Due to the low concentration of ²³Na in the human body, ²³Na-MRI suffers from long acquisition times T_aq, especially for high nominal spatial resolutions¹. The measurement time can be reduced by data undersampling and a consequent iterative Compressed Sensing (CS) reconstruction^2,3,4,5. Here we investigated the performance of homodyne data processing⁶ prior to iterative 3D-Dictionary-Learning Compressed Sensing reconstruction (3D-DLCS)⁷ for the reconstruction of asymmetrically undersampled radial ²³Na data, to take advantage of the point-symmetry in k-space.

Simulated T2-weighted and measured in-vivo 3D-radial ²³Na-MRI were reconstructed with a Nonuniform Fast Fourier Transform (NUFFT)⁸ and 3D-DLCS. In each case, three datasets were reconstructed: The first contains a homogeneous distribution of the projections with an undersampling factor (USF) of 10. In the second dataset 30% of the projections were omitted. The projections were distributed asymmetrically. 60% of the data are acquired with USF=10 and the remaining 40% with USF=40. The missing projections are filled with zeros. The third set contains the same data as the second, but reweighted according to a homodyne procedure adapted to suit 3D-radial data (see Fig. 1). The points in k-space for which the corresponding mirrored datapoints are not acquired are weighted with w=2. Since the data from the scanner are complex, the real part of the phase-corrected image has to be used. A low-resolution phase image obtained from interpolating the data up to the Nyquist limit for USF=40 is used for this purpose9. The phase information that is lost in this step can be recovered during the iterative reconstruction of the data. Due to the relatively high resolution of the data and the low SNR, blocks of size 6×6×6 and a dictionary size D=600 were used for the 3D-DLCS reconstruction algorithm.

Simulated and in-vivo data: The nominal resolution for the simulated/measured data was Δx³=1.3×1.3×1.3mm³, TE/TR=0.3/20ms, α=37°. Eight averages were simulated/measured; the total measurement time was Taq=32min for the full dataset and T_aq=23min for the asymmetrically sampled dataset. The image quality is assessed with the peak-signal-to-noise ratio (PSNR) in the reconstructions from simulated data. For the in-vivo-measurement, a healthy volunteer (male, age 23) was measured on a 7T whole-body MR system (Magnetom 7T, Siemens Healthcare, Erlangen, Germany). A double-resonant (1H: 297.2 MHz; 23Na: 78.6 MHz) quadrature birdcage coil (Rapid Biomed, Rimpar, Germany) was used. The data were acquired with a 3D density adapted radial sampling pattern¹⁰.

The 3D-DLCS reconstruction of asymmetrically undersampled data shows blurring in the head-feet direction if the missing data are filled with zeros (Figs. 2c and 3d). This blurring can be markedly reduced by homodyne processing of the data prior to the iterative reconstruction, resulting in a better quality of sagittal or coronal slices of the measured object, similar to the ones from the uniformly sampled dataset (Figs. 2d and 3e). This is also reflected by the PSNR values calculated for simulated data: PSNR for the reconstructions from homodyne processed data and for the zero filled data were 22.4 and 21.9 respectively. For the 3D-DLCS reconstruction of the uniformly sampled dataset PSNR=23.7, and for the NUFFT reconstruction PSNR=14.2. The phase information that is lost in the homodyne processing of the data can be recovered during the iterative 3D-DLCS reconstruction of the data (Fig. 4). The resulting phase images are in good accordance with the ones from the uniformly sampled dataset. The total reconstruction time for one 3D-DLCS reconstruction was approximately 5h on a standalone desktop PC (Intel Core i7-2600 CPU, 3.4 GHz, 16 GB memory).The blurring in head-feet direction induced by asymmetrically undersampling of 3D-radial ²³Na-MRI data can be efficiently reduced by homodyne processing prior to a reconstruction with the 3D-DLCS algorithm. After homodyne processing, the results from the 3D-DLCS reconstruction of the asymmetrically undersampled data with a nominal resolution of Δx³=1.3×1.3×1.3mm³ and N_proj=8750 are almost equivalent to those from of the uniformly sampled dataset with N_proj=12500, which corresponds to a measurement time reduction of 30%. The 3D-DLCS reconstructions of both the uniform and asymmetrical data greatly outperform the NUFFT reconstructions. The preservation of the image phase could be of interest e.g. for B₀- and B₁-correction of the image, which is a necessary step towards quantitative imaging.