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This page is meant to be a general introduction to parallel magnetic resonance imaging and to the analysis of coil performance. The idea is to periodically update the following list with new paragraphs in order to create a good reference for anyone interested in delving into these topics. - Magnetic Resonance Imaging (MRI)
Magnetic Resonance Imaging (MRI) MRI has become a widely accepted diagnostic tool for its ability to produce high quality internal anatomical images. In a magnetic resonance experiment, the signal, which relates to the physical and biochemical properties of the sample, is detected in the form of a radiofrequency (RF) voltage induced in a detector coil in response to the application of alternate magnetic fields to the object of interest. In order to understand how an image is formed in MRI, it is necessary to start with a brief introduction to nuclear magnetization. The term NMR describes a resonance phenomenon, first observed in 1946, which involves magnetic fields and nuclei, more specifically nuclei with an odd number of protons and/or odd number of neutrons. Atoms with this characteristic possess an angular momentum, or spin. In the absence of an external magnetic field, the spins inside any object are randomly oriented and the net macroscopic magnetic moment is zero. In the presence of an external magnetic field, the magnetic moment vectors of the protons tend to align in the direction of the magnetic field and create a net magnetization. Furthermore, the nuclear spins have a resonance at a well-defined frequency called the Larmor frequency. The Larmor frequency (ω = γ Bo) is proportional to the magnitude Bo of the magnetic field and to a constant, the gyromagnetic ratio γ, which is unique for each type of atom and describes the ratio of its magnetic moment to its angular momentum.
Placing an object inside a static magnetic field
where ћ is the Planck’s constant divided by 2π, Ns is the total number of spins in the sample, I is the nuclear spin quantum number, kB is the Boltzmann’s constant and Ts is the absolute temperature of the sample. If the equilibrium of the proton system is perturbed by an externally applied magnetic field B1, oscillating at the same frequency of the spins, the magnetization vector is rotated by a certain angle and starts precessing with angular frequency about the main magnetic field Bo. After the external excitation is removed, the system returns to its thermal equilibrium state. This process is characterized by a precession of M about Bo (free precession), the recovery of the longitudinal component of M (longitudinal relaxation) and the decay of the transverse component of M due to dephasing among the ensemble of precessing spins (transverse relaxation). The time-varying magnetization resulting from the just described NMR phenomenon can be detected using the well-known Faraday’s law of induction. Any loop of wires resonating at the radio frequency (RF) of the precessing magnetization can be used as a receiver coil. Its complex-valued detection sensitivity C(r) can be determined recalling the principle of reciprocity. According to Faraday’s law M is responsible for a magnetic flux Φ through the surface of the detector coil that induce an electromotive force given by:
In the case of biological specimens, the MR system is commonly tuned to the Larmor frequency of the hydrogen nucleus (H), with a single proton, as it is the most abundant nuclear species in biological tissue and gives rise to the largest signals. The sample is composed of many atoms and the nucleus of each atom can be seen as a small oscillator inducing a RF signal in the receiver coil. The signal described by Eq. [2] is then a sum of local signal from all these components. The aim of MRI is to spatially differentiate the various contributions to the signal and map the distribution of their amplitudes, in order to generate an image. In conventional MRI, this is done by superimposing magnetic field gradients on the main magnetic field. If a linear magnetic field gradient G with a general orientation r is applied together with Bo, the Larmor frequency at position r becomes:
As a consequence, the net magnetization density acquires a spatially varying phase and can be expressed as:
Introducing the wave vector k = t(γ/2π)G and using the symbol S for the voltage signal, Eq. [2] can be rewritten as:
The expression shows that, when linear gradients are used, the signal corresponds to the Fourier transform of the magnetization density. The spin density distribution is simply obtained by performing the inverse Fourier transform of the signal and it is then used to generate an image. For Cartesian sampling schemes, linear combinations of magnetic field gradients can be applied to acquire images in any specific image plane. The strategy is the same for any arbitrary plane. From Eq. [5] it is possible to map each gradient encoding step to a point in the discrete space defined by k. Points of k-space along a single axis (i.e. frequency-encoding axis) are acquired applying gradients at the same time the signal is being received (frequency-encoding readout). In order to change the k-space coordinates in the axes perpendicular to the frequency-encoding axis (i.e. phase-encoding axes), magnetic field gradients are applied before the frequency-encoding readout. For each increment in the phase-encoding direction, a complete frequency-encoding readout is necessary. Volumetric (3D) acquisitions use two phase-encoding directions, whereas 2D acquisitions use only one. Many MR imaging protocols sample k-space following non-Cartesian schemes. Selected References: • Lauterbur P. C. (1980). "Progress in n.m.r. zeugmatography imaging." Philos Trans R Soc Lond B Biol Sci 289(1037):483-487. • Mansfield P. and Maudsley A. A. (1977). "Medical imaging by NMR." Br J Radiol 50(591): 188-194. The description above highlights the advantages associated with the use of magnetic field gradients for encoding spatial information, but also the intrinsic limits of such a method. If, on one hand, the technique enables one to use any arbitrary image plane or slice thickness and it is suitable for any kind of application, on the other hand it can be very slow, especially when high-resolution images are needed. In fact, for each k-space position in the phase-encoding direction the complete gradient readout in the frequency-encoding direction must be re-applied. One approach to this problem would be to apply the magnetic field gradients at a higher rate and strength to speed up the acquisitions. This approach results in reduced signal-to-noise ratio (SNR), but SNR may in principle be increased by moving to higher-field MR systems. However, the switching-rate of the magnetic field gradients is limited by physiological constraints, as rapidly varying magnetic fields have the potential to induce currents in the human body. The use of high-field MR scanners, meanwhile, raises issue of human tissue heating. The limits on imaging speed were in part overcome only after the development in the late nineties of parallel MRI, a new technique for spatial encoding, which enables to generate an image combining partial information acquired in parallel by the different elements of an array of detector coils. An array of detectors consists of a number of overlapping coils, closely positioned and each associated with an independent preamplifier and receiver chain. Each element receives a separate signal for every k-space position and further information is needed to combine the various portions into a single signal. The sensitivity profile, which accounts for the spatial inhomogeneity of the coil response to the precessing spins, can be measured for each coil of the array and used to weight the individual signal contributions. For example, the total signal for each k-space point can be obtained as the sum of the signals from the single coils, each multiplied by the complex conjugate of the value of the respective sensitivity at the same position. Coil arrays were initially used to acquire large FOVs as they offer the SNR and the resolution of a small coil over an extended region. However, it was with the introduction of parallel MRI that coil arrays reached their greatest popularity. The basic idea of parallel MRI is to exploit the different reception patterns of the array elements to perform spatial encoding. Using the spatial information provided by coil sensitivities to substitute for time-consuming gradient encoding steps enables one to accelerate the image acquisition process. The discretized MR signal received by the l-th element of the detector coil array at k-space point km, in the case of an arbitrary two-dimensional slice, can be expressed as:
where Cl(rj) is the sensitivity function for coil l, M(rj) is the transverse magnetization and nl(km) is the time-dependent Gaussian white noise. If the coil sensitivities and the gradient-induced modulation are combined in a single encoding matrix B, then Eq. [6] can be re-written in the simplified matrix notation:
The encoding matrix has a number of rows equal to the number of array elements times the total acquired k-space points and a number of columns equal to the pixels in the image:
In order to reconstruct an image is sufficient to find the matrix inverse of B. Let us say there are N pixels in the image and L elements in the coil array. Assuming the coil sensitivities are known (e.g. from a calibration measurement), if a full set of N k-space points is acquired, then Eq. [7] is an over-determined system of L*N equations and N unknowns. It is therefore possible to speed up the acquisition by reducing the sampled positions in k-space by a factor R, so long as L*N/R ≥ N, i.e. the matrix B is still invertible. Typically R is smaller than the number of coils L and so there are different choices for the inverse matrix Binv. It has been shown that the solution that leads to the lowest noise amplification in the reconstructed image is the modified Moore-Penrose pseudoinverse:
where Y is the noise covariance matrix, which describes the time-averaged statistical properties of the noise received by the coils and B* is the conjugate transpose of B. Undersampling by a factor R is equivalent to reducing the FOV by the same factor. That causes aliasing, which results in a fold-over effect in the image. Practically, parallel MRI enables reconstruction of a full-FOV image by combining the aliased images detected by each element of the coil array. The value of each pixel in the aliased images corresponds to the superposition of the signals from R aliased positions. Applying the matrix in Eq. [8] to the set of folded images, the R signal contributions are separated so that each pixel of the aliased images resolves into R pixel values that contribute to form the full-FOV image (see figure).
The general formulation presented here represents a brief introduction to the concepts of parallel MRI, showing with a simple formalism how the sensitivity functions of the coil array elements are exploited to achieve additional spatial encoding. Many other parallel MRI techniques have been reported in the scientific literature, but they all originated from the same basic idea. Selected References: • Roemer P. B., Edelstein W. A., et al. (1990). "The NMR phased array." Magn Reson Med 16(2): 192-225. • Kwiat D., Einav S., et al. (1991). "Decoupled coil detector array for fast image acquisition in Magnetic-Resonance-Imaging." Med Phys 18(2): 251-265. • Carlson J. W. and Minemura T. (1993). "Imaging time reduction through multiple receiver coil data acquisition and image-reconstruction." Magn Reson Med 29(5): 681-688. • Ra J. B. and Rim C. Y. (1993). "Subencoding data sets from multiple detectors." Magn Reson Med 30(1): 142-145. • Sodickson D. K. and Manning W. J. (1997). "Simultaneous acquisition of spatial harmonics (SMASH): fast imaging with radiofrequency coil arrays." Magn Reson Med 38(4): 591-603. • Pruessmann K. P., Weiger M., et al. (1999). "SENSE: sensitivity encoding for fast MRI." Magn Reson Med 42(5): 952-962. • Sodickson D. K. (2000). "Tailored SMASH image reconstructions for robust in-vivo parallel MR imaging." Magn Reson Med 44(2): 243-251. • Kyriakos W. E., Panych L. P., et al. (2000). "Sensitivity profiles from an array of coils for encoding and reconstruction in parallel (SPACE RIP)." Magn Reson Med 44(2): 301-308. • Heidemann R. M., Griswold M. A., et al. (2001). "VD-AUTO-SMASH imaging." Magn Reson Med 45(6): 1066-1074. • Sodickson D. K. and McKenzie C. A. (2001). "A generalized approach to parallel magnetic resonance imaging." Med Phys 28(8): 1629-1643. • Griswold M. A., Jacob P. M., et al. (2002). "Generalized Autocalibrating Partially Parallel Acquisitions (GRAPPA)." Magn Reson Med 47(6): 1202-1210. • Bydder M., Larkman D. J., et al. (2002). "Generalized SMASH imaging." Magn Reson Med 47(1): 160-170. • Yeh E. N., McKenzie C. A., et al. (2005). "Parallel magnetic resonance imaging with adaptive radius in k-space (PARS): constrained image reconstruction using k-space locality in radiofrequency coil encoded data." Magn Reson Med 53(6): 1383-1392.
Selected References:
Ultimate Intrinsic Signal-to-Noise Ratio (SNR)
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