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## Coil sensitivity encoding for fast MRI. In: (1998)

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Venue: | Proceedings of the ISMRM 6th Annual Meeting, |

Citations: | 190 - 3 self |

### BibTeX

@INPROCEEDINGS{Pruessmann98coilsensitivity,

author = {Klaas P Pruessmann and Markus Weiger and Markus B Scheidegger and Peter Boesiger},

title = {Coil sensitivity encoding for fast MRI. In:},

booktitle = {Proceedings of the ISMRM 6th Annual Meeting,},

year = {1998},

pages = {579}

}

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### Abstract

New theoretical and practical concepts are presented for considerably enhancing the performance of magnetic resonance imaging (MRI) by means of arrays of multiple receiver coils. Sensitivity encoding (SENSE) is based on the fact that receiver sensitivity generally has an encoding effect complementary to Fourier preparation by linear field gradients. Thus, by using multiple receiver coils in parallel scan time in Fourier imaging can be considerably reduced. The problem of image reconstruction from sensitivity encoded data is formulated in a general fashion and solved for arbitrary coil configurations and k-space sampling patterns. Special attention is given to the currently most practical case, namely, sampling a common Cartesian grid with reduced density. For this case the feasibility of the proposed methods was verified both in vitro and in vivo. Scan time was reduced to one-half using a two-coil array in brain imaging. With an array of five coils double-oblique heart images were obtained in one-third of conventional scan time. Magn Reson Med 42:952-962, 1999. 1999 Wiley-Liss, Inc. Key words: MRI; sensitivity encoding; SENSE; fast imaging; receiver coil array Among today's many medical imaging techniques, MRI stands out by a rarely stated peculiarity: the size of the details resolved with MRI is much smaller than the wavelength of the radiation involved. The reason for this surprising ability is that the origin of a resonance signal is not determined by optical means such as focusing or collimation but by spectral analysis. The idea of Lauterbur (1) to encode object contrast in the resonance spectrum by a magnetic field gradient forms the exclusive basis of signal localization in Fourier imaging. However powerful, the gradient-encoding concept implies a fundamental restriction. Only one position in k-space can be sampled at a time, making k-space speed the crucial determinant of scan time. Accordingly, gradient performance has been greatly enhanced in the past, reducing minimum scan time drastically with respect to earlier stages of the technique. However, due to both physiological and technical concerns, inherent limits of k-space speed have almost been reached. An entirely different approach to sub-wavelength resolution in MRI is based on the fact that with a receiver placed near the object the contribution of a signal source to the induced voltage varies appreciably with its relative position. That is, knowledge of spatial receiver sensitivity implies information about the origin of detected MR signals, which may be utilized for image generation. Unlike position in k-space, sensitivity is a receiver property and does not refer to the state of the object under examination. Therefore, samples of distinct information content can be obtained at one time by using distinct receivers in parallel (2), implying the possibility of reducing scan time in Fourier imaging without having to travel faster in k-space. In 1988 Hutchinson and Raff (3) suggested dispensing entirely with phase encoding steps in Fourier imaging by using a very large number of receivers. Kwiat et al. (4) proposed a similar concept in 1991. In 1989 Kelton et al. In all contributions procedures for image reconstruction were derived. However, applications of the concepts noted have not been reported, reflecting the considerable practical challenges of sensitivity based imaging, including the signal-to-noise ratio (SNR) issue, sensitivity assessment, and hardware requirements. Only in 1997 did Sodickson et al. To overcome the restrictions of previously proposed methods, in this work we reformulate the problem of image reconstruction from multiple receiver data. Using the framework of linear algebra, two different reconstruction strategies have been derived. In their general forms the resulting formulae hold for arbitrary sampling patterns in k-space. A detailed discussion is dedicated to the most practical case, namely, sampling along a Cartesian grid in k-space corresponding to standard Fourier imaging with reduced FOV. Owing to the underlying principle, the concepts outlined in this work have been named SENSE, short for SENSitivity Encoding (8-10). Together with SENSE theory and methods, a detailed SNR analysis is presented as well as an experimental in vitro evaluation and a selection of in vivo examples. THEORY AND METHODS In this section SENSE theory is presented and methods for image reconstruction from sensitivity encoded data are derived. The theory addresses the most general case of combining gradient and sensitivity encoding. That is, no restrictions are made as to the coil configuration and the sampling pattern in k-space. Two reconstruction strategies are discussed. The first approach strictly aims at optimal voxel shape and is called strong reconstruction for convenience. In weak reconstruction, the voxel shape criterion is weaker in favor of the SNR. With both strategies the reconstruction algorithm is numerically demanding in the general case. This is mainly because with hybrid encoding the bulk of the work of reconstruction can usually not be done by fast Fourier transform (FFT). However, it is shown that in weak reconstruction FFT can still be applied if k-space is sampled in a regular Cartesian fashion. For this reason sensitivity encoding with Cartesian sampling is particularly feasible. Moreover, the reconstruction mechanism is relatively easily understood in this case. Therefore, the first part of this section gives a practical description of the Cartesian case. The following parts are dedicated to general theory, SNR and error considerations, and sensitivity assessment. Sensitivity Encoding With Cartesian Sampling of k-Space In two-dimensional (2D) Fourier imaging with common Cartesian sampling of k-space, sensitivity encoding by means of a receiver array permits reduction of the number of Fourier encoding steps. This is achieved by increasing the distance of sampling positions in k-space while maintaining the maximum k-values. Thus scan time is reduced at preserved spatial resolution. The factor by which the number of k-space samples is reduced is referred to as the reduction factor R. In standard Fourier imaging, reducing the sampling density results in the reduction of the FOV, causing aliasing. In fact, SENSE reconstruction in the Cartesian case is efficiently performed by first creating one such aliased image for each array element using discrete Fourier transform (DFT). The second step then is to create a full-FOV image from the set of intermediate images. To achieve this one must undo the signal superposition underlying the fold-over effect. That is, for each pixel in the reduced FOV the signal contributions from a number of positions in the full FOV need to be separated. As depicted in The key to signal separation lies in the fact that in each single-coil image signal superposition occurs with different weights according to local coil sensitivities. Consider one pixel in the reduced FOV and the corresponding set of pixels in the full FOV where the subscripts ␥, count the coils and the superimposed pixels, respectively, r denotes the position of the pixel , and s ␥ is the spatial sensitivity of the coil ␥. The sensitivity matrix is used to calculate the unfolding matrix U: where the superscript H indicates the transposed complex conjugate, and ⌿ is the n C ϫ n C receiver noise matrix (see Appendix A), which describes the levels and correlation of noise in the receiver channels. Using the unfolding matrix, signal separation is performed by where the resulting vector v has length n P and lists separated pixel values for the originally superimposed positions. By repeating this procedure for each pixel in the reduced FOV a non-aliased full-FOV image is obtained. Unfolding is possible as long as the inversions in Eq. [2] can be performed. In particular, the number of pixels to be separated, n P , must not exceed n C . In other words, the reduction factor is bound by the number of coils used. Note that n P does not need to be the same for all partial unfolding steps. Upon non-integer reduction the number of pixels actually superimposed may vary in the reduced FOV. Generally, the degree of aliasing plays an important role with respect to SNR. As a rule of thumb it can be said that local SNR improves when the degree of aliasing is reduced. Therefore it is beneficial to exclude a pixel from reconstruction when the corresponding volume contributes no signal, e.g., because it lies outside of the object. Formally this is done simply by removing the corresponding column in the sensitivity matrix and setting the excluded pixel to zero in the final image. Knowledge of which voxels may safely be excluded is obtained as a by-product of sensitivity assessment. In Eq. [2] receiver noise levels and correlation are considered for the sake of SNR optimization. Optionally, the assessment of receiver noise may be skipped and the represents the superposition of pixels forming a Cartesian grid. In this example four of these pixels are in the full FOV; thus the actual degree of aliasing is four. Fast MRI 953 matrix ⌿ replaced by identity. Then unfolding is still ensured, yet at an SNR penalty, which generally will be the more marked the less equivalent the receivers are with respect to load, gain, and mutual coupling. Equation SENSE: Sensitivity Encoding for General Theory of Sensitivity Encoding Consider an imaging experiment using an array of n C receiver coils. Fourier encoding is described by a set of n K sampling positions in k-space. Let the whole object be within the volume of interest (VOI). Then a sample value m obtained from the ␥-th coil at the -th position in k-space is given by where r denotes 3D position, is the net encoding function composed of harmonic modulation and the complex spatial sensitivity s ␥ of coil ␥, and c results from tissue and sequence parameters. The effects of non-uniform k-space weighting due to relaxation shall be neglected in the scope of this work. From the linearity of encoding it is clear that image reconstruction must essentially be linear as well. That is, each of n V image values is to be calculated as a linear combination of sample values: where counts the voxels to be resolved. The transform F shall be referred to as the reconstruction matrix. Its size is n V ϫ n C n K . Assembling sample and image values in vectors, image reconstruction may be rewritten in matrix notation: With such linear mapping the propagation of noise from sample values into image values is conveniently described by noise matrices. The -th diagonal entry of the image noise matrix X represents the noise variance in the -th image value while the off-diagonal entries reflect noise correlation between image values. The sample noise matrix ⌿ is defined accordingly in Appendix A. As shown there, these matrices fulfil the relation The central objective in choosing a reconstruction matrix is to make each image value selectively reflect signal from the voxel it represents. [9] The term in brackets describes the spatial weighting of signal in . It is therefore called the corresponding voxel function: F ,(␥,) e ␥, (r). [10] Hence, the matrix F has to be chosen such that the resulting voxel functions approximate the desired voxel shapes. Let i (r) denote an orthonormal set of ideal voxel shapes, e.g., box functions. The relation between ideal voxel shapes and encoding functions is described by the n C n K ϫ n V encoding matrix [11] There are many possible ways of approximating ideal voxels. Here we discuss two concepts. The first approach is to choose those voxel functions that exhibit the least square deviation from the ideal. This criterion entirely determines reconstruction; the approach is therefore referred to as the strong one. In Appendix B it is shown that it yields where C denotes the correlation matrix of the encoding functions. The image noise matrix [8] is then given by The second approach, dubbed the weak one, uses a different concept of similarity between real voxel functions and ideal shapes. It requires that each voxel function fulfil the orthonormality relations of its ideal counterpart: where Id n v denotes n V ϫ n V identity. By this condition the reconstruction matrix F is generally not yet entirely determined. It leaves n C n K -n V degrees of freedom per voxel, which may be utilized for SNR optimization. To that end each diagonal element of the image noise matrix X is minimized under condition . [16] 954 Pruessmann et al. In this case the image noise matrix reads The reconstruction formulae The limitations of weak reconstruction may be understood by considering Dirac distributions as ideal voxel functions: where r denotes the center of the -th voxel. The encoding matrix then reduces to In this case the weak criterion [15] may be restated as follows: each voxel function must be equal to one in the center of the voxel it belongs to and equal to zero in all other voxels' centers. A voxel function with this property will be acceptable only as long as it is well behaved between voxel centers. In this view, the criterion becomes unreliable when there are solutions that vary considerably within voxels and at the same time yield favorably low noise. The Dirac choice in the weak approach also is of great practical significance. It is with this choice that reconstruction in the Cartesian case can be performed in the practical fashion described at the beginning of this section. For the derivation see Appendix D. Noise in SENSE Images There are actually two kinds of noise that affect SENSE images, i.e., noise in sample values and noise in sensitivity data. The latter, however, can usually be reduced to a negligible level by smoothing. Then Eq. [8] for the calculation of image noise holds. This equation illustrates two important aspects of noise propagation in SENSE reconstruction. First, with multiple channels the diagonal entries in ⌿ vary from channel to channel and there is noise correlation between samples taken simultaneously, i.e., there are non-zero cross-terms. Second, unlike a matrix representation of FFT, a SENSE reconstruction matrix generally is not unitary. As a consequence, unlike standard Fourier images the noise level in a SENSE image varies from pixel to pixel and there is noise correlation between pixels. For similar reasons the noise level does not have the common square-root dependence on the number of samples taken. In the case of Cartesian sampling with reconstruction as initially explained, this can be made yet clearer. For one particular voxel we compare the noise levels as obtained with full and reduced Cartesian Fourier encoding. According to Appendix D the partial image noise matrix for the relevant unfolding step is [20] Let denote the index of the voxel under consideration within the set of voxels to be separated. With full Fourier encoding no aliasing occurs and the matrix S has only one column. Note that this single column is identical to the -th column of S in the case of reduced Fourier encoding. Thus, the ratio of the noise levels obtained in that voxel is given by where S corresponds to reduced Fourier encoding and R denotes the factor by which the number of samples is reduced with respect to full Fourier encoding: The rightmost square-root expression in Eq. [21] strongly depends on coil geometry and is thus called the local geometry factor g, which is always at least equal to one: Note that by virtue of condition [15] the voxel functions in the two reconstructions compared are both scaled to one in the voxel center. Therefore, the noise ratio in Eq. [21] reflects just the inverse of the SNR ratio, thus: This relation confirms an upper bound for SNR characterized by the square root of the number of samples acquired. The geometry factor describes the ability with the used coil configuration to separate pixels superimposed by aliasing. In practice it allows a priori SNR estimates and provides an important criterion for the design of dedicated coil arrays. Propagation of Systematic Error in SENSE Reconstruction In addition to noise a SENSE reconstructed image may be impaired by errors of systematic nature. Errors in sample values may be due, e.g., to tissue motion, main field inhomogeneity, eddy currents, or gradient non-linearity. The related artifacts are well known in standard imaging. A potential problem specific to sensitivity encoded imaging arises from errors in sensitivity values. The nature of artifacts in SENSE images generally is governed by error propagation in the reconstruction formu-SENSE: Sensitivity Encoding for Fast MRI 955 lae. More specific statements are possible in the case of Cartesian sampling with unfolding reconstruction. Here errors in sample values are first reflected in the intermediate images and then undergo mapping by the unfolding matrices U. These cause error cross-talk within sets of equidistant voxels, as depicted in Determination of Sensitivity Maps Sensitivity based reconstruction requires highly accurate sensitivity assessment. To this end concepts known from methods for intensity correction (11-13) have been extended (14). Reliable sensitivity information is obtained by reference measurements with the definitive set-up in addition to actual imaging. With this strategy it is not possible to assess absolute sensitivity values. However, according to Eq. [4], a spatial scaling of sensitivity values, as long as it is the same for all coils, is mapped onto the final image without further interference with reconstruction. Therefore it is sufficient to determine sensitivity maps with equal and appropriately homogeneous scaling. The first step in generating such maps is to acquire and reconstruct single-coil, full-FOV images of the slice of interest in a conventional manner. Division of each of these images by the ''sum-of-squares'' of the set yields sensitivity maps with scaling clear of modulus object contrast but still modulated by the ''sum-of-squares'' of absolute sensitivities. More homogeneous scaling is achieved by dividing by a body coil image. The ''sum-of-squares'' denominator is applicable only if the object phase is sufficiently smooth so as to endure map refinement largely unaltered. On the other hand, it offers the advantage of being more reliable in clearing modulus object contrast from raw maps. This is because a body coil reference cannot be acquired quite simultaneously with the array reference. In either case the raw maps obtained are impaired by noise where x, y are pixel indices, and P denotes the order of the polynomial. Minimizing the weighted square deviation from raw sensitivity values yields a set of (P ϩ 1) 2 equations: where s x,y denotes raw sensitivity values and w is the weighting function. Solving Eq. [26] yields the desired refined sensitivity value for the pixel x 0 , y 0 . The weighting function reflects the relative significance of s x,y for the refinement of the sensitivity value at x 0 , y 0 . It is the product of a distance Gaussian and reliability terms derived from analysis of error propagation in creating the raw map: where ⍀ is a parameter reflecting the degree of smoothing, d x,y denotes the pixel value at position x,y of the denominator image used for preparing the raw map, and ⌽ x,y is an ''object indicator'' map: ⌽ x,y ϭ 1 where object signal dominates noise in the denominator image, ⌽ x,y ϭ 0 elsewhere. ⌽ is determined from the denominator image by pixel-wise modulus discrimination with a threshold on the order of the noise level and further exclusion of pixels with then sparse neighborhood [29] the function ⌽ can serve as the basis for excluding voxels from reconstruction. Besides improved accuracy at object edges, the fitting approach has the advantage of not being restricted to regions yielding immediate sensitivity information. It permits extrapolation over a limited range necessary for dealing with slightly varying tissue configurations. Refined sensitivity values are calculated for all ''object'' pixels according to ⌽, plus an extrapolation zone [27] and [28] are dominated by the pixels near x 0 , y 0 , the number of considerable contributions depending on ⍀. In terms of complexity it is advantageous to consider only the significant terms in Eqs. [27] and [28] and restrict higher order fitting to border regions. RESULTS Sensitivity encoding using common Cartesian sampling of k-space and DFT-based reconstruction was performed in vitro and in vivo on a Philips Gyroscan ACS-NT15 at 1.5 T. Phantom Experiments A five-coil array was used in the set-up depicted in To illustrate the need for advanced sensitivity assessment, the fitting order P was varied in sensitivity map refinement. Images obtained at R ϭ 3.0 with P ϭ 0, P ϭ 1, and P ϭ 2 are shown in