Views and Reviews Is the Fastest MRI a Hologram? Michael Hutchinson, MD, PhD, Ulrich Raff, PhD From New York Core Neuroscience, 222 East 31st Street, New York, NY 10016, USA (MH); and Department of Physics and Medicine, University of Santiago de Chile, Santiago, Region ´ Metropolitana, Chile (UR).
ABSTRACT Real-time MR imaging might exert a profound influence on neuroscience in the future by enabling the direct visualization of neuronal interactions. At this time, however, all practical embodiments of MRI require at least some degree of gradient encoding, and this in turn sets a lower limit of about 100 ms for volume acquisition. A novel formulation of MRI is proposed here which is given the acronym ULTRA (Unlimited Trains of Radio Acquisitions). In the preferred embodiment ULTRA is completely free of gradient reversals, which allows for signal acquisition from the entire object volume simultaneously. This permits a rate of signal acquisition that is increased hundreds of times compared with existing techniques, with full 3-D imaging in as little as one millisecond. The proposed detector now resembles a holographic recording.
Keywords: Ultrafast, MRI, gradient-free, hologram, real-time. Acceptance: Received May 9, 2014. Accepted for publication May 11, 2014. Correspondence: Address correspondence to Michael Hutchinson, MD, PhD, New York Core Neuroscience, 222 East 31st Street, New York, NY 10016, USA. E-mail: hutchinson@ nyneuroscience.com. J Neuroimaging 2014;24:537-542. DOI: 10.1111/jon.12141
Introduction In the original formulation of MRI by Lauterbur,1 spatial encoding is achieved by means of a plurality of successively applied magnetic field gradients. Each new gradient is associated with a different radiofrequency (RF) excitation, and RF-induced echoes form a line in k-space, discretized into N elements, where N is the dimension of the image matrix. After N progressively increasing gradients, and N echoes, there are now N lines in k-space, and the N × N k-space matrix is subjected to a 2-D Fourier transform, rendering the N × N image matrix. In echoplanar imaging (EPI) Mansfield2 showed that spatial encoding could be rapidly achieved by means of trains of gradient reversals after a single RF excitation. On the other hand, parallel MRI (pMRI) uses the spatial sensitivities of multiple receiver coils arranged around the object to perform spatial encoding, thus reducing the number of gradient reversals and RF pulses. The first pMRI proposal by Hutchinson3,4 was a theoretical scheme for single shot imaging which ignored sources of noise in the surface coils. Subsequent analyses of detector arrays by Roemer5 and Ocali6 accounted for the effects of noise, with algorithms for conventional gradient encoding. This gave rise to the concept of “ultimate signal-to-noise,”6 by which was meant that, at least for gradient encoding, large numbers of small receiver coils—when properly configured—are more efficient than a single receiver. Parallel MRI underwent a substantial practical step forward when Sodickson,7 with SMASH, and Pruessmann,8 with SENSE, elegantly showed how gradient-encoding and spatial sensitivity encoding could be merged. (These techniques will
Copyright
henceforth be designated “hybrid” since they use both methods of encoding.) On the other hand this merging of the two fundamental ways of encoding comes at a considerable price, because for the hybrid techniques ultimate signal-to-noise ratio (SNR) is reached not with large numbers of coils, but with small numbers (typically 4).
Limitations of Parallel MRI SNR in hybrid pMRI is reduced by three unrelated sources: (i) electrical noise, (ii) reduced numbers of echoes, and (iii) a critical geometric factor, g, as discussed by Ohliger9 and Wiesiger.10 Sources of electrical noise include preamplifiers, coupling between adjacent coils, and body thermal noise. In addition, there is noise from the eddy currents in the body surface due not only to the gradient reversals but also to reradiation of signal by the coil, which is both a receiver and a dipole radiator. These sources of noise can be minimized, however, and for conventional gradient encoding multiple small coils have been shown to be more efficient than one large coil5,6 so there is no insurmountable noise problem inherent to small coils. When multiple coils are used to encode, however, there can be an additional problem due to the inefficiency of spatial sensitivity encoding, compared with gradient encoding, when the coils are large. This in turn is due to the slow spatial variations of the spatial sensitivity of each coil. In addition, since there are
◦ 2014 by the American Society of Neuroimaging C
537
Fig 1. Color-coded pixel-based spatial sensitivity map for a quadrant detector placed around the top right corner of the object matrix. Note the lack of granularity.
fewer phase-encoding steps SNR is further reduced. The SNR is now given by9,10 : SNRFull SNR = √ , Rg
(1)
where SNRFull is the SNR for gradient-only encoding, R is the acceleration factor (in this case the number of detectors) and g √ is the geometric factor. The loss factor R presents the reduced SNR due to the reduced number of phase encoding steps, and nothing can be done to mitigate this. The g-factor is intrinsic to the geometry of the coil array, and is a measure of the capacity of the array to compensate for reductions in gradient encoding. For very small numbers of detectors g is close to 1, but as the number of coils increases, and the number of phase-encoding steps correspondingly decreases, g suddenly becomes large and the images are degraded. It has generally been accepted that for all practical purposes this single limitation means that for hybrid techniques acceleration factors cannot be more than about 4, so that the rate of acquisition of SNR is about double that of nonparallel techniques. The reason g > 1 with increasing accelerations is that the transformations used to obtain the image in conventional pMRI are nonunitary, and the reason for this is that the solutions to Maxwell’s equations are “smooth,” by which is meant that for large detectors the spatial dependence of the radio field is not as granular as the spatial dependence of the gradients used for conventional encoding. This in turn means that large groups of adjacent pixels may have similar spatial sensitivity profiles. As R increases, and the number of gradient-encoding steps correspondingly decreases, more and more of the burden of encoding is shouldered by the receiver array. For R = N = 4, only 25% of the encoding is gradient-based. With more detectors there is an unsupportable reliance on spatial sensitivity encoding and this leads to rapid image degradation when R > 4. The lack of granularity of the spatial sensitivity map of a quadrant coil is illustrated in Figure 1. The spatial sensitivity
538
Fig 2. Pixel-based spatial sensitivity map for a point detector situated at the top right hand corner of the image matrix. Note the improved granularity of spatial sensitivity compared with the quadrant coil. This is illustrative only, since the details of granularity are obscured somewhat by the crudity of the color scale.
maps can be modeled for illustrative purposes by taking, as an approximation to the solutions of Maxwell’s equations, only the 1/r3 dependence of the near field dipole.
In this Case, Smaller May be Better This conventional view of an inherent limitation of all constructions of parallel MRI has been challenged recently by Keil and Wald.11 These authors observed that the widely accepted notion—that gradients must necessarily do a large part of the encoding—is becoming outdated. In this regard it should be recognized that the original concept for single shot, single slice, imaging free of gradient reversals3,4 was reduced to practice several years ago by McDougall and Wright,12 using a 64 channel coil with an acceleration factor of 64. This is, we believe, only possible because the detectors were small in comparison with the hybrid techniques, see below. Figure 2 demonstrates, as expected, that a point detector has a much more granular spatial sensitivity map than a quadrant detector. Note, also as expected, that for any given point detector the spatial sensitivity of a pixel in the center of the object matrix is much lower than for a quadrant detector. This explains the simplistic conventional view that small detectors are insensitive to deep tissue. This conventional view is, however, also incorrect11 because deep tissue is well represented in the set of all coils, and this compensates for the fact that its representation in any given coil is relatively small. Figure 3 illustrates the point. It is a representation of the pixel-based spatial sensitivity map for a single detector surrounding the matrix. The circular coil can be thought of as a large collection of very small coils. It therefore follows that for an array of very small surface coils, the spatial sensitivity of a central pixel is nearly the same as for a coil at the center. This is why small surface coils are sensitive to deep tissue, provided the whole massively parallel array is considered.
Journal of Neuroimaging Vol 24 No 6 November/December 2014
Fig 3. Pixel-based spatial sensitivity map for a single circular detector surrounding the image matrix. Note that for a pixel at the center of the matrix, spatial sensitivity is almost as great as for a pixel at the corner. This is because it is equally represented in all elements of the receiver coil, whereas a pixel near the edge is well-represented only in nearby detectors.
Comparison of Figures 2 and 1 illustrates how much more granular the spatial sensitivity map is for a point detector than for a quadrant coil. Also, taken together, Figures 2 and 3 show why there is no theoretical limit to the ability of small coils to encode deeply into tissue, as has been pointed out.11 At the same time, McDougall and Wright12 have confirmed this conjecture empirically, at least in the case of single slice imaging.
Methods In principle, current MRI techniques could acquire signal from the entire volume simultaneously, by successive application of gradients in the x, y, and z directions, and with the signal analyzed by means of a 3-dimensional Fourier transform. The problem with this in practice is that after each successive gradient in z the entire set of spins would have to be allowed to relax, and then reexcited for further signal acquisition, otherwise T2 dephasing would quickly eliminate the signal. Therefore, it would not be possible to obtain an entire image on millisecond timescales. Breaking out of k-space entirely, by dispensing with gradient reversals, would theoretically confer two large advantages. The first is speed. The second is counterintuitive, namely an increase in SNR. This is because when there are no gradients signal from the entire volume can be encoded simultaneously. This requires the use of massively parallel arrays of detectors arranged around the object. It is shown here how this leads not only to markedly reduced imaging times, but also to marked increases in signal per unit of time. In ULTRA, there is a single gradient employed throughout the acquisition, however, this is never switched or even varied, so that the technique is essentially noise-free. In practice however, in order to overcome the field inhomogeneities of B0 ULTRA will require the periodic imposition of 180° RF
pulses, approximately every 10 ms, so that the noise will be a low-frequency hum at approximately 100 Hz (see below). The key innovation is the introduction of an extremely large parallel array of detectors, N × M, arranged in cylindrical fashion around the object, with z axis parallel to B0 . Note that N is the 2-D image dimension in the (x, y) plane and M is an arbitrary number corresponding to the number of slices. If this array is flattened to make it 2-dimensional it would look like Figure 4, where each detector is depicted as a single loop. For a spatial resolution of 1 mm and a field of view of 250 mm, each of the loops is approximately 3 mm perpendicular to B0 . Such an array was not envisaged in the earliest description of pMRI,3,4 because it was thought at the time to be associated with lower signal. This is because the embodiment assumed that only a single slice would be excited at any given time, and it was not clear how to decode 3-D information simultaneously. The detector array then thought to be most consistent with maximal signal consisted of N long parallel loops on a cylinder,4 arranged with long axis parallel to B0 . It now appears that the assumption behind this idea was incorrect, because the signal per unit time is actually maximized by acquiring signal from the entire volume simultaneously. This requires that individual detectors are made much smaller, as illustrated in Figure 4. Because of the granularity of the spatial sensitivity profiles of small detectors (Fig 2), however, this in turn means that g-factor considerations are much less important than they are for conventional hybrid pMRI.
Image Acquisition For simplicity, and to illustrate the essence of the proposed ULTRA methodology, we first consider the case that T1 and T2 are infinite, so that there is no decay of signal and the image is defined only by the spin density ρ ijk , where (i, j, k) are the quantized (voxelated) coordinates corresponding to (x, y, z) in the object volume. 1. A single gradient is applied in a fixed direction, say bx , which remains fixed throughout acquisition. 2. All spins in the entire volume are excited simultaneously, with a single 90° RF pulse.
The signal at detector dmn due to volume element (i, j, k) is then just: Smn;i j k (t) = Rmn;i j k ρi j k e iωt ,
(2)
where ω = x bx , and R is the matrix representing detector spatial sensitivity, derived from Maxwell’s equations. Since ω is proportional to x, it can simply replace x and Eq. (2) can be rewritten: (ω)
(ω)
(ω)
Smn; j k (t) = Rmn; j k ρ j k e iωt ,
(3)
where the left-hand side is the amplitude in detector (m, n) due to the volume element at location (j, k) in the yz plane defined by frequency ω (or, in other words, the yz plane defined by fixed x). Assuming that voxelation of the object volume is
Hutchinson and Raff: Ultrafast MRI
539
.
B0 . . .
bx Fig 4. Schematic representation of the flattened ULTRA coil. Each detector is represented as a small loop. There are N detectors perpendicular to B0 and M detectors parallel to B0 .
N × N × M, then R(ω) mn is an NxM dimension matrix. We can now write: R(ω) mn; j k ρ (ω) j k e iωt . (4) S(ω) mn (t) = j,k
Let the one dimensional Fourier transform of Smn (t) be Then (ω) is the N × M dimensional matrix representing the ωth component of the set of Fourier transformed signals in all detectors.* With this definition, Eq. (4) can be rewritten as mn (ω) .
ρ (ω) = {R(ω) }−1 (ω).∗
(5)
In words this means the following: the 2-dimensional matrix representing the spin densities, in the plane defined by frequency ω, is the product of the inverse spatial sensitivity matrix and the matrix representing the Fourier transform of the time dependent signals in all detectors. Image reconstruction proceeds in the following steps: 1. Perform a 1-dimensional Fourier transform of the signal in each of the detectors. 2. For each ω, use the amplitude in each detector to construct the N × M matrix (ω). 3. Multiply this by the N × M inverse sensitivity matrix {R(ω)} −1 .
The result is the N × M matrix ρ (ω) representing the spin densities of all (y, z) pixels in the plane defined by x = ω. (In the special case M = N, then ρ (ω) is an N × N matrix and the spatial resolution along both y and z axes is the same.) *[The corrections to equation 5 and the text were made after the article was first published online on July 15, 2014].
540
This procedure is then repeated N times, for all values of x = ω, to obtain the voxelated spin densities of the entire volume. It can now be seen that the entire image can be constructed from as little as 1 ms of data. The entire volume is imaged with one excitation since each small detector loop receives signal from the entire volume. Therefore all of the signal, from all of the spins, all of the time, is detected by the entire set of N × M detectors. This maximizes the rate of signal acquisition and puts ULTRA in a unique category, since in all prior embodiments of MRI at any given time only a partial volume of spins gives rise to signal. It therefore follows that ULTRA is faster than all previous embodiments. Put another way, if the entire set of N × M detectors in ULTRA were joined to form one large conventional detector, this would capture a much greater signal than a large conventional detector employing gradient reversals, since all spins in the object volume would be contributing to the signal at all times. Nevertheless, there would be no way of making an image. ULTRA takes advantage of an inherently very high signal by distributing it throughout a very large number of small coils, and subsequently rechanneling this large signal into a single 3-dimensional image. With reference to Figure 4, where the matrix of coils is a 2-dimensional plane, it now becomes clear that the detector in ULTRA resembles a hologram. That is, the entire 3dimensional volume of spin density is completely and simultaneously encoded in a 2-dimensional plane. The signal in a particular detector is unusual in that it does not simply rise and fall, like a conventional echo, but rather oscillates periodically without further RF input. This is illustrated schematically in Figure 5.
Journal of Neuroimaging Vol 24 No 6 November/December 2014
S
t
Fig 5. Schematic representation of the signal S in any given detector, as a function of time t.
The first thing to note is that the signal goes on forever (for infinite T1 and T2), or at least until the quantum decay (spontaneous emission) of the excited state—which could be on the order of minutes. Nevertheless, the signal is periodic, and even without decay, and even without gradient switching, its amplitude must go to zero every τ = (FOV)/(256γ bx ) seconds, where FOV is the field of view, γ is the gyromagnetic ratio, bx is the magnetic field gradient in the x direction, and 256 is the image matrix dimension. Therefore, in this special case of infinite T1 and T2, the signal is a periodic function, such that within each period τ the amplitude must go to zero at least once, as dictated by gx . In reality of course, not only are T1 and T2 finite, but for short time scales the signal will decay because of inhomogeneities in B0 . We can make a rough estimate of the latter decay time by assuming an inhomogeneity of p parts per million. At 1.5 Tesla, where the frequency is 65 MHz, and for p = 1, the signal would begin to decay in a significant way after 1/65 seconds, or approximately 15 ms. In order to correct for this, a periodic 180° RF pulse would have to be applied, say, every 10 ms. For p = 2, the RF pulse would have to be applied approximately every 5 ms, etc. The imposition of such RF pulses would lead to a lowfrequency humming sound, which would be quite different from the high frequency, high decibel, sound of gradient reversals which characterizes current MRI technology. Note that these RF pulses are not used to encode, only to rephase any dephasing caused by inhomogeneity in B0 . It is of course also the case that the signal in each detector is small because of its small size. Nevertheless it is not N × N times smaller, but rather N times smaller, because there are of order N times as many spins at any time that are contributing to the signal in any given detector loop. Since whole volume image acquisition occurs on the order of 1 ms, the repetitive acquisition of signal over, say, N ms, with N averaged accumulations, means that the entire volume can be imaged with good signal strength on order of N ms. Thus, while a whole volume spin-density image is possible in 1 ms, a high-quality clinical whole volume spin-density image can be made in as little as 256 ms.
T1- and T2-Weighted Images The foregoing pertains to spin density images only. The clinical embodiment of MRI requires that images are sensitive also to the spin-lattice relaxation time, T1, and the spin-spin relaxation time, T2. While it is not intended here to give an exhaustive
account of how such images might be obtained using ULTRA, nevertheless this can be sketched. Note first that from each broad peak in Figure 3 an entire 3-dimensional image can be created. If T1 and T2 were infinite, then each broad peak would have the same height. For 10 peaks at a time, comprising say 10 ms of signal, the entire signal can be averaged into one image having high SNR. The next set of images, from 10 to 20 ms, can likewise be averaged into one image, and so on, until signal has been acquired for, say, 10 high-signal image sets spanning 100 ms. T2 can then be determined for each pixel within the volume by fitting it to an exponential decay generated from all image sets. Likewise, after 100 ms the spins can be rephased by a partial excitation, and the process repeated, giving yet more signal for the T2-weighted images, while at the same time allowing for the estimation of T1, by the relative amplitude of the broad peaks compared with the amplitude after the initial 90° RF excitation. The entire process can be repeated every 100 ms, so that after 1,000 ms all information pertaining to both T1 and T2 can be obtained with high SNR. Likewise, it would be a relatively trivial matter to create inversion-recovery images or diffusion-weighted images.
Discussion A formulation of parallel MRI is proposed (ULTRA) which greatly increases the number of detectors compared with conventional pMRI techniques, while at the same time reducing their size. Dispensing with gradient switching allows for imaging of the entire object volume in a single shot, and therefore to a large increase in the rate of acquisition of signal. Clinical imaging times are reduced from minutes to seconds, and an entire 3-D image can be made in as little as 1 millisecond, which will possibly allow for real-time MR imaging. In the preferred embodiment of ULTRA, there are no magnetic field gradient reversals. Encoding is assisted by means of a single constant field gradient applied in one direction only, throughout acquisition. The key innovation is the introduction of much smaller detectors, this paradoxically enabling a much higher rate of acquisition of signal. At any given time, the entire 3-D image is represented in a detector that can be made into a 2-dimensional sheet. In a sense, therefore, the proposed detector is a hologram. There are several advantages over conventional MRI. 1. Functional MRI is now possible on millisecond timescales. 2. All clinical imaging can now be completed in seconds, with good SNR. 3. Such capability is also consistent with inexpensive total-body scans. 4. ULTRA greatly reduces auditory noise and vibration.
Because the detectors encode the entire 3-D volume in a single shot, the technique can be considered holographic. Finally it must be emphasized that this is a theoretical proposal only. Practical implementation will be difficult, and although the theory may be mathematically correct we are keenly aware that any unforeseen engineering difficulty could render it impractical.
Hutchinson and Raff: Ultrafast MRI
541
References 1. Lauterbur PC. Image formation by induced local interactions. Nature 1973;242(1):190. 2. Mansfield P, Pykett IL. Biological and medical imaging by NMR. J Mag Res 1983;29(2):355-373. 3. Hutchinson M, Raff U. Subsecond Data Acquisition Using Multiple Detectors. Proceedings, Society for Magnetic Resonance in Medicine, 6th Annual Meeting, New York. 1987; Supplement S1: 459. 4. Hutchinson M, Raff U. Fast MRI Data Acquisition Using Multiple Detectors. Mag Res Med 1988;6:87-91. 5. Roemer PB, Edelstein WA, Hayes CE, et al. The NMR phased array. Mag Res Med 1990;16:192-225. 6. Ocali O, Atalar E. Ultimate Intrinsic Signal-to-Noise Ratio in MRI. Mag Res Med 1998;39:462-473.
542
7. Sodickson DK, Manning WJ. Simultaneous acquisition of spatial harmonics (SMASH): fast imaging with radiofrequency coils. Mag Res Med 1997;38(4):591-603. 8. Pruessman KP, Weiger M, Scheidegger MB, Boesiger P. SENSE: sensitivity encoding for fast MRI. Mag Res Med 1999;42(15):952-962. 9. Ohliger MA, Grant AK, Sodickson DK. Ultimate intrinsic signal-tonoise ratio for parallel MRI: Electromagnetic field considerations. 2003. Mag Res Med 2003;50:1018-1030. 10. Wiesiger F, Boesiger P, Pruessman K. Electrodynamics and ultimate SNR in parallel MR imaging. Mag Res Med 2004;52: 376-390. 11. Keil B, Wald LL. Massively parallel MRI detector arrays. J Mag Res 2013;229:75-89. 12. McDougall MP, Wright SM. 64 channel array coil for single echo acquisition. Mag Res Med 2005;54:386-392.
Journal of Neuroimaging Vol 24 No 6 November/December 2014
ABSTRACT Real-time MR imaging might exert a profound influence on neuroscience in the future by enabling the direct visualization of neuronal interactions. At this time, however, all practical embodiments of MRI require at least some degree of gradient encoding, and this in turn sets a lower limit of about 100 ms for volume acquisition. A novel formulation of MRI is proposed here which is given the acronym ULTRA (Unlimited Trains of Radio Acquisitions). In the preferred embodiment ULTRA is completely free of gradient reversals, which allows for signal acquisition from the entire object volume simultaneously. This permits a rate of signal acquisition that is increased hundreds of times compared with existing techniques, with full 3-D imaging in as little as one millisecond. The proposed detector now resembles a holographic recording.
Keywords: Ultrafast, MRI, gradient-free, hologram, real-time. Acceptance: Received May 9, 2014. Accepted for publication May 11, 2014. Correspondence: Address correspondence to Michael Hutchinson, MD, PhD, New York Core Neuroscience, 222 East 31st Street, New York, NY 10016, USA. E-mail: hutchinson@ nyneuroscience.com. J Neuroimaging 2014;24:537-542. DOI: 10.1111/jon.12141
Introduction In the original formulation of MRI by Lauterbur,1 spatial encoding is achieved by means of a plurality of successively applied magnetic field gradients. Each new gradient is associated with a different radiofrequency (RF) excitation, and RF-induced echoes form a line in k-space, discretized into N elements, where N is the dimension of the image matrix. After N progressively increasing gradients, and N echoes, there are now N lines in k-space, and the N × N k-space matrix is subjected to a 2-D Fourier transform, rendering the N × N image matrix. In echoplanar imaging (EPI) Mansfield2 showed that spatial encoding could be rapidly achieved by means of trains of gradient reversals after a single RF excitation. On the other hand, parallel MRI (pMRI) uses the spatial sensitivities of multiple receiver coils arranged around the object to perform spatial encoding, thus reducing the number of gradient reversals and RF pulses. The first pMRI proposal by Hutchinson3,4 was a theoretical scheme for single shot imaging which ignored sources of noise in the surface coils. Subsequent analyses of detector arrays by Roemer5 and Ocali6 accounted for the effects of noise, with algorithms for conventional gradient encoding. This gave rise to the concept of “ultimate signal-to-noise,”6 by which was meant that, at least for gradient encoding, large numbers of small receiver coils—when properly configured—are more efficient than a single receiver. Parallel MRI underwent a substantial practical step forward when Sodickson,7 with SMASH, and Pruessmann,8 with SENSE, elegantly showed how gradient-encoding and spatial sensitivity encoding could be merged. (These techniques will
Copyright
henceforth be designated “hybrid” since they use both methods of encoding.) On the other hand this merging of the two fundamental ways of encoding comes at a considerable price, because for the hybrid techniques ultimate signal-to-noise ratio (SNR) is reached not with large numbers of coils, but with small numbers (typically 4).
Limitations of Parallel MRI SNR in hybrid pMRI is reduced by three unrelated sources: (i) electrical noise, (ii) reduced numbers of echoes, and (iii) a critical geometric factor, g, as discussed by Ohliger9 and Wiesiger.10 Sources of electrical noise include preamplifiers, coupling between adjacent coils, and body thermal noise. In addition, there is noise from the eddy currents in the body surface due not only to the gradient reversals but also to reradiation of signal by the coil, which is both a receiver and a dipole radiator. These sources of noise can be minimized, however, and for conventional gradient encoding multiple small coils have been shown to be more efficient than one large coil5,6 so there is no insurmountable noise problem inherent to small coils. When multiple coils are used to encode, however, there can be an additional problem due to the inefficiency of spatial sensitivity encoding, compared with gradient encoding, when the coils are large. This in turn is due to the slow spatial variations of the spatial sensitivity of each coil. In addition, since there are
◦ 2014 by the American Society of Neuroimaging C
537
Fig 1. Color-coded pixel-based spatial sensitivity map for a quadrant detector placed around the top right corner of the object matrix. Note the lack of granularity.
fewer phase-encoding steps SNR is further reduced. The SNR is now given by9,10 : SNRFull SNR = √ , Rg
(1)
where SNRFull is the SNR for gradient-only encoding, R is the acceleration factor (in this case the number of detectors) and g √ is the geometric factor. The loss factor R presents the reduced SNR due to the reduced number of phase encoding steps, and nothing can be done to mitigate this. The g-factor is intrinsic to the geometry of the coil array, and is a measure of the capacity of the array to compensate for reductions in gradient encoding. For very small numbers of detectors g is close to 1, but as the number of coils increases, and the number of phase-encoding steps correspondingly decreases, g suddenly becomes large and the images are degraded. It has generally been accepted that for all practical purposes this single limitation means that for hybrid techniques acceleration factors cannot be more than about 4, so that the rate of acquisition of SNR is about double that of nonparallel techniques. The reason g > 1 with increasing accelerations is that the transformations used to obtain the image in conventional pMRI are nonunitary, and the reason for this is that the solutions to Maxwell’s equations are “smooth,” by which is meant that for large detectors the spatial dependence of the radio field is not as granular as the spatial dependence of the gradients used for conventional encoding. This in turn means that large groups of adjacent pixels may have similar spatial sensitivity profiles. As R increases, and the number of gradient-encoding steps correspondingly decreases, more and more of the burden of encoding is shouldered by the receiver array. For R = N = 4, only 25% of the encoding is gradient-based. With more detectors there is an unsupportable reliance on spatial sensitivity encoding and this leads to rapid image degradation when R > 4. The lack of granularity of the spatial sensitivity map of a quadrant coil is illustrated in Figure 1. The spatial sensitivity
538
Fig 2. Pixel-based spatial sensitivity map for a point detector situated at the top right hand corner of the image matrix. Note the improved granularity of spatial sensitivity compared with the quadrant coil. This is illustrative only, since the details of granularity are obscured somewhat by the crudity of the color scale.
maps can be modeled for illustrative purposes by taking, as an approximation to the solutions of Maxwell’s equations, only the 1/r3 dependence of the near field dipole.
In this Case, Smaller May be Better This conventional view of an inherent limitation of all constructions of parallel MRI has been challenged recently by Keil and Wald.11 These authors observed that the widely accepted notion—that gradients must necessarily do a large part of the encoding—is becoming outdated. In this regard it should be recognized that the original concept for single shot, single slice, imaging free of gradient reversals3,4 was reduced to practice several years ago by McDougall and Wright,12 using a 64 channel coil with an acceleration factor of 64. This is, we believe, only possible because the detectors were small in comparison with the hybrid techniques, see below. Figure 2 demonstrates, as expected, that a point detector has a much more granular spatial sensitivity map than a quadrant detector. Note, also as expected, that for any given point detector the spatial sensitivity of a pixel in the center of the object matrix is much lower than for a quadrant detector. This explains the simplistic conventional view that small detectors are insensitive to deep tissue. This conventional view is, however, also incorrect11 because deep tissue is well represented in the set of all coils, and this compensates for the fact that its representation in any given coil is relatively small. Figure 3 illustrates the point. It is a representation of the pixel-based spatial sensitivity map for a single detector surrounding the matrix. The circular coil can be thought of as a large collection of very small coils. It therefore follows that for an array of very small surface coils, the spatial sensitivity of a central pixel is nearly the same as for a coil at the center. This is why small surface coils are sensitive to deep tissue, provided the whole massively parallel array is considered.
Journal of Neuroimaging Vol 24 No 6 November/December 2014
Fig 3. Pixel-based spatial sensitivity map for a single circular detector surrounding the image matrix. Note that for a pixel at the center of the matrix, spatial sensitivity is almost as great as for a pixel at the corner. This is because it is equally represented in all elements of the receiver coil, whereas a pixel near the edge is well-represented only in nearby detectors.
Comparison of Figures 2 and 1 illustrates how much more granular the spatial sensitivity map is for a point detector than for a quadrant coil. Also, taken together, Figures 2 and 3 show why there is no theoretical limit to the ability of small coils to encode deeply into tissue, as has been pointed out.11 At the same time, McDougall and Wright12 have confirmed this conjecture empirically, at least in the case of single slice imaging.
Methods In principle, current MRI techniques could acquire signal from the entire volume simultaneously, by successive application of gradients in the x, y, and z directions, and with the signal analyzed by means of a 3-dimensional Fourier transform. The problem with this in practice is that after each successive gradient in z the entire set of spins would have to be allowed to relax, and then reexcited for further signal acquisition, otherwise T2 dephasing would quickly eliminate the signal. Therefore, it would not be possible to obtain an entire image on millisecond timescales. Breaking out of k-space entirely, by dispensing with gradient reversals, would theoretically confer two large advantages. The first is speed. The second is counterintuitive, namely an increase in SNR. This is because when there are no gradients signal from the entire volume can be encoded simultaneously. This requires the use of massively parallel arrays of detectors arranged around the object. It is shown here how this leads not only to markedly reduced imaging times, but also to marked increases in signal per unit of time. In ULTRA, there is a single gradient employed throughout the acquisition, however, this is never switched or even varied, so that the technique is essentially noise-free. In practice however, in order to overcome the field inhomogeneities of B0 ULTRA will require the periodic imposition of 180° RF
pulses, approximately every 10 ms, so that the noise will be a low-frequency hum at approximately 100 Hz (see below). The key innovation is the introduction of an extremely large parallel array of detectors, N × M, arranged in cylindrical fashion around the object, with z axis parallel to B0 . Note that N is the 2-D image dimension in the (x, y) plane and M is an arbitrary number corresponding to the number of slices. If this array is flattened to make it 2-dimensional it would look like Figure 4, where each detector is depicted as a single loop. For a spatial resolution of 1 mm and a field of view of 250 mm, each of the loops is approximately 3 mm perpendicular to B0 . Such an array was not envisaged in the earliest description of pMRI,3,4 because it was thought at the time to be associated with lower signal. This is because the embodiment assumed that only a single slice would be excited at any given time, and it was not clear how to decode 3-D information simultaneously. The detector array then thought to be most consistent with maximal signal consisted of N long parallel loops on a cylinder,4 arranged with long axis parallel to B0 . It now appears that the assumption behind this idea was incorrect, because the signal per unit time is actually maximized by acquiring signal from the entire volume simultaneously. This requires that individual detectors are made much smaller, as illustrated in Figure 4. Because of the granularity of the spatial sensitivity profiles of small detectors (Fig 2), however, this in turn means that g-factor considerations are much less important than they are for conventional hybrid pMRI.
Image Acquisition For simplicity, and to illustrate the essence of the proposed ULTRA methodology, we first consider the case that T1 and T2 are infinite, so that there is no decay of signal and the image is defined only by the spin density ρ ijk , where (i, j, k) are the quantized (voxelated) coordinates corresponding to (x, y, z) in the object volume. 1. A single gradient is applied in a fixed direction, say bx , which remains fixed throughout acquisition. 2. All spins in the entire volume are excited simultaneously, with a single 90° RF pulse.
The signal at detector dmn due to volume element (i, j, k) is then just: Smn;i j k (t) = Rmn;i j k ρi j k e iωt ,
(2)
where ω = x bx , and R is the matrix representing detector spatial sensitivity, derived from Maxwell’s equations. Since ω is proportional to x, it can simply replace x and Eq. (2) can be rewritten: (ω)
(ω)
(ω)
Smn; j k (t) = Rmn; j k ρ j k e iωt ,
(3)
where the left-hand side is the amplitude in detector (m, n) due to the volume element at location (j, k) in the yz plane defined by frequency ω (or, in other words, the yz plane defined by fixed x). Assuming that voxelation of the object volume is
Hutchinson and Raff: Ultrafast MRI
539
.
B0 . . .
bx Fig 4. Schematic representation of the flattened ULTRA coil. Each detector is represented as a small loop. There are N detectors perpendicular to B0 and M detectors parallel to B0 .
N × N × M, then R(ω) mn is an NxM dimension matrix. We can now write: R(ω) mn; j k ρ (ω) j k e iωt . (4) S(ω) mn (t) = j,k
Let the one dimensional Fourier transform of Smn (t) be Then (ω) is the N × M dimensional matrix representing the ωth component of the set of Fourier transformed signals in all detectors.* With this definition, Eq. (4) can be rewritten as mn (ω) .
ρ (ω) = {R(ω) }−1 (ω).∗
(5)
In words this means the following: the 2-dimensional matrix representing the spin densities, in the plane defined by frequency ω, is the product of the inverse spatial sensitivity matrix and the matrix representing the Fourier transform of the time dependent signals in all detectors. Image reconstruction proceeds in the following steps: 1. Perform a 1-dimensional Fourier transform of the signal in each of the detectors. 2. For each ω, use the amplitude in each detector to construct the N × M matrix (ω). 3. Multiply this by the N × M inverse sensitivity matrix {R(ω)} −1 .
The result is the N × M matrix ρ (ω) representing the spin densities of all (y, z) pixels in the plane defined by x = ω. (In the special case M = N, then ρ (ω) is an N × N matrix and the spatial resolution along both y and z axes is the same.) *[The corrections to equation 5 and the text were made after the article was first published online on July 15, 2014].
540
This procedure is then repeated N times, for all values of x = ω, to obtain the voxelated spin densities of the entire volume. It can now be seen that the entire image can be constructed from as little as 1 ms of data. The entire volume is imaged with one excitation since each small detector loop receives signal from the entire volume. Therefore all of the signal, from all of the spins, all of the time, is detected by the entire set of N × M detectors. This maximizes the rate of signal acquisition and puts ULTRA in a unique category, since in all prior embodiments of MRI at any given time only a partial volume of spins gives rise to signal. It therefore follows that ULTRA is faster than all previous embodiments. Put another way, if the entire set of N × M detectors in ULTRA were joined to form one large conventional detector, this would capture a much greater signal than a large conventional detector employing gradient reversals, since all spins in the object volume would be contributing to the signal at all times. Nevertheless, there would be no way of making an image. ULTRA takes advantage of an inherently very high signal by distributing it throughout a very large number of small coils, and subsequently rechanneling this large signal into a single 3-dimensional image. With reference to Figure 4, where the matrix of coils is a 2-dimensional plane, it now becomes clear that the detector in ULTRA resembles a hologram. That is, the entire 3dimensional volume of spin density is completely and simultaneously encoded in a 2-dimensional plane. The signal in a particular detector is unusual in that it does not simply rise and fall, like a conventional echo, but rather oscillates periodically without further RF input. This is illustrated schematically in Figure 5.
Journal of Neuroimaging Vol 24 No 6 November/December 2014
S
t
Fig 5. Schematic representation of the signal S in any given detector, as a function of time t.
The first thing to note is that the signal goes on forever (for infinite T1 and T2), or at least until the quantum decay (spontaneous emission) of the excited state—which could be on the order of minutes. Nevertheless, the signal is periodic, and even without decay, and even without gradient switching, its amplitude must go to zero every τ = (FOV)/(256γ bx ) seconds, where FOV is the field of view, γ is the gyromagnetic ratio, bx is the magnetic field gradient in the x direction, and 256 is the image matrix dimension. Therefore, in this special case of infinite T1 and T2, the signal is a periodic function, such that within each period τ the amplitude must go to zero at least once, as dictated by gx . In reality of course, not only are T1 and T2 finite, but for short time scales the signal will decay because of inhomogeneities in B0 . We can make a rough estimate of the latter decay time by assuming an inhomogeneity of p parts per million. At 1.5 Tesla, where the frequency is 65 MHz, and for p = 1, the signal would begin to decay in a significant way after 1/65 seconds, or approximately 15 ms. In order to correct for this, a periodic 180° RF pulse would have to be applied, say, every 10 ms. For p = 2, the RF pulse would have to be applied approximately every 5 ms, etc. The imposition of such RF pulses would lead to a lowfrequency humming sound, which would be quite different from the high frequency, high decibel, sound of gradient reversals which characterizes current MRI technology. Note that these RF pulses are not used to encode, only to rephase any dephasing caused by inhomogeneity in B0 . It is of course also the case that the signal in each detector is small because of its small size. Nevertheless it is not N × N times smaller, but rather N times smaller, because there are of order N times as many spins at any time that are contributing to the signal in any given detector loop. Since whole volume image acquisition occurs on the order of 1 ms, the repetitive acquisition of signal over, say, N ms, with N averaged accumulations, means that the entire volume can be imaged with good signal strength on order of N ms. Thus, while a whole volume spin-density image is possible in 1 ms, a high-quality clinical whole volume spin-density image can be made in as little as 256 ms.
T1- and T2-Weighted Images The foregoing pertains to spin density images only. The clinical embodiment of MRI requires that images are sensitive also to the spin-lattice relaxation time, T1, and the spin-spin relaxation time, T2. While it is not intended here to give an exhaustive
account of how such images might be obtained using ULTRA, nevertheless this can be sketched. Note first that from each broad peak in Figure 3 an entire 3-dimensional image can be created. If T1 and T2 were infinite, then each broad peak would have the same height. For 10 peaks at a time, comprising say 10 ms of signal, the entire signal can be averaged into one image having high SNR. The next set of images, from 10 to 20 ms, can likewise be averaged into one image, and so on, until signal has been acquired for, say, 10 high-signal image sets spanning 100 ms. T2 can then be determined for each pixel within the volume by fitting it to an exponential decay generated from all image sets. Likewise, after 100 ms the spins can be rephased by a partial excitation, and the process repeated, giving yet more signal for the T2-weighted images, while at the same time allowing for the estimation of T1, by the relative amplitude of the broad peaks compared with the amplitude after the initial 90° RF excitation. The entire process can be repeated every 100 ms, so that after 1,000 ms all information pertaining to both T1 and T2 can be obtained with high SNR. Likewise, it would be a relatively trivial matter to create inversion-recovery images or diffusion-weighted images.
Discussion A formulation of parallel MRI is proposed (ULTRA) which greatly increases the number of detectors compared with conventional pMRI techniques, while at the same time reducing their size. Dispensing with gradient switching allows for imaging of the entire object volume in a single shot, and therefore to a large increase in the rate of acquisition of signal. Clinical imaging times are reduced from minutes to seconds, and an entire 3-D image can be made in as little as 1 millisecond, which will possibly allow for real-time MR imaging. In the preferred embodiment of ULTRA, there are no magnetic field gradient reversals. Encoding is assisted by means of a single constant field gradient applied in one direction only, throughout acquisition. The key innovation is the introduction of much smaller detectors, this paradoxically enabling a much higher rate of acquisition of signal. At any given time, the entire 3-D image is represented in a detector that can be made into a 2-dimensional sheet. In a sense, therefore, the proposed detector is a hologram. There are several advantages over conventional MRI. 1. Functional MRI is now possible on millisecond timescales. 2. All clinical imaging can now be completed in seconds, with good SNR. 3. Such capability is also consistent with inexpensive total-body scans. 4. ULTRA greatly reduces auditory noise and vibration.
Because the detectors encode the entire 3-D volume in a single shot, the technique can be considered holographic. Finally it must be emphasized that this is a theoretical proposal only. Practical implementation will be difficult, and although the theory may be mathematically correct we are keenly aware that any unforeseen engineering difficulty could render it impractical.
Hutchinson and Raff: Ultrafast MRI
541
References 1. Lauterbur PC. Image formation by induced local interactions. Nature 1973;242(1):190. 2. Mansfield P, Pykett IL. Biological and medical imaging by NMR. J Mag Res 1983;29(2):355-373. 3. Hutchinson M, Raff U. Subsecond Data Acquisition Using Multiple Detectors. Proceedings, Society for Magnetic Resonance in Medicine, 6th Annual Meeting, New York. 1987; Supplement S1: 459. 4. Hutchinson M, Raff U. Fast MRI Data Acquisition Using Multiple Detectors. Mag Res Med 1988;6:87-91. 5. Roemer PB, Edelstein WA, Hayes CE, et al. The NMR phased array. Mag Res Med 1990;16:192-225. 6. Ocali O, Atalar E. Ultimate Intrinsic Signal-to-Noise Ratio in MRI. Mag Res Med 1998;39:462-473.
542
7. Sodickson DK, Manning WJ. Simultaneous acquisition of spatial harmonics (SMASH): fast imaging with radiofrequency coils. Mag Res Med 1997;38(4):591-603. 8. Pruessman KP, Weiger M, Scheidegger MB, Boesiger P. SENSE: sensitivity encoding for fast MRI. Mag Res Med 1999;42(15):952-962. 9. Ohliger MA, Grant AK, Sodickson DK. Ultimate intrinsic signal-tonoise ratio for parallel MRI: Electromagnetic field considerations. 2003. Mag Res Med 2003;50:1018-1030. 10. Wiesiger F, Boesiger P, Pruessman K. Electrodynamics and ultimate SNR in parallel MR imaging. Mag Res Med 2004;52: 376-390. 11. Keil B, Wald LL. Massively parallel MRI detector arrays. J Mag Res 2013;229:75-89. 12. McDougall MP, Wright SM. 64 channel array coil for single echo acquisition. Mag Res Med 2005;54:386-392.
Journal of Neuroimaging Vol 24 No 6 November/December 2014
