short as 1 us, with accurate and reproducible shape, at frequencies up to ~1 GHz. In solid-state NMR applications, very high power pulses with a very high duty cycle may be used for proton decoupling.
(ii) In the case of MRI, up to 30–60 kW of power must be available without the performance varying over time owing to any component heating, as well as the amplifiers being capable of high duty cycles for RF-intensive sequences, such as turbo spin-echoes.
The design of RF amplifiers is described in detail in Chapter 6.
1.4 Precession
When the RF pulse is turned off, the component of magnetization in the transverse plane precesses around B0, as shown in Figure 1.7(a). The precession frequency, ωprecession, is exactly the same as the frequency of irradiation:
ωprecession = ω0 = γB0 (1.12)
The concept of a rotating reference frame is very useful in analyzing the behaviour of the net magnetization, and is shown in Figure 1.7(b). The rotating reference frame (x'y') is denned as rotating around B0 at an angular frequency ω0.
In fact, the exact precession frequencies of different nuclei within a molecule are determined by two other factors, chemical shift and scalar coupling, which are outlined in the following sections.
1.4.1 Chemical Shift
The term "chemical shift" refers to the fact that protons in different chemical environments within a molecule resonate at slightly different frequencies: their precession frequencies are shifted (with respect to a reference, discussed below) with the magnitude of the shift depending on their particular chemical environment. The cause of this shift is that the exact magnetic field experienced by each proton in the molecule is slightly lower than B0 owing to the shielding effects of the electron cloud surrounding each proton. Electrons have a magnetic moment, which is opposite in sign to that of the proton, and so the effective magnetic field experienced by the proton is reduced. As an example, consider the lactic acid molecule shown in Figure 1.8. There are four different proton groups (CH3, CH, OH, COOH) in this molecule: each of these proton groups has a slightly different precession frequency since they experience a slightly different magnetic field. The effective magnetic field, Beff, experienced by a proton is given by:
Beff = B0(1 - σ) (1.13)
where σ is called the shielding constant, and is related to the electronic environment surrounding the nucleus. The resonant frequency of the proton is given by:
ω = γBeff = γB0(1 - σ) (1.14)
One of the main factors that determines the value of σ is the electroneg-ativity of the atoms connected to the protons: the proton in an –OH group has a lower shielding constant than those in a –CH2– group since oxygen is more electronegative than carbon and pulls electrons away from the proton, thus reducing the shielding. The resonant frequency of the proton of an –OH group is therefore higher than that of the protons in a –CH2– group. The chemical shift (δ), in units of parts-per-million (ppm), is defined as:
δ(ppm) = 106 f-fref/fref (1.15)
where fref is the resonant frequency of the protons in tetramethylsilane (TMS), which acts as a reference for proton NMR spectra. Figure 1.9 shows approximate chemical shift ranges for protons in different chemical environments. For the lactic acid molecule, the order of resonance frequencies is: CH3< CH < OH < COOH.
1.4.2 Scalar Coupling
Consider the protons in the molecule shown in Figure 1.10. There are three chemically distinct protons (HA, HB, and HC), which have three different chemical shifts and corresponding resonant frequencies. However, there is also an interaction between the two protons HA and HB, which are separated by three chemical bonds (HA-to-C-to-C-to-HB). The magnetic field experienced by HA depends upon whether HB is in the a (parallel) or β (anti-parallel) state, and vice versa: this phenomenon is termed scalar coupling. Therefore, there are four different energy levels for this coupled two-proton system: αα, αβ, βα, and ββ, as shown in Figure 1.10(b). The exact value of the scalar coupling constant/ depends upon a number of different factors, including the particular type of chemical bond (single, double or triple bond), as well as the angle subtended between the C–H bonds (the Karplus angle), but in general the values are between 1 and 7 Hz. The greater the number of bonds between the protons the weaker the coupling, and so for the molecule in Figure 1.10(a) there is effectively no coupling between H and H since there are five chemical bonds between the protons.
For the coupled HA–HB system, the four different transition energies corresponding to the energy level diagram in Figure 1.10(b) are given by:
[MATHEMATICAL EXPRESSION OMITTED] (1.16)
Figure 1.11 shows the evolution of the precessing magnetization for the three different protons. From Figure 1.9 the chemical shift of HC has the lowest value, followed by HA and the highest value for HB. HC precesses at a single frequency, whereas protons HA and HB are "split" into two frequencies by the scalar coupling.
1.4.3 Relaxation Processes
The final effect that must be considered in terms of the precession of the net magnetization is relaxation. After the RF pulse has been turned off, each of the magnetization components Mz, Mx and My returns to their thermal equilibrium values, with the time-evolution determined by specific time-constants.
The time-evolutions of Mz, Mx and My are characterized by differential equations known as the Bloch equations:
[MATHEMATICAL EXPRESSION OMITTED] (1.17)
The return of Mz to its equilibrium value of M0 is governed by the spin-lattice (T1) relaxation time, and the return of Mx and My, to their thermal equilibrium value of zero by the spin–spin (T2) relaxation time. It should be noted that the relative values of T1 and T2 can be very different for different types of sample, but T1 is always greater or equal to T2. For non-viscous liquids used in high-resolution NMR, the values of T1 and T2 are very similar. In contrast, in solid samples T2 can be up to six orders shorter than T1, and for human MRI the T2 of many soft tissues is between one and two orders of magnitude smaller than T1.
Solving the Bloch equations for the Mx and My components of magnetization gives:
[MATHEMATICAL EXPRESSION OMITTED] (1.18)
(Continues...)
Excerpted from Magnetic Resonance Technology by Andrew G Webb. Copyright © 2016 The Royal Society of Chemistry. Excerpted by permission of The Royal Society of Chemistry.
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