Sodium (23Na) is the second most abundant nucleus observable with magnetic resonance in biological tissues, surpassed only by the hydrogen nucleus (1H). Sodium plays a pivotal role in the human body and has an important function in maintaining the homeostasis of organisms through osmoregulation and pH regulation.1 In addition, it is involved in cell physiology through the regulation of sodium–potassium pumps, whose main purpose is to maintain the transmembrane sodium and potassium gradients by extruding three sodium ions from the cell while transferring two potassium ions into the cell and simultaneously consuming energy provided by adenosine triphosphate (ATP) hydrolysis.2 The impairment of tissue energy metabolism or the disruption of cell membrane integrity causes an increase in intracellular sodium concentration (10–15 mmol/L), while the extracellular sodium concentration (140–150 mmol/L) remains constant due to tissue perfusion.3-5 Therefore, a disturbance in the balance between intra- and extracellular sodium concentration is considered to be a sensitive early indicator of some diseases.4 For example, when the activity of the sodium–potassium pumps is reduced due to insufficient ATP supply, for example, during ischemia,6 the pumps cannot expel the influxive sodium properly, and thus an increase in intracellular sodium concentration can be observed.

Sodium magnetic resonance imaging (MRI) is the only noninvasive imaging technique that enables the absolute spatial quantification of sodium concentration in living tissues. It can provide direct biochemical information for cell integrity and tissue viability with little or no macroscopic alterations, making it useful for tracking temporal changes in tissue viability during a course of treatment and giving it the potential to become a biomarker for early preventive diagnosis in clinical practice.4 Initial demonstrations of the feasibility of sodium MRI in the human body date back to the early 1980s.7, 8 With the availability of stronger magnetic fields (≥3 T) as well as advancements in acquisition strategies and hardware, the potential of sodium MRI has been investigated in many recent studies across a variety of diseases, ranging from brain tumors,9, 10 multiple sclerosis,11, 12 stroke,13 osteoarthritis,14 and breast cancer,15, 16 to nephropathy,17 and others.4, 5

However, compared to conventional hydrogen MRI, sodium MRI has to surmount a number of hurdles if it is to enter routine clinical practice. In particular, the interrelated issues of relatively low signal-to-noise ratio (SNR) and long measurement times (usually exceeding 10 minutes) due to the low relative sensitivity of sodium nuclear magnetic resonance (NMR) compared to hydrogen (approximately 9.2%)3 remain problematic, even when imaging is conducted at ultra-high fields.18 These issues emerge from the fact that the gyromagnetic ratio of sodium is approximately 4-fold lower than that of hydrogen; the nuclear spin of sodium takes a value of 3/2 compared to 1/2 for hydrogen and hence exhibits a nuclear quadrupolar moment, and sodium concentration in vivo is approximately 2000 times lower than that of hydrogen. Furthermore, the interaction of the nuclear quadrupolar moment with the electric field gradients originating from the electronic distribution surrounding the nucleus in biological tissues results in a biexponential relaxation behavior, causing relatively fast decay of the sodium NMR signal.19, 20 A short transversal relaxation component, typically less than 5 msec, commonly constitutes about 60% of the signal, while the long component, typically ranging from 15 msec to 30 msec, contributes about 40% of the signal, favoring ultra-short echo time (UTE) imaging techniques for the detection of both components.4 In contrast, the quadrupolar interaction is averaged to zero in a homogenous environment such as a fluid, and therefore the transverse relaxation of sodium NMR signal proceeds as a relatively slow mono-exponential decay.4, 19, 20

Fortunately, a variety of highly efficient acquisition techniques and delicate reconstruction approaches have been developed to enhance image quality and/or reduce the acquisition times of MRI scans by means of k-space undersampling. For example, parallel imaging can accelerate measurements with multichannel receiver coils by utilizing coil sensitivity variations in conjunction with a smaller number of gradient encoding steps.21-23 However, the wide application of parallel imaging in sodium MRI is currently hindered by the limited availability of phased-array sodium-tuned coils. An alternative method applicable for accelerating single-channel coil scans is compressed sensing (CS), which is based on the principle that an image with a sparse representation in a known transform domain can be recovered from incoherently undersampled k-space data by means of a nonlinear iterative reconstruction.24, 25 MRI agrees well with this principle as MRI scanners naturally acquire the Fourier-encoded raw data instead of pixel samples and MRI images are naturally compressible in some transform domains. More importantly, due to the biexponential relaxation behavior with a fast T2 value typically less than 5 msec in biological tissues, sodium MRI usually employs non-Cartesian UTE sampling schemes, such as radial26 or spiral27, 28 acquisitions, achieving the incoherent undersampling required by CS. Ever since Madelin et al demonstrated the applicability of CS in sodium MRI in a study of human knee cartilage,29 CS has been increasingly applied to sodium imaging of the brain,30-32 skeletal muscle,33, 34 breasts,15, 35 and human torso.36 Recent efforts have been made to further advance CS sodium MRI by incorporating methods such as dictionary-based learning,36, 37 prior hydrogen anatomical constraint,32 parallel imaging,38 or deep learning.39 As an emerging technique, CS has great potential in further facilitating the clinical applicability of sodium MRI by, for example, applying advanced incoherent undersampling methods,27, 40, 41 or by accelerating intracellular sodium mapping,42, 43 quantitative relaxometry,9, 10, 44 and dynamic sodium MRI.45, 46

Sodium MRI methods and applications have been extensively reviewed elsewhere,4, 5, 47, 48 and there are multiple reviews on the CS techniques and applications in hydrogen MRI.49-53 However, to the best of the authors' knowledge, sodium MRI with the incorporation of CS has not yet been thoroughly reviewed. In light of the above, this article offers a review of CS-based sodium MRI over the last decade, focusing on advanced techniques, clinical applications, and potential future research prospects.

Basic Principles for the Application of Compressed Sensing to Sodium MRI Biomedical images, for example, sodium MRI images, are naturally compressible with little or no perceptual loss of information. Conventionally, image compression is performed following image acquisition in order to save storage space and transfer time. The image content is transformed into a vector of sparse coefficients by compression tools, such as discrete cosine transform (DCT) or wavelet transform, and a standard compression strategy is used to encode the few significant coefficients and discard the most negligible or unimportant coefficients, thus enabling near-perfect reconstruction of the original data. As sodium MRI suffers from relatively low image quality and long measurement times, reducing measurement time without significant degradation of the image quality is crucial for sodium MRI to become clinically feasible. This naturally raises the following question: Can one directly measure only the compressed information in sodium MRI, while maintaining most of the reconstructed image quality? Note that the MRI system naturally acquires Fourier-encoded coefficients (i.e. k-space samples) rather than pixels, DCT, or wavelet coefficients. The acquired demodulated signal in the MRI system is subject to the following form of Fourier integral: (1)where is the acquired MRI signal after demodulation, is the coordinate of the spatial domain, is the transverse magnetization of the object immediately after radiofrequency pulse excitation, and is the k-space coordinate.

Therefore, the above question can be restated: Is it possible to reconstruct sodium MRI images without significant visual loss by measuring a small subset of the k-space? Fortunately, the mathematical theory of CS published by Donoho54 and Candès et al24, 25 clears the path for accelerated sodium MRI. According to the mathematical results, three ingredients are required for the application of CS to sodium MRI: 1) transform sparsity to separate and remove the noise from the desired image content, 2) incoherent undersampling to speed up data acquisition and avoid distinct aliasing artifacts, and 3) nonlinear iterative reconstruction to balance sparse representation of the desired image and data consistency of the acquired k-space data. As shown in Fig. 1, a sparse representation can be obtained by applying a sparsifying transform exemplified by a wavelet transform. A nonlinear iterative reconstruction is performed by leveraging, for example, the nonlinear conjugate gradient approach.55 An image is updated in the (i + 1)th iteration by feeding a conjugate gradient, which contains the information of the ith sparse domain as well as the difference between the measured k-space and the ith k-space, thus promoting image sparsity and data consistency. When the conjugate gradient or the number of iterations reaches the stopping criteria set by the user, the iteration loop is broken, and the final image is produced. The following subsections discuss the three fundamental requirements for the implementation of CS to sodium MRI in more detail.

A simplified schematic of the fundamentals of compressed sensing. Randomly undersampled k-space datasets are acquired (top left); their inverse Fourier transformation results in an image with incoherent artifacts. A sparse representation is obtained by employing wavelet transform. The gray box shows a nonlinear iterative reconstruction using a nonlinear conjugate gradient method. An image is updated by feeding a conjugate gradient, which is calculated based on k-space consistency and sparse domain information. When the conjugate gradient or the number of iterations reaches a set stopping criteria, the iteration loop is broken, and the final image is produced. : undersampled Fourier transform operator, : inverse undersampled Fourier transform operator, : sparsifying transform operator.

Source: Figure reproduced from reference 55, with permission from John Wiley and Sons (License No. 5147660427238).

Transform Sparsity A vector can be said to be “sparse” provided that most of its coefficients are equal to zero and only a few coefficients contain all of the information. From a signal processing perspective, most energy from a sparse signal is contained within a few measurements, while the remaining measurements are zero or negligible. In mathematical terms, transform sparsity produces a sparse vector after a specific mathematical transformation and can be defined as follows: If an unknown signal with samples is a vector, , which can be expressed in terms of an orthonormal basis set (: i = 1, …, m) for (e.g. orthonormal wavelet basis), as follows: (2)where is the transform coefficient set of ; the orthonormal basis set (: i = 1, …, m) in matrix form, , is also called sparsifying transform operator. Then, the signal, , is said to be -sparse if only elements of are nonzeroes , while the remaining elements are zeroes (as shown in Fig. 2a). Basic principles of sparse representations. (a) Transform sparsity produces a set of sparse transform coefficients, , with non-zero elements after a sparsifying transform, , operating on a signal, . (b) An -sparse signal can be transformed into an set of measurements, , through an sampling matrix, A (). The colors represent the values of the elements in the matrices. The elements of in white are zeros.

Sparsifying transform projects an image into a sparse domain based on a number of transform coefficients, thus suggesting the number of measurements required for an exact reconstruction. However, choosing the right transform tool to exploit sparsity for a particular class of MRI images is a challenging task and an ongoing research area. Fixed sparsifying transform operators are frequently employed in sodium MRI, such as wavelets,30 finite differences,15, 33, 37 or the orthogonal DCT.30 Moreover, it is possible to perform sparse representation of sodium images based on a trained dictionary,33, 34, 36, 37 or others.33, 39

Incoherent Undersampling

An essential requirement for CS is that the aliasing artifacts produced by k-space undersampling due to the Nyquist constraint violation are incoherent (i.e. noise-like) in the image domain since it is impossible to distinguish between signal and aliasing in the case of the undersampling not being random. In light of the fast biexponential relaxation behavior of the sodium nucleus, non-Cartesian UTE k-space trajectories are usually employed in sodium MRI, such as 3D radial,56 stack of spirals,57 density-adapted 3D projection,58 twisted projection imaging (TPI),40, 41 flexible TPI,59 3D cones,60 and Fermat looped, and orthogonally encoded trajectories (FLORET).27 Generally, incoherent undersampling is achieved by randomly skipping a subset of phase-encode lines in Cartesian sampling or projections in non-Cartesian sampling (as shown in Fig. 3), such as variable-density sampling schemes,61, 62 leading to a reduction in MRI scan times. The artifacts produced by incoherent undersampling show a noise-like behavior in the image domain and even more so in an appropriately selected sparse domain. It has been reported that 3D non-Cartesian sampling, such as FLORET (Fig. 3f), provides better sparsity and hence yields greater CS performance than traditional 2D Cartesian sampling with incoherently undersampled phase-encodes and fully sampled readouts (Fig. 3a). This makes 3D non-Cartesian sampling an excellent candidate for CS-based reconstruction.49, 55

k-space trajectories with 2-fold incoherent undersampling. Phase-encodes or projections in gray are skipped, while red ones are sampled. (a) 2D Cartesian trajectories with incoherently undersampled phase-encodes and fully sampled readouts. (b–f) 3D non-Cartesian UTE k-space trajectories often used in sodium MRI: (b) 3D radial, (c) stack of spirals, (d) 3D cones, (e) twisted projection imaging, and (f) Fermat looped, orthogonally encoded trajectories.

Nonlinear Iterative Reconstruction In the case of the undersampling described above, only linear measurements (n < m) of the unknown signal, , with samples, are acquired and can be expressed in the following form: (3)where is a known sampling vector in the ith measurement; is the sampling matrix of dimension ; and is the measured dataset with samples from measurements by applying the sampling matrix, . A more intuitive interpretation of undersampling is shown in Fig. 2b. Of particular interest is the exact recovery of signal in the vastly undersampled case, where the number of unknowns () is much larger than that of the observations (), which might seem impossible at first glance. Candès et al24, 25, 63 proposed that the signal of interest, , can be exactly recovered by solving the -convex problem: (4)where the -norm is the sum of the magnitudes of vector , provided that the sampling matrix, , has the restricted isometry property.25 More specifically, for a given restricted isometry constant, , a sampling matrix, , is said to have a -restricted isometry property if it satisfies the following condition for all -sparse vectors and for : (5)It has been found that there is a high probability of satisfying the requirements of the restricted isometry property if the undersampling pattern is incoherent and if the number of measurements is above a given constant that is determined according to the number of samples, , and the sparsity value, .63 In clinical practice, the signal is usually projected into a specific domain to increase its sparsity as much as possible, thereby reducing the number of unknowns to achieve near-optimum image reconstruction, as shown in the following unconstrained formula adapted from Eq. 4: (6)where is the iteratively generated image, is the reconstructed image, is the acquired k-space data, is the Fourier transform operator, is the sparsifying transform operator such that becomes sparse, and is the regularization parameter to balance the -norm and -norm. Minimizing the -norm of the transform coefficients promotes sparsity, while the -norm constraint of the measured data ensures data consistency. In other words, out of all the potential solutions consistent with the measured data, Eq. 6 finds one that is compressible by transform . In addition, it is well-documented that the total variation (TV) norm, which is essentially the -norm of the variations of neighboring pixels or voxels, can promote image restoration because the finite difference operator can play a role as an edge-preserving filter to smooth regions with constant intensity.64 Thus, to enforce the image sparsity both in the transform domain and in the finite-difference domain, a certain amount of TV penalty can be added to Eq. 6, as follows: (7)where is the weighing factor for the TV penalty.

In addition to the aforementioned three fundamental requirements, there are several practical limitations and/or considerations when applying CS to sodium MRI. One of the limitations is that sodium MRI has relatively weak image sparsity due to the high noise contamination of sodium images. The concept of compressibility is introduced to quantify image sparsity, which is defined as the percentage of specified transform coefficients required to generate an image comparable to the fully sampled original image. It has been reported that the hydrogen images of the human head can be compressed up to 1.6%, while the corresponding sodium images only provide a compressibility of 72%.65 In light of this, advanced techniques are often employed to promote the sparsity of sodium images. Second, center-out non-Cartesian sampling schemes often used in sodium MRI for incoherent undersampling require rapid switching of gradients and are therefore sensitive to hardware system imperfection. This can result in eddy currents, gradient delays, and related artifacts. Finally, CS has the potential to improve SNR and/or image resolution, whereas this may result in the loss of low-contrast features. Hence, the trade-off between these two factors must be taken into consideration.

Historical Milestones

Five years after Lustig et al demonstrated the applicability of CS in hydrogen MRI for the first time,55 Madelin et al proved the applicability of CS in sodium MRI by successfully halving the scan time required for imaging knee cartilage at 7 T without significant loss of accuracy in total sodium concentration (TSC).29 Although CS sodium MRI is still considered an emerging technique, it has made significant progress in the past decade, as shown in the list of publications given in Table 1, which is, to the best of our knowledge, a complete list of all publications relating to CS-based sodium MRI to this date. Following the work of Madelin et al,29 Gnahm et al optimized the CS sodium MRI technique by combining its TV regularization with prior high-resolution anatomical information from hydrogen MRI, resulting in a substantial increase in SNR and enhanced contrast of structures in the sodium MRI images.31, 32 In addition to the standard -norm and TV penalty, several innovative sparsity regularizations have been employed for the application of CS to sodium MRI, such as second-order TV and dictionary-based learning.15, 37, 68, 70 Lachner et al pioneered the combination of parallel imaging with CS sodium MRI in a study on female breast imaging using a multichannel phased-array sodium/hydrogen double-tuned coil and found that the incorporation of parallel imaging improved CS reconstruction with higher image quality.38 More recently, Adlung et al provided proof for the first time that convolutional neural networks in the field of deep learning were able to reconstruct 4-fold undersampled sodium MRI images with loss functions acting as regularizations while maintaining SNR and TSC quantification accuracy for ischemic stroke patients.39 The majority of the research relating to CS-based sodium MRI have been conducted at an ultra-high field strength of 7 T using various forms of 3D radial sampling schemes with undersampling factors (USFs) ranging from 2 to 10. The effects of sparse reconstruction on the quantitation of TSC and relaxometry have been investigated in multiple studies.30, 34, 39, 68-70 Further to UTE sequences using n