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Principles of magnetic resonance imaging

Principles of magnetic resonance imaging

Taken from Mike Puddephat's PhD, this article describes the principles of magnetic resonance imaging.

Nuclear magnetic resonance (NMR) is a non-invasive means of obtaining clinical images and of studying tissue metabolism in vivo. Bloch and Purcell independently discovered NMR in 1946 (Bloch (1946), Bloch et al. (1946) and Purcell et al. (1946)). Six years later they were awarded the Nobel Prize for their achievements. Since then, the development of NMR spectrometers and NMR scanners has led to the opening up of whole new branches of physics, chemistry, biology and medicine.

Jadetzky and Wertz (1956) and Odeblad et al. (1956) employed NMR in chemical analysis for the investigation of molecular structure and molecular motion in solids and liquids. Moon and Richards (1973) and Hoult et al. (1974) were able to obtain biochemical information from cells and tissues. The process of detecting metabolites by NMR is known as magnetic resonance spectroscopy (MRS) and the first MR spectrum of a human tumour was obtained by Griffiths et al. (1981) from a rhabdomyosarcoma on the dorsum of the hand. The process of acquiring two and 3D images by NMR, known as magnetic resonance imaging (MRI), was first illustrated by Lauterbur (1973) who produced a 2D MR image of a phantom. Over the last 20 years, Fourier transform imaging techniques have tremendously accelerated the development of MRI (Kumar et al., 1978).

The main aim of this article is to provide an overview of the principles of NMR. For a more detailed account, refer to a book such as "NMR and its applications to living systems" by Gadian (1995). Many of the figures in this article are based on illustrations from "Basic Principles of MR Imaging", written by Keller (1988).

Nuclei with an odd number of protons and neutrons possess a property called spin. In quantum mechanics spin is represented by a magnetic spin quantum number. Spin can be visualised as a rotating motion of the nucleus about its own axis. As atomic nuclei are charged, the spinning motion causes a magnetic moment in the direction of the spin axis. This phenomenon is shown in Figure 1. The strength of the magnetic moment is a property of the type of nucleus. Hydrogen nuclei (1H), as well as possessing the strongest magnetic moment, are in high abundance in biological material. Consequently hydrogen imaging is the most widely used MRI procedure.

Figure 1

Figure 1: A charged, spinning nucleus creates a magnetic moment which acts like a bar magnet (dipole).

Consider a collection of 1H nuclei (spinning protons) as in Figure 2(a). In the absence of an externally applied magnetic field, the magnetic moments have random orientations. However, if an externally supplied magnetic field B0 is imposed, the magnetic moments have a tendency to align with the external field (see Figure 2(b)).

Figure 2
                       (a)                                                         (b)

Figure 2: (a) A collection of 1H nuclei (spinning protons) in the absence of an externally applied magnetic field. The magnetic moments have random orientations. (b) An external magnetic field B0 is applied which causes the nuclei to align themselves in one of two orientations with respect to B0 (denoted parallel and anti-parallel).

The magnetic moments or spins are constrained to adopt one of two orientations with respect to B0, denoted parallel and anti-parallel. The angles subtended by these orientations and the direction of B0 are labelled theta in Figure 3(a). The spin axes are not exactly aligned with B0, they precess around B0 with a characteristic frequency as shown in Figure 3(b). This is analogous to the motion of a spinning top precessing in the earth's gravitational field. Atomic nuclei with the same magnetic spin quantum number as 1H will exhibit the same effects - spins adopt one of two orientations in an externally applied magnetic field. Elements whose nuclei have the same magnetic spin quantum number include 13C, 19E and 31P. Nuclei with higher magnetic spin quantum number will adopt more than two orientations.

Figure 3
                       (a)                                             (b)

Figure 3: (a) In the presence of an externally applied magnetic field, B0, nuclei are constrained to adopt one of two orientations with respect to B0. As the nuclei possess spin, these orientations are not exactly at 0 and 180 degrees to B0. (b) A magnetic moment precessing around B0. Its path describes the surface of a cone.

The Larmor equation expresses the relationship between the strength of a magnetic field, B0, and the precessional frequency, F, of an individual spin.
Equation
The proportionality constant to the left of B0 is known as the gyromagnetic ratio of the nucleus. The precessional frequency, F, is also known as the Larmor frequency. For a hydrogen nucleus, the gyromagnetic ratio is 4257 Hz/Gauss. Thus at 1.5 Tesla (15,000 Gauss), F = 63.855 Megahertz.

Radiofrequency field and MR signal

For a collection of 1H nuclei, let the number of spins adopting the parallel and anti-parallel states be P1 and P2 respectively, with corresponding energy levels E1 and E2. E2 is greater than E1 causing P1 to be greater than P2. An obvious question is why do spins adopt the higher energy anti-parallel state? The answer is that spins of P2 may move to P1 if the exact amount of energy, delta(E) = E2 - E1 is supplied to the system. If the temperature of the system were absolute zero, all spins would adopt the parallel orientation. Thermal energy will cause P2 to be populated. At room temperature in a 1.5 Tesla magnetic field, there will typically be a population ratio P2:P1 equal to 100,000:100,006.

At any given instant, the magnetic moments of a collection of 1H nuclei can be represented as vectors, as shown in Figure 4. Every vector can be described by its components perpendicular to and parallel to B0. For a large enough number of spins distributed on the surface of the cone, individual components perpendicular to B0 cancel, leaving only components in the direction parallel to B0. As most spins adopt the parallel rather than the antiparallel state, the net magnetisation M is in the direction of the B0 field.

Figure 4

Figure 4: A collection of spins at any given instant in an external magnetic field, B0. A small net magnetisation, M, is detectable in the direction of B0.

Suppose the direction of B0 is aligned with the z-axis of Euclidean 3-space. The plane perpendicular to B0 contains the x and y-axes. In order to detect a signal from 1H nuclei, radio frequency (RF) energy must be applied. RF energy at the Larmor frequency causes nuclear spins to swap between parallel and anti-parallel states. This has an oscillatory effect on the component of M parallel to the z-axis. RF energy, like all electromagnetic radiation, has electric and magnetic field components. Suppose the magnetic field component is represented by B1 and lies in the x-y plane. The x-y components of M will be made coherent by the B1 field giving a net x-y component to M and hence effectively cause M to tilt from the z direction into the x-y plane. This phenomenon is described further in Figure 5.

The angle through which M has rotated away from the z-axis is known as the flip angle. The strength and duration of B1 determine the amount of energy available to achieve spin transitions between parallel and anti-parallel states. Thus, the flip angle is proportional to the strength and duration of B1. After pulses of 90 degrees and 270 degrees, M has no z component and the population ratio P2:P1 is exactly one. A pulse of 180 degrees rotates M into a position directly opposite to B0, with greater numbers of spins adopting anti-parallel (rather than parallel) states. If the B1 field is applied indefinitely, M tilts away from the z-axis, through the x-y plane towards the negative z direction, and finally back towards the x-y plane and z-axis (where the process begins again).

Figure 5

Figure 5: (top) The effect of RF radiation on the net magnetisation M is to produce a second magnetic field Mx-y. M is tilted from its original longitudinal z-axis orientation, along the direction of the external magnetic field B0, into the transverse x-y plane. (bottom) An illustration of flip angle, which is the angle through which M has rotated away from the z-axis.

Figure 6(a) shows the situation after an RF pulse is applied that causes the net magnetisation vector M to flip by 90 degrees. M lies in the x-y plane and begins to precess about the B0 axis. M will induce an electromotive force in a receiver coil according to Faraday's law of magnetic induction. This is the principle of NMR signal detection. It is from this received RF signal that an MR image can be constructed. Figure 6(b) shows a graph of the voltage or signal induced in a receiver coil verses time. Such a graph, or waveform, is termed a free induction decay (FID). The magnitude of the generated signal depends on the number of nuclei contributing to produce the transverse magnetisation and on the relaxation times (see next section).

Figure 6
                       (a)                                                     (b)

Figure 6: (a) After a 90 degrees RF pulse, M lies in the x-y plane and rotates about the z-axis. The component of M in the x-y plane decays over time. An alternating current, shown in Figure (b), is induced in the receiver coil.

Relaxation Processes

The return of M to its equilibrium state (the direction of the z-axis) is known as relaxation. There are three factors that influence the decay of M: magnetic field inhomogeneity, longitudinal T1 relaxation and transverse T2 relaxation. T1 relaxation (also known as spin-lattice relaxation) is the realignment of spins (and so of M) with the external magnetic field B0 (z-axis). T2 relaxation (also known as T2 decay, transverse relaxation or spin-spin relaxation) is the decrease in the x-y component of magnetisation.

Magnet Inhomogeneity

It is virtually impossible to construct an NMR magnet with perfectly uniform magnetic field strength, B0. Much additional hardware is supplied with NMR machines to assist in normalising the B0 field. However, it is inevitable that an NMR sample will experience different B0's across its body so that nuclei comprising the sample (that exhibit spin) will have different precessional frequencies (according to the Larmor equation). Immediately following a 90 degree pulse, a sample will have Mx-y coherent. However, as time goes on, phase differences at various points across the sample will occur due to nuclei precessing at different frequencies. These phase differences will increase with time and the vector addition of these phases will reduce Mx-y with time.

T1 Relaxation

Following termination of an RF pulse, nuclei will dissipate their excess energy as heat to the surrounding environment (or lattice) and revert to their equilibrium position. Realignment of the nuclei along B0, through a process known as recovery, leads to a gradual increase in the longitudinal magnetisation. The time taken for a nucleus to relax back to its equilibrium state depends on the rate that excess energy is dissipated to the lattice. Let M-0-long be the amount of magnetisation parallel with B0 before an RF pulse is applied. Let M-long be the z component of M at time t, following a 90 degree pulse at time t = 0. It can be shown that the process of equilibrium restoration is described by the equation
Equation
where T1 is the time taken for approximately 63% of the longitudinal magnetisation to be restored following a 90 degree pulse.

T2 Relaxation

While nuclei dissipate their excess energy to the lattice following an RF pulse, the magnetic moments interact with each other causing a decrease in transverse magnetisation. This effect is similar to that produced by magnet inhomogeneity, but on a smaller scale. The decrease in transverse magnetisation (which does not involve the emission of energy) is called decay. The rate of decay is described by a time constant, T2*, that is the time it takes for the transverse magnetisation to decay to 37% of its original magnitude. T2* characterises dephasing due to both B0 inhomogeneity and transverse relaxation. Let M-0-trans be the amount of transverse magnetisation (Mx-y) immediately following an RF pulse. Let M-trans be the amount of transverse magnetisation at time t, following a 90 degree pulse at time t = 0. It can be shown that
Equation

In order to obtain signal with a T2 dependence rather than a T2* dependence, a pulse sequence known as the spin-echo has been devised which reduces the effect of B0 inhomogeneity on Mx-y. A pulse sequence is an appropriate combination of one or more RF pulses and gradients (see next section) with intervening periods of recovery. A pulse sequence consists of several components, of which the main ones are the repetition time (TR), the echo time (TE), flip angle, the number of excitations (NEX), bandwidth and acquisition matrix.

Figures 7 and 8 show pictorially how the spin echo pulse sequence works. Figure 7 is a graph of pulsed RF and received signal verses time, while Figure 8 is a phase diagram of the magnetisation vector M. After a 90 degree pulse, a signal is formed which decays with T2* characteristics. This is illustrated by the top right ellipse in Figure 8 which shows three spins at different phases due to their different precessional frequencies. The fastest spin is labelled f and the slowest s. At time TE/2, a 180 degree pulse is applied to the sample (see bottom left ellipse in Figure 8) which causes the three spins to invert. After inversion, the order of the spins is reversed with the fastest lagging behind the others. At time TE, the spins become coherent again so that a signal (known as the spin echo) is produced.

If a further 180 degree pulse is applied at time TE/2 after the peak signal of the first spin echo, then a second spin echo signal will form at time TE after the first spin echo. The peak signal amplitude of each spin echo is reduced from its previous peak amplitude due to T2 dephasing which cannot be rephased by the 180 degree pulses. Figure 9 shows how the signal from a spin echo sequence decays over time. A line drawn through the peak amplitude of a large number of spin echoes describes the T2 decay, while individual spin echoes exhibit T2* decay.

Signal strength decays with time to varying degrees depending on the different materials in the sample. Different organs have different T1s and T2s and hence different rates of decay of signal. When imaging anatomy, some degree of control of the contrast of different organs or parts of organs is possible by varying TR and TE. The intensity of a spin echo signal, I, can be approximated as
Equation
where N(H) is the proton density and f(V) is a function of flow.

Figure 7

Figure 7: Formation of a spin echo at time TE after a 90 degree pulse.

Figure 8

Figure 8: Dephasing of the magnetisation vector by T2* and rephasing by a 180 degree pulse to form a spin echo.

Figure 9

Figure 9: Decay of signal with time in a spin echo sequence.

Magnetic Resonance Imaging

The actual location within the sample from which the RF signal was emitted is determined by superimposing magnetic field gradients on the magnet generating the otherwise homogeneous external magnetic field B0. For example, a magnetic field gradient can be superimposed by placing two coils of wire (wound in opposite directions) around the B0 field with longitudinal axis orientated in the z direction and then by passing direct current through the coils. The magnetic field from the coil pair adds to the B0 field, with the result that one end of the magnet has a higher field strength than the other. According to the Larmor equation, the magnetic field gradient causes identical nuclei to precess at different Larmor frequencies. The frequency deviation is proportional to the distance of the nuclei from the centre of the gradient coil and the current flowing through the coil.

Slice Selection

If with the above gradient switched on, a single frequency RF pulse is applied to the whole sample, only a narrow plane perpendicular to the longitudinal axis at the centre of the sample will absorb the RF energy. Everywhere else in the sample is receiving the wrong frequency of excitation for resonance to occur. This technique allows a slice, with thickness determined by the magnetic field gradient strength, to be selected from a sample.

Frequency Encoding

Three magnetic field gradients, placed orthogonally to one another inside the bore of the magnet, are required to encode information in three dimensions. With a slice selected and excited as described in the previous paragraph, current is switched to one of the two remaining gradient coils (referred to as the frequency encoding gradient). This has the effect of spatially encoding the excited slice along one axis, so that columns of spins perpendicular to the axis precess at slightly different Larmor frequencies. For a homogeneous sample, the intensity of the signal at each frequency is proportional to the number of protons in the corresponding column.

The frequency encoding gradient is turned on just before the receiver is gated on and is left on while the signal is sampled or read out. For this reason the frequency encoding gradient is also known as the readout gradient. The resulting FID is a graph of signal (formed from the interference pattern of the different frequencies) induced in the receiver verses time. If the FID is subjected to Fourier transform, a conventional spectrum in which signal amplitude is plotted as a function of frequency can be obtained. Thus, a graph of signal verses frequency is obtained which corresponds to a series of lines or views representing columns of spins in the slice. Figure 10 shows two simple FIDs and their Fourier transforms.

Figure 10

Figure 10

Figure 10: Two FIDs and their Fourier transforms.

Phase Encoding

Suppose a slice through a homogeneous sample has been selected and excited as described in Slice Selection section, and then frequency encoded according to the previous section. After a short time, the phase of the spins at one end of the gradient leads those at the other end because they are precessing faster. If the frequency encoding gradient is switched off, spins precess (once more) at the same angular velocity but with a retained phase difference. This phenomenon is known as phase memory.

A phase encoding gradient is applied orthogonally to the other two gradients after slice selection and excitation, but before frequency encoding. The phase encoding gradient does not change the frequency of the received signal because it is not on during signal acquisition. It serves as a phase memory, remembering relative phase throughout the slice.

To construct a 256 x 256 pixels image a pulse sequence is repeated 256 times with only the phase encoding gradient changing. The change occurs in a stepwise fashion, with field strength decreasing until it reaches zero, then increasing in the opposite direction until it reaches its original amplitude. At the end of the scan, 256 lines (one for each phase encoding step) comprising 256 samples of frequency are produced. A Fourier transformation allows phase information to be extracted so that a pixel (x, y) in the slice can be assigned the intensity of signal which has the correct phase and frequency corresponding to the appropriate volume element. The signal intensity is then converted to a grey scale to form an image.

MRI Sequences

MRI signal intensity depends on many parameters, including proton density, T1 and T2 relaxation times. Different pathologies can be selected by the proper choice of pulse sequence parameters. Repetition time (TR) is the time between two consecutive RF pulses measured in milliseconds. For a given type of nucleus in a given environment, TR determines the amount of T1 relaxation. The longer the TR, the more the longitudinal magnetisation is recovered. Tissues with short T1 have greater signal intensity than tissues with a longer T1 at a given TR. A long TR allows more magnetisation to recover and thus reduces differences in the T1 contribution in the image contrast. Echo time (TE) is the time from the application of an RF pulse to the measurement of the MR signal. TE determines how much decay of the transverse magnetisation is allowed to occur before the signal is read. It therefore controls the amount of T2 relaxation. The application of RF pulses at different TRs and the receiving of signals at different TEs produces variation in contrast in MR images. Next some common MRI sequences are described.

Spin Echo Pulse Sequence

The spin echo (SE) sequence is the most commonly used pulse sequence in clinical imaging. The sequence comprises two radiofrequency pulses - the 90 degree pulse that creates the detectable magnetisation and the 180 degree pulse that refocuses it at TE. The selection of TE and TR determines resulting image contrast. In T1-weighted images, tissues that have short T1 relaxation times (such as fat) present as bright signal. Tissues with long T1 relaxation times (such as cysts, cerebrospinal fluid and edema) show as dark signal. In T2-weighted images, tissues that have long T2 relaxation times (such as fluids) appear bright.

In cerebral tissue, differences in T1 relaxation times between white and grey matter permit the differentiation of these tissues on heavily T1-weighted images. Proton density-weighted images also allow distinction of white and grey matter, with tissue signal intensities mirroring those obtained on T2-weighted images. In general, T1-weighted images provide excellent anatomic detail, while T2-weighted images are often superior for detecting pathology.

Gradient Recalled Echo Pulse Sequences

Gradient recalled echo (GRE) sequences, which are significantly faster than SE sequences, differ from SE sequences in that there is no 180 degree refocusing RF pulse. In addition, the single RF pulse in a GRE sequence is usually switched on for less time than the 90 degree pulse used in SE sequences. The scan time can be reduced by using a shorter TR, but this is at the expense of the signal to noise ratio (SNR) which drops due to magnetic susceptibility between tissues. At the interface of bone and tissue or air and tissue, there is an apparent loss of signal that is heightened as TE is increased. Therefore it is usually inappropriate to acquire T2-weighted images with the use of GRE sequences. Nevertheless, GRE sequences are widely used for obtaining T1-weighted images for a large number of slices or a volume of tissue in order to keep scanning times to a minimum. GRE sequences are often used to acquire T1-weighted 3D volume data that can be reformatted to display image sections in any plane. However, the reformatted data will not have the same in-plane resolution as the original images unless the voxel dimensions are the same in all three dimensions.

MRI Artefacts

The term artefact refers to the occurrence of undesired image distortions, which can lead to misinterpretation of MRI data. The theoretical limit of the precision of measurements obtained from medical images is determined by the point spread function of the imaging device (Rossmann (1969) and Robson et al. (1997)). However, in practice, the limit is determined by the physiological movements of a living subject (e.g. respiration, heartbeat, twitching or tremor). The finite thickness of the slice of tissue imaged may also represent a constraint. If the signals arising from different tissue compartments cannot be separated within each voxel, then an artefact known as partial voluming is produced. This uncertainty in the exact contents of any voxel is an inherent property of the discretised image and would even exist if the contrast between tissues were infinite. Furthermore, chemical shift and susceptibility artefacts (Schenck (1996)), magnetic field and radio frequency non-uniformity, and Field of View and slice thickness calibration inaccuracies can all compromise the accuracy with which quantitative information can be obtained for a structure of interest in the living human body. A detailed analysis of all these effects is, however, beyond the scope of this article.

References

BLOCH, F. Nuclear induction. Phys. Rev., 70, 460 (1946).

BLOCH, F., HANSEN, W. W. and PACKARD, M. The nuclear induction experiment. Phys. Rev., 70, 474 (1946).

GADIAN, D. G. Nuclear magnetic resonance and its applications to living systems. Oxford University Press, New York (1995).

GRIFFITHS, J. R., STEVENS, A. N., IIES, R. A., GORDON, R. F. and SHAW, D. 31P NMR investigation of solid tumours in the living rat. Bioscience Rep., 1, 319-325 (1981).

HOULT, D. I., BUSBY, S. J. W., GADIAN, D. G., RICHARDS, R. E. and SEELEY, P. H. Observation of tissue metabolites using 31P nuclear magnetic resonance. Nature, 252, 285-287 (1974).

JADETZKY, O. and WERTZ, J. F. Detection of sodium complexes by nuclear spin resonance. Am. J. Physiol., 187, 608 (1956).

KELLER, P. Basic Principles of MR Imaging. GE Medical Systems, Milwaukee (1988).

KUMAR, A., WELTI, D. and RENST, R. R. NMR Fourier zeugmatography. J. Magn. Reson., 18, 69-83 (1978).

LAUTERBUR, P. C. Image formation by induced local interactions; examples employing nuclear magnetic resonance. Nature, 242, 190-191 (1973).

MOON, R. B. and RICHARDS, J. H. Determination of intracellular pH by 31P magnetic resonance. J. Biol. Chem., 248, 7276-7278 (1973).

ODEBLAD, K., BHAR, N. B. and LINDSTROM, G. Proton magnetic resonance of human red blood cells in heavy water exchange experiments. Arch. Biochem. Biophys., 63, 221-225 (1956).

PURCELL, E. M., TORREY, H. C. and POUND, R.

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