Ultrafast CEST imaging

Journal of Magnetic Resonance 243 (2014) 47–53

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Communication

Ultrafast CEST imaging Jörg Döpfert a, Moritz Zaiss b, Christopher Witte a, Leif Schröder a,⇑ a b

Leibniz-Institut für Molekulare Pharmakologie, Robert-Rössle-Str. 10, 13125 Berlin, Germany German Cancer Research Center (DKFZ), Im Neuenheimer Feld 280, 69120 Heidelberg, Germany

a r t i c l e

i n f o

Article history: Received 11 February 2014 Revised 5 March 2014 Available online 20 March 2014 Keywords: Ultrafast CEST imaging PARACEST agents Chemical exchange High-throughput screening

a b s t r a c t We describe a new MR imaging method for the rapid characterization or screening of chemical exchange saturation transfer (CEST) contrast agents. It is based on encoding the chemical shift dimension with an additional gradient as proposed in previous ultrafast CEST spectroscopy approaches, but extends these with imaging capabilities. This allows us to investigate multiple compounds simultaneously with an arbitrary sample tube arrangement. The technique requires a fast multislice readout to ensure the saturation is not lost during data acquisition due to T 1 relaxation. We therefore employ radial subsampling, acquiring only 10 projections per CEST image with a 128  128 matrix. To recover the images, we use a heuristic reconstruction algorithm that incorporates low rank and limited object support as prior knowledge. This way, we are able to acquire a spectral CEST data set consisting of 15 saturation offsets more than 16 times faster than compared with conventional CEST imaging. Ó 2014 Elsevier Inc. All rights reserved.

1. Introduction Chemical exchange saturation transfer (CEST) is a relatively new technique for generating switchable MRI contrast which has shown great potential for molecular imaging [1]. A special feature of CEST is its ability to retrieve spectral information, which enables further characterization of CEST contrast agents or the detection of different agents in the same sample. However, the acquisition of such CEST spectra is inherently slow since each spectral data point requires (1) a separate preparation with a different saturation pulse frequency offset and (2) a sufficiently long waiting time to guarantee relaxation of the magnetization back to equilibrium before the next data point is acquired. Recently, ultrafast methods were proposed to drastically speed up the acquisition of CEST spectra in phantom experiments, and applied to both 1H [2,3] and hyperpolarized 129Xe nuclei [4,5]. They are based on encoding spectral information along a spatial dimension (e.g. along the z-direction) using a magnetic field gradient, hence requiring only a single saturation pulse for the entire spectrum. In the very first approach, only a single sample could be investigated at a time, since no spatial information was resolved [2]. By including slice selection and hence exploiting a further spatial dimension, we could extend the method to acquire spectra of multiple samples, as shown in Ref. [3]. However, the sample tubes ⇑ Corresponding author. E-mail addresses: [email protected] (J. Döpfert), [email protected] (L. Schröder). http://dx.doi.org/10.1016/j.jmr.2014.03.008 1090-7807/Ó 2014 Elsevier Inc. All rights reserved.

had to be arranged in a special way such that the slices did not overlap, which might be tedious especially when many samples are involved. Here, we further extend these ultrafast approaches with imaging (ultrafast CEST imaging, UCI), where in addition to the spectral information also the spatial information in the x–y plane is fully recovered. Hence, in principle any arrangement of multiple sample tubes becomes feasible. This work thus represents a combination of ultrafast CEST methods and high-throughput CEST imaging methods presented in Ref. [6] and merges both their advantages: First, fast spectral encoding by using a magnetic field gradient, and second, simultaneous investigation of many agents by using imaging. As in previous ultrafast approaches, the proposed method employs a CEST gradient in the z-direction to encode the spectral information during the saturation period. However, no read gradient is subsequently applied in the same direction to obtain the spectrum, but instead multiple slices are imaged rapidly along the CEST gradient’s direction. Thus, each slice represents a distinct saturation frequency offset (see Fig. 1A and B). This way, the acquisition can be accelerated (since only one saturation is needed for multiple offsets) while the spatial information in the x–y plane is preserved. The method hence effectively exploits all three spatial dimensions. One prerequisite for UCI is that the slices have to be recorded very rapidly since the saturated magnetization relaxes back to its equilibrium (T 1 -relaxation). We meet this requirement by heavily subsampling k-space. UCI hence consists of three

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Communication / Journal of Magnetic Resonance 243 (2014) 47–53

scheme. We recover the image data by employing a novel heuristic reconstruction algorithm that uses low-rank [7] and finite support [8] as prior information. This way, we achieve a reduction in scan time by a factor of more than 16 compared to a conventionally aquired, fully sampled data set. 2. Methods 2.1. Principle of ultrafast CEST imaging UCI is very simple to implement if a multislice CEST imaging sequence already exists. The only modification that has to be done is the implementation of an additional CEST gradient Gc with magnitude Gc in the slice direction during the saturation period (see Fig. 1A). This causes the spin packets to precess at different frequencies along this direction, and hence each slice experiences a different saturation offset (see Fig. 1B). Therefore, an entire spectral image series can be acquired after a single saturation preparation, provided the readout is fast enough to neglect relaxation. This scan is henceforth called Son . Since the initial signal strength is likely to vary from slice to slice due to e.g. inhomogeneous coil sensitivities, an additional reference scan without saturation pulse is required, henceforth called Soff . Dividing Son by Soff element-wise, the pure spectral information can be recovered, similarly to the previous ultrafast CEST approaches without imaging capabilities. 2.2. Implementation details Let X ¼ cB0 denote the Larmor frequency of the abundant pool (e.g. bulk water), where c is the gyromagnetic ratio of the observed nuclei (c  267:5  106 rad=T s for protons) and B0 the static magnetic field. The offset of any frequency x is given by dx ¼ x  X. The number of saturation offsets in ultrafast CEST imaging is obviously given by the number of slices, N s . The desired range of saturation offsets, i.e. the CEST bandwidth Dx, is generated by the CEST gradient Gc over the spatial extend d of the slice package (see Fig. 1B):

Dx ¼ c Gc d:

Fig. 1. (A) Radial UCI pulse sequence with N p projections and N s slices. Instead of the radial gradient echo readout following the saturation period, in principle any fast multislice imaging sequence could be used. (B) Principle of UCI. Due to the CEST gradient Gc , the sample experiences the saturation pulse at different off-resonant conditions dx along the slice or z-direction. Hence, each slice Si corresponds to a different saturation offset dxi . (C) Radial trajectory of each slice Si employing a Golden Angle increment of /  111:25 . In this example N p ¼ 4 projections were used per slice. The indices next to each projection do not reflect the order of the acquisition, but the order of the angle increments. (D) Creating a binary signal mask by thresholding (simulation). Upper row: Reconstructing a single undersampled slice with FT leads to artefacts. Lower row: by merging all slices from the Soff scan, a densely sampled k-space can be obtained, leading to an artefact-free reconstruction and an acceptable signal mask.

If the slice package is centered in the z-direction of the gradient’s coordinate system, the desired center frequency of the resulting CEST spectrum dx0 is adjusted by simply setting the saturation pulse frequency dxs to this value. If however the slice package is off-centered by a distance Dd, the following correction has to be applied:

dxs ¼ dx0 

Dd Dx: d

ð2Þ

Since the slice thickness s cannot be made infinitesimally small, a slice does not represent exactly a single saturation offset but rather an average over a range of saturation frequencies given by cGc s. We approximated the saturation offsets by calculating the frequency that would be achieved by an infinitesimally thin slice at the center of each ‘‘real’’ slice (see also Supporting Information):

dxi ¼ dx0 þ components: (a) gradient-encoding of the CEST frequencies, (b) multislice imaging and (c) subsampling. In this article, we demonstrate UCI using a phantom consisting of three NMR tubes containing a PARACEST agent (Eu-DOTA-4AmC) in different concentrations. After outlining and verifying the principle of the method, we then accelerate the acquisition process by heavily subsampling k-space based on a radial Golden Angle sampling

ð1Þ

 Dx Dx  s  þ ði  1ÞDs ; 2 2 d

ð3Þ

where the interslice distance Ds is given by

Ds ¼

ds : Ns  1

ð4Þ

Note that magnetic field inhomogeneities can lead to incorrect frequency assignments.

Communication / Journal of Magnetic Resonance 243 (2014) 47–53

2.3. Fast image acquisition and constrained reconstruction

3. Results and discussion

As mentioned in the introduction, the multislice UCI acquisition should be run as fast as possible, since T 1 relaxation degrades the saturation preparation. Considering the T 1 values of around 3 s of the PARACEST samples in our experiments, the acquisition time t acq should clearly be shorter than 1 s. A conventional Cartesian readout using a 128  128 matrix with a repetition time of TR = 3 ms per line would take around 380 ms for recording a single slice. This allows one to acquire at best two slices (i.e. two saturation offsets) after each saturation, which does not represent a huge speedup compared to conventional CEST imaging. Going to a very small matrix of 16  16 would permit one to acquire up to 15 slices in an acceptable time of 720 ms after each saturation, but obviously at the expense of a very low resolution. To preserve a high resolution, we instead used undersampling to strongly accelerate the acquisition of the slices. To enable an efficient reconstruction of the undersampled data, the k-space trajectory should be as irregular as possible which results in most of the undersampling artefacts being incoherent [9]. The incoherence achievable with Cartesian trajectories is relatively low, since only the phase encoding dimension can be efficiently subsampled. Radial trajectories are a powerful alternative, enabling incoherent subsampling in multiple directions [10] while inherently oversampling of the k-space center (see Fig. 1A for a diagram of the radial gradient echo sequence we utilized throughout this work). To achieve incoherence not only in the spatial but also in the spectral domain, we applied radial sampling based on a Golden Angle ordering scheme [10–12] that extends over the whole package of slices. Hence, the angular increment of consecutive spokes (or ffiffi   111:25 (see Fig. 1C), leading projections) is given by / ¼ p360 5þ1 to unique trajectories for each spectral image (that is, each slice Si). When employing a simple Fourier transform (FT) to reconstruct the radially undersampled raw data, artefacts occur in the resulting images due to violation of the Nyquist criterion (see first two images in the upper row of Fig. 1D). To remove these artefacts and to achieve a meaningful reconstruction, prior knowledge can be incorporated. For instance, widely used compressed sensing algorithms exploit the sparsity of the resulting images in a given transform domain [9]. Other examples include assuming the set of images to be low-rank [13–16] or defining regions in the image where the signal must vanish (limited support (LS) constraint [8,17]). Here, we combined the latter two approaches (low-rank and LS) resulting in an heuristic algorithm (see Appendix for a detailed description) based on the singular value tresholding (SVT) algorithm proposed in Ref. [7]. This allowed us to reconstruct heavily undersampled spectral data sets with a 128  128 matrix size using only N p ¼ 10 projections per image. At a TR of 3 ms per projection, we could therefore record N ¼ 15 saturation offsets (i.e. slices) in an acceptable time of tacq ¼ 462 ms. For the LS part of the reconstruction to work, information about the object support is required; in other words, one needs to know which regions in the image may contain non-zero MR signal and which regions must not. Usually, this is done by additionally acquiring a fully sampled image to generate a binary signal mask by e.g. thresholding. Undersampled images do not serve this purpose, since the undersampling artefacts after FT lead to errors in the signal mask (see Fig. 1D, upper row). However, we can save the time required to acquire a fully sampled image by considering the Soff scan, which contains undersampled data for all slices. Since no saturation pulse was applied, all slices should more or less represent the same image, that is, the axial section of the phantom (see Fig. 1B). By merging the k-space data for all slices Si, one obtains a more densely sampled k-space which can be FT-reconstructed with many less artefacts, leading to a useful signal mask upon thresholding (see Fig. 1D, bottom row).

3.1. Proof of principle

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As a first proof of principle, we verified the first two fundamental components of UCI: gradient-encoding of the CEST frequencies combined with multislice imaging. The third and final component, i.e. subsampling, was omitted here to make the verification independent from possible errors in the constrained reconstruction. The sample under investigation consisted of three NMR tubes arranged as in Fig. 1B and filled with aqueous solutions containing the PARACEST agent EuDOTA-4AmC in different concentrations. We recorded N = 15 saturation offsets at saturation power and duration of B1 ¼ 15 lT and t sat ¼ 3 s, respectively. Hence, N s ¼ 15 slices had to be acquired during the multislice readout (see Supporting Information, Fig. S1 for the slice geometry). Full radial sampling of a 128  128 image matrix requires the acquisition of N p ¼ p2  128  200 projections [18]. Since the acquisition time t acq after each saturation should be well below 1 s due to T 1 relaxation, not all the 15 slices could be acquired at once (t acq would have summed up to N s  N p  TR  15  200  3 ms ¼ 9 s). We therefore had to reapply the saturation after recording 10 projections of each slice (t acq  15  10  3 ms ¼ 450 ms). This rendered the proposed method rather slow (remember that no subsampling is applied yet) and hence the term ‘‘ultrafast’’ is at first somewhat misleading. To avoid confusion however, we will refer to it as ‘‘fully sampled UCI’’ in this section. Fig. 2 shows selected images from both the fully sampled UCI Son and Soff data set. For comparison, a standard CEST data set is depicted as well, where the 15 different saturation offsets were achieved by changing the saturation frequency prior to each image acquisition. We observed similar tendencies for the Son images and the standard CEST images, for instance both show direct saturation at around 0 ppm and a CEST pool saturation at around 50 ppm. The Soff images show no saturation as expected, but represent slight signal variations between the slices. To compare the spectral information present in the data sets, we first of all normalized them as follows: We divided all standard CEST images pixel-wise by an additionally acquired off resonant image (without saturation), and divided the fully sampled UCI scan Son pixel-wise by Soff . We then placed regions of interest (ROIs) in the resulting normalized images to calculate the mean signal over each sample tube for each saturation offset. The resulting CEST spectra on the right of Fig. 2 show a good agreement between fully sampled UCI and standard CEST. This demonstrates that the proposed method is indeed suitable to measure reliable CEST spectra. As mentioned above, the acquisition of fully sampled UCI scans implies frequent saturation and wait periods, rendering the method in fact slower than standard CEST imaging. However, the proposed technique would be much faster than the conventional one if all 15 slices could be acquired after only a single saturation period, hence justifying the term ‘‘ultrafast’’. This can be achieved by the third component of UCI, subsampling, as shown in the next section.

3.2. UCI with radial undersampling In this section, we use radial subsampling to reduce the amount of projections per slice to 10, which permits the acquisition of all 15 slices after a single saturation in tacq ¼ 462 ms. This allowed us to cut down the total scan time for a UCI data set consisting of the Son and Soff scan separated by a 10 s waiting time to only 17 s. To verify the performance of the reconstruction of the strongly undersampled data, Fig. 3 depicts selected images from the fully sampled and FT reconstructed Son scan from Fig. 2 and

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Fig. 2. Comparison of a fully sampled standard CEST data set to a fully sampled UCI data set (N p ¼ 200 projections). The conventionally acquired images (top row) are comparable to the UCI Son scan (middle row). Note that the saturation offsets (denoted in units of ppm at the bottom left of each image) differ slightly, since both sequences shared the same bandwidth Dx ¼ 92:5 ppm over which the offsets were distributed equidistantly for standard CEST, but the UCI offsets were calculated according to Eq. (3). The Soff scan (bottom row) does represent variations in signal intensities from slice to slice, due to e.g. coil sensitivities. The plot on the right compares CEST spectra for UCI (obtained by dividing Son by Soff element-wise) with conventional CEST spectra (obtained by dividing the standard CEST images by an off resonant image without saturation). We calculated the spectra on the basis of three ROIs covering each of the three sample tubes.

selected images from the subsampled Son scan reconstructed with the proposed SVT-LS algorithm. As shown in the difference images, the undersampled data matches the fully sampled data quite well. Apart from slight streaking artefacts, most errors occur on the edges of the sample tubes, probably due to the LS constraint, which enforces an abrupt signal drop outside the signal mask. Note also that the circularly-shaped aliasing artefact on the border of the fully sampled images is suppressed in the SVT-LS reconstruction, again due to the LS constraint. The complete set of images can be found in the Supporting Information (Fig. S3) together with the corresponding Matlab code. To verify the spectral information, we first normalized the UCI data and the conventionally acquired data as described in the previous section and in the caption of Fig. 4. The resulting CEST spectra for each sample tube are shown in Fig. 4 for three different saturation powers (7.5 lT, 15 lT and 22 lT). The spectra obtained conventionally by sweeping the saturation frequency are in good agreement with the subsampled UCI spectra. The applied sampling scheme and reconstruction thus obviously preserve most of the relevant spectral and spatial information, despite the heavy undersampling. The acquisition times for the UCI spectra and the conventional spectra were 1.3 min and 21.3 min, respectively, corresponding to an acceleration factor of more than 16.

Fig. 3. Performance of the proposed SVT-LS reconstruction. Selected images from a fully sampled, FT reconstructed UCI scan (200 projections) at B1 ¼ 15 lT compared to the corresponding subsampled, SVT-LS reconstructed UCI scan (10 projections). The difference images (displayed oversaturated to enhance the contrast, see scale bar) demonstrate that both data sets agree reasonably well, with errors being present mainly on the sample’s edges.

These results suggest that UCI may be suitable for fast and efficient screening of large sets of CEST contrast agents. In this study, only three NMR tubes were investigated simultaneously for a first demonstration, but in principle, the number of samples is mainly limited by the inner diameter of the probe. The number of saturation offsets that can be acquired with UCI after a single saturation is not only limited by the acquisition time that should be much faster than the relaxation process (as discussed above), but also by the active area of the receiver/transmit coil: All slices have to fit into that area. Hence, thin slices and a small interslice distance are desirable. However, SNR and gradient limitations do not allow the slices to be arbitrarily thin, and RF spillover from adjacent slices requires a reasonably sized interslice distance. The chosen number of slices is hence a trade-off between all those parameters. SNR should also be sufficient to allow the high degree of undersampling that is needed for fast image acquisition. Furthermore, the shape of the sample tubes should be homogeneous along the CEST gradient’s direction. If this is not the case (for example, if the tubes are slightly tilted), UCI in principle still works, but (1) the low-rank prerequisite for the SVT part of the reconstruction might be violated, (2) a separate signal mask would have to be generated for each slice for the LS part, and (3) separate ROIs would have to be defined for each slice to obtain the CEST spectra. Note also that the efficiency of the LS part of the reconstruction depends on the size of the void space in the final images. In the extreme case of no void space, i.e. when the signal mask covers the entire image, the LS reconstruction is profitless. However, other priors such as sparsity in the wavelet domain or knowledge about the shape of the CEST spectra [19] could be incorporated into the reconstruction. Note also that radial trajectories are quite sensitive to off-resonances, which makes their use problematic for the screening of CEST agents that change the bulk water frequency in significantly different ways. However, UCI can also be implemented with more robust Cartesian readouts, albeit possibly with a smaller degree of undersampling. Instead of the multislice readout used in this work, 3D MRI could be considered to acquire volumetric image data, possibly enabling efficient acceleration with compressed sensing and parallel imaging [10]. Undersampling can also be employed to accelerate conventional CEST imaging: Varma et al. used the keyhole technique to accelerate the acquisition of CEST spectra by a factor of 4, recording only 32 lines to fill a 128  128 matrix [20]. This work showed that the acquisition of 10 projections was sufficient to successfully reconstruct spectral CEST images with the same matrix size. Hence, the use of a radial trajectory and the proposed reconstruction algorithm could also enable a further acceleration of conventional

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Fig. 4. Validation of subsampled UCI spectra for ROIs in the three sample tubes at three different B1 strengths. Apart from UCI data (circles), the diagram shows CEST spectra obtained with the standard approach (crosses and dotted lines) by sweeping the saturation frequency for comparison. For all B1 , the spectra show good agreement. The total scan time for all three data sets was 1.3 min for UCI and 21.3 min for standard CEST imaging, corresponding to an acceleration factor of more than 16. We additionally normalized the UCI spectra by multiplying them by a factor determined by the mean signal intensity of the points at dx ¼ 33:7 ppm where the offset of UCI and the offset of standard CEST coincide (see arrow and Supporting Information). This accounted for intensity differences between the Son and Soff scans, which were probably mostly introduced by the reconstruction algorithm.

CEST. However, the high degree of undersampling achieved here might not be directly applicable to in vivo conditions. Also agents exhibiting a relatively small chemical shift distance from the bulk water resonance (e.g. DIACEST agents) may be in principle readily investigated with UCI. If it is desired in this context to sample the offsets around the CEST resonance more densely than the offsets around the bulk water pool, a separate UCI acquisition for each of the two pools would be required. If SNR is sufficiently high, UCI may also be directly applied for the fast characterization of CEST agents based on hyperpolarized xenon (Hyper-CEST [21]). In this case, T 1 relaxation is typically around 100 s, which permits the use of long readout times after a single delivery of hyperpolarized xenon and subsequent saturation. Therefore, Hyper-UCI could be performed without the need for strong undersampling. 4. Conclusion In this work, we propose ultrafast CEST imaging, a method that extends gradient-based ultrafast CEST spectroscopy [2,3] with imaging capabilities. It is easily implementable into any existing multislice CEST imaging sequence by adding a CEST gradient during the saturation period. We verified the applicability of UCI using a phantom consisting of three NMR tubes containing a PARACEST agent in different concentrations. To accelerate multislice acquisition, we used radial subsampling and a constrained reconstruction algorithm. The obtained UCI spectra showed excellent agreement to CEST spectra acquired the conventional way, but were acquired 16 times as fast. The method might hence enable efficient screening or characterization of multiple CEST samples, which is particularly relevant for contrast agent development. 5. Experimental All MRI experiments were performed at room temperature (T  292 K) on a 9.4 T Avance 400 NMR spectrometer (Bruker Biospin, Ettlingen, Germany) equipped with gradient coils for imaging. A probe with an inner diameter of 15 mm was used for excitation and detection. Parameters for all radial imaging sequences were: matrix 128  128, echo time TE = 1.6 ms, repetition time per projection TR = 3 ms, readout bandwidth = 101 kHz, slice thickness s = 0.45 mm, field of view = 15  15 mm, Gaussian excitation pulse with a duration of 1 ms and a flip angle of 12.5°. After application of the continuous wave saturation pulse with a duration of tsat ¼ 3 s and the subsequent acquisition of the desired amount of

projections, we waited for 10 s to allow the magnetization to recover prior to the next saturation period. The additional parameters for the UCI sequences were: N = 15 slices, total slice package extent d = 11.65 mm, interslice distance Ds ¼ 0:8 mm, Dx ¼ 92:5 ppm, center frequency dx0 ¼ 33:7 ppm. In the pulse sequence, the slice loop is placed inside of the projections loop, i.e. initially, the first projection of each slice is acquired, and then the second projection of each slice and so on. For the standard CEST images, the saturation and wait periods were applied every 100 projections (which needed t acq ¼ 308 ms to acquire), leading to a total acquisition time for the 16 images (15 saturation offsets plus one off resonant image) of 7.1 min. FT of radial raw data was performed using non-uniform Fourier transform (NUFFT) as implemented by Jeffrey Fessler [22] in Matlab (Mathworks, Natick, USA) and adapted by Michael Lustig in his SparseMRI software package [9]. The image reconstruction algorithm was written in Matlab based on the SVT algorithm in Ref. [7] (see Appendix A). Reconstruction of a subsampled UCI data set (including on and off resonant scans) took approximately 2 min on a standard desktop computer. All PARACEST samples were prepared by diluting Eu-DOTA-4AmC (see Supporting Information, Fig. S5) with H2O to approximately the following concentrations: 7.5 mM, 15 mM and 30 mM, corresponding to the lower, upper left and upper right tube in Fig. 2, respectively. Acknowledgments The authors would like to thank Stefan Klippel, Dr. Simon Konstandin, Martin Kunth and Matthias Schnurr for valuable discussions. This work has been supported by the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007–2013)/ERC Grant Agreement No. 242710 and the Human Frontier Science Program. Appendix A. Reconstruction algorithm The image reconstruction problem can be described as [18]

AðXÞ ¼ Y:

ðA:1Þ

where Y is the measured undersampled k-space data, X is the (unknown) full image series (in real space) to be reconstructed, and A is a linear operator consisting of the calculation of the FT of X and the subsequent evaluation at the k-space trajectory’s positions. Since the data is sub-sampled, Y is usually smaller than X, making the problem ill-posed and not solvable by simply inverting Eq. (A.1).

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Instead, we try to estimate X in an iterative fashion by applying the following prior knowledge: 1. assuming that the CEST image series X can be approximated by a low-rank matrix [16] 2. driving the solution X towards zero in areas where no sample is present (limited object support (LS), see for example Refs. [8,17]) While low-rank and LS constraints have been successfully applied for the reconstruction of MR images separately [16,8], the combination of both is novel (Refs. [13–15] used a combination of low-rank and transform domain sparsity as priors). Easily implementable algorithms exist for the incorporation of low-rank constraints (e.g. singular value thresholding (SVT) [7]) and LS constraints (e.g. projection onto convex sets (POCS) [8]). To apply both constraints simultaneously, we started from the original SVT approach and ‘‘merged in’’ the LS constraint in a POCS type of fashion, resulting in the following heuristic reconstruction algorithm, SVT-LS reconstruction algorithm 1: Y0 ¼ 0 .initialization 2: for k ¼ 1; k 6 kmax ; k þ þ do 3: Xk1 ¼ A ðYk1 Þ 4: 5: Zk ¼ Xk1  M .LS constraint 6: Xk ¼ Xk1 þ kðZk  Xk1 Þ 7: 8: Xk ¼ shrinkðXk ; sÞ .shrink singular values 9: 10: Yk ¼ Yk1 þ dðY  AðXk ÞÞ .data fidelity 11: 12: if kðY  AðXk ÞÞk2 6 tol then break 13: end for

where this notation was used: A : Adjoint of A : Element-wise multiplication Xk : k-th guess for image series Yk : Auxiliary variable for data fidelity Zk : Auxiliary variable for LS constraint M: Binary signal mask, matrix containing ones in regions with signal and zeros otherwise (see Fig. 1D) k: Regulates the LS constraint s: Regulates the singular value shrinkage d: Regulates the enforcement of data fidelity kmax : Maximum number of iterations tol: Tolerance level shrinkðXk ; sÞ: Singular value shrinkage operation defined in Section B and Ref. [7] kXk2 : ‘2 -norm of X Note that the algorithm reduces to the conventional SVT algorithm (Eq. 3.3 in Ref. [7]) if k ¼ 0 or if lines 5 and 6 are removed. Line 5 is the projection of the image onto the convex set of images that have a limited object support (see Table 1 of Ref. [8]) and line 6 corresponds to the relaxed projection operator (see Eq. 1 in the same reference). A reconstruction using SVT-LS is compared to reconstructions using either LS or SVT alone in the Supporting Information, Fig. S4. A Matlab implementation of the algorithm together with an example data set is available for download, see link in the Support-

ing Information. The parameters used in this work were kmax ¼ 200, s ¼ 2  105 ; k ¼ 0:76; d ¼ 0:8 and tol = 0.09. Appendix B. Singular value shrinkage operation Consider the singular value decomposition of a complex m  n matrix X

X ¼ URV ;

ðB:1Þ

where U and V are m  m and n  n unitary matrices, respectively, denotes the conjugate transpose operation, and R is a rectangular m  n diagonal matrix whose entries Ri;i ¼ ri are called singular values of X. For each s P 0, define ⁄

b ; shrinkðXÞ ¼ U RV

ðB:2Þ

b being a rectangular m  n diagonal matrix with entries with R

b i;i ¼ maxð0; ri  sÞ: R

ðB:3Þ

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