Density Adapted Stack of Stars Sequence for 23Na using Dictionary Learning Compressed Sensing Reconstruction
Fabian J. Kratzer1, Sebastian Flassbeck1, Armin M. Nagel1,2, Peter Bachert1, Mark E. Ladd1, and Nicolas G. R. Behl1

1German Cancer Research Center (DKFZ), Heidelberg, Germany, 2University Hospital Erlangen, Erlangen, Germany


Sodium plays important roles in many cellular processes, which motivates imaging of the 23Na distribution. Short relaxation times and low in-vivo signal result in the need of sequences with short echo times and techniques to improve the SNR. Therefore, we present a stack of stars (SOS) sequence with density adapted readout gradients to increase SNR. We combine this sequence with an anisotropic dictionary learning compressed sensing reconstruction to further reduce noise in the images.


Sodium is involved in many cellular processes, therefore 23Na imaging enables insight in many physiological processes. However, low NMR sensitivity of sodium combined with low concentrations result in a signal about 104 lower than the 1H signal.

Furthermore, short relaxation times of sodium result in the need of sequences with short echo times. Hence, often 3D spiral techniques or radial sequences are used. The use of density adapted gradients ensures a mostly homogenous k-space coverage in the latter, resulting in an increased signal-to-noise ratio (SNR)1,2. Nevertheless, 3D measurements can suffer from long measurement times because of their non-selective excitation, especially if only a small number of slices is of interest. Therefore, a 2D density adapted radial sequence was introduced2.

Here we present a 3D slab selective density adapted stack of stars sequence, enabling 3D sodium imaging with adjustable field-of-view in z-direction and anisotropic voxel size3.

To improve image quality on the reconstruction side of sodium imaging, Dictionary Learning Compressed Sensing4 with isotropic dictionary entries (atoms) has shown to be useful. However, anisotropic resolution can lead to faster changes in slice direction than in-plane in the image. This can lead to smearing between the slices, resulting in artifacts when atoms with isotropic pixel dimensions are used to represent anisotropic regions in the images. Therefore, we present the combination of a 3D density adapted stack of stars sequence with Dictionary Learning Compressed Sensing using anisotropic atoms.


A stack of stars sequence with density adapted readout gradients was developed and its SNR efficiency investigated3. The gradient amplitude is reduced during readout; hence, the distance in k-space between two subsequent measurement points along each spoke decreases. This can compensate for the increased distance between the spokes, enabling a more homogenous k-space sampling and therefore leading to an increased SNR. A sequence diagram can be seen in Fig.1.

To improve imaging on the reconstruction side a Compressed Sensing (CS) algorithm4 is used. Within the algorithm the following cost function is minimized.

$$$ \{\alpha_{ijk}, \hat{D},\hat{x}\} = \arg \min_{\alpha_{ijk},x} \left \{ \lambda \Vert Fx-y\Vert_2^2 + \mu TV(x) + \nu \sum_{i,j,k} \Vert D \alpha_{ijk} - R_{ijk}x\Vert_2^2 + \sum_{i,j,k} \Vert \alpha_{ijk}\Vert_0 \right \}$$$

The cost function consists of a λ-weighted data consistency term, a µ-weighted total variation term, a ν-weighted dictionary term, and a sparsifying dictionary term. To avoid smearing between slices, anisotropic atoms were implemented.

To evaluate the general performance of the combination of the SOS sequence with the Compressed Sensing algorithm, the image acquisition was numerically simulated, including relaxation effects as proposed by Lommen et al.5; hence, the ground truth was known. The simulated data were reconstructed with a Non-Uniform Fast Fourier Transform (NUFFT) with and without Hann filter. Furthermore, they were reconstructed with Compressed Sensing using a variety of parameters. For comparison with the ground truth, peak signal-to-noise ratio (PSNR) and structural similarity (SSIM) were determined.

The measurements were performed on a 7T whole body MR-system (Siemens) using a birdcage coil.


CS reconstruction took 7-25 minutes on a standalone PC (Intel Xeon E5 1620 v4, RAM: 64GB) for reconstruction matrix size 180pixel*180pixel*8pixel. The use of anisotropic atom sizes (5pixel*5pixel*1pixel / 4pixel*4pixel*2pixel) resulted in reconstructions without visible smearing between neighboring slices, contrary to isotropic atom sizes (3pixel*3pixel*3pixel / 4pixel*4pixel*4pixel)(Fig.2).

To evaluate the reconstruction performance, simulated data reconstructed with NUFFT and CS were compared to the ground truth (Fig.3). The Compressed Sensing reconstruction resulted in PSNR=22.7dB, SSIM was found to be 0.72, whereas a NUFFT reconstruction combined with a Hann filter reached PSNR=21.3dB and SSIM=0.34.

To validate the combination of the SOS sequence and the CS reconstruction in vivo, Fig.4 shows a NUFFT, a filtered NUFFT, and a CS reconstruction of a healthy volunteer (male, 24 years) from the same data set. CS was able to reduce noise significantly.

Discussion and Conclusion

In this a work, a sodium stack of stars sequence was developed and combined with a Dictionary Learning Compressed Sensing algorithm, leading to more flexibility compared to 3D radial acquisition schemes.

Reconstruction of anisotropic images with CS using isotropic dictionary entries can result in smearing artifacts. The use of anisotropic atoms was able to suppress this effect.

To evaluate the performance of the CS reconstruction, numerically simulated data were reconstructed with CS and NUFFT and compared to the ground truth, where CS achieved higher PSNR and SSIM values. Furthermore, Compressed Sensing reconstructions showed significantly reduced noise in in-vivo images compared to NUFFT reconstructions.

A potential further development could be the use of a multi-channel coil to additionally implement SENSE in the Compressed Sensing Reconstruction, as well as the application of this sequence/reconstruction combination to quantitative sodium imaging.


No acknowledgement found.


1. Nagel et al., Magn Reson Med, 2009 62: 1565-1573

2. Konstandin et al., Magn Reson Med, 2011 65(4): 1090-1096

3. Kratzer et al., Proc. X-Nuclei Workshop ISMRM 2018: #3

4. Behl et al., Magn Reson Med, 2016 75(4): 1605-1616

5. Lommen et al., Proc. ISMRM 2017: 5628


Figure 1: a) Sequence diagram of the density adapted stack of stars sequence. The density readout gradient (Gxy) changes amplitude and therefore speed through the k-space, resulting in a decreasing distance between subsequent measurement points along the spokes when going to higher frequencies, as shown in b). This compensates the increasing distance between the spokes, leading to a more homogenously covered k-space and therefore higher SNR compared to conventional trapezoidal readout gradients. The full k-space trajectory is shown in c).

Figure 2: Dictionary Compressed Sensing reconstructions of simulated data with anisotropic voxel size (1.5*1.5*5mm3) with a) isotropic (λ = 0.005, µ = 1, ν = 0, atom size B = 3pixel*3pixel*3pixel) and b) anisotropic ( λ = 0.01, µ = 0, ν = 1, atom size B = 5pixel*5pixel*1pixel) atom size. Smearing artifacts appear in the reconstruction with isotropic atom size, i.e. CSF from the partitions below, as indicated with the red arrow. This can be avoided by the use of anisotropic block sizes.

Figure 3: Different reconstructions of simulated brain data (8 partitions, TR = 37.5 ms, TE = 1.01 ms, 2*2*5mm3, Nspokes/star = 200). a) Ground truth, b) NUFFT, c) NUFFT + Hann filter, and d) Compressed Sensing reconstructions (λ = 0.02, µ = 0, ν = 1, atom size B = 4pixel*4pixel*2pixel). To evaluate reconstruction performance PSNR and SSIM were determined compared to the ground truth.

Figure 4: In-vivo data set (FA = 35°, TR = 31.9ms, TE = 1.32ms, 8 partitions, 25% slice oversampling, Nspokes/star = 128, 2*2*5mm3, 21 averages, Tacq = 14:18) reconstructed with a) NUFFT, b) NUFFT + Hamm filter and c) with dictionary learning Compressed Sensing (λ = 0.005, µ = 7*10-7, ν = 1, atom size B = 5pixel*5pixel*1pixel).

Proc. Intl. Soc. Mag. Reson. Med. 27 (2019)